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0000000000000000000000000000000000000000..9dcb8cafaeaaff1797d7ac50fc74804ad6708bfe --- /dev/null +++ b/parse/train/9sF3n8eAco/9sF3n8eAco.md @@ -0,0 +1,187 @@ +# ALL-YOU-CAN-FIT 8-BIT FLEXIBLE FLOATINGPOINT FORMAT FOR ACCURATE AND MEMORYEFFICIENT INFERENCE OF DEEP NEURAL NETWORKS + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Modern deep neural network (DNN) models generally require a huge amount of weight and activation values to achieve good inference outcomes. Those data inevitably demand a massive off-chip memory capacity/bandwidth, and the situation gets even worse if they are represented in high-precision floating-point formats. Effort has been made for representing those data in different 8-bit floating-point formats, nevertheless, a notable accuracy loss is still unavoidable. In this paper we introduce an extremely flexible 8-bit floating-point (FFP8) format whose defining factors – the bit width of exponent/fraction field, the exponent bias, and even the presence of the sign bit – are all configurable. We also present a methodology to properly determine those factors so that the accuracy of model inference can be maximized. The foundation of this methodology is based on a key observation – both the maximum magnitude and the value distribution are quite dissimilar between weights and activations in most DNN models. Experimental results demonstrate that the proposed FFP8 format achieves an extremely low accuracy loss of $0 . 1 \% \sim 0 . 3 \%$ for several representative image classification models even without the need of model retraining. Besides, it is easy to turn a classical floating-point processing unit into an FFP8-compliant one, and the extra hardware cost is minor. + +# 1 INTRODUCTION + +With the rapid progress of deep neural network (DNN) techniques, innovative applications of deep learning in various domains, such as computer vision and natural language processing (NLP), are getting more mature and powerful (Huang et al., 2017; Vaswani et al., 2017; Szegedy et al., 2015; Howard et al., 2017; He et al., 2017; Krizhevsky et al., 2012). + +To improve the model accuracy, one of the most commonly used strategies is to add more layers into a network, which inevitably increases the number of weight parameters and activation values of a model. Today, it is typical to store weights and activations in the 32-bit IEEE single-precision floating-point format (FP32). Those 32-bit data accesses thus become an extremely heavy burden on the memory subsystem in a typical edge or AIoT device, which often has very limited memory capacity and bandwidth. Even for high-end GPU or dedicated network processing unit (NPU) based computing platforms, off-chip DRAM bandwidth is still a major performance bottleneck. + +To relieve the issue of memory bandwidth bottleneck, several attempts of various aspects have been made including (but not limited to) weight pruning (Li et al., 2017; Han et al., 2016), weight/activation quantization (Courbariaux et al., 2015; Hubara et al., 2017), and probably the most straightforward way: storing weights and activations in a shorter format (Koster et al., 2017). ¨ One trivial way to do so is to adopt the 16-bit IEEE half-precision floating-point format (FP16). An FP16 number consists of 1 sign bit, 5 exponent bits, and 10 fraction bits. In addition, Google proposed another 16-bit format, named Brain Floating-Point Format (BFP16), simply by truncating the lower half of the FP32 format (Kalamkar et al., 2019). Compared with FP16, BFP16 allows a significantly wider dynamic value range at the cost of 3-bit precision loss. Note that the exponent bias in all of the above formats is not a free design parameter. Conventionally, the value is solely determined by the exponent size. For example, for FP16 with 5-bit exponent, the exponent bias is automatically fixed to 15 $^ { \prime } ( 2 ^ { 5 - 1 } - 1 )$ . + +To make the data even shorter, 8-bit fixed-point signed/unsigned integer formats (INT8 and UINT8) are also broadly adopted. However, the 8-bit fixed-point format inherently has a narrower dynamic value range so that the model accuracy loss is usually not negligible even after extra symmetric or asymmetric quantization. As a consequence, there are a number of attempts concentrating on utilizing mixed-precision or pure 8-bit floating-point numbers in deep learning applications. + +Various techniques have been developed for mixed-precision training (Banner et al., 2018; Micikevicius et al., 2018; Das et al., 2018; Zhou et al., 2016). Moreover, recent studies proposed several training frameworks that produces weights only in 8-bit floating-point formats (Wang & Choi, 2018; Cambier et al., 2020; Sun et al., 2019). In these studies, the underlying 8-bit floating-point numbers in training and inference are represented in the format of FP8(1, 5, 2) or FP8(1, 4, 3), where the enclosed three parameters indicate the bit length of sign, exponent, and fraction, respectively. Note that 4 or 5 bits are essential for the exponent in their frameworks, or the corresponding dynamic range may not cover both weight and activation values well. Consequently, only 2 or 3 bits are available for fraction, which inevitably leads to lower accuracy. + +In this paper, we present an extremely flexible 8-bit floating-point (FFP8) number format. In FFP8, all parameters – the bit width of exponent/fraction, the exponent bias, and the presence of the sign bit – are configurable. Three major features of our inference methodology associated with the proposed FFP8 format are listed as follows. First, it is observed that both the maximum magnitude and the value distribution are quite dissimilar between weights and activations in most DNNs. It suggests the best exponent size and exponent bias for weights should be different from those for activations to achieve higher accuracy. Second, a large set of commonly-used activation functions always produce nonnegative outputs (e.g., ReLU). It implies that activations are actually unsigned if one of those activation functions is in use. Hence, it implies the sign bit is not required for those activations, which makes either exponent or fraction 1-bit longer. Note that even one bit can make a big impact since only 8 bits are available. Third, all aforementioned studies require their own sophisticated training frameworks to produce 8-bit floating-point models. Our flow does not. Our flow simply takes a model generated by any conventional FP32 training framework as the input. Then, it simply converts the given pre-trained FP32 model into an FFP8 model. + +The rest of this paper is organized as follows. Section 2 briefly introduces related work. In Section 3, we elaborate more on the proposed FFP8 format and how to properly convert a pre-trained FP32 model into an FFP8 one. Section 4 demonstrates the experimental results on various DNN models. The system and hardware design issues for the support of FFP8 numbers are discussed in Section 5. Finally, the concluding remarks are given in Section 6. + +# 2 RELATED WORK + +DNNs are becoming larger and more complicated, which means they require a bigger memory space and consume more energy during inference. As a result, it is getting harder to deploy them on systems with limited memory capacity and power budget (e.g., edge devices). (Han et al., 2016; Zhao et al., 2020; Horowitz, 2014) also demonstrated that off-chip DRAM access is responsible for a significantly big share of system power consumption. Hence, it remains an active research topic about how to reduce the memory usage for weights and activations. As mentioned in the previous section, one way to do so is to use short 8-bit floating-point number formats. + +Wang & Choi (2018) introduced a DNN training methodology using 8-bit floating-point numbers in FP8(1, 5, 2) format. The methodology features chunk-based accumulation and stochastic rounding methods for accuracy loss minimization. Besides, to achieve a better trade-off between precision and dynamic range during model training, Sun et al. (2019) proposed an improved methodology that utilizes two different 8-bit floating-point formats $- \operatorname { F P 8 } ( 1 , 4 , 3 )$ for forward propagation and $\mathrm { F P } 8 ( 1 , 5 , 2 )$ for backward propagation. Nevertheless, both methodologies fail to make a DNN model entirely in 8-bit numbers: the first and the last layers of the given model are still in 16-bit floating-point numbers; otherwise, the model suffers about $2 \%$ accuracy degradation. Cambier et al. (2020) then proposed the S2FP8 format, which allows a DNN model represented in 8-bit floating point numbers completely. By adding a scaling factor and a shifting factor, data can thus be well represented in $\mathrm { F P } 8 ( 1 , 5 , 2 )$ after proper shifting and squeezing operations, which eliminates the need of 16-bit floating point numbers. However, the S2FP8 format still results in about $1 \%$ accuracy drop in ResNet-50 (He et al., 2018). + +![](images/a4ce14514a2f36bf632014b4a5f9b2fc8fa4038370d1f491a4f9b8da40b70b60.jpg) +Figure 1: Various floating-point formats: (a) conventional ones, and (b) the proposed FFP8 format + +# 3 FLEXIBLE 8-BIT FLOATING-POINT (FFP8) FORMAT + +# 3.1 DEFINITION OF THE FFP8 FORMAT + +In this subsection, a flexible 8-bit floating-point (FFP8) format, which leads to more accurate inference outcomes of deep neural networks, is presented. A typical floating-point format consists of sign (s), exponent (e), and fraction $( f )$ fields, and the bit length of each field is specified as $x , y , z$ , respectively. Besides, one more parameter, exponent bias $( b )$ , is required to completely specify a floating-point number. Conventionally, $b$ is always implicitly set to $2 ^ { \overset { \cdot } { y } - 1 } - 1$ . In this paper, an $n$ -bit floating-point format is denoted as $( x , y , z , b )$ or $( x , y , z )$ , where $n = x + y + z$ . If $b$ is missing, the default value is implicitly used. + +Figure 1(a) illustrates one 16-bit FP16 format and two commonly used 8-bit formats: (1, 5, 2, 15) and (1, 4, 3, 7). Note that there is actually only one parameter, the bit width of the exponent $( y )$ , that can be freely chosen when defining a new $n$ -bit conventional floating-point format since $x$ is always 1, $z$ is always $n - y - 1$ , and $b$ is always $2 ^ { y - 1 } - 1$ . The above fact motivated us to think out-of-the-box and thus develop a more flexible format, named flexible 8-bit floating-point (FFP8) format and shown in Figure 1(b). In addition to the size of exponent $( y )$ , the FFP8 format offers two more parameters. First, the exponent bias $( b )$ is not necessarily equal to $2 ^ { y - 1 } - 1$ and can be set to any integer, which helps cover the value distributions of both weights and activations well in a shorter exponent size $( y )$ . Second, the sign bit is present or not (i.e., $x$ can be 0 or 1). Without the sign bit $( x = 0$ ), unsigned activations can be better represented in higher precision. To be more precise, there are only two restrictions on an $n$ -bit FFP8 format: 1) $x + y + z = n ;$ 2) $x$ must be 0 or 1. Next, we are about to show how the proposed FFP8 format improves the inference accuracy. + +# 3.2 WEIGHT DISTRIBUTION AND THE WAYS VARIOUS 8-BIT FORMATS COPE WITH IT + +Overall speaking, the magnitudes of weights in most DNN models are usually small. Take a popular image classification model VGG-16 (Simonyan & Zisserman, 2015) as an example, the maximum magnitude of weights in the whole model is less than 2. Figure 2 gives the overall weight distribution (in log scale) of VGG-16 trained via a conventional FP32 framework. Figure 2(a) illustrates how the conventional FP8(1, 4, 3) copes with those weights. The rectangle in red is called the range window, which specifies the value range that $\mathrm { F P } 8 ( 1 , 4 , 3 )$ can represent. The purple vertical dash line further partitions the window into the norm region (right side) and the denorm region (left side). The star marks the position where the weight of the maximum magnitude locates. Note that virtually the entire right half of the range window in Figure 2(a) covers no weights, while $9 . 6 \%$ of leftmost weights cannot be included in the range window. In order to contain almost every weight in a range window, FP8(1, 5, 2) can be selected alternatively, as shown in Figure 2(b). A bigger exponent size (4 to 5) results in a larger range window. However, there are only 4 $( 2 ^ { 2 } )$ instead of 8 $( 2 ^ { 3 } )$ representative values available for each $x$ in the norm region since the fraction size decreases from 3 to 2, which potentially results in a lower accuracy. Note that $\mathrm { F P } 8 ( 1 , 4 , 3 )$ and $\mathrm { F P } 8 ( 1 , 5 , 2 )$ are two most commonly used 8-bit floating point formats in the previous studies. + +With the help of the proposed FFP8 format, things can change a lot. Figure 2(c) shows what happens if FFP8(1, 4, 3, 15) is in use. The range window is of the same size as that in Figure 2(a) but is leftshifted by 8 positions due to the exponent bias is set to 15 instead of the default value 7. It is obvious that FFP8(1, 4, 3, 15) can better cope with weights in VGG-16 than $\mathrm { F P } 8 ( 1 , 4 , 3 )$ and FP(1, 5, 2). Furthermore, Figure 2(d) shows what if FFP8(1, 3, 4, 7) is in use. Comparing FFP8(1, 3, 4, 7) against FFP8(1, 4, 3, 15), weights in the norm region (over $3 5 \%$ ) are represented in 1-bit higher precision whereas $4 . 6 \%$ of leftmost weights are not included in the range window. However, those out-of-the-window weights can be regarded as some sort of pruned weights. That is, more investigation should be further conducted to determine whether FFP8(1, 3, 4, 7) or FFP8(1, 4, 3, 15) is better for the weights in VGG-16. One way or another, it is clear that the proposed FFP8 formats can achieve certain improvement that the conventional 8-bit formats cannot. + +![](images/7dd6076e42799f8a1b210cc7bfec813e0f774d3f5f676dc0da06c1295c39223b.jpg) +Figure 2: The range windows of various 8-bit floating-point formats versus the weight distribution of VGG-16; (a) in FP8(1, 4, 3), (b) in FP8(1, 5, 2), (c) in FFP8(1, 4, 3, 15), (d) in FFP8(1, 3, 4, 7) + +# 3.3 FFP8-BASED INFERENCE FRAMEWORK AND BEST-FIT FORMAT SELECTION + +In the proposed memory-efficient inference framework, weights and activations stored in off-chip memory are in FFP8 formats while operations inside the computing engine are still in FP32, which is a common strategy widely adopted in current state-of-the-art computing platforms (Intel, 2018; Nvidia, 2017). Hence, a systematic process, shown in Figure 3, is required to transform FP32 numbers into FFP8 ones in the following two scenarios: 1) quantizing pre-trained FP32 weights to FFP8 ones in advance, and 2) converting FP32 output activations into FFP8 ones before writing them back to off-chip memory. The process first prepares a list of 256 values that can be precisely represented by the specified FFP8 format. Then, the given FP32 inputs (weights or activations) are converted into FFP8 outputs using the round-to-nearest-even method. + +In order to explore best-fit 8-bit formats for weights and activations in VGG-16, we have made a set of attempts, and the results are summarized in Table 1. First, it is unwise to represent weights in $\mathrm { F P } ( 1 , 5 , 2 )$ . Though Figure 2(b) shows FP(1, 5, 2) can cover virtually all weights, the accuracy loss is serious due to its 2-bit extremely low precision. Next, $\mathrm { F P } ( 1 , 4 , 3 )$ performs much better than $\mathrm { F P } ( 1 , 5 , 2 )$ , which suggests one extra precision bit does make a notable improvement here though $9 . 6 \%$ smallest weights are clear to 0 (i.e., pruned away), as depicted in Figure 2(a). Moreover, the proposed FFP8 format can surely do better. $\mathrm { F F P 8 } ( 1 , 4 , 3 , 1 5 )$ further outperforms FP(1, 4, 3) because weights are better covered by its left-shifted range window, as shown in Figure 2(c). + +Inspired by the fact that $\mathrm { F P } ( 1 , 4 , 3 )$ outperforms $\mathrm { F P } ( 1 , 5 , 2 )$ , we further examine whether an even smaller exponent size can help or not. First, it is unwise to represent weights in $\mathrm { F P } 8 ( 1 , 3 , 4 )$ without modifying the default exponent bias $_ { ( = 3 ) }$ . Though the fraction size increases from 3 to 4, $6 4 . 1 \%$ leftmost weights are out of the range window, which is way too much. However, by setting the exponent bias to 7, which is equivalent to sliding the range window to the left by 4, the resultant FFP8(1, 3, 4, 7) successfully achieves a higher accuracy since it has a bigger 4-bit fraction size and only prunes away $4 . 6 \%$ smallest weights, as illustrated in Figure 2(d). + +![](images/ee11dd4faa248729c765769c1ef8aaf5238614624c00d21f44d0bc584667709b.jpg) +Figure 3: The process converting FP32 inputs into FFP8 outputs with the given format parameters + +Table 1: Top-1 and Top-5 accuracy of VGG-16 on ImageNet dataset (Deng et al., 2009) in various number formats; delta $( \Delta )$ indicates the accuracy drop as compared to FP32 + +
WeightActivationTop-1 (△)Top-5 (△)
FP32FP3271.59%90.38%
(1,5,2,15)(1,4,3,7)69.89% (-1.70%)89.29% (-1.09%)
(1,4,3,7)(1,4,3,7)70.86% (-0.73%)90.02% (-0.36%)
(1,4,3,15)(1,4,3,7)70.96% (-0.63%)90.10% (-0.28%)
(1,3,4,3)(1,4,3,7)70.18% (-1.41%)89.56% (-0.82%)
(1,3,4,7)(1,4,3,7)71.19% (-0.40%)90.12% (-0.26%)
(1,3,4,7)(1,4,3,7)+(0,4,4,7)71.19% (-0.40%)90.14% (-0.24%)
+ +In addition to weights, it is certainly worth finding out best-fit formats for activations as well. For those attempts made above, activations are always in FFP8(1, 4, 3, 7) for two reasons: 1) the overall distribution of activations is wider than that of weights, and 2) the maximum magnitude of activations is much bigger than that of weights. The detailed distribution of activations will be given in Section 3.4 later. Meanwhile, it is also worth noting that a large set of commonly used activation functions always produce nonnegative outputs, e.g., ReLU, ReLU6, and sigmoid. That is, if those outputs are represented in any signed format, a half of the code space is actually wasted. It may not be a problem for 32-bit and 16-bit formats with long enough fraction bits; however, it is indeed a serious issue for any 8-bit format, which merely has 256 available codes in total. Since VGG-16 utilizes ReLU as its activation function, it is feasible to select signed FFP8(1, 4, 3, 7) for the first layer and unsigned FFP8(0, 4, 4, 7) for all succeeding layers. It implies that 256 instead of 128 codes are available to represent those unsigned activations in all layers except for the first one. With no surprise, the accuracy is further improved since the range window remains untouched while the fraction size gains one extra bit. In the last configuration, the Top-1 accuracy loss is only $0 . 4 \%$ when compared against the FP32 baseline, as indicated in Table 1. + +# 3.4 LAYER-WISE OPTIMIZATION + +Figure 4 and Figure 5 illustrate several distributions of weights and activations in VGG-16, respectively. Each figure includes the distributions of one whole model and three selected individual layers. In Section 3.3, the best-fit FFP8 format is determined by the overall distribution of the whole model. Here we have three key observations from those distributions: 1) the distributions of weights are quite dissimilar to those of activations, 2) even the distributions across different layers are dissimilar for both weights and activations, and 3) the distribution of an individual layer is narrower than that of the whole model. The above observations clearly suggest that applying layer-wise optimization (LWO) properly on number format selection is very likely to improve the accuracy further. + +For instance, after comparing Figure $5 ( \mathrm { a } ) \& ( \mathrm { b } )$ , it is found that the distribution of the whole model is wider than that of the first layer, and the maximum log magnitudes of the whole model and the first layer are 8 and 1, respectively. As a consequence, selecting $\mathrm { F F P } 8 ( 1 , 3 , 4 , 6 )$ instead of FFP8(1, 4, 3, 7) for activations in the first layer can further increase the Top-1 accuracy by $0 . 1 9 \%$ (from $7 1 . 1 9 \%$ to $7 1 . 3 8 \%$ ), as indicated in Table 2. Next, we examine a new configuration that makes weights of all layers are in FFP8(1, 2, 5, 3). It is obvious an unwise attempt since $3 7 . 6 \%$ leftmost weights are out of the range window according to Figure 4(a). However, it may not be a bad idea to use FFP8(1, 2, 5, 5) in Layer 6 and FFP8(1, 2, 5, 6) in the last layer because only $6 . 9 \%$ and $5 . 3 \%$ smallest weights are out of the range window respectively according to Figure $4 ( \mathrm { c } ) \& ( \mathrm { d } )$ . Therefore, we examine another new configuration that makes weights of all layers in $\mathrm { F F P } 8 ( 1 , 2 , 5 , * )$ . Here the asterisk $( \ast )$ represents the largest possible exponent bias, which ensures that the maximum weight is still inside the range window. This LWO on weights successfully increases the Top-1 accuracy by $0 . 1 \%$ (from $7 1 . 3 8 \%$ to $7 1 . 4 8 \%$ ). Similarly, we can apply the LWO on activations, which again raises the Top-5 accuracy by $0 . 0 5 \%$ . The selected FFP8 format of each layer after LWO is reported in Table 3. + +Previous studies usually choose $\mathrm { F P } 8 ( 1 , 4 , 3 )$ or $\mathrm { F P } 8 ( 1 , 5 , 2 )$ formats because the exponent size has to be large enough to cover most weights and activations at the cost of an even smaller fraction size. However, the exponent size for weights can be as small as 2 after LWO in our flow. Consequently, the larger 5-bit fraction size does help in accuracy improvement, as indicated in Table 2. + +Note that the Top-1 accuracy achieved by the final configuration, which applies layer-wise optimization on both weights and activations, is merely $0 . 1 1 \%$ lower as compared to that of the FP32 baseline $( 7 1 . 4 8 \%$ vs. $7 1 . 5 9 \%$ ). More importantly, model retraining is not applied yet. In other words, all the accuracy improvements made so far are simply from properly re-expressing weights and activations of a pre-trained FP32 model in their best-fit FFP8 formats. + +![](images/21188150164f237a39626163cff2bf1d4af3aeae56669848881ed879c8bd5df6.jpg) +Figure 4: Weight distributions of (a) whole model, (b) first layer, (c) Layer 6, (d) last layer + +![](images/4db1795b0969e5913dd60342cf777d83c28f7e085bf9522442eca015a01afa91.jpg) +Figure 5: Activation distributions of (a) whole model, (b) first layer, (c) Layer 6, (d) last layer + +Table 2: Top-1 and Top-5 accuracy of VGG-16 after layer-wise optimization + +
WeightActivationTop-1 (△)Top-5 (△)
FP32FP3271.59%90.38%
(1,3,4,7)(1,4,3,7)+(0,4,4,7)71.19% (-0.40%)90.14% (-0.24%)
(1,3,4,7)(1,3,4,6)+(0,4,4,7)71.38% (-0.21%)90.33% (-0.05%)
(1,2,5,3)(1,3,4,6)+(0,4,4,7)71.24% (-0.35%)90.22% (-0.16%)
(1,2,5,*)(1,3,4,6)+(0,4,4,7)71.48% (-0.11%)90.27% (-0.11%)
(1,2,5,*)(1,3,4,6)+(0,4,4,*)71.48% (-0.11%)90.32% (-0.06%)
+ +Table 3: The FFP8 format of each layer in VGG-16 after layer-wise optimization + +
LayerWeightActivationLayerWeightActivation
1(1,2,5,3)(1,3,4,6)8(1,2,5,5)(0,4,4,8)
2(1,2,5,4)(0,4,4,11)9(1,2,5,5)(0,4,4,8)
3(1,2,5,4)(0,4,4,10)10(1,2,5,6)(0,4,4,8)
4(1,2,5,5)(0,4,4,10)11(1,2,5,5)(0,4,4,7)
5(1,2,5,4)(0,4,4,9)12(1,2,5,5)(0,4,4,7)
6(1,2,5,5)(0,4,4,9)13(1,2,5,6)(0,4,4,8)
7(1,2,5,4)(0,4,4,9)
+ +# 3.5 ACCURACY IMPROVEMENT VIA MODEL RETRAINING + +All FFP8 weights in our previous experiments are simply converted from pre-trained weights generated from a typical FP32 training framework. However, it is reported that quantization-aware model retraining can usually improve the accuracy (Jacob et al., 2017), which motivates us to check whether it can successfully apply to our work. We first train the ResNet-18 model in an FP32 framework and the corresponding Top-1 accuracy is $6 9 . 7 6 \%$ . Next, an FFP8-based inference is performed under the following settings: 1) all weights are in $\mathrm { F F P } 8 ( 1 , 3 , 4 , 7 ) , 2 )$ activations of the first layer are in FFP8(1, 3, 4, 6), and 3) activations of the other layers are in $\mathrm { F F P 8 } ( 0 , 4 , 4 , 7 )$ . The Top-1 accuracy for the above configuration is down to $6 9 . 4 4 \%$ . Then, a quantization-aware retraining process of 15 epochs is applied and the resultant Top-1 accuracy goes back to $6 9 . 7 6 \%$ . Note that the retraining is done simply in a typical FP32 framework without the need of special training skills. + +# 4 EXPERIMENTAL RESULTS ON VARIOUS DNN MODELS + +In this section, we intend to demonstrate that the proposed FFP8 format constantly performs well in various DNN models in addition to VGG-16. Table 4 reports the accuracy results of various models under different format configurations. Configuration A gives the results of the FP32 baseline. Configuration B utilizes a conventional FP8(1, 4, 3) format with a default exponent bias (7), which incurs a penalty of roughly $1 \%$ drop on Top-1 accuracy. With the help of the FFP8 format, Configuration C adopts the best-fit format for weights, which effectively reduces the Top-1 accuracy loss to $0 . 4 { \sim } 0 . 7 5 \%$ . Configuration D further selects two best-fit formats for activations (the signed one for the first layer and the unsigned one for the rest), which minimizes the Top-1 accuracy loss to $0 . 3 \%$ . Note that neither layer-wise optimization nor model retraining is even applied for this achievement. + +Table 4: Accuracy results of various models under different format configurations + +
Cfg.WeightActivationVGG-16ResNet-50ResNet-34ResNet-18
Top-1/Top-5Top-1/Top-5Top-1/Top-5Top-1/Top-5
AFP32FP3271.59 /90.3876.13/92.8673.31/91.4269.76/89.08
B(1,4,3,7)(1,4,3,7)70.86 /90.0275.24 /92.5272.39 /90.9768.70 / 88.50
C(1,3,4,7)(1,4,3,7)71.19 /90.1275.38 /92.6872.81 /91.1669.25 /88.80
D(1,3,4,7)(1,3,4,6)+(0,4,4,7)71.38 / 90.3375.85 /92.8173.12 /91.3369.44 / 88.93
+ +![](images/a47b50b7569e8f7830a2fdff1eda89b1c2a135c7cc9df8ab55b141b9c3e88e1a.jpg) +Figure 6: Memory-efficient system architectures: off-chip external data (a) in BFP16, (b) in FFP8; and (c) an FFP8 to FP32 hardware converter + +In addition to image classification, we also want to know whether the proposed FFP8 format performs equally well in other application domains. Here, we examine two more applications: semantic segmentation and ECG check. First, FCN32s is a popular CNN model for semantic segmentation (Long et al., 2014). If FCN32s is in FP32, the mIOU is $6 3 . 6 3 \%$ using the VOC2011 dataset (Everingham et al.). Alternatively, if all the weights are in FFP8(1, 4, 3, 14) and all the activations are in FFP8(1, 4, 3, 2), the resultant mIOU would be $6 3 . 4 5 \%$ , a slight drop of $0 . 1 8 \%$ . Second, an LSTM model for ECG check (Physionet, 2017), has also been tested. If the LSTM model is in FP32, the check accuracy is $8 1 . 1 2 \%$ . If all the weights are in $\mathrm { F F P } 8 ( 1 , 3 , 4 , 6 )$ , the first-layer activations are in FFP8(1, 3, 4, 5), and activations in the other layer are in FFP8(1, 4, 3, 16), the resultant accuracy would be $8 1 . 5 3 \%$ , an accuracy gain of $0 . 4 1 \%$ . The experimental results once again demonstrate that FFP8 performs very well in these two categories as well. + +# 5 ASPECTS OF SYSTEM AND HARDWARE + +The target of this work is to develop a memory-efficient inference system, especially for those with limited memory capacity and bandwidth. A current state-of-the-art system, proposed by Nvidia and Intel, has successfully cut the required memory size and traffic through representing external data (weights or activations) in BFP16/INT8 instead of FP32, as illustrated in Figure 6(a) (Intel, 2018; 2019; Nvidia, 2017; 2020). It not only reduces the memory size, alleviates the performance bottleneck due to memory bandwidth limitation but also saves a significant amount of energy due to fewer power-consuming external memory access operations. To preserve the computation accuracy at the same time, external BFP16 data are converted to FP32 data right before entering the FP32 fused-multiply-add (FMA) unit. That is, internal computations can all be in FP32. Those FP32 data are converted back to BFP16 only if they are about to be written back to external memory. + +In this work, we propose a system architecture that is very similar to the previous one, as shown in Figure 6(b). The key difference is that external data are in FFP8 instead of BFP16, which implies the proposed system merely demands a quarter of the memory size and throughput required by today’s FP32-based counterparts. + +It is easy to convert a BFP16 number into its FP32 equivalent by concatenating a 16-bit pattern of all 0s at its least significant end. In fact, it is also easy to transform an FFP8 number to its FP32 equivalent via a converter, as depicted in Figure 6(c). Few register bits are allocated to store the current format settings of $x , y ,$ , and $b$ within the converter. Then, $x$ is used to recover the sign bit; the biased exponent and fraction can be extracted via $x$ and $y$ ; finally the exponent can be further corrected by the exponent bias $b$ . Hence, it is apparent that the required hardware logic for the converter is indeed minimal and the extra hardware cost is truly minor as well. Furthermore, updates of those format registers are extremely infrequent. Even the layer-wise optimization is applied, those registers are only modified at the start of each layer. In other words, virtually no runtime overhead is imposed due to those register updates. + +# 6 CONCLUSION + +In this work, we propose the flexible 8-bit floating-point (FFP8) format for accurate and memoryefficient inference of deep neural networks. Our FFP8 format offers three adjustable options: 1) the size of exponent/fraction field, 2) the value of exponent bias, and 3) the presence of the sign bit, whereas those rigid conventional formats simply leave nothing. In this paper, we explain how the exponent size and bias jointly define the representable value range of a given FFP8 format. We also demonstrate how to explore the best-fit signed/unsigned FFP8 formats for weights and activations to achieve more accurate inference outcomes. Besides, a layer-wise optimization flow, which discovers the best-fit formats for each individual layer, is presented to further improve the accuracy. A model retraining methodology that can be carried out in typical FP32 frameworks without the need of special training skills is also introduced. The experimental results on various DNN models show that the proposed FFP8-based inference flow achieves an extremely low accuracy loss of $0 . 1 \% \sim 0 . 3 \%$ as compared to the FP32 baseline even without model retraining. Moreover, we also show that the extra hardware for supporting the FFP8 format is minimal. Therefore, it is conclusive that the proposed FFP8-based inference framework should be a better solution for those computing systems with limited memory capacity and bandwidth, e.g., edge and AIoT devices. + +# REFERENCES + +Ron Banner, Itay Hubara, Elad Hoffer, and Daniel Soudry. Scalable methods for 8-bit training of neural networks. In Advances in Neural Information Processing Systems, 2018. + +Leopold Cambier, Anahita Bhiwandiwalla, Ting Gong, Mehran Nekuii, Oguz HElibol, and Hanlin Tang. Shifted and squeezed 8-bit floating point format for low-precision training of deep neural networks. In International Conference on Learning Representations, 2020. + +Matthieu Courbariaux, Yoshua Bengio, and Jean-Pierre David. Binaryconnect: Training deep neural networks with binary weights during propagations. In Advances in neural information processing systems, 2015. + +Dipankar Das, Naveen Mellempudi, Dheevatsa Mudigere, Dhiraj Kalamkar, Sasikanth Avancha, Kunal Banerjee, Srinivas Sridharan, and et al. Mixed precision training of convolutional neural networks using integer operations. In International Conference on Learning Representations, 2018. + +Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Fei-Fei Li. Imagenet: A large-scale hierarchical image database. In Conference on Computer Vision and Pattern Recognition, 2009. + +M. Everingham, L. Van Gool, C. K. I. Williams, J. Winn, and A. Zisserman. The PASCAL Visual Object Classes Challenge 2011 (VOC2011) Results. http://www.pascal-network. org/challenges/VOC/voc2011/workshop/index.html. + +Song Han, Mao Huizi, and Dally William J. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. In International Conference on Learning Representations, 2016. + +Song Han, Xingyu Liu, Huizi Mao, Jing Pu, Ardavan Pedram, Mark A. Horowitz, and William J. Dally. EIE: efficient inference engine on compressed deep neural network. arXiv e-prints, art. arXiv:1602.01528, February 2016. + +Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Conference on Computer Vision and Pattern Recognition, 2018. + +Xiangnan He, Lizi Liao, Hanwang Zhang, Liqiang Nie, Xia Hu, and Tat-Seng Chua. Neural collaborative filtering. arXiv e-prints, art. arXiv:1708.05031, August 2017. + +M. Horowitz. Computing’s energy problem (and what we can do about it). In 2014 IEEE International Solid-State Circuits Conference Digest of Technical Papers (ISSCC), pp. 10–14, 2014. doi: 10.1109/ISSCC.2014.6757323. + +Andrew G. Howard, Menglong Zhu, Bo Chen, Dmitry Kalenichenko, Weijun Wang, Tobias Weyand, Marco Andreetto, and Hartwig Adam. MobileNets: efficient convolutional neural networks for mobile vision applications. arXiv e-prints, art. arXiv:1704.04861, April 2017. + +Gao Huang, Liu Zhuang, Laurens van der Maaten, and Kilian Q. Weinberger. Densely connected convolutional networks. In Conference on Computer Vision and Pattern Recognition, 2017. + +Itay Hubara, Matthieu Courbariaux, Daniel Soudry, Ran El-Yaniv, and Yoshua Bengio. Quantized neural networks: Training neural networks with low precision weights and activations. In The Journal of Machine Learning Research, 2017. + +Intel. Bfloat16 – hardware numerics definition white paper. 2018. + +Intel. Leadership performance with 2nd-generation intel xeon scalable processors, 2019. + +Benoit Jacob, Skirmantas Kligys, Bo Chen, Menglong Zhu, Matthew Tang, Andrew Howard, Hartwig Adam, and Dmitry Kalenichenko. Quantization and training of neural networks for efficient integer-arithmetic-only inference. arXiv e-prints, art. arXiv:1712.05877, December 2017. + +Dhiraj Kalamkar, Dheevatsa Mudigere, Naveen Mellempudi, Dipankar Das, Kunal Banerjee, Sasikanth Avancha, Dharma Teja Vooturi, Nataraj Jammalamadaka, Jianyu Huang, Hector Yuen, Jiyan Yang, Jongsoo Park, Alexander Heinecke, Evangelos Georganas, Sudarshan Srinivasan, Abhisek Kundu, Misha Smelyanskiy, Bharat Kaul, and Pradeep Dubey. A study of bfloat16 for deep learning training. arXiv e-prints, art. arXiv:1905.12322, May 2019. + +Urs Koster, Tristan Webb, Xin Wang, Marcel Nassar, Arjun K Bansal, William Constable, Oguz ¨ Elibol, Scott Gray, Stewart Hall, and Luke et al Hornof. Flexpoint: An adaptive numerical format for efficient training of deep neural networks. In Advances in neural information processing systems, 2017. + +Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, 2012. + +Hao Li, Asim Kadav, Igor Durdanovic, Graf. Samet, and Hans Pete. Pruning filters for effecient convnets. In International Conference on Learning Representations, 2017. + +Jonathan Long, Evan Shelhamer, and Trevor Darrell. Fully convolutional networks for semantic segmentation. arXiv e-prints, art. arXiv:1411.4038, November 2014. + +Paulius Micikevicius, Sharan Narang, Jonah Alben, Gregory Diamos, Erich Elsen, David Garcia, and Boris et al Ginsburg. Mixed precision training. In International Conference on Learning Representations, 2018. + +Nvidia. Nvidia tesla v100 gpu architecture, 2017. + +Nvidia. Nvidia a100 tensor core gpu architecture, 2020. + +Physionet. Af classification from a short single lead ecg recording - the physionet computing in cardiology challenge. https://physionet.org/content/challenge-2017/1.0. 0/, 2017. + +Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. In International Conference on Learning Representations, 2015. + +Xiao Sun, Jungwook Choi, Chia-Yu Chen, Naigang Wang, Swagath Venkataramani, Vijayalakshmi Srinivasan, Xiaodong Cui, Wei Zhang, and Kailash Gopalakrishnan. Hybrid 8-bit floating point (hfp8) training and inference for deep neural networks. In Neural Information Processing Systems, 2019. + +Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In Computer Vision and Pattern Recognition, 2015. + +Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Erhan, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, 2017. + +Naigang Wang and Jungw Choi. Training deep neural networks with 8-bit floating point numbers. In Advances in Neural Information Processing Systems, 2018. + +Yang Zhao, Xiaohan Chen, Yue Wang, Chaojian Li, Haoran You, Yonggan Fu, Yuan Xie, Zhangyang Wang, and Yingyan Lin. Smartexchange: Trading higher-cost memory storage/access for lowercost computation. In International Symposium on Computer Architecture (ISCA), 2020. + +Shuchang Zhou, Zekun Ni, He Wen, Yuxin Wu, and Yuheng Zou. Dorefa-net: Training low bitwidth convolutional neural networks with low bitwidth gradients. In CoRR, abs/1606.06160, 2016. \ No newline at end of file diff --git a/parse/train/9sF3n8eAco/9sF3n8eAco_content_list.json b/parse/train/9sF3n8eAco/9sF3n8eAco_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..07b118d8cd5b562547414e4c0f690a3ef97cdc6e --- /dev/null +++ b/parse/train/9sF3n8eAco/9sF3n8eAco_content_list.json @@ -0,0 +1,1050 @@ +[ + { + "type": "text", + "text": "ALL-YOU-CAN-FIT 8-BIT FLEXIBLE FLOATINGPOINT FORMAT FOR ACCURATE AND MEMORYEFFICIENT INFERENCE OF DEEP NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 195, + 398, + 223 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 260, + 544, + 275 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Modern deep neural network (DNN) models generally require a huge amount of weight and activation values to achieve good inference outcomes. Those data inevitably demand a massive off-chip memory capacity/bandwidth, and the situation gets even worse if they are represented in high-precision floating-point formats. Effort has been made for representing those data in different 8-bit floating-point formats, nevertheless, a notable accuracy loss is still unavoidable. In this paper we introduce an extremely flexible 8-bit floating-point (FFP8) format whose defining factors – the bit width of exponent/fraction field, the exponent bias, and even the presence of the sign bit – are all configurable. We also present a methodology to properly determine those factors so that the accuracy of model inference can be maximized. The foundation of this methodology is based on a key observation – both the maximum magnitude and the value distribution are quite dissimilar between weights and activations in most DNN models. Experimental results demonstrate that the proposed FFP8 format achieves an extremely low accuracy loss of $0 . 1 \\% \\sim 0 . 3 \\%$ for several representative image classification models even without the need of model retraining. Besides, it is easy to turn a classical floating-point processing unit into an FFP8-compliant one, and the extra hardware cost is minor. ", + "bbox": [ + 233, + 291, + 764, + 527 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 178, + 558, + 336, + 573 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "With the rapid progress of deep neural network (DNN) techniques, innovative applications of deep learning in various domains, such as computer vision and natural language processing (NLP), are getting more mature and powerful (Huang et al., 2017; Vaswani et al., 2017; Szegedy et al., 2015; Howard et al., 2017; He et al., 2017; Krizhevsky et al., 2012). ", + "bbox": [ + 174, + 589, + 823, + 645 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To improve the model accuracy, one of the most commonly used strategies is to add more layers into a network, which inevitably increases the number of weight parameters and activation values of a model. Today, it is typical to store weights and activations in the 32-bit IEEE single-precision floating-point format (FP32). Those 32-bit data accesses thus become an extremely heavy burden on the memory subsystem in a typical edge or AIoT device, which often has very limited memory capacity and bandwidth. Even for high-end GPU or dedicated network processing unit (NPU) based computing platforms, off-chip DRAM bandwidth is still a major performance bottleneck. ", + "bbox": [ + 174, + 652, + 823, + 750 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To relieve the issue of memory bandwidth bottleneck, several attempts of various aspects have been made including (but not limited to) weight pruning (Li et al., 2017; Han et al., 2016), weight/activation quantization (Courbariaux et al., 2015; Hubara et al., 2017), and probably the most straightforward way: storing weights and activations in a shorter format (Koster et al., 2017). ¨ One trivial way to do so is to adopt the 16-bit IEEE half-precision floating-point format (FP16). An FP16 number consists of 1 sign bit, 5 exponent bits, and 10 fraction bits. In addition, Google proposed another 16-bit format, named Brain Floating-Point Format (BFP16), simply by truncating the lower half of the FP32 format (Kalamkar et al., 2019). Compared with FP16, BFP16 allows a significantly wider dynamic value range at the cost of 3-bit precision loss. Note that the exponent bias in all of the above formats is not a free design parameter. Conventionally, the value is solely determined by the exponent size. For example, for FP16 with 5-bit exponent, the exponent bias is automatically fixed to 15 $^ { \\prime } ( 2 ^ { 5 - 1 } - 1 )$ . ", + "bbox": [ + 174, + 757, + 825, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To make the data even shorter, 8-bit fixed-point signed/unsigned integer formats (INT8 and UINT8) are also broadly adopted. However, the 8-bit fixed-point format inherently has a narrower dynamic value range so that the model accuracy loss is usually not negligible even after extra symmetric or asymmetric quantization. As a consequence, there are a number of attempts concentrating on utilizing mixed-precision or pure 8-bit floating-point numbers in deep learning applications. ", + "bbox": [ + 174, + 103, + 823, + 174 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Various techniques have been developed for mixed-precision training (Banner et al., 2018; Micikevicius et al., 2018; Das et al., 2018; Zhou et al., 2016). Moreover, recent studies proposed several training frameworks that produces weights only in 8-bit floating-point formats (Wang & Choi, 2018; Cambier et al., 2020; Sun et al., 2019). In these studies, the underlying 8-bit floating-point numbers in training and inference are represented in the format of FP8(1, 5, 2) or FP8(1, 4, 3), where the enclosed three parameters indicate the bit length of sign, exponent, and fraction, respectively. Note that 4 or 5 bits are essential for the exponent in their frameworks, or the corresponding dynamic range may not cover both weight and activation values well. Consequently, only 2 or 3 bits are available for fraction, which inevitably leads to lower accuracy. ", + "bbox": [ + 174, + 180, + 825, + 305 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we present an extremely flexible 8-bit floating-point (FFP8) number format. In FFP8, all parameters – the bit width of exponent/fraction, the exponent bias, and the presence of the sign bit – are configurable. Three major features of our inference methodology associated with the proposed FFP8 format are listed as follows. First, it is observed that both the maximum magnitude and the value distribution are quite dissimilar between weights and activations in most DNNs. It suggests the best exponent size and exponent bias for weights should be different from those for activations to achieve higher accuracy. Second, a large set of commonly-used activation functions always produce nonnegative outputs (e.g., ReLU). It implies that activations are actually unsigned if one of those activation functions is in use. Hence, it implies the sign bit is not required for those activations, which makes either exponent or fraction 1-bit longer. Note that even one bit can make a big impact since only 8 bits are available. Third, all aforementioned studies require their own sophisticated training frameworks to produce 8-bit floating-point models. Our flow does not. Our flow simply takes a model generated by any conventional FP32 training framework as the input. Then, it simply converts the given pre-trained FP32 model into an FFP8 model. ", + "bbox": [ + 174, + 313, + 825, + 506 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The rest of this paper is organized as follows. Section 2 briefly introduces related work. In Section 3, we elaborate more on the proposed FFP8 format and how to properly convert a pre-trained FP32 model into an FFP8 one. Section 4 demonstrates the experimental results on various DNN models. The system and hardware design issues for the support of FFP8 numbers are discussed in Section 5. Finally, the concluding remarks are given in Section 6. ", + "bbox": [ + 176, + 513, + 823, + 583 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 606, + 344, + 622 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "DNNs are becoming larger and more complicated, which means they require a bigger memory space and consume more energy during inference. As a result, it is getting harder to deploy them on systems with limited memory capacity and power budget (e.g., edge devices). (Han et al., 2016; Zhao et al., 2020; Horowitz, 2014) also demonstrated that off-chip DRAM access is responsible for a significantly big share of system power consumption. Hence, it remains an active research topic about how to reduce the memory usage for weights and activations. As mentioned in the previous section, one way to do so is to use short 8-bit floating-point number formats. ", + "bbox": [ + 174, + 638, + 825, + 736 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Wang & Choi (2018) introduced a DNN training methodology using 8-bit floating-point numbers in FP8(1, 5, 2) format. The methodology features chunk-based accumulation and stochastic rounding methods for accuracy loss minimization. Besides, to achieve a better trade-off between precision and dynamic range during model training, Sun et al. (2019) proposed an improved methodology that utilizes two different 8-bit floating-point formats $- \\operatorname { F P 8 } ( 1 , 4 , 3 )$ for forward propagation and $\\mathrm { F P } 8 ( 1 , 5 , 2 )$ for backward propagation. Nevertheless, both methodologies fail to make a DNN model entirely in 8-bit numbers: the first and the last layers of the given model are still in 16-bit floating-point numbers; otherwise, the model suffers about $2 \\%$ accuracy degradation. Cambier et al. (2020) then proposed the S2FP8 format, which allows a DNN model represented in 8-bit floating point numbers completely. By adding a scaling factor and a shifting factor, data can thus be well represented in $\\mathrm { F P } 8 ( 1 , 5 , 2 )$ after proper shifting and squeezing operations, which eliminates the need of 16-bit floating point numbers. However, the S2FP8 format still results in about $1 \\%$ accuracy drop in ResNet-50 (He et al., 2018). ", + "bbox": [ + 174, + 743, + 825, + 922 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/a4ce14514a2f36bf632014b4a5f9b2fc8fa4038370d1f491a4f9b8da40b70b60.jpg", + "image_caption": [ + "Figure 1: Various floating-point formats: (a) conventional ones, and (b) the proposed FFP8 format " + ], + "image_footnote": [], + "bbox": [ + 174, + 99, + 820, + 194 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 FLEXIBLE 8-BIT FLOATING-POINT (FFP8) FORMAT ", + "text_level": 1, + "bbox": [ + 176, + 246, + 637, + 262 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 DEFINITION OF THE FFP8 FORMAT ", + "text_level": 1, + "bbox": [ + 176, + 279, + 459, + 292 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this subsection, a flexible 8-bit floating-point (FFP8) format, which leads to more accurate inference outcomes of deep neural networks, is presented. A typical floating-point format consists of sign (s), exponent (e), and fraction $( f )$ fields, and the bit length of each field is specified as $x , y , z$ , respectively. Besides, one more parameter, exponent bias $( b )$ , is required to completely specify a floating-point number. Conventionally, $b$ is always implicitly set to $2 ^ { \\overset { \\cdot } { y } - 1 } - 1$ . In this paper, an $n$ -bit floating-point format is denoted as $( x , y , z , b )$ or $( x , y , z )$ , where $n = x + y + z$ . If $b$ is missing, the default value is implicitly used. ", + "bbox": [ + 174, + 306, + 825, + 404 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Figure 1(a) illustrates one 16-bit FP16 format and two commonly used 8-bit formats: (1, 5, 2, 15) and (1, 4, 3, 7). Note that there is actually only one parameter, the bit width of the exponent $( y )$ , that can be freely chosen when defining a new $n$ -bit conventional floating-point format since $x$ is always 1, $z$ is always $n - y - 1$ , and $b$ is always $2 ^ { y - 1 } - 1$ . The above fact motivated us to think out-of-the-box and thus develop a more flexible format, named flexible 8-bit floating-point (FFP8) format and shown in Figure 1(b). In addition to the size of exponent $( y )$ , the FFP8 format offers two more parameters. First, the exponent bias $( b )$ is not necessarily equal to $2 ^ { y - 1 } - 1$ and can be set to any integer, which helps cover the value distributions of both weights and activations well in a shorter exponent size $( y )$ . Second, the sign bit is present or not (i.e., $x$ can be 0 or 1). Without the sign bit $( x = 0$ ), unsigned activations can be better represented in higher precision. To be more precise, there are only two restrictions on an $n$ -bit FFP8 format: 1) $x + y + z = n ;$ 2) $x$ must be 0 or 1. Next, we are about to show how the proposed FFP8 format improves the inference accuracy. ", + "bbox": [ + 173, + 410, + 825, + 577 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 WEIGHT DISTRIBUTION AND THE WAYS VARIOUS 8-BIT FORMATS COPE WITH IT ", + "text_level": 1, + "bbox": [ + 176, + 597, + 779, + 612 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Overall speaking, the magnitudes of weights in most DNN models are usually small. Take a popular image classification model VGG-16 (Simonyan & Zisserman, 2015) as an example, the maximum magnitude of weights in the whole model is less than 2. Figure 2 gives the overall weight distribution (in log scale) of VGG-16 trained via a conventional FP32 framework. Figure 2(a) illustrates how the conventional FP8(1, 4, 3) copes with those weights. The rectangle in red is called the range window, which specifies the value range that $\\mathrm { F P } 8 ( 1 , 4 , 3 )$ can represent. The purple vertical dash line further partitions the window into the norm region (right side) and the denorm region (left side). The star marks the position where the weight of the maximum magnitude locates. Note that virtually the entire right half of the range window in Figure 2(a) covers no weights, while $9 . 6 \\%$ of leftmost weights cannot be included in the range window. In order to contain almost every weight in a range window, FP8(1, 5, 2) can be selected alternatively, as shown in Figure 2(b). A bigger exponent size (4 to 5) results in a larger range window. However, there are only 4 $( 2 ^ { 2 } )$ instead of 8 $( 2 ^ { 3 } )$ representative values available for each $x$ in the norm region since the fraction size decreases from 3 to 2, which potentially results in a lower accuracy. Note that $\\mathrm { F P } 8 ( 1 , 4 , 3 )$ and $\\mathrm { F P } 8 ( 1 , 5 , 2 )$ are two most commonly used 8-bit floating point formats in the previous studies. ", + "bbox": [ + 174, + 625, + 825, + 833 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "With the help of the proposed FFP8 format, things can change a lot. Figure 2(c) shows what happens if FFP8(1, 4, 3, 15) is in use. The range window is of the same size as that in Figure 2(a) but is leftshifted by 8 positions due to the exponent bias is set to 15 instead of the default value 7. It is obvious that FFP8(1, 4, 3, 15) can better cope with weights in VGG-16 than $\\mathrm { F P } 8 ( 1 , 4 , 3 )$ and FP(1, 5, 2). Furthermore, Figure 2(d) shows what if FFP8(1, 3, 4, 7) is in use. Comparing FFP8(1, 3, 4, 7) against FFP8(1, 4, 3, 15), weights in the norm region (over $3 5 \\%$ ) are represented in 1-bit higher precision whereas $4 . 6 \\%$ of leftmost weights are not included in the range window. However, those out-of-the-window weights can be regarded as some sort of pruned weights. That is, more investigation should be further conducted to determine whether FFP8(1, 3, 4, 7) or FFP8(1, 4, 3, 15) is better for the weights in VGG-16. One way or another, it is clear that the proposed FFP8 formats can achieve certain improvement that the conventional 8-bit formats cannot. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/7dd6076e42799f8a1b210cc7bfec813e0f774d3f5f676dc0da06c1295c39223b.jpg", + "image_caption": [ + "Figure 2: The range windows of various 8-bit floating-point formats versus the weight distribution of VGG-16; (a) in FP8(1, 4, 3), (b) in FP8(1, 5, 2), (c) in FFP8(1, 4, 3, 15), (d) in FFP8(1, 3, 4, 7) " + ], + "image_footnote": [], + "bbox": [ + 171, + 99, + 820, + 371 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 448, + 825, + 517 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 FFP8-BASED INFERENCE FRAMEWORK AND BEST-FIT FORMAT SELECTION ", + "text_level": 1, + "bbox": [ + 176, + 545, + 738, + 560 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the proposed memory-efficient inference framework, weights and activations stored in off-chip memory are in FFP8 formats while operations inside the computing engine are still in FP32, which is a common strategy widely adopted in current state-of-the-art computing platforms (Intel, 2018; Nvidia, 2017). Hence, a systematic process, shown in Figure 3, is required to transform FP32 numbers into FFP8 ones in the following two scenarios: 1) quantizing pre-trained FP32 weights to FFP8 ones in advance, and 2) converting FP32 output activations into FFP8 ones before writing them back to off-chip memory. The process first prepares a list of 256 values that can be precisely represented by the specified FFP8 format. Then, the given FP32 inputs (weights or activations) are converted into FFP8 outputs using the round-to-nearest-even method. ", + "bbox": [ + 174, + 575, + 825, + 702 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In order to explore best-fit 8-bit formats for weights and activations in VGG-16, we have made a set of attempts, and the results are summarized in Table 1. First, it is unwise to represent weights in $\\mathrm { F P } ( 1 , 5 , 2 )$ . Though Figure 2(b) shows FP(1, 5, 2) can cover virtually all weights, the accuracy loss is serious due to its 2-bit extremely low precision. Next, $\\mathrm { F P } ( 1 , 4 , 3 )$ performs much better than $\\mathrm { F P } ( 1 , 5 , 2 )$ , which suggests one extra precision bit does make a notable improvement here though $9 . 6 \\%$ smallest weights are clear to 0 (i.e., pruned away), as depicted in Figure 2(a). Moreover, the proposed FFP8 format can surely do better. $\\mathrm { F F P 8 } ( 1 , 4 , 3 , 1 5 )$ further outperforms FP(1, 4, 3) because weights are better covered by its left-shifted range window, as shown in Figure 2(c). ", + "bbox": [ + 174, + 708, + 823, + 819 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Inspired by the fact that $\\mathrm { F P } ( 1 , 4 , 3 )$ outperforms $\\mathrm { F P } ( 1 , 5 , 2 )$ , we further examine whether an even smaller exponent size can help or not. First, it is unwise to represent weights in $\\mathrm { F P } 8 ( 1 , 3 , 4 )$ without modifying the default exponent bias $_ { ( = 3 ) }$ . Though the fraction size increases from 3 to 4, $6 4 . 1 \\%$ leftmost weights are out of the range window, which is way too much. However, by setting the exponent bias to 7, which is equivalent to sliding the range window to the left by 4, the resultant FFP8(1, 3, 4, 7) successfully achieves a higher accuracy since it has a bigger 4-bit fraction size and only prunes away $4 . 6 \\%$ smallest weights, as illustrated in Figure 2(d). ", + "bbox": [ + 174, + 825, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/ee11dd4faa248729c765769c1ef8aaf5238614624c00d21f44d0bc584667709b.jpg", + "image_caption": [ + "Figure 3: The process converting FP32 inputs into FFP8 outputs with the given format parameters " + ], + "image_footnote": [], + "bbox": [ + 200, + 103, + 797, + 164 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/e4d6b151d68afae8892706607f8670d80bdf80f4a7f32cfd9c8fe5d6241db613.jpg", + "table_caption": [ + "Table 1: Top-1 and Top-5 accuracy of VGG-16 on ImageNet dataset (Deng et al., 2009) in various number formats; delta $( \\Delta )$ indicates the accuracy drop as compared to FP32 " + ], + "table_footnote": [], + "table_body": "
WeightActivationTop-1 (△)Top-5 (△)
FP32FP3271.59%90.38%
(1,5,2,15)(1,4,3,7)69.89% (-1.70%)89.29% (-1.09%)
(1,4,3,7)(1,4,3,7)70.86% (-0.73%)90.02% (-0.36%)
(1,4,3,15)(1,4,3,7)70.96% (-0.63%)90.10% (-0.28%)
(1,3,4,3)(1,4,3,7)70.18% (-1.41%)89.56% (-0.82%)
(1,3,4,7)(1,4,3,7)71.19% (-0.40%)90.12% (-0.26%)
(1,3,4,7)(1,4,3,7)+(0,4,4,7)71.19% (-0.40%)90.14% (-0.24%)
", + "bbox": [ + 250, + 251, + 745, + 367 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In addition to weights, it is certainly worth finding out best-fit formats for activations as well. For those attempts made above, activations are always in FFP8(1, 4, 3, 7) for two reasons: 1) the overall distribution of activations is wider than that of weights, and 2) the maximum magnitude of activations is much bigger than that of weights. The detailed distribution of activations will be given in Section 3.4 later. Meanwhile, it is also worth noting that a large set of commonly used activation functions always produce nonnegative outputs, e.g., ReLU, ReLU6, and sigmoid. That is, if those outputs are represented in any signed format, a half of the code space is actually wasted. It may not be a problem for 32-bit and 16-bit formats with long enough fraction bits; however, it is indeed a serious issue for any 8-bit format, which merely has 256 available codes in total. Since VGG-16 utilizes ReLU as its activation function, it is feasible to select signed FFP8(1, 4, 3, 7) for the first layer and unsigned FFP8(0, 4, 4, 7) for all succeeding layers. It implies that 256 instead of 128 codes are available to represent those unsigned activations in all layers except for the first one. With no surprise, the accuracy is further improved since the range window remains untouched while the fraction size gains one extra bit. In the last configuration, the Top-1 accuracy loss is only $0 . 4 \\%$ when compared against the FP32 baseline, as indicated in Table 1. ", + "bbox": [ + 174, + 398, + 825, + 608 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.4 LAYER-WISE OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 176, + 626, + 411, + 640 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Figure 4 and Figure 5 illustrate several distributions of weights and activations in VGG-16, respectively. Each figure includes the distributions of one whole model and three selected individual layers. In Section 3.3, the best-fit FFP8 format is determined by the overall distribution of the whole model. Here we have three key observations from those distributions: 1) the distributions of weights are quite dissimilar to those of activations, 2) even the distributions across different layers are dissimilar for both weights and activations, and 3) the distribution of an individual layer is narrower than that of the whole model. The above observations clearly suggest that applying layer-wise optimization (LWO) properly on number format selection is very likely to improve the accuracy further. ", + "bbox": [ + 173, + 651, + 825, + 763 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For instance, after comparing Figure $5 ( \\mathrm { a } ) \\& ( \\mathrm { b } )$ , it is found that the distribution of the whole model is wider than that of the first layer, and the maximum log magnitudes of the whole model and the first layer are 8 and 1, respectively. As a consequence, selecting $\\mathrm { F F P } 8 ( 1 , 3 , 4 , 6 )$ instead of FFP8(1, 4, 3, 7) for activations in the first layer can further increase the Top-1 accuracy by $0 . 1 9 \\%$ (from $7 1 . 1 9 \\%$ to $7 1 . 3 8 \\%$ ), as indicated in Table 2. Next, we examine a new configuration that makes weights of all layers are in FFP8(1, 2, 5, 3). It is obvious an unwise attempt since $3 7 . 6 \\%$ leftmost weights are out of the range window according to Figure 4(a). However, it may not be a bad idea to use FFP8(1, 2, 5, 5) in Layer 6 and FFP8(1, 2, 5, 6) in the last layer because only $6 . 9 \\%$ and $5 . 3 \\%$ smallest weights are out of the range window respectively according to Figure $4 ( \\mathrm { c } ) \\& ( \\mathrm { d } )$ . Therefore, we examine another new configuration that makes weights of all layers in $\\mathrm { F F P } 8 ( 1 , 2 , 5 , * )$ . Here the asterisk $( \\ast )$ represents the largest possible exponent bias, which ensures that the maximum weight is still inside the range window. This LWO on weights successfully increases the Top-1 accuracy by $0 . 1 \\%$ (from $7 1 . 3 8 \\%$ to $7 1 . 4 8 \\%$ ). Similarly, we can apply the LWO on activations, which again raises the Top-5 accuracy by $0 . 0 5 \\%$ . The selected FFP8 format of each layer after LWO is reported in Table 3. ", + "bbox": [ + 174, + 770, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Previous studies usually choose $\\mathrm { F P } 8 ( 1 , 4 , 3 )$ or $\\mathrm { F P } 8 ( 1 , 5 , 2 )$ formats because the exponent size has to be large enough to cover most weights and activations at the cost of an even smaller fraction size. However, the exponent size for weights can be as small as 2 after LWO in our flow. Consequently, the larger 5-bit fraction size does help in accuracy improvement, as indicated in Table 2. ", + "bbox": [ + 174, + 151, + 825, + 208 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Note that the Top-1 accuracy achieved by the final configuration, which applies layer-wise optimization on both weights and activations, is merely $0 . 1 1 \\%$ lower as compared to that of the FP32 baseline $( 7 1 . 4 8 \\%$ vs. $7 1 . 5 9 \\%$ ). More importantly, model retraining is not applied yet. In other words, all the accuracy improvements made so far are simply from properly re-expressing weights and activations of a pre-trained FP32 model in their best-fit FFP8 formats. ", + "bbox": [ + 173, + 215, + 825, + 285 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/21188150164f237a39626163cff2bf1d4af3aeae56669848881ed879c8bd5df6.jpg", + "image_caption": [ + "Figure 4: Weight distributions of (a) whole model, (b) first layer, (c) Layer 6, (d) last layer " + ], + "image_footnote": [], + "bbox": [ + 173, + 301, + 818, + 570 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/4db1795b0969e5913dd60342cf777d83c28f7e085bf9522442eca015a01afa91.jpg", + "image_caption": [ + "Figure 5: Activation distributions of (a) whole model, (b) first layer, (c) Layer 6, (d) last layer " + ], + "image_footnote": [], + "bbox": [ + 173, + 622, + 820, + 895 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/1fb4c61126695337d0ba2747d540fe13ea12d7deb54557382bd0aed875f17150.jpg", + "table_caption": [ + "Table 2: Top-1 and Top-5 accuracy of VGG-16 after layer-wise optimization " + ], + "table_footnote": [], + "table_body": "
WeightActivationTop-1 (△)Top-5 (△)
FP32FP3271.59%90.38%
(1,3,4,7)(1,4,3,7)+(0,4,4,7)71.19% (-0.40%)90.14% (-0.24%)
(1,3,4,7)(1,3,4,6)+(0,4,4,7)71.38% (-0.21%)90.33% (-0.05%)
(1,2,5,3)(1,3,4,6)+(0,4,4,7)71.24% (-0.35%)90.22% (-0.16%)
(1,2,5,*)(1,3,4,6)+(0,4,4,7)71.48% (-0.11%)90.27% (-0.11%)
(1,2,5,*)(1,3,4,6)+(0,4,4,*)71.48% (-0.11%)90.32% (-0.06%)
", + "bbox": [ + 254, + 132, + 741, + 234 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/2047fba06027d960656a886dc31e42a70a05b81e6799fc01b933816bf692af83.jpg", + "table_caption": [ + "Table 3: The FFP8 format of each layer in VGG-16 after layer-wise optimization " + ], + "table_footnote": [], + "table_body": "
LayerWeightActivationLayerWeightActivation
1(1,2,5,3)(1,3,4,6)8(1,2,5,5)(0,4,4,8)
2(1,2,5,4)(0,4,4,11)9(1,2,5,5)(0,4,4,8)
3(1,2,5,4)(0,4,4,10)10(1,2,5,6)(0,4,4,8)
4(1,2,5,5)(0,4,4,10)11(1,2,5,5)(0,4,4,7)
5(1,2,5,4)(0,4,4,9)12(1,2,5,5)(0,4,4,7)
6(1,2,5,5)(0,4,4,9)13(1,2,5,6)(0,4,4,8)
7(1,2,5,4)(0,4,4,9)
", + "bbox": [ + 272, + 292, + 728, + 409 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.5 ACCURACY IMPROVEMENT VIA MODEL RETRAINING ", + "text_level": 1, + "bbox": [ + 174, + 436, + 584, + 452 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "All FFP8 weights in our previous experiments are simply converted from pre-trained weights generated from a typical FP32 training framework. However, it is reported that quantization-aware model retraining can usually improve the accuracy (Jacob et al., 2017), which motivates us to check whether it can successfully apply to our work. We first train the ResNet-18 model in an FP32 framework and the corresponding Top-1 accuracy is $6 9 . 7 6 \\%$ . Next, an FFP8-based inference is performed under the following settings: 1) all weights are in $\\mathrm { F F P } 8 ( 1 , 3 , 4 , 7 ) , 2 )$ activations of the first layer are in FFP8(1, 3, 4, 6), and 3) activations of the other layers are in $\\mathrm { F F P 8 } ( 0 , 4 , 4 , 7 )$ . The Top-1 accuracy for the above configuration is down to $6 9 . 4 4 \\%$ . Then, a quantization-aware retraining process of 15 epochs is applied and the resultant Top-1 accuracy goes back to $6 9 . 7 6 \\%$ . Note that the retraining is done simply in a typical FP32 framework without the need of special training skills. ", + "bbox": [ + 173, + 462, + 825, + 602 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 EXPERIMENTAL RESULTS ON VARIOUS DNN MODELS ", + "text_level": 1, + "bbox": [ + 176, + 621, + 658, + 638 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we intend to demonstrate that the proposed FFP8 format constantly performs well in various DNN models in addition to VGG-16. Table 4 reports the accuracy results of various models under different format configurations. Configuration A gives the results of the FP32 baseline. Configuration B utilizes a conventional FP8(1, 4, 3) format with a default exponent bias (7), which incurs a penalty of roughly $1 \\%$ drop on Top-1 accuracy. With the help of the FFP8 format, Configuration C adopts the best-fit format for weights, which effectively reduces the Top-1 accuracy loss to $0 . 4 { \\sim } 0 . 7 5 \\%$ . Configuration D further selects two best-fit formats for activations (the signed one for the first layer and the unsigned one for the rest), which minimizes the Top-1 accuracy loss to $0 . 3 \\%$ . Note that neither layer-wise optimization nor model retraining is even applied for this achievement. ", + "bbox": [ + 173, + 652, + 825, + 779 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/6b8c3d8ff57968b6442f66e62e35287b0dd19cb39d69e88b00f49b98dcbe0531.jpg", + "table_caption": [ + "Table 4: Accuracy results of various models under different format configurations " + ], + "table_footnote": [], + "table_body": "
Cfg.WeightActivationVGG-16ResNet-50ResNet-34ResNet-18
Top-1/Top-5Top-1/Top-5Top-1/Top-5Top-1/Top-5
AFP32FP3271.59 /90.3876.13/92.8673.31/91.4269.76/89.08
B(1,4,3,7)(1,4,3,7)70.86 /90.0275.24 /92.5272.39 /90.9768.70 / 88.50
C(1,3,4,7)(1,4,3,7)71.19 /90.1275.38 /92.6872.81 /91.1669.25 /88.80
D(1,3,4,7)(1,3,4,6)+(0,4,4,7)71.38 / 90.3375.85 /92.8173.12 /91.3369.44 / 88.93
", + "bbox": [ + 178, + 815, + 825, + 922 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/a47b50b7569e8f7830a2fdff1eda89b1c2a135c7cc9df8ab55b141b9c3e88e1a.jpg", + "image_caption": [ + "Figure 6: Memory-efficient system architectures: off-chip external data (a) in BFP16, (b) in FFP8; and (c) an FFP8 to FP32 hardware converter " + ], + "image_footnote": [], + "bbox": [ + 186, + 102, + 816, + 369 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In addition to image classification, we also want to know whether the proposed FFP8 format performs equally well in other application domains. Here, we examine two more applications: semantic segmentation and ECG check. First, FCN32s is a popular CNN model for semantic segmentation (Long et al., 2014). If FCN32s is in FP32, the mIOU is $6 3 . 6 3 \\%$ using the VOC2011 dataset (Everingham et al.). Alternatively, if all the weights are in FFP8(1, 4, 3, 14) and all the activations are in FFP8(1, 4, 3, 2), the resultant mIOU would be $6 3 . 4 5 \\%$ , a slight drop of $0 . 1 8 \\%$ . Second, an LSTM model for ECG check (Physionet, 2017), has also been tested. If the LSTM model is in FP32, the check accuracy is $8 1 . 1 2 \\%$ . If all the weights are in $\\mathrm { F F P } 8 ( 1 , 3 , 4 , 6 )$ , the first-layer activations are in FFP8(1, 3, 4, 5), and activations in the other layer are in FFP8(1, 4, 3, 16), the resultant accuracy would be $8 1 . 5 3 \\%$ , an accuracy gain of $0 . 4 1 \\%$ . The experimental results once again demonstrate that FFP8 performs very well in these two categories as well. ", + "bbox": [ + 174, + 438, + 825, + 590 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 ASPECTS OF SYSTEM AND HARDWARE ", + "text_level": 1, + "bbox": [ + 176, + 612, + 529, + 628 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The target of this work is to develop a memory-efficient inference system, especially for those with limited memory capacity and bandwidth. A current state-of-the-art system, proposed by Nvidia and Intel, has successfully cut the required memory size and traffic through representing external data (weights or activations) in BFP16/INT8 instead of FP32, as illustrated in Figure 6(a) (Intel, 2018; 2019; Nvidia, 2017; 2020). It not only reduces the memory size, alleviates the performance bottleneck due to memory bandwidth limitation but also saves a significant amount of energy due to fewer power-consuming external memory access operations. To preserve the computation accuracy at the same time, external BFP16 data are converted to FP32 data right before entering the FP32 fused-multiply-add (FMA) unit. That is, internal computations can all be in FP32. Those FP32 data are converted back to BFP16 only if they are about to be written back to external memory. ", + "bbox": [ + 174, + 645, + 825, + 784 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this work, we propose a system architecture that is very similar to the previous one, as shown in Figure 6(b). The key difference is that external data are in FFP8 instead of BFP16, which implies the proposed system merely demands a quarter of the memory size and throughput required by today’s FP32-based counterparts. ", + "bbox": [ + 174, + 791, + 825, + 847 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "It is easy to convert a BFP16 number into its FP32 equivalent by concatenating a 16-bit pattern of all 0s at its least significant end. In fact, it is also easy to transform an FFP8 number to its FP32 equivalent via a converter, as depicted in Figure 6(c). Few register bits are allocated to store the current format settings of $x , y ,$ , and $b$ within the converter. Then, $x$ is used to recover the sign bit; the biased exponent and fraction can be extracted via $x$ and $y$ ; finally the exponent can be further corrected by the exponent bias $b$ . Hence, it is apparent that the required hardware logic for the converter is indeed minimal and the extra hardware cost is truly minor as well. Furthermore, updates of those format registers are extremely infrequent. Even the layer-wise optimization is applied, those registers are only modified at the start of each layer. In other words, virtually no runtime overhead is imposed due to those register updates. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 194, + 318, + 210 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this work, we propose the flexible 8-bit floating-point (FFP8) format for accurate and memoryefficient inference of deep neural networks. Our FFP8 format offers three adjustable options: 1) the size of exponent/fraction field, 2) the value of exponent bias, and 3) the presence of the sign bit, whereas those rigid conventional formats simply leave nothing. In this paper, we explain how the exponent size and bias jointly define the representable value range of a given FFP8 format. We also demonstrate how to explore the best-fit signed/unsigned FFP8 formats for weights and activations to achieve more accurate inference outcomes. Besides, a layer-wise optimization flow, which discovers the best-fit formats for each individual layer, is presented to further improve the accuracy. A model retraining methodology that can be carried out in typical FP32 frameworks without the need of special training skills is also introduced. The experimental results on various DNN models show that the proposed FFP8-based inference flow achieves an extremely low accuracy loss of $0 . 1 \\% \\sim 0 . 3 \\%$ as compared to the FP32 baseline even without model retraining. Moreover, we also show that the extra hardware for supporting the FFP8 format is minimal. Therefore, it is conclusive that the proposed FFP8-based inference framework should be a better solution for those computing systems with limited memory capacity and bandwidth, e.g., edge and AIoT devices. 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", + "bbox": [ + 176, + 295, + 821, + 324 + ], + "page_idx": 10 + } +] \ No newline at end of file diff --git a/parse/train/9sF3n8eAco/9sF3n8eAco_middle.json b/parse/train/9sF3n8eAco/9sF3n8eAco_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..d567212377a130edf7f33487c35fd7518faac740 --- /dev/null +++ b/parse/train/9sF3n8eAco/9sF3n8eAco_middle.json @@ -0,0 +1,26343 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 77, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 77, + 505, + 97 + ], + "score": 1.0, + "content": "ALL-YOU-CAN-FIT 8-BIT FLEXIBLE FLOATING-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 98, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 505, + 117 + ], + "score": 1.0, + "content": "POINT FORMAT FOR ACCURATE AND MEMORY-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 119, + 501, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 501, + 136 + ], + "score": 1.0, + "content": "EFFICIENT INFERENCE OF DEEP NEURAL NETWORKS", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 112, + 155, + 244, + 177 + ], + "lines": [ + { + "bbox": [ + 113, + 156, + 201, + 167 + ], + "spans": [ + { + "bbox": [ + 113, + 156, + 201, + 167 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 112, + 166, + 245, + 178 + ], + "spans": [ + { + "bbox": [ + 112, + 166, + 245, + 178 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 278, + 206, + 333, + 218 + ], + "lines": [ + { + "bbox": [ + 276, + 206, + 335, + 219 + ], + "spans": [ + { + "bbox": [ + 276, + 206, + 335, + 219 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 143, + 231, + 468, + 418 + ], + "lines": [ + { + "bbox": [ + 141, + 231, + 470, + 245 + ], + "spans": [ + { + "bbox": [ + 141, + 231, + 470, + 245 + ], + "score": 1.0, + "content": "Modern deep neural network (DNN) models generally require a huge amount of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 243, + 469, + 255 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 469, + 255 + ], + "score": 1.0, + "content": "weight and activation values to achieve good inference outcomes. 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Those data in-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 254, + 469, + 266 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 469, + 266 + ], + "score": 1.0, + "content": "evitably demand a massive off-chip memory capacity/bandwidth, and the situation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "score": 1.0, + "content": "gets even worse if they are represented in high-precision floating-point formats.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 470, + 288 + ], + "score": 1.0, + "content": "Effort has been made for representing those data in different 8-bit floating-point", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 287, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 287, + 470, + 300 + ], + "score": 1.0, + "content": "formats, nevertheless, a notable accuracy loss is still unavoidable. In this paper we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 298, + 469, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 469, + 311 + ], + "score": 1.0, + "content": "introduce an extremely flexible 8-bit floating-point (FFP8) format whose defining", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 309, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 142, + 309, + 469, + 321 + ], + "score": 1.0, + "content": "factors – the bit width of exponent/fraction field, the exponent bias, and even the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 320, + 470, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 470, + 333 + ], + "score": 1.0, + "content": "presence of the sign bit – are all configurable. We also present a methodology to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 331, + 470, + 343 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 470, + 343 + ], + "score": 1.0, + "content": "properly determine those factors so that the accuracy of model inference can be", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 342, + 469, + 354 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 469, + 354 + ], + "score": 1.0, + "content": "maximized. The foundation of this methodology is based on a key observation –", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 353, + 469, + 365 + ], + "spans": [ + { + "bbox": [ + 141, + 353, + 469, + 365 + ], + "score": 1.0, + "content": "both the maximum magnitude and the value distribution are quite dissimilar be-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 364, + 469, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 364, + 469, + 376 + ], + "score": 1.0, + "content": "tween weights and activations in most DNN models. Experimental results demon-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 375, + 470, + 387 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 470, + 387 + ], + "score": 1.0, + "content": "strate that the proposed FFP8 format achieves an extremely low accuracy loss of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 384, + 470, + 399 + ], + "spans": [ + { + "bbox": [ + 142, + 385, + 199, + 397 + ], + "score": 0.95, + "content": "0 . 1 \\% \\sim 0 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 384, + 470, + 399 + ], + "score": 1.0, + "content": "for several representative image classification models even without", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 396, + 469, + 409 + ], + "spans": [ + { + "bbox": [ + 141, + 396, + 469, + 409 + ], + "score": 1.0, + "content": "the need of model retraining. Besides, it is easy to turn a classical floating-point", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 408, + 469, + 420 + ], + "spans": [ + { + "bbox": [ + 141, + 408, + 469, + 420 + ], + "score": 1.0, + "content": "processing unit into an FFP8-compliant one, and the extra hardware cost is minor.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 14, + "bbox_fs": [ + 141, + 231, + 470, + 420 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 442, + 206, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 208, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 208, + 457 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 504, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "With the rapid progress of deep neural network (DNN) techniques, innovative applications of deep", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "learning in various domains, such as computer vision and natural language processing (NLP), are", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 490, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 501 + ], + "score": 1.0, + "content": "getting more mature and powerful (Huang et al., 2017; Vaswani et al., 2017; Szegedy et al., 2015;", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 499, + 354, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 354, + 513 + ], + "score": 1.0, + "content": "Howard et al., 2017; He et al., 2017; Krizhevsky et al., 2012).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 466, + 506, + 513 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 504, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "score": 1.0, + "content": "To improve the model accuracy, one of the most commonly used strategies is to add more layers", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 529, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 541 + ], + "score": 1.0, + "content": "into a network, which inevitably increases the number of weight parameters and activation values", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "of a model. 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However, the 8-bit fixed-point format inherently has a narrower dynamic", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "value range so that the model accuracy loss is usually not negligible even after extra symmetric", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "or asymmetric quantization. As a consequence, there are a number of attempts concentrating on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 474, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 474, + 139 + ], + "score": 1.0, + "content": "utilizing mixed-precision or pure 8-bit floating-point numbers in deep learning applications.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "Various techniques have been developed for mixed-precision training (Banner et al., 2018; Micike-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "vicius et al., 2018; Das et al., 2018; Zhou et al., 2016). Moreover, recent studies proposed several", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "training frameworks that produces weights only in 8-bit floating-point formats (Wang & Choi, 2018;", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "Cambier et al., 2020; Sun et al., 2019). In these studies, the underlying 8-bit floating-point numbers", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "in training and inference are represented in the format of FP8(1, 5, 2) or FP8(1, 4, 3), where the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "enclosed three parameters indicate the bit length of sign, exponent, and fraction, respectively. Note", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "that 4 or 5 bits are essential for the exponent in their frameworks, or the corresponding dynamic", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "score": 1.0, + "content": "range may not cover both weight and activation values well. Consequently, only 2 or 3 bits are", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 229, + 361, + 245 + ], + "spans": [ + { + "bbox": [ + 104, + 229, + 361, + 245 + ], + "score": 1.0, + "content": "available for fraction, which inevitably leads to lower accuracy.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "In this paper, we present an extremely flexible 8-bit floating-point (FFP8) number format. In FFP8,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "all parameters – the bit width of exponent/fraction, the exponent bias, and the presence of the sign bit", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "– are configurable. Three major features of our inference methodology associated with the proposed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "FFP8 format are listed as follows. First, it is observed that both the maximum magnitude and the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "value distribution are quite dissimilar between weights and activations in most DNNs. It suggests", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "the best exponent size and exponent bias for weights should be different from those for activations to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "achieve higher accuracy. Second, a large set of commonly-used activation functions always produce", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 337 + ], + "score": 1.0, + "content": "nonnegative outputs (e.g., ReLU). It implies that activations are actually unsigned if one of those", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "activation functions is in use. Hence, it implies the sign bit is not required for those activations,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 346, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 506, + 359 + ], + "score": 1.0, + "content": "which makes either exponent or fraction 1-bit longer. Note that even one bit can make a big impact", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "since only 8 bits are available. Third, all aforementioned studies require their own sophisticated", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 367, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 382 + ], + "score": 1.0, + "content": "training frameworks to produce 8-bit floating-point models. Our flow does not. Our flow simply", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "takes a model generated by any conventional FP32 training framework as the input. Then, it simply", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 362, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 362, + 402 + ], + "score": 1.0, + "content": "converts the given pre-trained FP32 model into an FFP8 model.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 108, + 407, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "The rest of this paper is organized as follows. Section 2 briefly introduces related work. In Section 3,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "we elaborate more on the proposed FFP8 format and how to properly convert a pre-trained FP32", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "model into an FFP8 one. Section 4 demonstrates the experimental results on various DNN models.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "The system and hardware design issues for the support of FFP8 numbers are discussed in Section 5.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 450, + 327, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 327, + 464 + ], + "score": 1.0, + "content": "Finally, the concluding remarks are given in Section 6.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 480, + 211, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 479, + 213, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 213, + 495 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "score": 1.0, + "content": "DNNs are becoming larger and more complicated, which means they require a bigger memory", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "space and consume more energy during inference. As a result, it is getting harder to deploy them", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "on systems with limited memory capacity and power budget (e.g., edge devices). (Han et al., 2016;", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "Zhao et al., 2020; Horowitz, 2014) also demonstrated that off-chip DRAM access is responsible for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "a significantly big share of system power consumption. Hence, it remains an active research topic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "about how to reduce the memory usage for weights and activations. As mentioned in the previous", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 414, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 414, + 585 + ], + "score": 1.0, + "content": "section, one way to do so is to use short 8-bit floating-point number formats.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "Wang & Choi (2018) introduced a DNN training methodology using 8-bit floating-point numbers in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "FP8(1, 5, 2) format. The methodology features chunk-based accumulation and stochastic rounding", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "methods for accuracy loss minimization. Besides, to achieve a better trade-off between precision", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 620, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 636 + ], + "score": 1.0, + "content": "and dynamic range during model training, Sun et al. (2019) proposed an improved methodology", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 327, + 646 + ], + "score": 1.0, + "content": "that utilizes two different 8-bit floating-point formats", + "type": "text" + }, + { + "bbox": [ + 327, + 633, + 385, + 644 + ], + "score": 0.54, + "content": "- \\operatorname { F P 8 } ( 1 , 4 , 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "for forward propagation and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 159, + 655 + ], + "score": 0.3, + "content": "\\mathrm { F P } 8 ( 1 , 5 , 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "for backward propagation. Nevertheless, both methodologies fail to make a DNN", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "model entirely in 8-bit numbers: the first and the last layers of the given model are still in 16-bit", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 341, + 679 + ], + "score": 1.0, + "content": "floating-point numbers; otherwise, the model suffers about", + "type": "text" + }, + { + "bbox": [ + 341, + 666, + 356, + 676 + ], + "score": 0.85, + "content": "2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 664, + 505, + 679 + ], + "score": 1.0, + "content": "accuracy degradation. Cambier et al.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "(2020) then proposed the S2FP8 format, which allows a DNN model represented in 8-bit floating", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "point numbers completely. By adding a scaling factor and a shifting factor, data can thus be well", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 104, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 164, + 712 + ], + "score": 1.0, + "content": "represented in", + "type": "text" + }, + { + "bbox": [ + 165, + 699, + 213, + 710 + ], + "score": 0.44, + "content": "\\mathrm { F P } 8 ( 1 , 5 , 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "after proper shifting and squeezing operations, which eliminates the need", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 431, + 723 + ], + "score": 1.0, + "content": "of 16-bit floating point numbers. However, the S2FP8 format still results in about", + "type": "text" + }, + { + "bbox": [ + 432, + 710, + 446, + 720 + ], + "score": 0.85, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "accuracy drop", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 720, + 232, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 232, + 733 + ], + "score": 1.0, + "content": "in ResNet-50 (He et al., 2018).", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 47 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 138 + ], + "lines": [ + { + "bbox": [ + 107, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 107, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "To make the data even shorter, 8-bit fixed-point signed/unsigned integer formats (INT8 and UINT8)", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "are also broadly adopted. However, the 8-bit fixed-point format inherently has a narrower dynamic", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "value range so that the model accuracy loss is usually not negligible even after extra symmetric", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "or asymmetric quantization. As a consequence, there are a number of attempts concentrating on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 474, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 474, + 139 + ], + "score": 1.0, + "content": "utilizing mixed-precision or pure 8-bit floating-point numbers in deep learning applications.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 83, + 506, + 139 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "Various techniques have been developed for mixed-precision training (Banner et al., 2018; Micike-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "vicius et al., 2018; Das et al., 2018; Zhou et al., 2016). Moreover, recent studies proposed several", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "training frameworks that produces weights only in 8-bit floating-point formats (Wang & Choi, 2018;", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "Cambier et al., 2020; Sun et al., 2019). In these studies, the underlying 8-bit floating-point numbers", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "in training and inference are represented in the format of FP8(1, 5, 2) or FP8(1, 4, 3), where the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "enclosed three parameters indicate the bit length of sign, exponent, and fraction, respectively. Note", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "that 4 or 5 bits are essential for the exponent in their frameworks, or the corresponding dynamic", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "score": 1.0, + "content": "range may not cover both weight and activation values well. Consequently, only 2 or 3 bits are", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 229, + 361, + 245 + ], + "spans": [ + { + "bbox": [ + 104, + 229, + 361, + 245 + ], + "score": 1.0, + "content": "available for fraction, which inevitably leads to lower accuracy.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 142, + 506, + 245 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "In this paper, we present an extremely flexible 8-bit floating-point (FFP8) number format. In FFP8,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "all parameters – the bit width of exponent/fraction, the exponent bias, and the presence of the sign bit", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "– are configurable. Three major features of our inference methodology associated with the proposed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "FFP8 format are listed as follows. First, it is observed that both the maximum magnitude and the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "value distribution are quite dissimilar between weights and activations in most DNNs. It suggests", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "the best exponent size and exponent bias for weights should be different from those for activations to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "achieve higher accuracy. Second, a large set of commonly-used activation functions always produce", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 337 + ], + "score": 1.0, + "content": "nonnegative outputs (e.g., ReLU). It implies that activations are actually unsigned if one of those", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "activation functions is in use. Hence, it implies the sign bit is not required for those activations,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 346, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 506, + 359 + ], + "score": 1.0, + "content": "which makes either exponent or fraction 1-bit longer. Note that even one bit can make a big impact", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "since only 8 bits are available. Third, all aforementioned studies require their own sophisticated", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 367, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 382 + ], + "score": 1.0, + "content": "training frameworks to produce 8-bit floating-point models. Our flow does not. Our flow simply", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "takes a model generated by any conventional FP32 training framework as the input. Then, it simply", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 362, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 362, + 402 + ], + "score": 1.0, + "content": "converts the given pre-trained FP32 model into an FFP8 model.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 247, + 506, + 402 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 407, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "The rest of this paper is organized as follows. Section 2 briefly introduces related work. In Section 3,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "we elaborate more on the proposed FFP8 format and how to properly convert a pre-trained FP32", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "model into an FFP8 one. Section 4 demonstrates the experimental results on various DNN models.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "The system and hardware design issues for the support of FFP8 numbers are discussed in Section 5.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 450, + 327, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 327, + 464 + ], + "score": 1.0, + "content": "Finally, the concluding remarks are given in Section 6.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 407, + 506, + 464 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 480, + 211, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 479, + 213, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 213, + 495 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 520 + ], + "score": 1.0, + "content": "DNNs are becoming larger and more complicated, which means they require a bigger memory", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "space and consume more energy during inference. As a result, it is getting harder to deploy them", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "on systems with limited memory capacity and power budget (e.g., edge devices). (Han et al., 2016;", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "Zhao et al., 2020; Horowitz, 2014) also demonstrated that off-chip DRAM access is responsible for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "a significantly big share of system power consumption. Hence, it remains an active research topic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "about how to reduce the memory usage for weights and activations. As mentioned in the previous", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 414, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 414, + 585 + ], + "score": 1.0, + "content": "section, one way to do so is to use short 8-bit floating-point number formats.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 505, + 505, + 585 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "Wang & Choi (2018) introduced a DNN training methodology using 8-bit floating-point numbers in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "FP8(1, 5, 2) format. The methodology features chunk-based accumulation and stochastic rounding", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "methods for accuracy loss minimization. Besides, to achieve a better trade-off between precision", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 620, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 636 + ], + "score": 1.0, + "content": "and dynamic range during model training, Sun et al. (2019) proposed an improved methodology", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 327, + 646 + ], + "score": 1.0, + "content": "that utilizes two different 8-bit floating-point formats", + "type": "text" + }, + { + "bbox": [ + 327, + 633, + 385, + 644 + ], + "score": 0.54, + "content": "- \\operatorname { F P 8 } ( 1 , 4 , 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "for forward propagation and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 159, + 655 + ], + "score": 0.3, + "content": "\\mathrm { F P } 8 ( 1 , 5 , 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "for backward propagation. 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If", + "type": "text" + }, + { + "bbox": [ + 439, + 298, + 444, + 308 + ], + "score": 0.81, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 297, + 505, + 311 + ], + "score": 1.0, + "content": "is missing, the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 309, + 234, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 234, + 321 + ], + "score": 1.0, + "content": "default value is implicitly used.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 325, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "Figure 1(a) illustrates one 16-bit FP16 format and two commonly used 8-bit formats: (1, 5, 2, 15)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 488, + 349 + ], + "score": 1.0, + "content": "and (1, 4, 3, 7). Note that there is actually only one parameter, the bit width of the exponent", + "type": "text" + }, + { + "bbox": [ + 488, + 338, + 501, + 348 + ], + "score": 0.77, + "content": "( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 336, + 505, + 349 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 299, + 360 + ], + "score": 1.0, + "content": "that can be freely chosen when defining a new", + "type": "text" + }, + { + "bbox": [ + 299, + 349, + 306, + 358 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 348, + 487, + 360 + ], + "score": 1.0, + "content": "-bit conventional floating-point format since", + "type": "text" + }, + { + "bbox": [ + 487, + 350, + 494, + 358 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 357, + 507, + 371 + ], + "spans": [ + { + "bbox": [ + 104, + 357, + 147, + 371 + ], + "score": 1.0, + "content": "always 1,", + "type": "text" + }, + { + "bbox": [ + 147, + 361, + 154, + 369 + ], + "score": 0.65, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 357, + 196, + 371 + ], + "score": 1.0, + "content": "is always", + "type": "text" + }, + { + "bbox": [ + 196, + 359, + 239, + 370 + ], + "score": 0.91, + "content": "n - y - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 357, + 261, + 371 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 261, + 359, + 267, + 369 + ], + "score": 0.71, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 357, + 308, + 371 + ], + "score": 1.0, + "content": "is always", + "type": "text" + }, + { + "bbox": [ + 309, + 358, + 348, + 369 + ], + "score": 0.92, + "content": "2 ^ { y - 1 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 357, + 507, + 371 + ], + "score": 1.0, + "content": ". The above fact motivated us to think", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "out-of-the-box and thus develop a more flexible format, named flexible 8-bit floating-point (FFP8)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 392, + 393 + ], + "score": 1.0, + "content": "format and shown in Figure 1(b). In addition to the size of exponent", + "type": "text" + }, + { + "bbox": [ + 392, + 381, + 405, + 392 + ], + "score": 0.8, + "content": "( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 380, + 505, + 393 + ], + "score": 1.0, + "content": ", the FFP8 format offers", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 104, + 392, + 296, + 404 + ], + "score": 1.0, + "content": "two more parameters. First, the exponent bias", + "type": "text" + }, + { + "bbox": [ + 297, + 392, + 308, + 403 + ], + "score": 0.57, + "content": "( b )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 392, + 417, + 404 + ], + "score": 1.0, + "content": "is not necessarily equal to", + "type": "text" + }, + { + "bbox": [ + 418, + 391, + 457, + 402 + ], + "score": 0.91, + "content": "2 ^ { y - 1 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 392, + 506, + 404 + ], + "score": 1.0, + "content": "and can be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 104, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "set to any integer, which helps cover the value distributions of both weights and activations well in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 413, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 203, + 425 + ], + "score": 1.0, + "content": "a shorter exponent size", + "type": "text" + }, + { + "bbox": [ + 203, + 414, + 216, + 425 + ], + "score": 0.74, + "content": "( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 413, + 396, + 425 + ], + "score": 1.0, + "content": ". Second, the sign bit is present or not (i.e.,", + "type": "text" + }, + { + "bbox": [ + 397, + 415, + 404, + 424 + ], + "score": 0.63, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 413, + 506, + 425 + ], + "score": 1.0, + "content": "can be 0 or 1). Without", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 156, + 437 + ], + "score": 1.0, + "content": "the sign bit", + "type": "text" + }, + { + "bbox": [ + 156, + 425, + 183, + 435 + ], + "score": 0.85, + "content": "( x = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "), unsigned activations can be better represented in higher precision. To be more", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 287, + 448 + ], + "score": 1.0, + "content": "precise, there are only two restrictions on an", + "type": "text" + }, + { + "bbox": [ + 287, + 437, + 294, + 445 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 435, + 378, + 448 + ], + "score": 1.0, + "content": "-bit FFP8 format: 1)", + "type": "text" + }, + { + "bbox": [ + 378, + 436, + 444, + 447 + ], + "score": 0.87, + "content": "x + y + z = n ;", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 435, + 455, + 448 + ], + "score": 1.0, + "content": "2)", + "type": "text" + }, + { + "bbox": [ + 455, + 437, + 462, + 446 + ], + "score": 0.6, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "must be 0", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 446, + 498, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 498, + 459 + ], + "score": 1.0, + "content": "or 1. Next, we are about to show how the proposed FFP8 format improves the inference accuracy.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 473, + 477, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 479, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 479, + 486 + ], + "score": 1.0, + "content": "3.2 WEIGHT DISTRIBUTION AND THE WAYS VARIOUS 8-BIT FORMATS COPE WITH IT", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "Overall speaking, the magnitudes of weights in most DNN models are usually small. Take a popular", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 506, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 520 + ], + "score": 1.0, + "content": "image classification model VGG-16 (Simonyan & Zisserman, 2015) as an example, the maximum", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "score": 1.0, + "content": "magnitude of weights in the whole model is less than 2. Figure 2 gives the overall weight distribution", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "(in log scale) of VGG-16 trained via a conventional FP32 framework. Figure 2(a) illustrates how", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 538, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 554 + ], + "score": 1.0, + "content": "the conventional FP8(1, 4, 3) copes with those weights. The rectangle in red is called the range", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 291, + 563 + ], + "score": 1.0, + "content": "window, which specifies the value range that", + "type": "text" + }, + { + "bbox": [ + 291, + 550, + 341, + 561 + ], + "score": 0.4, + "content": "\\mathrm { F P } 8 ( 1 , 4 , 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "can represent. The purple vertical dash", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "line further partitions the window into the norm region (right side) and the denorm region (left side).", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "The star marks the position where the weight of the maximum magnitude locates. Note that virtually", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 434, + 595 + ], + "score": 1.0, + "content": "the entire right half of the range window in Figure 2(a) covers no weights, while", + "type": "text" + }, + { + "bbox": [ + 435, + 583, + 457, + 594 + ], + "score": 0.86, + "content": "9 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "of leftmost", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "weights cannot be included in the range window. In order to contain almost every weight in a range", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "window, FP8(1, 5, 2) can be selected alternatively, as shown in Figure 2(b). A bigger exponent", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 615, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 415, + 628 + ], + "score": 1.0, + "content": "size (4 to 5) results in a larger range window. However, there are only 4", + "type": "text" + }, + { + "bbox": [ + 415, + 616, + 431, + 627 + ], + "score": 0.61, + "content": "( 2 ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 615, + 486, + 628 + ], + "score": 1.0, + "content": "instead of 8", + "type": "text" + }, + { + "bbox": [ + 487, + 615, + 504, + 627 + ], + "score": 0.67, + "content": "( 2 ^ { 3 } )", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 266, + 639 + ], + "score": 1.0, + "content": "representative values available for each", + "type": "text" + }, + { + "bbox": [ + 266, + 629, + 273, + 637 + ], + "score": 0.71, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "in the norm region since the fraction size decreases from", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 356, + 651 + ], + "score": 1.0, + "content": "3 to 2, which potentially results in a lower accuracy. Note that", + "type": "text" + }, + { + "bbox": [ + 356, + 638, + 405, + 649 + ], + "score": 0.5, + "content": "\\mathrm { F P } 8 ( 1 , 4 , 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 637, + 423, + 651 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 638, + 471, + 649 + ], + "score": 0.66, + "content": "\\mathrm { F P } 8 ( 1 , 5 , 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "are two", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 649, + 398, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 398, + 661 + ], + "score": 1.0, + "content": "most commonly used 8-bit floating point formats in the previous studies.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "With the help of the proposed FFP8 format, things can change a lot. Figure 2(c) shows what happens", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "if FFP8(1, 4, 3, 15) is in use. The range window is of the same size as that in Figure 2(a) but is left-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "shifted by 8 positions due to the exponent bias is set to 15 instead of the default value 7. 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A typical floating-point format consists of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 263, + 507, + 280 + ], + "spans": [ + { + "bbox": [ + 104, + 263, + 249, + 280 + ], + "score": 1.0, + "content": "sign (s), exponent (e), and fraction", + "type": "text" + }, + { + "bbox": [ + 249, + 266, + 263, + 276 + ], + "score": 0.83, + "content": "( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 263, + 475, + 280 + ], + "score": 1.0, + "content": "fields, and the bit length of each field is specified as", + "type": "text" + }, + { + "bbox": [ + 475, + 267, + 501, + 276 + ], + "score": 0.87, + "content": "x , y , z", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 263, + 507, + 280 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 344, + 289 + ], + "score": 1.0, + "content": "respectively. Besides, one more parameter, exponent bias", + "type": "text" + }, + { + "bbox": [ + 344, + 276, + 356, + 287 + ], + "score": 0.65, + "content": "( b )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 276, + 505, + 289 + ], + "score": 1.0, + "content": ", is required to completely specify a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 286, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 264, + 300 + ], + "score": 1.0, + "content": "floating-point number. 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In this paper, an", + "type": "text" + }, + { + "bbox": [ + 483, + 289, + 490, + 297 + ], + "score": 0.78, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 286, + 506, + 300 + ], + "score": 1.0, + "content": "-bit", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 246, + 311 + ], + "score": 1.0, + "content": "floating-point format is denoted as", + "type": "text" + }, + { + "bbox": [ + 246, + 298, + 288, + 309 + ], + "score": 0.93, + "content": "( x , y , z , b )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 297, + 301, + 311 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 301, + 298, + 334, + 310 + ], + "score": 0.93, + "content": "( x , y , z )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 297, + 365, + 311 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 365, + 299, + 425, + 309 + ], + "score": 0.9, + "content": "n = x + y + z", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 297, + 438, + 311 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 439, + 298, + 444, + 308 + ], + "score": 0.81, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 297, + 505, + 311 + ], + "score": 1.0, + "content": "is missing, the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 309, + 234, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 234, + 321 + ], + "score": 1.0, + "content": "default value is implicitly used.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 242, + 507, + 321 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 325, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "Figure 1(a) illustrates one 16-bit FP16 format and two commonly used 8-bit formats: (1, 5, 2, 15)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 488, + 349 + ], + "score": 1.0, + "content": "and (1, 4, 3, 7). Note that there is actually only one parameter, the bit width of the exponent", + "type": "text" + }, + { + "bbox": [ + 488, + 338, + 501, + 348 + ], + "score": 0.77, + "content": "( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 336, + 505, + 349 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 299, + 360 + ], + "score": 1.0, + "content": "that can be freely chosen when defining a new", + "type": "text" + }, + { + "bbox": [ + 299, + 349, + 306, + 358 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 348, + 487, + 360 + ], + "score": 1.0, + "content": "-bit conventional floating-point format since", + "type": "text" + }, + { + "bbox": [ + 487, + 350, + 494, + 358 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 357, + 507, + 371 + ], + "spans": [ + { + "bbox": [ + 104, + 357, + 147, + 371 + ], + "score": 1.0, + "content": "always 1,", + "type": "text" + }, + { + "bbox": [ + 147, + 361, + 154, + 369 + ], + "score": 0.65, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 357, + 196, + 371 + ], + "score": 1.0, + "content": "is always", + "type": "text" + }, + { + "bbox": [ + 196, + 359, + 239, + 370 + ], + "score": 0.91, + "content": "n - y - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 357, + 261, + 371 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 261, + 359, + 267, + 369 + ], + "score": 0.71, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 357, + 308, + 371 + ], + "score": 1.0, + "content": "is always", + "type": "text" + }, + { + "bbox": [ + 309, + 358, + 348, + 369 + ], + "score": 0.92, + "content": "2 ^ { y - 1 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 357, + 507, + 371 + ], + "score": 1.0, + "content": ". The above fact motivated us to think", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "out-of-the-box and thus develop a more flexible format, named flexible 8-bit floating-point (FFP8)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 392, + 393 + ], + "score": 1.0, + "content": "format and shown in Figure 1(b). In addition to the size of exponent", + "type": "text" + }, + { + "bbox": [ + 392, + 381, + 405, + 392 + ], + "score": 0.8, + "content": "( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 380, + 505, + 393 + ], + "score": 1.0, + "content": ", the FFP8 format offers", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 104, + 392, + 296, + 404 + ], + "score": 1.0, + "content": "two more parameters. First, the exponent bias", + "type": "text" + }, + { + "bbox": [ + 297, + 392, + 308, + 403 + ], + "score": 0.57, + "content": "( b )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 392, + 417, + 404 + ], + "score": 1.0, + "content": "is not necessarily equal to", + "type": "text" + }, + { + "bbox": [ + 418, + 391, + 457, + 402 + ], + "score": 0.91, + "content": "2 ^ { y - 1 } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 392, + 506, + 404 + ], + "score": 1.0, + "content": "and can be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 104, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "set to any integer, which helps cover the value distributions of both weights and activations well in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 413, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 203, + 425 + ], + "score": 1.0, + "content": "a shorter exponent size", + "type": "text" + }, + { + "bbox": [ + 203, + 414, + 216, + 425 + ], + "score": 0.74, + "content": "( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 413, + 396, + 425 + ], + "score": 1.0, + "content": ". Second, the sign bit is present or not (i.e.,", + "type": "text" + }, + { + "bbox": [ + 397, + 415, + 404, + 424 + ], + "score": 0.63, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 413, + 506, + 425 + ], + "score": 1.0, + "content": "can be 0 or 1). Without", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 156, + 437 + ], + "score": 1.0, + "content": "the sign bit", + "type": "text" + }, + { + "bbox": [ + 156, + 425, + 183, + 435 + ], + "score": 0.85, + "content": "( x = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "), unsigned activations can be better represented in higher precision. To be more", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 287, + 448 + ], + "score": 1.0, + "content": "precise, there are only two restrictions on an", + "type": "text" + }, + { + "bbox": [ + 287, + 437, + 294, + 445 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 435, + 378, + 448 + ], + "score": 1.0, + "content": "-bit FFP8 format: 1)", + "type": "text" + }, + { + "bbox": [ + 378, + 436, + 444, + 447 + ], + "score": 0.87, + "content": "x + y + z = n ;", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 435, + 455, + 448 + ], + "score": 1.0, + "content": "2)", + "type": "text" + }, + { + "bbox": [ + 455, + 437, + 462, + 446 + ], + "score": 0.6, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "must be 0", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 446, + 498, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 498, + 459 + ], + "score": 1.0, + "content": "or 1. Next, we are about to show how the proposed FFP8 format improves the inference accuracy.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 326, + 507, + 459 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 473, + 477, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 479, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 479, + 486 + ], + "score": 1.0, + "content": "3.2 WEIGHT DISTRIBUTION AND THE WAYS VARIOUS 8-BIT FORMATS COPE WITH IT", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "Overall speaking, the magnitudes of weights in most DNN models are usually small. Take a popular", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 506, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 520 + ], + "score": 1.0, + "content": "image classification model VGG-16 (Simonyan & Zisserman, 2015) as an example, the maximum", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "score": 1.0, + "content": "magnitude of weights in the whole model is less than 2. Figure 2 gives the overall weight distribution", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "(in log scale) of VGG-16 trained via a conventional FP32 framework. Figure 2(a) illustrates how", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 538, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 554 + ], + "score": 1.0, + "content": "the conventional FP8(1, 4, 3) copes with those weights. The rectangle in red is called the range", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 291, + 563 + ], + "score": 1.0, + "content": "window, which specifies the value range that", + "type": "text" + }, + { + "bbox": [ + 291, + 550, + 341, + 561 + ], + "score": 0.4, + "content": "\\mathrm { F P } 8 ( 1 , 4 , 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "can represent. The purple vertical dash", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "line further partitions the window into the norm region (right side) and the denorm region (left side).", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "The star marks the position where the weight of the maximum magnitude locates. Note that virtually", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 434, + 595 + ], + "score": 1.0, + "content": "the entire right half of the range window in Figure 2(a) covers no weights, while", + "type": "text" + }, + { + "bbox": [ + 435, + 583, + 457, + 594 + ], + "score": 0.86, + "content": "9 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "of leftmost", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "weights cannot be included in the range window. In order to contain almost every weight in a range", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "window, FP8(1, 5, 2) can be selected alternatively, as shown in Figure 2(b). A bigger exponent", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 615, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 415, + 628 + ], + "score": 1.0, + "content": "size (4 to 5) results in a larger range window. However, there are only 4", + "type": "text" + }, + { + "bbox": [ + 415, + 616, + 431, + 627 + ], + "score": 0.61, + "content": "( 2 ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 615, + 486, + 628 + ], + "score": 1.0, + "content": "instead of 8", + "type": "text" + }, + { + "bbox": [ + 487, + 615, + 504, + 627 + ], + "score": 0.67, + "content": "( 2 ^ { 3 } )", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 266, + 639 + ], + "score": 1.0, + "content": "representative values available for each", + "type": "text" + }, + { + "bbox": [ + 266, + 629, + 273, + 637 + ], + "score": 0.71, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "in the norm region since the fraction size decreases from", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 356, + 651 + ], + "score": 1.0, + "content": "3 to 2, which potentially results in a lower accuracy. Note that", + "type": "text" + }, + { + "bbox": [ + 356, + 638, + 405, + 649 + ], + "score": 0.5, + "content": "\\mathrm { F P } 8 ( 1 , 4 , 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 637, + 423, + 651 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 638, + 471, + 649 + ], + "score": 0.66, + "content": "\\mathrm { F P } 8 ( 1 , 5 , 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "are two", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 649, + 398, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 398, + 661 + ], + "score": 1.0, + "content": "most commonly used 8-bit floating point formats in the previous studies.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 495, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "With the help of the proposed FFP8 format, things can change a lot. Figure 2(c) shows what happens", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "if FFP8(1, 4, 3, 15) is in use. The range window is of the same size as that in Figure 2(a) but is left-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "shifted by 8 positions due to the exponent bias is set to 15 instead of the default value 7. It is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 413, + 712 + ], + "score": 1.0, + "content": "obvious that FFP8(1, 4, 3, 15) can better cope with weights in VGG-16 than", + "type": "text" + }, + { + "bbox": [ + 414, + 699, + 462, + 710 + ], + "score": 0.33, + "content": "\\mathrm { F P } 8 ( 1 , 4 , 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "and FP(1,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "5, 2). Furthermore, Figure 2(d) shows what if FFP8(1, 3, 4, 7) is in use. Comparing FFP8(1, 3,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 381, + 733 + ], + "score": 1.0, + "content": "4, 7) against FFP8(1, 4, 3, 15), weights in the norm region (over", + "type": "text" + }, + { + "bbox": [ + 381, + 721, + 401, + 731 + ], + "score": 0.84, + "content": "3 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 720, + 506, + 733 + ], + "score": 1.0, + "content": ") are represented in 1-bit", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 354, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 208, + 368 + ], + "score": 1.0, + "content": "higher precision whereas", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 209, + 356, + 231, + 366 + ], + "score": 0.86, + "content": "4 . 6 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 231, + 354, + 506, + 368 + ], + "score": 1.0, + "content": "of leftmost weights are not included in the range window. However,", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "those out-of-the-window weights can be regarded as some sort of pruned weights. That is, more", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "score": 1.0, + "content": "investigation should be further conducted to determine whether FFP8(1, 3, 4, 7) or FFP8(1, 4, 3, 15)", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "is better for the weights in VGG-16. 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However,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "those out-of-the-window weights can be regarded as some sort of pruned weights. That is, more", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 390 + ], + "score": 1.0, + "content": "investigation should be further conducted to determine whether FFP8(1, 3, 4, 7) or FFP8(1, 4, 3, 15)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "is better for the weights in VGG-16. One way or another, it is clear that the proposed FFP8 formats", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 399, + 412, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 412, + 412 + ], + "score": 1.0, + "content": "can achieve certain improvement that the conventional 8-bit formats cannot.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 432, + 452, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 454, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 454, + 445 + ], + "score": 1.0, + "content": "3.3 FFP8-BASED INFERENCE FRAMEWORK AND BEST-FIT FORMAT SELECTION", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "In the proposed memory-efficient inference framework, weights and activations stored in off-chip", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "memory are in FFP8 formats while operations inside the computing engine are still in FP32, which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "is a common strategy widely adopted in current state-of-the-art computing platforms (Intel, 2018;", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "Nvidia, 2017). Hence, a systematic process, shown in Figure 3, is required to transform FP32", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "numbers into FFP8 ones in the following two scenarios: 1) quantizing pre-trained FP32 weights", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "to FFP8 ones in advance, and 2) converting FP32 output activations into FFP8 ones before writing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "them back to off-chip memory. The process first prepares a list of 256 values that can be precisely", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "score": 1.0, + "content": "represented by the specified FFP8 format. Then, the given FP32 inputs (weights or activations) are", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 544, + 385, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 385, + 557 + ], + "score": 1.0, + "content": "converted into FFP8 outputs using the round-to-nearest-even method.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 504, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "score": 1.0, + "content": "In order to explore best-fit 8-bit formats for weights and activations in VGG-16, we have made a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "set of attempts, and the results are summarized in Table 1. First, it is unwise to represent weights", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 117, + 596 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 583, + 162, + 594 + ], + "score": 0.31, + "content": "\\mathrm { F P } ( 1 , 5 , 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 582, + 505, + 596 + ], + "score": 1.0, + "content": ". Though Figure 2(b) shows FP(1, 5, 2) can cover virtually all weights, the accuracy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 351, + 606 + ], + "score": 1.0, + "content": "loss is serious due to its 2-bit extremely low precision. Next,", + "type": "text" + }, + { + "bbox": [ + 352, + 594, + 395, + 605 + ], + "score": 0.43, + "content": "\\mathrm { F P } ( 1 , 4 , 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "performs much better than", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 151, + 616 + ], + "score": 0.35, + "content": "\\mathrm { F P } ( 1 , 5 , 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 604, + 506, + 618 + ], + "score": 1.0, + "content": ", which suggests one extra precision bit does make a notable improvement here though", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 129, + 627 + ], + "score": 0.84, + "content": "9 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 615, + 506, + 630 + ], + "score": 1.0, + "content": "smallest weights are clear to 0 (i.e., pruned away), as depicted in Figure 2(a). Moreover,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 303, + 639 + ], + "score": 1.0, + "content": "the proposed FFP8 format can surely do better.", + "type": "text" + }, + { + "bbox": [ + 303, + 627, + 374, + 638 + ], + "score": 0.31, + "content": "\\mathrm { F F P 8 } ( 1 , 4 , 3 , 1 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "further outperforms FP(1, 4, 3)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 639, + 478, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 478, + 650 + ], + "score": 1.0, + "content": "because weights are better covered by its left-shifted range window, as shown in Figure 2(c).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 206, + 667 + ], + "score": 1.0, + "content": "Inspired by the fact that", + "type": "text" + }, + { + "bbox": [ + 207, + 655, + 252, + 666 + ], + "score": 0.5, + "content": "\\mathrm { F P } ( 1 , 4 , 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 655, + 305, + 667 + ], + "score": 1.0, + "content": "outperforms", + "type": "text" + }, + { + "bbox": [ + 305, + 655, + 350, + 666 + ], + "score": 0.47, + "content": "\\mathrm { F P } ( 1 , 5 , 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 655, + 506, + 667 + ], + "score": 1.0, + "content": ", we further examine whether an even", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 422, + 678 + ], + "score": 1.0, + "content": "smaller exponent size can help or not. 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Hence, a systematic process, shown in Figure 3, is required to transform FP32", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "numbers into FFP8 ones in the following two scenarios: 1) quantizing pre-trained FP32 weights", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "to FFP8 ones in advance, and 2) converting FP32 output activations into FFP8 ones before writing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "them back to off-chip memory. 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WeightActivationTop-1 (△)Top-5 (△)
FP32FP3271.59%90.38%
(1,5,2,15)(1,4,3,7)69.89% (-1.70%)89.29% (-1.09%)
(1,4,3,7)(1,4,3,7)70.86% (-0.73%)90.02% (-0.36%)
(1,4,3,15)(1,4,3,7)70.96% (-0.63%)90.10% (-0.28%)
(1,3,4,3)(1,4,3,7)70.18% (-1.41%)89.56% (-0.82%)
(1,3,4,7)(1,4,3,7)71.19% (-0.40%)90.12% (-0.26%)
(1,3,4,7)(1,4,3,7)+(0,4,4,7)71.19% (-0.40%)90.14% (-0.24%)
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The detailed distribution of activations will be given in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "Section 3.4 later. Meanwhile, it is also worth noting that a large set of commonly used activation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "score": 1.0, + "content": "functions always produce nonnegative outputs, e.g., ReLU, ReLU6, and sigmoid. That is, if those", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "outputs are represented in any signed format, a half of the code space is actually wasted. It may not", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 394, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 406 + ], + "score": 1.0, + "content": "be a problem for 32-bit and 16-bit formats with long enough fraction bits; however, it is indeed a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "serious issue for any 8-bit format, which merely has 256 available codes in total. Since VGG-16", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "score": 1.0, + "content": "utilizes ReLU as its activation function, it is feasible to select signed FFP8(1, 4, 3, 7) for the first", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "layer and unsigned FFP8(0, 4, 4, 7) for all succeeding layers. It implies that 256 instead of 128", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 438, + 504, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 504, + 449 + ], + "score": 1.0, + "content": "codes are available to represent those unsigned activations in all layers except for the first one. With", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 449, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 460 + ], + "score": 1.0, + "content": "no surprise, the accuracy is further improved since the range window remains untouched while the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 457, + 472 + ], + "score": 1.0, + "content": "fraction size gains one extra bit. In the last configuration, the Top-1 accuracy loss is only", + "type": "text" + }, + { + "bbox": [ + 458, + 460, + 480, + 470 + ], + "score": 0.84, + "content": "0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "when", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 471, + 350, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 350, + 482 + ], + "score": 1.0, + "content": "compared against the FP32 baseline, as indicated in Table 1.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 496, + 252, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 254, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 254, + 509 + ], + "score": 1.0, + "content": "3.4 LAYER-WISE OPTIMIZATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 516, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "Figure 4 and Figure 5 illustrate several distributions of weights and activations in VGG-16, respec-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 528, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 539 + ], + "score": 1.0, + "content": "tively. Each figure includes the distributions of one whole model and three selected individual layers.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "score": 1.0, + "content": "In Section 3.3, the best-fit FFP8 format is determined by the overall distribution of the whole model.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "Here we have three key observations from those distributions: 1) the distributions of weights are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "score": 1.0, + "content": "quite dissimilar to those of activations, 2) even the distributions across different layers are dissimilar", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 573, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 583 + ], + "score": 1.0, + "content": "for both weights and activations, and 3) the distribution of an individual layer is narrower than that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "of the whole model. The above observations clearly suggest that applying layer-wise optimization", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 594, + 469, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 469, + 606 + ], + "score": 1.0, + "content": "(LWO) properly on number format selection is very likely to improve the accuracy further.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 254, + 623 + ], + "score": 1.0, + "content": "For instance, after comparing Figure", + "type": "text" + }, + { + "bbox": [ + 254, + 611, + 290, + 622 + ], + "score": 0.53, + "content": "5 ( \\mathrm { a } ) \\& ( \\mathrm { b } )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 611, + 505, + 623 + ], + "score": 1.0, + "content": ", it is found that the distribution of the whole model is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "wider than that of the first layer, and the maximum log magnitudes of the whole model and the first", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 343, + 645 + ], + "score": 1.0, + "content": "layer are 8 and 1, respectively. As a consequence, selecting", + "type": "text" + }, + { + "bbox": [ + 344, + 633, + 407, + 644 + ], + "score": 0.27, + "content": "\\mathrm { F F P } 8 ( 1 , 3 , 4 , 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "instead of FFP8(1, 4, 3,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 643, + 504, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 417, + 657 + ], + "score": 1.0, + "content": "7) for activations in the first layer can further increase the Top-1 accuracy by", + "type": "text" + }, + { + "bbox": [ + 418, + 644, + 445, + 655 + ], + "score": 0.88, + "content": "0 . 1 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 643, + 471, + 657 + ], + "score": 1.0, + "content": "(from", + "type": "text" + }, + { + "bbox": [ + 472, + 644, + 504, + 654 + ], + "score": 0.85, + "content": "7 1 . 1 9 \\%", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 116, + 667 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 655, + 149, + 666 + ], + "score": 0.85, + "content": "7 1 . 3 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "), as indicated in Table 2. Next, we examine a new configuration that makes weights of all", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 377, + 678 + ], + "score": 1.0, + "content": "layers are in FFP8(1, 2, 5, 3). It is obvious an unwise attempt since", + "type": "text" + }, + { + "bbox": [ + 378, + 666, + 405, + 676 + ], + "score": 0.87, + "content": "3 7 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "leftmost weights are out", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "of the range window according to Figure 4(a). However, it may not be a bad idea to use FFP8(1, 2,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 372, + 700 + ], + "score": 1.0, + "content": "5, 5) in Layer 6 and FFP8(1, 2, 5, 6) in the last layer because only", + "type": "text" + }, + { + "bbox": [ + 372, + 688, + 395, + 698 + ], + "score": 0.88, + "content": "6 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 687, + 412, + 700 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 413, + 688, + 435, + 698 + ], + "score": 0.88, + "content": "5 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "smallest weights", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 363, + 712 + ], + "score": 1.0, + "content": "are out of the range window respectively according to Figure", + "type": "text" + }, + { + "bbox": [ + 363, + 699, + 399, + 710 + ], + "score": 0.65, + "content": "4 ( \\mathrm { c } ) \\& ( \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 698, + 506, + 712 + ], + "score": 1.0, + "content": ". Therefore, we examine", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 352, + 722 + ], + "score": 1.0, + "content": "another new configuration that makes weights of all layers in", + "type": "text" + }, + { + "bbox": [ + 353, + 710, + 416, + 721 + ], + "score": 0.7, + "content": "\\mathrm { F F P } 8 ( 1 , 2 , 5 , * )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 709, + 491, + 722 + ], + "score": 1.0, + "content": ". 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WeightActivationTop-1 (△)Top-5 (△)
FP32FP3271.59%90.38%
(1,5,2,15)(1,4,3,7)69.89% (-1.70%)89.29% (-1.09%)
(1,4,3,7)(1,4,3,7)70.86% (-0.73%)90.02% (-0.36%)
(1,4,3,15)(1,4,3,7)70.96% (-0.63%)90.10% (-0.28%)
(1,3,4,3)(1,4,3,7)70.18% (-1.41%)89.56% (-0.82%)
(1,3,4,7)(1,4,3,7)71.19% (-0.40%)90.12% (-0.26%)
(1,3,4,7)(1,4,3,7)+(0,4,4,7)71.19% (-0.40%)90.14% (-0.24%)
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For", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "those attempts made above, activations are always in FFP8(1, 4, 3, 7) for two reasons: 1) the overall", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "score": 1.0, + "content": "distribution of activations is wider than that of weights, and 2) the maximum magnitude of activa-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "tions is much bigger than that of weights. The detailed distribution of activations will be given in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "Section 3.4 later. Meanwhile, it is also worth noting that a large set of commonly used activation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "score": 1.0, + "content": "functions always produce nonnegative outputs, e.g., ReLU, ReLU6, and sigmoid. That is, if those", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "outputs are represented in any signed format, a half of the code space is actually wasted. It may not", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 394, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 406 + ], + "score": 1.0, + "content": "be a problem for 32-bit and 16-bit formats with long enough fraction bits; however, it is indeed a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "serious issue for any 8-bit format, which merely has 256 available codes in total. Since VGG-16", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "score": 1.0, + "content": "utilizes ReLU as its activation function, it is feasible to select signed FFP8(1, 4, 3, 7) for the first", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "layer and unsigned FFP8(0, 4, 4, 7) for all succeeding layers. It implies that 256 instead of 128", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 438, + 504, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 504, + 449 + ], + "score": 1.0, + "content": "codes are available to represent those unsigned activations in all layers except for the first one. With", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 449, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 460 + ], + "score": 1.0, + "content": "no surprise, the accuracy is further improved since the range window remains untouched while the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 457, + 472 + ], + "score": 1.0, + "content": "fraction size gains one extra bit. In the last configuration, the Top-1 accuracy loss is only", + "type": "text" + }, + { + "bbox": [ + 458, + 460, + 480, + 470 + ], + "score": 0.84, + "content": "0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "when", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 471, + 350, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 350, + 482 + ], + "score": 1.0, + "content": "compared against the FP32 baseline, as indicated in Table 1.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 317, + 506, + 482 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 496, + 252, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 254, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 254, + 509 + ], + "score": 1.0, + "content": "3.4 LAYER-WISE OPTIMIZATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 516, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "Figure 4 and Figure 5 illustrate several distributions of weights and activations in VGG-16, respec-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 528, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 539 + ], + "score": 1.0, + "content": "tively. Each figure includes the distributions of one whole model and three selected individual layers.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "score": 1.0, + "content": "In Section 3.3, the best-fit FFP8 format is determined by the overall distribution of the whole model.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "Here we have three key observations from those distributions: 1) the distributions of weights are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "score": 1.0, + "content": "quite dissimilar to those of activations, 2) even the distributions across different layers are dissimilar", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 573, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 583 + ], + "score": 1.0, + "content": "for both weights and activations, and 3) the distribution of an individual layer is narrower than that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "of the whole model. The above observations clearly suggest that applying layer-wise optimization", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 594, + 469, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 469, + 606 + ], + "score": 1.0, + "content": "(LWO) properly on number format selection is very likely to improve the accuracy further.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 517, + 506, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 254, + 623 + ], + "score": 1.0, + "content": "For instance, after comparing Figure", + "type": "text" + }, + { + "bbox": [ + 254, + 611, + 290, + 622 + ], + "score": 0.53, + "content": "5 ( \\mathrm { a } ) \\& ( \\mathrm { b } )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 611, + 505, + 623 + ], + "score": 1.0, + "content": ", it is found that the distribution of the whole model is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "wider than that of the first layer, and the maximum log magnitudes of the whole model and the first", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 343, + 645 + ], + "score": 1.0, + "content": "layer are 8 and 1, respectively. As a consequence, selecting", + "type": "text" + }, + { + "bbox": [ + 344, + 633, + 407, + 644 + ], + "score": 0.27, + "content": "\\mathrm { F F P } 8 ( 1 , 3 , 4 , 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "instead of FFP8(1, 4, 3,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 643, + 504, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 417, + 657 + ], + "score": 1.0, + "content": "7) for activations in the first layer can further increase the Top-1 accuracy by", + "type": "text" + }, + { + "bbox": [ + 418, + 644, + 445, + 655 + ], + "score": 0.88, + "content": "0 . 1 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 643, + 471, + 657 + ], + "score": 1.0, + "content": "(from", + "type": "text" + }, + { + "bbox": [ + 472, + 644, + 504, + 654 + ], + "score": 0.85, + "content": "7 1 . 1 9 \\%", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 116, + 667 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 655, + 149, + 666 + ], + "score": 0.85, + "content": "7 1 . 3 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "), as indicated in Table 2. Next, we examine a new configuration that makes weights of all", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 377, + 678 + ], + "score": 1.0, + "content": "layers are in FFP8(1, 2, 5, 3). It is obvious an unwise attempt since", + "type": "text" + }, + { + "bbox": [ + 378, + 666, + 405, + 676 + ], + "score": 0.87, + "content": "3 7 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "leftmost weights are out", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "of the range window according to Figure 4(a). However, it may not be a bad idea to use FFP8(1, 2,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 372, + 700 + ], + "score": 1.0, + "content": "5, 5) in Layer 6 and FFP8(1, 2, 5, 6) in the last layer because only", + "type": "text" + }, + { + "bbox": [ + 372, + 688, + 395, + 698 + ], + "score": 0.88, + "content": "6 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 687, + 412, + 700 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 413, + 688, + 435, + 698 + ], + "score": 0.88, + "content": "5 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "smallest weights", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 363, + 712 + ], + "score": 1.0, + "content": "are out of the range window respectively according to Figure", + "type": "text" + }, + { + "bbox": [ + 363, + 699, + 399, + 710 + ], + "score": 0.65, + "content": "4 ( \\mathrm { c } ) \\& ( \\mathrm { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 698, + 506, + 712 + ], + "score": 1.0, + "content": ". Therefore, we examine", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 352, + 722 + ], + "score": 1.0, + "content": "another new configuration that makes weights of all layers in", + "type": "text" + }, + { + "bbox": [ + 353, + 710, + 416, + 721 + ], + "score": 0.7, + "content": "\\mathrm { F F P } 8 ( 1 , 2 , 5 , * )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 709, + 491, + 722 + ], + "score": 1.0, + "content": ". Here the asterisk", + "type": "text" + }, + { + "bbox": [ + 491, + 710, + 504, + 721 + ], + "score": 0.61, + "content": "( \\ast )", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "represents the largest possible exponent bias, which ensures that the maximum weight is still inside", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 456, + 95 + ], + "score": 1.0, + "content": "the range window. This LWO on weights successfully increases the Top-1 accuracy by", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 456, + 83, + 479, + 93 + ], + "score": 0.85, + "content": "0 . 1 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 479, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "(from", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 138, + 104 + ], + "score": 0.87, + "content": "7 1 . 3 8 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 139, + 93, + 150, + 107 + ], + "score": 1.0, + "content": "to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 150, + 94, + 183, + 104 + ], + "score": 0.85, + "content": "7 1 . 4 8 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 183, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "). Similarly, we can apply the LWO on activations, which again raises the Top-5", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 483, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 156, + 117 + ], + "score": 1.0, + "content": "accuracy by", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 157, + 105, + 183, + 115 + ], + "score": 0.86, + "content": "0 . 0 5 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 184, + 104, + 483, + 117 + ], + "score": 1.0, + "content": ". The selected FFP8 format of each layer after LWO is reported in Table 3.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 611, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 456, + 95 + ], + "score": 1.0, + "content": "the range window. This LWO on weights successfully increases the Top-1 accuracy by", + "type": "text" + }, + { + "bbox": [ + 456, + 83, + 479, + 93 + ], + "score": 0.85, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "(from", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 138, + 104 + ], + "score": 0.87, + "content": "7 1 . 3 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 93, + 150, + 107 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 150, + 94, + 183, + 104 + ], + "score": 0.85, + "content": "7 1 . 4 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "). Similarly, we can apply the LWO on activations, which again raises the Top-5", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 483, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 156, + 117 + ], + "score": 1.0, + "content": "accuracy by", + "type": "text" + }, + { + "bbox": [ + 157, + 105, + 183, + 115 + ], + "score": 0.86, + "content": "0 . 0 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 104, + 483, + 117 + ], + "score": 1.0, + "content": ". The selected FFP8 format of each layer after LWO is reported in Table 3.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 120, + 505, + 165 + ], + "lines": [ + { + "bbox": [ + 105, + 121, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 236, + 133 + ], + "score": 1.0, + "content": "Previous studies usually choose", + "type": "text" + }, + { + "bbox": [ + 236, + 122, + 286, + 133 + ], + "score": 0.44, + "content": "\\mathrm { F P } 8 ( 1 , 4 , 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 122, + 298, + 133 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 298, + 121, + 348, + 133 + ], + "score": 0.35, + "content": "\\mathrm { F P } 8 ( 1 , 5 , 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "formats because the exponent size has", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "to be large enough to cover most weights and activations at the cost of an even smaller fraction size.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "However, the exponent size for weights can be as small as 2 after LWO in our flow. Consequently,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 460, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 460, + 166 + ], + "score": 1.0, + "content": "the larger 5-bit fraction size does help in accuracy improvement, as indicated in Table 2.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 106, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "Note that the Top-1 accuracy achieved by the final configuration, which applies layer-wise optimiza-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 182, + 504, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 291, + 195 + ], + "score": 1.0, + "content": "tion on both weights and activations, is merely", + "type": "text" + }, + { + "bbox": [ + 291, + 182, + 318, + 193 + ], + "score": 0.87, + "content": "0 . 1 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 183, + 504, + 195 + ], + "score": 1.0, + "content": "lower as compared to that of the FP32 baseline", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 109, + 193, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 109, + 193, + 142, + 204 + ], + "score": 0.83, + "content": "( 7 1 . 4 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 193, + 157, + 206 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 157, + 193, + 191, + 204 + ], + "score": 0.86, + "content": "7 1 . 5 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 193, + 505, + 206 + ], + "score": 1.0, + "content": "). More importantly, model retraining is not applied yet. In other words, all the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 218 + ], + "score": 1.0, + "content": "accuracy improvements made so far are simply from properly re-expressing weights and activations", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 342, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 342, + 227 + ], + "score": 1.0, + "content": "of a pre-trained FP32 model in their best-fit FFP8 formats.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "image", + "bbox": [ + 106, + 239, + 501, + 452 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 239, + 501, + 452 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 239, + 501, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 501, + 452 + ], + "score": 0.972, + "type": "image", + "image_path": 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(1,3,4,7)(1,4,3,7)+(0,4,4,7)71.19% (-0.40%)90.14% (-0.24%)
(1,3,4,7)(1,3,4,6)+(0,4,4,7)71.38% (-0.21%)90.33% (-0.05%)
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(1,2,5,*)(1,3,4,6)+(0,4,4,7)71.48% (-0.11%)90.27% (-0.11%)
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However, it is reported that quantization-aware", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "model retraining can usually improve the accuracy (Jacob et al., 2017), which motivates us to check", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "whether it can successfully apply to our work. We first train the ResNet-18 model in an FP32 frame-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 291, + 423 + ], + "score": 1.0, + "content": "work and the corresponding Top-1 accuracy is", + "type": "text" + }, + { + "bbox": [ + 291, + 411, + 323, + 421 + ], + "score": 0.88, + "content": "6 9 . 7 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 411, + 505, + 423 + ], + "score": 1.0, + "content": ". Next, an FFP8-based inference is performed", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 420, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 303, + 435 + ], + "score": 1.0, + "content": "under the following settings: 1) all weights are in", + "type": "text" + }, + { + "bbox": [ + 303, + 422, + 379, + 433 + ], + "score": 0.34, + "content": "\\mathrm { F F P } 8 ( 1 , 3 , 4 , 7 ) , 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 420, + 506, + 435 + ], + "score": 1.0, + "content": "activations of the first layer are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 355, + 446 + ], + "score": 1.0, + "content": "in FFP8(1, 3, 4, 6), and 3) activations of the other layers are in", + "type": "text" + }, + { + "bbox": [ + 356, + 433, + 419, + 444 + ], + "score": 0.34, + "content": "\\mathrm { F F P 8 } ( 0 , 4 , 4 , 7 )", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 432, + 505, + 446 + ], + "score": 1.0, + "content": ". The Top-1 accuracy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 261, + 456 + ], + "score": 1.0, + "content": "for the above configuration is down to", + "type": "text" + }, + { + "bbox": [ + 261, + 444, + 293, + 454 + ], + "score": 0.88, + "content": "6 9 . 4 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 443, + 506, + 456 + ], + "score": 1.0, + "content": ". Then, a quantization-aware retraining process of 15", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 363, + 468 + ], + "score": 1.0, + "content": "epochs is applied and the resultant Top-1 accuracy goes back to", + "type": "text" + }, + { + "bbox": [ + 363, + 455, + 395, + 465 + ], + "score": 0.88, + "content": "6 9 . 7 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 454, + 506, + 468 + ], + "score": 1.0, + "content": ". Note that the retraining is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 465, + 443, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 443, + 478 + ], + "score": 1.0, + "content": "done simply in a typical FP32 framework without the need of special training skills.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 492, + 403, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 404, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 404, + 507 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL RESULTS ON VARIOUS DNN MODELS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "In this section, we intend to demonstrate that the proposed FFP8 format constantly performs well in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "various DNN models in addition to VGG-16. Table 4 reports the accuracy results of various mod-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "els under different format configurations. Configuration A gives the results of the FP32 baseline.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 551, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 562 + ], + "score": 1.0, + "content": "Configuration B utilizes a conventional FP8(1, 4, 3) format with a default exponent bias (7), which", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 216, + 574 + ], + "score": 1.0, + "content": "incurs a penalty of roughly", + "type": "text" + }, + { + "bbox": [ + 216, + 561, + 231, + 572 + ], + "score": 0.85, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "drop on Top-1 accuracy. 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(1,3,4,7)(1,3,4,6)+(0,4,4,7)71.38% (-0.21%)90.33% (-0.05%)
(1,2,5,3)(1,3,4,6)+(0,4,4,7)71.24% (-0.35%)90.22% (-0.16%)
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4(1,2,5,5)(0,4,4,10)11(1,2,5,5)(0,4,4,7)
5(1,2,5,4)(0,4,4,9)12(1,2,5,5)(0,4,4,7)
6(1,2,5,5)(0,4,4,9)13(1,2,5,6)(0,4,4,8)
7(1,2,5,4)(0,4,4,9)
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However, it is reported that quantization-aware", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "model retraining can usually improve the accuracy (Jacob et al., 2017), which motivates us to check", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "whether it can successfully apply to our work. We first train the ResNet-18 model in an FP32 frame-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 291, + 423 + ], + "score": 1.0, + "content": "work and the corresponding Top-1 accuracy is", + "type": "text" + }, + { + "bbox": [ + 291, + 411, + 323, + 421 + ], + "score": 0.88, + "content": "6 9 . 7 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 411, + 505, + 423 + ], + "score": 1.0, + "content": ". Next, an FFP8-based inference is performed", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 420, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 303, + 435 + ], + "score": 1.0, + "content": "under the following settings: 1) all weights are in", + "type": "text" + }, + { + "bbox": [ + 303, + 422, + 379, + 433 + ], + "score": 0.34, + "content": "\\mathrm { F F P } 8 ( 1 , 3 , 4 , 7 ) , 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 420, + 506, + 435 + ], + "score": 1.0, + "content": "activations of the first layer are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 355, + 446 + ], + "score": 1.0, + "content": "in FFP8(1, 3, 4, 6), and 3) activations of the other layers are in", + "type": "text" + }, + { + "bbox": [ + 356, + 433, + 419, + 444 + ], + "score": 0.34, + "content": "\\mathrm { F F P 8 } ( 0 , 4 , 4 , 7 )", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 432, + 505, + 446 + ], + "score": 1.0, + "content": ". The Top-1 accuracy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 261, + 456 + ], + "score": 1.0, + "content": "for the above configuration is down to", + "type": "text" + }, + { + "bbox": [ + 261, + 444, + 293, + 454 + ], + "score": 0.88, + "content": "6 9 . 4 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 443, + 506, + 456 + ], + "score": 1.0, + "content": ". Then, a quantization-aware retraining process of 15", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 363, + 468 + ], + "score": 1.0, + "content": "epochs is applied and the resultant Top-1 accuracy goes back to", + "type": "text" + }, + { + "bbox": [ + 363, + 455, + 395, + 465 + ], + "score": 0.88, + "content": "6 9 . 7 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 454, + 506, + 468 + ], + "score": 1.0, + "content": ". Note that the retraining is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 465, + 443, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 443, + 478 + ], + "score": 1.0, + "content": "done simply in a typical FP32 framework without the need of special training skills.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 366, + 506, + 478 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 492, + 403, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 404, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 404, + 507 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL RESULTS ON VARIOUS DNN MODELS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "In this section, we intend to demonstrate that the proposed FFP8 format constantly performs well in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "various DNN models in addition to VGG-16. Table 4 reports the accuracy results of various mod-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "els under different format configurations. Configuration A gives the results of the FP32 baseline.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 551, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 562 + ], + "score": 1.0, + "content": "Configuration B utilizes a conventional FP8(1, 4, 3) format with a default exponent bias (7), which", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 216, + 574 + ], + "score": 1.0, + "content": "incurs a penalty of roughly", + "type": "text" + }, + { + "bbox": [ + 216, + 561, + 231, + 572 + ], + "score": 0.85, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "drop on Top-1 accuracy. With the help of the FFP8 format, Configu-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "score": 1.0, + "content": "ration C adopts the best-fit format for weights, which effectively reduces the Top-1 accuracy loss to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 154, + 594 + ], + "score": 0.9, + "content": "0 . 4 { \\sim } 0 . 7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 583, + 505, + 596 + ], + "score": 1.0, + "content": ". 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Cfg.WeightActivationVGG-16ResNet-50ResNet-34ResNet-18
Top-1/Top-5Top-1/Top-5Top-1/Top-5Top-1/Top-5
AFP32FP3271.59 /90.3876.13/92.8673.31/91.4269.76/89.08
B(1,4,3,7)(1,4,3,7)70.86 /90.0275.24 /92.5272.39 /90.9768.70 / 88.50
C(1,3,4,7)(1,4,3,7)71.19 /90.1275.38 /92.6872.81 /91.1669.25 /88.80
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The experimental results once again demonstrate that", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 456, + 335, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 335, + 469 + ], + "score": 1.0, + "content": "FFP8 performs very well in these two categories as well.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 485, + 324, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 325, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 325, + 500 + ], + "score": 1.0, + "content": "5 ASPECTS OF SYSTEM AND HARDWARE", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "The target of this work is to develop a memory-efficient inference system, especially for those with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "limited memory capacity and bandwidth. 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The experimental results once again demonstrate that", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 456, + 335, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 335, + 469 + ], + "score": 1.0, + "content": "FFP8 performs very well in these two categories as well.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 347, + 506, + 469 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 485, + 324, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 325, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 325, + 500 + ], + "score": 1.0, + "content": "5 ASPECTS OF SYSTEM AND HARDWARE", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "The target of this work is to develop a memory-efficient inference system, especially for those with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "limited memory capacity and bandwidth. A current state-of-the-art system, proposed by Nvidia", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "and Intel, has successfully cut the required memory size and traffic through representing external", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "data (weights or activations) in BFP16/INT8 instead of FP32, as illustrated in Figure 6(a) (Intel,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "2018; 2019; Nvidia, 2017; 2020). It not only reduces the memory size, alleviates the performance", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "score": 1.0, + "content": "bottleneck due to memory bandwidth limitation but also saves a significant amount of energy due to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "fewer power-consuming external memory access operations. To preserve the computation accuracy", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "at the same time, external BFP16 data are converted to FP32 data right before entering the FP32", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 600, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 611 + ], + "score": 1.0, + "content": "fused-multiply-add (FMA) unit. That is, internal computations can all be in FP32. Those FP32 data", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 609, + 468, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 468, + 623 + ], + "score": 1.0, + "content": "are converted back to BFP16 only if they are about to be written back to external memory.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 511, + 506, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "In this work, we propose a system architecture that is very similar to the previous one, as shown in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "Figure 6(b). The key difference is that external data are in FFP8 instead of BFP16, which implies the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "proposed system merely demands a quarter of the memory size and throughput required by today’s", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 659, + 210, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 210, + 672 + ], + "score": 1.0, + "content": "FP32-based counterparts.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 627, + 505, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "It is easy to convert a BFP16 number into its FP32 equivalent by concatenating a 16-bit pattern of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "all 0s at its least significant end. 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Few register bits are allocated to store the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 212, + 722 + ], + "score": 1.0, + "content": "current format settings of", + "type": "text" + }, + { + "bbox": [ + 212, + 711, + 230, + 721 + ], + "score": 0.32, + "content": "x , y ,", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 709, + 251, + 722 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 252, + 710, + 257, + 720 + ], + "score": 0.72, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 709, + 371, + 722 + ], + "score": 1.0, + "content": "within the converter. Then,", + "type": "text" + }, + { + "bbox": [ + 372, + 712, + 379, + 720 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "is used to recover the sign bit;", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 325, + 733 + ], + "score": 1.0, + "content": "the biased exponent and fraction can be extracted via", + "type": "text" + }, + { + "bbox": [ + 325, + 723, + 332, + 730 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 721, + 351, + 733 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 352, + 722, + 358, + 732 + ], + "score": 0.79, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "; finally the exponent can be further", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 236, + 95 + ], + "score": 1.0, + "content": "corrected by the exponent bias", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 236, + 83, + 242, + 92 + ], + "score": 0.55, + "content": "b", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 243, + 83, + 505, + 95 + ], + "score": 1.0, + "content": ". Hence, it is apparent that the required hardware logic for the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "converter is indeed minimal and the extra hardware cost is truly minor as well. Furthermore, updates", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "of those format registers are extremely infrequent. Even the layer-wise optimization is applied, those", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "registers are only modified at the start of each layer. In other words, virtually no runtime overhead", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 271, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 271, + 140 + ], + "score": 1.0, + "content": "is imposed due to those register updates.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 676, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 236, + 95 + ], + "score": 1.0, + "content": "corrected by the exponent bias", + "type": "text" + }, + { + "bbox": [ + 236, + 83, + 242, + 92 + ], + "score": 0.55, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 83, + 505, + 95 + ], + "score": 1.0, + "content": ". Hence, it is apparent that the required hardware logic for the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "converter is indeed minimal and the extra hardware cost is truly minor as well. Furthermore, updates", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "of those format registers are extremely infrequent. Even the layer-wise optimization is applied, those", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "registers are only modified at the start of each layer. In other words, virtually no runtime overhead", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 271, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 271, + 140 + ], + "score": 1.0, + "content": "is imposed due to those register updates.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 107, + 154, + 195, + 167 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 197, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 197, + 170 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 181, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "score": 1.0, + "content": "In this work, we propose the flexible 8-bit floating-point (FFP8) format for accurate and memory-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "efficient inference of deep neural networks. Our FFP8 format offers three adjustable options: 1) the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 201, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 215 + ], + "score": 1.0, + "content": "size of exponent/fraction field, 2) the value of exponent bias, and 3) the presence of the sign bit,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "whereas those rigid conventional formats simply leave nothing. In this paper, we explain how the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 224, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 506, + 237 + ], + "score": 1.0, + "content": "exponent size and bias jointly define the representable value range of a given FFP8 format. We also", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 236, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 247 + ], + "score": 1.0, + "content": "demonstrate how to explore the best-fit signed/unsigned FFP8 formats for weights and activations to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 245, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 259 + ], + "score": 1.0, + "content": "achieve more accurate inference outcomes. Besides, a layer-wise optimization flow, which discovers", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 258, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 269 + ], + "score": 1.0, + "content": "the best-fit formats for each individual layer, is presented to further improve the accuracy. A model", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 268, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 280 + ], + "score": 1.0, + "content": "retraining methodology that can be carried out in typical FP32 frameworks without the need of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "score": 1.0, + "content": "special training skills is also introduced. The experimental results on various DNN models show that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 504, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 446, + 302 + ], + "score": 1.0, + "content": "the proposed FFP8-based inference flow achieves an extremely low accuracy loss of", + "type": "text" + }, + { + "bbox": [ + 447, + 289, + 504, + 301 + ], + "score": 0.92, + "content": "0 . 1 \\% \\sim 0 . 3 \\%", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "as compared to the FP32 baseline even without model retraining. Moreover, we also show that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "the extra hardware for supporting the FFP8 format is minimal. Therefore, it is conclusive that the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "score": 1.0, + "content": "proposed FFP8-based inference framework should be a better solution for those computing systems", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 333, + 408, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 408, + 347 + ], + "score": 1.0, + "content": "with limited memory capacity and bandwidth, e.g., edge and AIoT devices.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 107, + 362, + 175, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 176, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 176, + 375 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 505, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "Ron Banner, Itay Hubara, Elad Hoffer, and Daniel Soudry. 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Top-1/Top-5Top-1/Top-5Top-1/Top-5Top-1/Top-5
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sha256:224bd08da0de71e9cee24a0ae57b1751ac8d8d37a382d577f212190cc04bd571 +size 42173 diff --git a/parse/train/gwPPcc_M0lv/gwPPcc_M0lv.md b/parse/train/gwPPcc_M0lv/gwPPcc_M0lv.md new file mode 100644 index 0000000000000000000000000000000000000000..88a32f7dc506d56b95dd0ecf26895993bbdebe35 --- /dev/null +++ b/parse/train/gwPPcc_M0lv/gwPPcc_M0lv.md @@ -0,0 +1,481 @@ +# Learning to recover orientations from projections in single-particle cryo-EM + +Anonymous Author(s) +Affiliation +Address +email + +# Abstract + +1 A major challenge in single-particle cryo-electron microscopy (cryo-EM) is that +2 the orientations adopted by the 3D particles prior to imaging are unknown; yet, this +3 knowledge is essential for high-resolution reconstruction. We present a method +4 to recover these orientations directly from the acquired set of 2D projections. +5 Our approach consists of two steps: (i) the estimation of distances between pairs +6 of projections, and (ii) the recovery of the orientation of each projection from +7 these distances. In step (i), pairwise distances are estimated by a Siamese neural +8 network trained on synthetic cryo-EM projections from resolved bio-structures. +9 In step (ii), orientations are recovered by minimizing the difference between +10 the distances estimated from the projections and the distances induced by the +11 recovered orientations. We evaluated the method on synthetic cryo-EM datasets. +12 Current results demonstrate that orientations can be accurately recovered from +13 projections that are shifted and corrupted with a high level of noise. The accuracy +14 of the recovery depends on the accuracy of the distance estimator. While not +15 yet deployed in a real experimental setup, the proposed method offers a novel +16 learning-based take on orientation recovery in SPA. Our code is available at https: +17 //github.com/anonymous/protein-reconstruction. + +# 18 1 Introduction + +19 Single-particle cryo-electron microscopy (cryo-EM) has revolutionized the field of structural biology +20 over the last decades [1, 2, 3]. The use of electron beams to image ice-embedded samples has +21 permitted the recovery of 3D bio-structures at unprecedented resolution. This “resolution revolution” +22 has had a tremendous impact in biomedical research, providing invaluable insights into the biological +23 processes that underlie many current diseases. +24 In single-particle cryo-EM, every 3D particle adopts a random orientation $\theta _ { i }$ in the ice layer before +25 being imaged. Hence, the projection geometry associated to each acquired 2D projection (Figure 1) +26 is unknown. Yet, this knowledge is essential for the tomographic reconstruction of bio-structures [4]. +27 We consider that a cryo-EM measurement (i.e., a projection) $\mathbf { p } _ { i } \in \mathbb { R } ^ { n _ { p } }$ is acquired through + +$$ +\mathbf { p } _ { i } = \mathbf { C } _ { \varphi } \mathbf { S _ { t } } _ { i } \mathbf { P } _ { \theta _ { i } } \mathbf { x } + \mathbf { n } , +$$ + +28 where $\mathbf { x } \in \mathbb { R } ^ { n _ { x } }$ is the unknown 3D density map [5] (Coulomb potential). The operator $\mathbf { P } _ { \pmb { \theta } _ { i } } : \mathbb { R } ^ { n _ { x } } $ +29 $\mathbb { R } ^ { n _ { p } }$ is the projection along the orientation $\theta _ { i }$ (i.e., the $\mathbf { X }$ -ray transform). The operator $\mathbf { S } _ { \mathbf { t } _ { i } } : \mathbb { R } ^ { n _ { p } } $ +30 $\mathbb { R } ^ { n _ { p } }$ is a shift of the projection by $\mathbf { t } _ { i } = ( t _ { i _ { 1 } } , t _ { i _ { 2 } } )$ . The convolution operator $\mathbf { C } _ { \varphi } : \mathbb { R } ^ { n _ { p } } \mathbb { R } ^ { n _ { p } }$ models +31 the microscope point-spread function (PSF) with parameters $\varphi = ( d _ { 1 } , d _ { 2 } , \alpha _ { \mathrm { a s t } } )$ , where $d _ { 1 }$ is the +32 defocus-major, $d _ { 2 }$ is the defocus-minor, and $\alpha _ { \mathrm { a s t } }$ is the angle of astigmatism [6, 7]. Finally, $\mathbf { n } \in \mathbb { R } ^ { n _ { p } }$ +33 represents additive noise. Figure 11 illustrates the effect of projection, shift, and noise. The challenge +34 is then to reconstruct $\mathbf { x }$ from a set of projections $\{ { \bf p } _ { i } \} _ { i = 1 } ^ { P }$ acquired along unknown orientations. +35 A popular approach is to alternatively refine the 3D structure and estimated orientations [8, 9, 10, 11, +36 12, 13]. Yet, the outcome of these iterative-refinement procedures is often predicated on the quality +37 of the initial reconstruction, or, equivalently, on the initial estimation of the orientations [14, 15]. +38 Several methods have been designed to produce a first rough ab initio structure for the refinement +39 procedure [16]. Moment-matching techniques [17, 18, 19, 20] reconstruct an initial structure such +40 that the first few moments of the distribution of its theoretical measurements match the ones of +41 its experimental projections; however, they typically remain sensitive to error in data and can +42 require relatively high computational complexity. Based on the central-slice theorem, common-lines +43 methods [21, 8, 22, 23, 24, 25, 26] aim at uniquely determining the orientations of each projection by +44 identifying the common-lines between triplets of projections—a real challenge given the massive +45 amount of noise. Alternatively, the marginalized maximum likelihood (ML) formulation of the +46 reconstruction problem [11]—classically used for the iterative-refinement procedures themselves— +47 can be minimized using stochastic gradient descent [27]. This permits to avoid the need for an initial +48 volume estimate, at the possible cost of greater convergence instability. +49 More recently, the recovery of geometrical information from unknown view tomography of 2D point +50 sources has been proposed [28], but the extension to 3D cryo-EM tomography is not straightforward. +51 Finally, [29] proposed to recover the in-plane rotations by learning to embed projections in an +52 appropriate latent space, but only after directions had been estimated through three rounds of 2D +53 classification in RELION. +54 Despite the aforementioned advances, providing a robust initial volume remains a challenge due to +55 the high-dimensionality and ill-posedness of the underlying optimization problem. On the other hand, +56 the remarkable ability of convolutional neural networks to capture relevant representations of images +57 has had a profound influence in imaging [30]. In this work, we present a learning-based approach to +58 recover the unknown orientations directly from the acquired set of projections—without the need for +59 an intermediate reconstruction procedure or an initial volume estimate. + +![](images/574fcacd9c7d71546e39eb9b29cfe0b129fe830c192894eca371c94a60073732.jpg) +Figure 1: Geometry of the imaging model defined in (1). The 3D density $\mathbf { x }$ in the coordinate system $( x _ { 1 } , x _ { 2 } , x _ { 3 } )$ is imaged along the orientation $\pmb \theta$ to produce the 2D projection $\mathbf { p }$ in the coordinate system $( y _ { 1 } , y _ { 2 } )$ of the microscope’s detector plane. The orientation $\pmb { \theta } = ( \theta _ { 3 } , \theta _ { 2 } , \theta _ { 1 } )$ is decomposed as the direction $( \theta _ { 2 } , \theta _ { 1 } ) \in [ 0 , \pi ] \times$ $[ 0 , 2 \pi [$ (parameterizing the sphere $\mathbb { S } ^ { 2 }$ ) and the in-plane rotation $\theta _ { 3 } \in [ 0 , 2 \pi [$ (parameterizing the circle $\mathbb { S } ^ { 1 }$ ). In our work, we represent the orientation $\pmb { \theta }$ as a unit quaternion $q$ . + +![](images/571ec2e85d672c9aec24a91431b51abf95fd62226de8c5f704170d6b9f6c194b.jpg) +Figure 2: Single-particle cryo-EM produces $P$ projections (with $P$ in the order of $1 0 ^ { 5 }$ ) from unknown orientations: $\{ ( \mathbf { p } _ { i } , q _ { i } ) \} _ { i = 1 } ^ { P }$ . Observing that distances between orientations constrain the latter, we aim to recover the orientations $\left\{ q _ { i } \right\}$ from $\{ d _ { q } ( q _ { i } , q _ { j } ) \}$ , where $d _ { q } ( q _ { i } , q _ { j } )$ is the distance (angle) between orientations $q _ { i }$ and $q _ { j }$ . Observing that the similarity between projections depends on their relative orientation, we aim to estimate the distance $d _ { q } ( q _ { i } , q _ { j } )$ from the projections $( \mathbf { p } _ { i } , \mathbf { p } _ { j } )$ . + +# 60 2 Method + +61 Our approach relies on two observations (Figure 2), yielding two steps (Figure 3). First, the more similar two projections 62 $( \mathbf { p } _ { i } , \mathbf { p } _ { j } )$ , the more likely they originated from two particles that adopted close + +![](images/3f3b0db6773005f59b0f2a35d0d58aab70fa3e035961632f38d44e661423dc1b.jpg) +Figure 3: Our method consists of two steps. First, we estimate distances between pairs of projections. Second, we recover the orientation of each projection from these distances. + +63 orientations $( q _ { i } , q _ { j } )$ in the ice prior to imaging;1 this observation guides a number of applications in +64 the field [2]. Hence, we aim to estimate distances between orientations $d _ { q } ( q _ { i } , q _ { j } )$ from the projections +65 as $\widehat { d } _ { p } ( \mathbf { p } _ { i } , \mathbf { p } _ { j } )$ , which we discuss in $\ S 2 . 2$ . Second, an orientation $q$ is constrained by the distances +66 between itself and the other orientations $\{ d ( q , q _ { j } ) \}$ . Hence, we aim to recover orientations $\left\{ { \widehat { q } } _ { k } \right\}$ such +67 that the induced distances $\{ d _ { q } ( \widehat { q } _ { i } , \widehat { q } _ { j } ) \}$ are close to the estimated distances $\{ \widehat { d _ { p } } ( \mathbf { p } _ { i } , \mathbf { p } _ { j } ) \}$ , which we +68 discuss in $\ S 2 . 3$ b b. All in all, from a set of projections $\left\{ \mathbf { p } _ { k } \right\}$ , we aim to recover their orientations $\left\{ { \widehat { q _ { k } } } \right\}$ +69 such that $d _ { q } ( \widehat { q _ { i } } , \widehat { q _ { j } } ) \approx \widehat { d _ { p } } ( \mathbf { p } _ { i } , \mathbf { p } _ { j } ) \approx d _ { q } ( q _ { i } , q _ { j } )$ , with equality if $\widehat { d } _ { p }$ and $\left\{ { \widehat { q } } _ { k } \right\}$ are perfectly estimated. +70 Our approach is similar to [31]. While the authors reconstruct 2D images from 1D projections, they +71 rely on the same two-step approach: they (i) estimate distances as $\hat { \hat { d _ { p } } } ( \mathbf { p } _ { i } , \mathbf { p } _ { j } ) = \vert \vert \mathbf { p } _ { i } - \mathbf { p } _ { j } \vert \vert _ { 2 }$ then +72 (ii) recover the orientations by spectrally embedding that distance graph. The Euclidean distance is +73 however not robust to perturbations: for example, two projections that only differ by a shift $\bf { S _ { t } }$ of one +74 pixel would be considered far apart while their orientations are the same. They noted that issue and +75 we observed it too (Appendix E). To circumvent this, we propose to learn $\widehat { d } _ { p }$ from examples $( \ S 2 . 2 )$ . + +# 76 2.1 Representation of orientations with quaternions + +77 The orientation of a 3D particle with respect to the microscope’s detector plane is a rotation relative to +78 a reference orientation (Figure 1). The group of all 3D rotations under composition is identified with +79 SO(3), the group of $3 \times 3$ orthogonal matrices with determinant 1 under matrix multiplication. A +80 rotation matrix $\mathbf { R } _ { \theta } \in \mathbf { S O } ( 3 )$ can be decomposed as a product of ${ \binom { 3 } { 2 } } = 3$ independent rotations, for +81 example as $\mathbf { R } _ { \theta } = \mathbf { R } _ { \theta _ { 3 } } \mathbf { R } _ { \theta _ { 2 } } \mathbf { R } _ { \theta _ { 1 } }$ , where $\pmb { \theta } = ( \theta _ { 3 } , \theta _ { 2 } , \theta _ { 1 } ) \in [ 0 , 2 \pi [ \times \overline { { [ 0 , \pi ] } } \times [ 0 , 2 \pi [$ [ are the (extrinsic +82 and proper) Euler angles in the $Z Y Z$ convention (a common parameterization in cryo-EM) [32]. +83 While Euler angles are a concise representation of orientation (3 numbers for 3 degrees of freedom), +84 they suffer from a topological constraint—there is no covering map from the 3-torus to SO(3)— +85 which manifests itself in the gimbal lock, the loss of one degree of freedom when $\theta _ { 2 } = 0$ . This makes +86 their optimization by gradient descent $( \ S 2 . 3 )$ problematic. On the other hand, optimizing rotation +87 matrices (made of 9 numbers) would require computationally costly constraints (orthogonality and +88 determinant 1) to reduce the degrees of freedom to 3. Moreover, the distance between orientations +89 cannot be directly computed from Euler angles and is costly (30 multiplications) to compute from +90 rotation matrices [33]. We solve both problems by representing orientations with unit quaternions. +91 Quaternions $q \in \mathbb { H }$ are an extension of complex numbers2 of the form $q = a + b i + c j + d \pmb { k }$ where +92 $a , b , c , d \in \mathbb { R }$ . Unit quaternions $q \in \mathbb { S } ^ { 3 }$ , where $\mathbb { S } ^ { 3 } = \{ q \in \mathbb { H } : | q | \lceil = 1 \}$ is the 3-sphere (with +93 the additional group structure inherited from quaternion multiplication), concisely and elegantly +94 represent a rotation of angle $\theta$ about axis $( x _ { 1 } , x _ { 2 } , x _ { 3 } )$ as $q = \cos ( \bar { \theta } / 2 ) + x _ { 1 } \sin ( \theta / 2 ) i \dot { + } x _ { 2 } \sin ( \bar { \theta / 2 } ) \dot { { \bf j } _ { + } }$ +95 $\bar { x _ { 3 } } \sin ( \theta / 2 ) k$ . They parameterize rotation matrices as + +$$ +\mathbf { R } _ { q } = \left( \begin{array} { c c c } { a ^ { 2 } + b ^ { 2 } - c ^ { 2 } - d ^ { 2 } } & { 2 b c - 2 a d } & { 2 b d + 2 a c } \\ { 2 b c + 2 a d } & { a ^ { 2 } - b ^ { 2 } + c ^ { 2 } - d ^ { 2 } } & { 2 c d - 2 a b } \\ { 2 b d - 2 a c } & { 2 c d + 2 a b } & { a ^ { 2 } - b ^ { 2 } - c ^ { 2 } + d ^ { 2 } } \end{array} \right) . +$$ + +![](images/a4869dab607e0b19aefdd984dbad627ea589878c7313f5aba9d2ea397257420b.jpg) +Figure 4: Distance learning. We are looking for a distance $\widehat { d } _ { p }$ between projections that is an accurate estimator of the distance $d _ { q }$ between their orientations. We propose to parameterize $\widehat { d } _ { p }$ as a Siamese neural network (SNN), trained on a synthetic dataset of projections with associated orientation. + +96 Note that $\mathbb { S } ^ { 3 } \to { \bf S O } ( 3 )$ is a two-to-one mapping (a double cover) as $q$ and $- q$ represent the same +97 orientation. Unlike Euler angles, ${ \mathbb S } ^ { 3 }$ is isomorphic to the universal cover of $\mathbf { S O } ( 3 )$ . Hence, the +98 distance between two orientations, i.e., the length of the geodesic between them on $\mathbf { \bar { S } O ( 3 ) }$ , is + +$$ +\begin{array} { c } { d _ { q } : \mathbb { S } ^ { 3 } \times \mathbb { S } ^ { 3 } \to [ 0 , \pi ] , } \\ { d _ { q } ( q _ { i } , q _ { j } ) = 2 \operatorname { a r c c o s } \left( | \langle q _ { i } , q _ { j } \rangle | \right) , } \end{array} +$$ + +where 99 $\langle \cdot , \cdot \rangle$ is the inner product, and the absolute value $\left. \cdot \right.$ ensures that $d _ { q } ( q _ { i } , q _ { j } ) = d _ { q } ( q _ { i } , - q _ { j } )$ . The distance 00 $d _ { q } ( q _ { i } , q _ { j } )$ corresponds to the magnitude of the rotation $\mathbf { R } _ { * }$ such that ${ \bf R } _ { q _ { i } } = { \bf R } _ { * } { \bf R } _ { q _ { j } }$ [33]. + +# 2.2 Distance learning + +102 We aim to estimate a function $\widehat { d } _ { p }$ such that $\widehat { d } _ { p } ( { \bf p } _ { i } , { \bf p } _ { j } ) \approx d _ { q } ( q _ { i } , q _ { j } )$ . While we could in principle +103 design $\widehat { d } _ { p }$ , that would be intricate—if not impossible—partly because the invariants are difficult +104 to specify. We instead opt to learn $\widehat { d } _ { p }$ , capitalizing on (i) the powerful function approximation +105 capabilities of neural networks, and (ii) the possibility to generate realistic datasets supported by the +106 availability of numerous 3D atomic models3 and our ability to model the cryo-EM imaging procedure. + +From a training dataset 107 $\left\{ { \bf p } _ { i } , q _ { i } \right\} _ { i = 1 } ^ { P }$ , we learn the projection distance + +$$ +\widehat { d } _ { p } = \mathop { \mathrm { a r g } } \underset { d _ { p } } { \mathrm { a r g } } \mathrm { m i n } L _ { \mathrm { D E } } , \quad \mathrm { w h e r e } \quad L _ { \mathrm { D E } } = \sum _ { i , j } \left| d _ { p } \big ( \mathbf { p } _ { i } , \mathbf { p } _ { j } \big ) - d _ { q } \big ( q _ { i } , q _ { j } \big ) \right| ^ { 2 } +$$ + +108 is the loss and $d _ { q }$ is defined in (2). The $d _ { p }$ is parameterized as the Siamese neural network (SNN) [34] + +$$ +d _ { p } ( \mathbf { p } _ { i } , \mathbf { p } _ { j } ) = d _ { f } ( \mathcal { G } _ { w } ( \mathbf { p } _ { i } ) , \mathcal { G } _ { w } ( \mathbf { p } _ { j } ) ) , +$$ + +109 where $\mathcal { G } _ { w }$ is a convolutional neural network with weights $w$ that is trained to extract the most relevant +110 features $\mathbf { f } _ { i } \in \mathbb { R } ^ { n _ { f } }$ from a projection $\mathbf { p } _ { i }$ . SNNs, also termed “twin networks”, are commonly used in +111 the field of deep metric learning to learn similarity functions [35]. We set the feature space distance +112 $d _ { f }$ as the cosine distance to facilitate the learning of a $\widehat { d } _ { p }$ that respects the elliptic geometry of $\mathbb { S } ^ { 3 }$ +113 (Appendix F). Figure 4 illustrates the proposed learning paradigm. +114 As evaluating a sum over $P ^ { 2 }$ pairs is computationally intractable for cryo-EM datasets with typically +115 $P$ in the order of $1 0 ^ { 5 }$ projections, we sample the sum and minimize (3) with stochastic gradient +116 descent (SGD) over small batches of pairs. The weights $w$ are updated by back-propagation. +117 The architecture of $\mathcal { G } _ { w }$ is described in Appendix G. When designing the architecture, we constrain +118 the functional space from which the trained $\mathcal { G } _ { w }$ is drawn and express our prior expert knowledge. For +119 example, we realize shift invariance, i.e., a guarantee that a shift $\bf { S _ { t } }$ does not change our estimated +120 distances and orientations, with a fully convolutional architecture. Size invariance, i.e., taking +121 projections $\mathbf { p }$ of varying sizes $n _ { p }$ while yielding a representation f of a fixed size $n _ { f }$ , is realized by a +122 final average pooling layer. As we do not (yet) know how to realize an invariance to noise or PSF, we +123 resort to data augmentation, i.e., training on perturbed projections. In $\ S 3 . 4$ , we show that a built-in +124 invariance (shift) is far preferable to one learned through augmentation (noise). Finally, as projections + +125 are made by integrating through the 3D volume, projections from opposed directions are mirrors of each other.4126 That is another kind of physical knowledge that should ideally be built into our method. + +127 One could hope to train $\mathcal { G } _ { w }$ to directly map projections to orientations as ${ \widehat { q _ { i } } } = \mathbf { f } _ { i } = { \mathcal { G } } _ { w } ( \mathbf { p } _ { i } )$ . While +128 that would avoid the orientation recovery step, a space of ${ n } _ { f } = 4$ bdimensions does not have room for +129 $\mathcal { G } _ { w }$ to represent the other factors of variation in $\mathbf { p }$ , such as different noise levels, PSFs, or proteins. +130 We tested that hypothesis in Appendix F. + +# 2.3 Orientation recovery + +132 The task of recovering points based on their relative distances has been extensively studied. Many +133 methods aim at mapping high-dimensional data onto a lower-dimensional space while preserving +134 distances, primarily for dimensionality reduction and data visualization. Well-known examples +135 include MDS [36], Isomap [37], LLE [38], Laplacian eigenmaps [39], t-SNE [40], and UMAP [41]. +136 The embedding of distance matrices in Euclidean space (given by their eigenvectors) is especially +137 well-described. In particular, the framework of Euclidean distance matrices (EDMs) [42] provides +138 theoretical guarantees on the recovery of points from distances. +139 We however aim to embed the orientations $q$ in $\mathbb { S } ^ { 3 }$ (§2.1), a setting for which we are unaware of any +140 theoretical characterization (e.g., on the shape of the loss function or its behavior when distances are +141 missing or noisy). The fact that ${ \mathbb S } ^ { 3 }$ is locally Euclidean does however offer some hope. Indeed, despite +142 the non-convexity and the lack of theoretical guarantees, we are able to appropriately minimize our +143 loss function, as we experimentally demonstrate in Appendix D. + +We recover the orientations of a set of projections 144 $\left\{ { \bf p } _ { k } \right\} _ { k = 1 } ^ { P }$ through + +$$ +\left\{ \widehat { q } _ { k } \right\} _ { k = 1 } ^ { P } = \underset { \left\{ q _ { k } \in \mathbb { S } ^ { 3 } \right\} } { \arg \operatorname* { m i n } } L _ { \mathrm { O R } } , \quad \mathrm { w h e r e } \quad L _ { \mathrm { O R } } = \sum _ { i , j } \left| \widehat { d } _ { p } \left( \mathbf { p } _ { i } , \mathbf { p } _ { j } \right) - d _ { q } \left( q _ { i } , q _ { j } \right) \right| ^ { 2 } +$$ + +145 is the loss and $\widehat { d } _ { p }$ is the estimator trained in (3). Note that the sole difference with (3) is that the +146 minimization is performed over the orientations $q$ rather than the distance $d _ { p }$ . Here again, we sample +147 the sum in practice and minimize (4) with mini-batch SGD. Sampling the sum amounts to building a +148 sparse (instead of complete) distance graph before embedding, a common strategy. + +# 149 2.4 Evaluation + +150 151 Unfortunately, we cannand the true orientations y take the difference between the recovered orientations as orientations are rotations up to an arbitrary reference or $\{ \widehat { q _ { k } } \} _ { k = 1 } ^ { P }$ +$\{ q _ { k } \} _ { k = 1 } ^ { P }$ +153 Any global rotation or reflection of the recovered orientations is as valid as any other, i.e., $d _ { q } ( q _ { i } , q _ { j } ) =$ +154 $d _ { q } ( \mathbf { T } q _ { i } , \mathbf { T } q _ { j } ) \ \forall \mathbf { T } \in \mathbf { O } ( 4 )$ , where $\mathbf { O } ( 4 )$ is the group of $4 \times 4$ orthogonal matrices. Hence, we align +155 the sets of orientations and compute the mean orientation recovery error as + +$$ +E _ { \mathrm { O R } } = \operatorname* { m i n } _ { \mathbf { T } \in \mathbf { O } ( 4 ) } \frac { 1 } { P } \sum _ { i = 1 } ^ { P } \left| d _ { q } \left( q _ { i } , \mathbf { T } \widehat { q _ { i } } \right) \right| . +$$ + +We implement 156 $\mathbf { T }$ as a product of ${ \binom { 4 } { 2 } } = 6$ independent rotations and an optional reflection: + +$$ +\mathbf { T } = \left[ \begin{array} { l l } { m } & { \mathbf { 0 } } \\ { \mathbf { 0 } } & { \mathbf { I } } \end{array} \right] \prod _ { \substack { 1 \leq i < j \leq 4 } } \mathbf { T } _ { \theta _ { i j } } , \quad m \in \{ - 1 , 1 \} , \ \theta _ { i j } \in [ 0 , 2 \pi [ , +$$ + +where 157 $\mathbf { T } _ { \theta _ { i j } } \in \mathbf { S O } ( 4 )$ is a rotation by angle $\theta _ { i j }$ on the $( x _ { i } , x _ { j } )$ plane. + +158 In practice, we again minimize (5) with mini-batch SGD. Because $\mathbf { O } ( 4 )$ is disconnected, we optimize +159 the 6 angles separately for $m = 1$ (proper rotations) and $m = - 1$ (improper rotations). Figure 15 +160 shows an alignment to $E _ { \mathrm { O R } } = 0$ after a perfect recovery. +162 We first evaluated whether orientation recovery through (4) was feasible assuming perfect distances, +163 and how it was affected by errors in the distances (§3.2). We then learned to estimate the distances +164 through (3), and evaluated the accuracy of this procedure (§3.3) and its robustness to perturbations of +165 the projections (§3.4). Finally, we ran the whole machinery on a synthetic dataset to assess how well +166 orientations could be recovered from estimated distances (§3.5). + +# 3.1 Experimental conditions + +168 Density maps. We considered two proteins (Figure 10): the $\beta$ -galactosidase, a protein with a +169 dihedral (D2) symmetry, and the lambda excision HJ intermediate (HJI), an asymmetric protein +170 with local cyclic (C1) symmetry. Their deposited PDB atomic models are 5a1a [43] and $5 \mathrm { j } 0 \mathrm { n }$ [44], +171 respectively. From these atomic models, we generated the density maps in Chimera [45] by fitting the +172 models with a $1 \mathring \mathrm { A }$ map for 5a1a and a $3 . 6 7 \mathring \mathrm { A }$ map for $5 \mathrm { j } 0 \mathrm { n }$ ; this gave us a volume of $1 1 0 \times 1 5 5 \times 1 9 9$ +173 voxels for 5a1a and one of $6 9 \times 5 7 \times 7 5$ voxels for $5 \mathrm { j } 0 \mathrm { n }$ . +174 Protein symmetries. Symmetries are problematic when learning distances: two projections can +175 be identical while not originating from the same orientation, which breaks an axiom of distance +176 functions (identity of indiscernibles). Figure 16b illustrates this problem. To capture only one of four +177 identical projections of 5a1a, we restricted directions to $( \theta _ { 2 } , \theta _ { 1 } ) \in [ 0 , \pi [ \times [ 0 , \frac { \pi } { 2 } [$ (a quarter of the +178 sphere, illustrated in Figure 12a) for that protein. This treatment of symmetries is incomplete5 but +179 sufficient for a proof-of-concept. +180 Projections. Using the ASTRA projector [46], we generated $P = 5 , 0 0 0$ synthetic projections +181 of $2 7 5 \times 2 7 5$ pixels (downsampled to $1 1 6 \times 1 1 6 )$ for 5a1a and $1 1 6 \times 1 1 6$ pixels for $5 { \dot { \jmath } } 0 \mathbf { n }$ , taken +182 from uniformly sampled orientations. 6 We then perturbed the measurements with different levels of +183 additive Gaussian noise [47, 48] and off-centering shifts. Figure 11 displays some samples. + +Datasets. For each protein, we split the projections into training, validation, and test subsets, and created disjoint pairs of projections from each (Table 1). The training and validation sets were used to train and evaluate the SNN, while the test set was used to evaluate orientation recovery given a trained SNN. Sampling orientations (mostly) uniformly induces a distribution of distances that is skewed towards larger distances (shown in Figure 12b). As this would skew $L _ { \mathrm { D E } }$ and bias $\widehat { d } _ { p }$ , we further sampled $1 \%$ of the training and validation pairs to make the distribution of distances uniform—for $\widehat { d } _ { p }$ to be uniformly accurate over the whole $[ 0 , \pi ]$ range of distances (see Appendix B for further illustrations). While 1, 650 projections were enough to perfectly reconstruct the density maps (as shown in Figures 9e and 9j), our method is not limited by the number of projections as optimization is done per batch. Optimization settings are described in Appendix C. + +# 3.2 Sensitivity of orientation recovery to errors in distance estimation + +We first evaluated the feasibility of orientation recovery assuming that the exact distances were known. +The method successfully recovers the orientation of every projection in this case (see Appendix D). + +To evaluate the robustness of (4), we perturbed the distances prior to recovery with an error sampled from a Gaussian distribution with mean 0 and variances $\sigma ^ { \hat { 2 } } \in [ 0 . 0 , 0 . 8 ]$ . Figure 5 shows that the recovery error $E _ { \mathrm { O R } }$ is a monotonic function of the error in distances: from $E _ { \mathrm { O R } } = 0$ with exact distances to $E _ { 0 \mathrm { R } } \approx 0 . 2$ radians $( \approx 1 1 . 5 ^ { \circ } )$ for $\sigma ^ { 2 } = 0 . 8$ . + +These results demonstrate that the performance of orientation recovery (4) depends on the quality of the estimated distances, which advocates for a proper and extensive training of the SNN. Moreover, we observe that $L _ { \mathrm { O R } }$ is a reliable proxy for $E _ { \mathrm { O R } }$ , allowing us to assess recovery performance in the absence of ground-truth orientations (i.e., when recovering the orientations of real projections). + +Table 1: Split of $P = 5 , 0 0 0$ projections in training, validation, and test subsets. + +
DatasetPp2Used pairs
Training2,512 (50%)6,310,14463,101
Validation838 (17%)702,2447,022
Test1,650 (33%)2,722,5002,722,500
+ +![](images/2aaf4e1ffebb6061b0d70680344575ada494cd691329db794acfa53f635c55b1.jpg) + +![](images/8cec7c08a80f0bbcb089809f087d6f046a01157d520bd8c120d20494ab91474b.jpg) +Figure 5: Orientation recovery from perturbed distances on $5 { \dot { \jmath } } 0 \mathbf { n }$ (left) and 5a1a (right). + +![](images/705437b5f98dff2a7bfcb3af8a130419f1d633b057aa6fcecdae2727f00411f2.jpg) +Figure 6: Distance learning. + +(b) Relationship between $\widehat { d _ { p } }$ and $d _ { q }$ on 1, 000 pairs from the test sets of $5 { \dot { \jmath } } 0 \mathbf { n }$ (left) and 5a1a (right). + +# 205 3.3 Learning to estimate distances + +06 We evaluated the ability of the SNN to learn to approximate the orientation distance $d _ { q }$ . For +7 comparison, we evaluated a baseline, the Euclidean distance $\widehat { d } _ { p } ( \mathbf { p } _ { i } , \mathbf { p } _ { j } ) = \| \mathbf { p } _ { i } , \mathbf { p } _ { j } \| _ { 2 }$ , in Appendix $\mathrm { E }$ +8 Figure 6a shows the convergence of $L _ { \mathrm { D E } }$ , reached in about 50 epochs. Figure 6b shows the relationship +09 between the distance $\widehat { d } _ { p }$ estimated from projections and the true distance $d _ { q }$ . The outliers for 5a1a +10 are explained by our incomplete treatment of its symmetry. While our learned distance function +is a much better estimator than the Euclidean distance—compare Figure 6b with Figure 16—they +2 share one characteristic: both plateau and underestimate the largest distances. We did attenuate +the phenomenon by sampling training distances uniformly (see $\ S 3 . 1 \ r ,$ ), and the issue is much less +4 severe than with the Euclidean distance. An alternative could be to only rely on smaller distances for +15 recovery. That would however require the addition of a spreading term in (4) to prevent the recovered +16 orientations to collapse. +217 These results confirm that a SNN is able to estimate differences in orientations from projections alone, +218 even though much has yet to be gained from improving upon the rather primitive SNN architecture +219 we are currently using. The use of additional training data should help further diminish overfitting. + +# 220 3.4 Sensitivity of distance learning to perturbations in the projections + +221 We first demonstrated that the learning of distances is insensible to off-centering shifts (Figure 7a), +222 which is expected given that shift invariance is built in our SNN (see $\ S 2 . 2 \AA ,$ ). + +As we cannot—or do not yet know how to—build noise invariance in the SNN architecture, we trained the SNN on noisy projections and evaluated whether it could learn to treat noise as an irrelevant information. Figure 7b shows $E _ { \mathrm { O R } } \approx 0 . 1 6$ radians $( \approx 9 ^ { \circ } )$ for noiseless projections and $E _ { \mathrm { O R } } \approx 0 . 4 2$ radians $( \approx 2 4 ^ { \circ } )$ for a more realistic noise variance of $\sigma ^ { 2 } = 1 6$ (with signal-to-noise ratio of $- 1 2 \ \mathrm { d B }$ ). Whereas a naive distance function (e.g., an Euclidean distance) would be extremely sensitive to noise, the SNN mostly learned to discard it. Moreover, the observed overfitting indicates that more training data should further decrease the sensitivity of the SNN to noise. + +230 Note that we did not evaluate sensitivity to the PSF at this stage but expect a similar behavior. + +231 Here again (§3.2), we observed that (i) the estimation of more accurate distances (a smaller $L _ { \mathrm { D E } } ,$ ) +232 leads to the recovery of more accurate orientations (a smaller $L _ { \mathrm { O R } }$ and $E _ { \mathrm { O R } } \mathrm { , }$ ), and that (ii) an higher +233 recovery loss $L _ { \mathrm { O R } }$ induces an higher error $E _ { \mathrm { O R } }$ . + +![](images/6f70f3284125a2c36d4a356143a7dcfa7b58837a136ef24c27fd3c69e4865116.jpg) + +(a) Learning from shifted projections $\{ \mathbf { S } _ { \mathbf { t } _ { i } } \mathbf { P } _ { \pmb { \theta } _ { i } } \mathbf { x } \}$ , with shifts $t _ { i _ { 1 } }$ and $t _ { i _ { 2 } }$ sampled from a triangular distribution with mean 0 and of increasing limits. + +![](images/fa385fb38fc1189c00191e90224e286cf12a4a8e9dac8951664eec3eb9e6ac1a.jpg) +(b) Learning from noisy projections $\{ \mathbf { P } _ { \pmb { \theta } _ { i } } \mathbf { x } + \mathbf { n } \}$ , with white noise $\mathbf { n } \sim { \mathcal { N } } ( 0 , \sigma ^ { 2 } \mathbf { I } )$ of increasing variance $\sigma ^ { 2 }$ . + +![](images/0264fa45e6883ef7433b143093a4bdda85e2307012d7e4a9d89d7791417030a3.jpg) +Figure 7: Sensitivity of distance learning to perturbations in the projections of $5 \mathrm { j } 0 \mathrm { n }$ . The box plots show the distance learning loss $L _ { \mathrm { D E } }$ (the distribution is taken over epochs). Boxes show the orientation recovery loss $L _ { \mathrm { O R } }$ and error $E _ { \mathrm { O R } }$ . +Figure 8: Distance learning and orientation recovery from estimated distances. The green and orange boxes show $L _ { \mathrm { D E } }$ (3) on the training and validation sets. The blue curve shows the evolution of the recovery loss until convergence, with the minimum $L _ { \mathrm { O R } }$ (4) highlighted. The red histogram shows the errors in the recovered orientations $\{ d _ { q } ( q _ { i } , \mathbf { T } \widehat { q _ { i } } ) \}$ , with the mean $E _ { \mathrm { O R } }$ (5) highlighted. + +# 234 3.5 Orientation recovery and reconstruction of density maps + +As a proof-of-concept, we attempted to solve the full inverse problem posed by (1), i.e., to reconstruct the density maps $\widehat { \mathbf { x } }$ from sets of projections $\{ { \bf { p } } _ { i } \}$ and their orientations $\left\{ { \widehat { q } } _ { i } \right\}$ recovered through the b bproposed method. It is worth noting that, at this stage of development, we only trained the SNN on projections originating from the protein we were attempting to reconstruct. In addition, reconstruction was performed with a direct reconstruction algorithm (ASTRA’s GPU implementation of the CGLS algorithm) rather than with a robuster iterative method. This is a specific experimental case that only partially shines light on the applicability of the method in real situations; this is discussed in $\ S 4$ . + +Figure 8a shows the recovery of orientations from distances that were estimated from noiseless projections of $5 \mathrm { j } 0 \mathrm { n }$ . A mean error of $E _ { 0 \mathrm { R } } \approx 0 . 2 0$ radians $( \approx 1 1 ^ { \circ } )$ in the recovered orientations led to a reconstruction with a resolution of $1 2 . 2 \mathring \mathrm { A }$ at a Fourier shell coefficient (FSC) of 0.5, shown in Figure ${ 9 \mathrm { c } }$ . As predicted by our other experiments, corrupting the projections with noise ( $\sigma ^ { 2 } = 1 6$ ) negatively impacts the quality of the recovered orientations (Figure 8b); the obtained mean error is then $E _ { 0 \mathrm { R } } \approx 0 . 2 5$ radians $( \approx 1 4 ^ { \circ } )$ ). Unsurprisingly, this leads to a reconstruction with a lower resolution of $1 5 . 2 \mathring \mathrm { A }$ , shown in Figure 9d. (Note that reconstruction was here obtained from the noiseless projections, the goal being to evaluate only the impact of orientation mis-estimation.) + +Finally, Figures 8c,d show the recovery of orientations from noiseless and noisy projections of 5a1a. A mean error of $E _ { \mathrm { O R } } \approx 0 . 1 3$ radians $( \approx 7 ^ { \circ } )$ in both cases led to reconstructions with resolutions of $8 . 0 \mathring \mathrm { A }$ and $9 . 6 \mathring \mathrm { A }$ , shown in Figures 9h,i. Distance estimation, orientation recovery, and reconstruction performed better on 5a1a than $5 \mathrm { j } 0 \mathrm { n }$ because its ground-truth density is of higher resolution. + +These results tend to indicate that a reasonable first structure can be reconstructed from projections whose orientations have been recovered through our method. + +# 4 Discussion + +257 In this work, we explored the use of distance learning between pairs of 2D cryo-EM projections +258 from a 3D protein structure to infer the unknown orientation at which each projection was imaged +259 from. Our two-step method relies on the estimation of pairwise distances between unseen projections, +260 followed by the recovery of the orientations from these distances. + +![](images/be11a1e7a160d661e763531e4c3e3700d0c00e6a7b41db7a15cda69b0a0bdc59.jpg) +Figure 9: Density maps $\widehat { \mathbf { x } }$ reconstructed from (a,f) ground-truth orientations, (b,g) random orientations, b(c,h) orientations recovered from noiseless projections, and (d,i) orientations recovered from noisy projections. The Fourier shell correlation (FSC) curves in (e,j) indicate the resolutions of the densities (w.r.t. ground-truth densities, shown in Figures 10b,d). + +The method has been evaluated on synthetic datasets for two different proteins. The results provide key insights on the viability of the proposed scheme. First, they demonstrate that a SNN can learn a distance function between projections that estimates the difference in their orientation $( \ S 3 . 3 )$ and that is invariant to shifts and robust to increasing levels of noise (§3.4)—an important condition in cryo-EM. Second, they demonstrate that an accurate estimation of distances leads to an accurate recovery of orientations $( \ S 3 . 2 , \ \ S 3 . 4 )$ . Finally, our method was able to recover orientations with an error of 0.12 to 0.25 radians (7 to $1 4 ^ { \circ }$ )—leading to an initial volume with a resolution of 8 to $1 5 \mathring \mathrm { A }$ (§3.5). In summary, the more accurate the estimated distances, the more precise the recovered orientations, and, ultimately, the higher-resolution the reconstructed volume. + +270 While the method is not yet ready to be deployed in practice, we believe that a series of developments +271 could make it relevant for single-particle cryo-EM reconstruction. 7 As previously discussed, the +272 results underline the importance of learning an accurate distance estimator. In this regard, the +273 performance of the SNN could be improved. First, the architecture of the twin convolutional neural +274 networks should be expanded and tuned. Second, training could be improved, perhaps by providing +275 more supervision by separately predicting the differences in direction $( \theta _ { 2 } , \theta _ { 1 } )$ and in-plane angle $\theta _ { 3 }$ . + +Importantly, the SNN would be better trained on a more diverse cryo-EM dataset. Indeed, its success as a faithful estimator eventually relies on our capacity to generate a synthetic training dataset whose data distribution is diverse enough to cover that of unseen projection datasets. Such realistic cryo-EM projections could be generated by relying on a more expressive formulation of the cryo-EM physics and taking advantage of the thousands of atomic models available in the PDB. In particular, a necessary extension will be to include the effects of the PSF and to evaluate its impact. + +A final phase of tests before deploying the method on real cryo-EM measurements will be to extensively test the method on “unseen proteins”, i.e., proteins whose simulated projections have never been seen by the SNN. In this regard, an interesting aspect of our method is that the twin networks within the SNN intrinsically predict the relationship between projections, allowing the SNN as a whole to abstract the particular volume. Learning should benefit from the profound structural similarity shared by proteins—after all, they are all derived from the same 21 building blocks. + +Training our 4.5M parameter model (see Appendices G and C) has the following negative environmental impact: it consumes 13 kWh of energy, which produces 6.36 lbs of $\mathrm { C O _ { 2 } }$ on average [49]. + +91 [1] J. Dubochet, M. Adrian, J.-J. Chang, J.-C. Homo, J. Lepault, A. W. McDowall, and P. Schultz, +92 “Cryo-electron microscopy of vitrified specimens,” Quarterly Reviews of Biophysics, vol. 21, +93 no. 2, pp. 129–228, 1988. +94 [2] J. Frank, Three-dimensional electron microscopy of macromolecular assemblies: Visualization +95 of biological molecules in their native state. Oxford University Press, 2006. +96 [3] Nature, “Method of the year 2015,” Nature Methods, vol. 13, no. 1, 2016. +97 [4] F. Natterer, The mathematics of computerized tomography. Society for Industrial and Applied +98 Mathematics, jan 2001. +99 [5] F. 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McMahan, D. Golovin, S. Chikkerur, D. Liu, M. Wattenberg, A. M. Hrafnkelsson, T. Boulos, J. Kubica, G. Holt, D. Sculley, M. Young, D. Ebner, J. Grady, L. Nie, T. Phillips, and E. Davydov, “Ad click prediction: a view from the trenches,” in Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD ’13. ACM Press, 2013, p. 1222. [Online]. Available: http://dl.acm.org/citation.cfm?doid $\mathbf { \Psi } =$ 2487575.2488200 + +# 34 Checklist + +1. For all authors... + +(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] +(b) Did you describe the limitations of your work? [Yes] See $\ S 4$ . +(c) Did you discuss any potential negative societal impacts of your work? [Yes] We didn’t identify any potential risk for improving protein imaging. Moreover, our work only addresses a small step in a huge pipeline. We however mentioned the environmental impact of training our model (see $\ S 4$ ). +(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] + +2. If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] + +3. If you ran experiments... + +(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We include a URL in the Abstract to a git repository that includes code, data, and instructions to reproduce our results. Moreover, notebooks and an interactive website are provided to further play with the method. +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See $\ S 3 . 1$ (including Table 1) for the preparation of data and how they were split. See Appendix C for the hyperparameters. +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] When there was variance, e.g., on Figure 7 and Figure 17b. +(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix C. + +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... + +(a) If your work uses existing assets, did you cite the creators? [Yes] We used proteins from the publicly available Protein Data Bank (PDB) and cited the ones we used, see $\ S 3 . 1$ . We also used and cited the ASTRA toolbox in the same section. +(b) Did you mention the license of the assets? [Yes] The license of our code is mentioned in the README.md and included in a LICENSE.txt file in our git repository. PDB data are free of all copyright restrictions and made fully and freely available for both non-commercial and commercial use. +(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] As a URL in the Abstract. +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] Our data are proteins. +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] Our data are proteins. + +5. If you used crowdsourcing or conducted research with human subjects... + +(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] +(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] +(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] \ No newline at end of file diff --git a/parse/train/gwPPcc_M0lv/gwPPcc_M0lv_content_list.json b/parse/train/gwPPcc_M0lv/gwPPcc_M0lv_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..f7030fb81cdd61a6f7d8f2885672dc9665f4cd50 --- /dev/null +++ b/parse/train/gwPPcc_M0lv/gwPPcc_M0lv_content_list.json @@ -0,0 +1,1267 @@ +[ + { + "type": "text", + "text": "Learning to recover orientations from projections in single-particle cryo-EM ", + "text_level": 1, + "bbox": [ + 267, + 123, + 728, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ", + "bbox": [ + 423, + 226, + 580, + 281 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 318, + 535, + 334 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 A major challenge in single-particle cryo-electron microscopy (cryo-EM) is that \n2 the orientations adopted by the 3D particles prior to imaging are unknown; yet, this \n3 knowledge is essential for high-resolution reconstruction. We present a method \n4 to recover these orientations directly from the acquired set of 2D projections. \n5 Our approach consists of two steps: (i) the estimation of distances between pairs \n6 of projections, and (ii) the recovery of the orientation of each projection from \n7 these distances. In step (i), pairwise distances are estimated by a Siamese neural \n8 network trained on synthetic cryo-EM projections from resolved bio-structures. \n9 In step (ii), orientations are recovered by minimizing the difference between \n10 the distances estimated from the projections and the distances induced by the \n11 recovered orientations. We evaluated the method on synthetic cryo-EM datasets. \n12 Current results demonstrate that orientations can be accurately recovered from \n13 projections that are shifted and corrupted with a high level of noise. The accuracy \n14 of the recovery depends on the accuracy of the distance estimator. While not \n15 yet deployed in a real experimental setup, the proposed method offers a novel \n16 learning-based take on orientation recovery in SPA. Our code is available at https: \n17 //github.com/anonymous/protein-reconstruction. ", + "bbox": [ + 148, + 349, + 767, + 584 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "18 1 Introduction ", + "text_level": 1, + "bbox": [ + 148, + 609, + 312, + 627 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "19 Single-particle cryo-electron microscopy (cryo-EM) has revolutionized the field of structural biology \n20 over the last decades [1, 2, 3]. The use of electron beams to image ice-embedded samples has \n21 permitted the recovery of 3D bio-structures at unprecedented resolution. This “resolution revolution” \n22 has had a tremendous impact in biomedical research, providing invaluable insights into the biological \n23 processes that underlie many current diseases. \n24 In single-particle cryo-EM, every 3D particle adopts a random orientation $\\theta _ { i }$ in the ice layer before \n25 being imaged. Hence, the projection geometry associated to each acquired 2D projection (Figure 1) \n26 is unknown. Yet, this knowledge is essential for the tomographic reconstruction of bio-structures [4]. \n27 We consider that a cryo-EM measurement (i.e., a projection) $\\mathbf { p } _ { i } \\in \\mathbb { R } ^ { n _ { p } }$ is acquired through ", + "bbox": [ + 147, + 641, + 826, + 710 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 717, + 825, + 773 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/381f84970a920193721cd01b02b71e6ee1b6d25a604f5dd23379b11fab3ca56a.jpg", + "text": "$$\n\\mathbf { p } _ { i } = \\mathbf { C } _ { \\varphi } \\mathbf { S _ { t } } _ { i } \\mathbf { P } _ { \\theta _ { i } } \\mathbf { x } + \\mathbf { n } ,\n$$", + "text_format": "latex", + "bbox": [ + 419, + 780, + 576, + 797 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "28 where $\\mathbf { x } \\in \\mathbb { R } ^ { n _ { x } }$ is the unknown 3D density map [5] (Coulomb potential). The operator $\\mathbf { P } _ { \\pmb { \\theta } _ { i } } : \\mathbb { R } ^ { n _ { x } } $ \n29 $\\mathbb { R } ^ { n _ { p } }$ is the projection along the orientation $\\theta _ { i }$ (i.e., the $\\mathbf { X }$ -ray transform). The operator $\\mathbf { S } _ { \\mathbf { t } _ { i } } : \\mathbb { R } ^ { n _ { p } } $ \n30 $\\mathbb { R } ^ { n _ { p } }$ is a shift of the projection by $\\mathbf { t } _ { i } = ( t _ { i _ { 1 } } , t _ { i _ { 2 } } )$ . The convolution operator $\\mathbf { C } _ { \\varphi } : \\mathbb { R } ^ { n _ { p } } \\mathbb { R } ^ { n _ { p } }$ models \n31 the microscope point-spread function (PSF) with parameters $\\varphi = ( d _ { 1 } , d _ { 2 } , \\alpha _ { \\mathrm { a s t } } )$ , where $d _ { 1 }$ is the \n32 defocus-major, $d _ { 2 }$ is the defocus-minor, and $\\alpha _ { \\mathrm { a s t } }$ is the angle of astigmatism [6, 7]. Finally, $\\mathbf { n } \\in \\mathbb { R } ^ { n _ { p } }$ \n33 represents additive noise. Figure 11 illustrates the effect of projection, shift, and noise. The challenge \n34 is then to reconstruct $\\mathbf { x }$ from a set of projections $\\{ { \\bf p } _ { i } \\} _ { i = 1 } ^ { P }$ acquired along unknown orientations. \n35 A popular approach is to alternatively refine the 3D structure and estimated orientations [8, 9, 10, 11, \n36 12, 13]. Yet, the outcome of these iterative-refinement procedures is often predicated on the quality \n37 of the initial reconstruction, or, equivalently, on the initial estimation of the orientations [14, 15]. \n38 Several methods have been designed to produce a first rough ab initio structure for the refinement \n39 procedure [16]. Moment-matching techniques [17, 18, 19, 20] reconstruct an initial structure such \n40 that the first few moments of the distribution of its theoretical measurements match the ones of \n41 its experimental projections; however, they typically remain sensitive to error in data and can \n42 require relatively high computational complexity. Based on the central-slice theorem, common-lines \n43 methods [21, 8, 22, 23, 24, 25, 26] aim at uniquely determining the orientations of each projection by \n44 identifying the common-lines between triplets of projections—a real challenge given the massive \n45 amount of noise. Alternatively, the marginalized maximum likelihood (ML) formulation of the \n46 reconstruction problem [11]—classically used for the iterative-refinement procedures themselves— \n47 can be minimized using stochastic gradient descent [27]. This permits to avoid the need for an initial \n48 volume estimate, at the possible cost of greater convergence instability. \n49 More recently, the recovery of geometrical information from unknown view tomography of 2D point \n50 sources has been proposed [28], but the extension to 3D cryo-EM tomography is not straightforward. \n51 Finally, [29] proposed to recover the in-plane rotations by learning to embed projections in an \n52 appropriate latent space, but only after directions had been estimated through three rounds of 2D \n53 classification in RELION. \n54 Despite the aforementioned advances, providing a robust initial volume remains a challenge due to \n55 the high-dimensionality and ill-posedness of the underlying optimization problem. On the other hand, \n56 the remarkable ability of convolutional neural networks to capture relevant representations of images \n57 has had a profound influence in imaging [30]. In this work, we present a learning-based approach to \n58 recover the unknown orientations directly from the acquired set of projections—without the need for \n59 an intermediate reconstruction procedure or an initial volume estimate. ", + "bbox": [ + 147, + 803, + 825, + 901 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/574fcacd9c7d71546e39eb9b29cfe0b129fe830c192894eca371c94a60073732.jpg", + "image_caption": [ + "Figure 1: Geometry of the imaging model defined in (1). The 3D density $\\mathbf { x }$ in the coordinate system $( x _ { 1 } , x _ { 2 } , x _ { 3 } )$ is imaged along the orientation $\\pmb \\theta$ to produce the 2D projection $\\mathbf { p }$ in the coordinate system $( y _ { 1 } , y _ { 2 } )$ of the microscope’s detector plane. The orientation $\\pmb { \\theta } = ( \\theta _ { 3 } , \\theta _ { 2 } , \\theta _ { 1 } )$ is decomposed as the direction $( \\theta _ { 2 } , \\theta _ { 1 } ) \\in [ 0 , \\pi ] \\times$ $[ 0 , 2 \\pi [$ (parameterizing the sphere $\\mathbb { S } ^ { 2 }$ ) and the in-plane rotation $\\theta _ { 3 } \\in [ 0 , 2 \\pi [$ (parameterizing the circle $\\mathbb { S } ^ { 1 }$ ). In our work, we represent the orientation $\\pmb { \\theta }$ as a unit quaternion $q$ . " + ], + "image_footnote": [], + "bbox": [ + 218, + 88, + 439, + 258 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/571ec2e85d672c9aec24a91431b51abf95fd62226de8c5f704170d6b9f6c194b.jpg", + "image_caption": [ + "Figure 2: Single-particle cryo-EM produces $P$ projections (with $P$ in the order of $1 0 ^ { 5 }$ ) from unknown orientations: $\\{ ( \\mathbf { p } _ { i } , q _ { i } ) \\} _ { i = 1 } ^ { P }$ . Observing that distances between orientations constrain the latter, we aim to recover the orientations $\\left\\{ q _ { i } \\right\\}$ from $\\{ d _ { q } ( q _ { i } , q _ { j } ) \\}$ , where $d _ { q } ( q _ { i } , q _ { j } )$ is the distance (angle) between orientations $q _ { i }$ and $q _ { j }$ . Observing that the similarity between projections depends on their relative orientation, we aim to estimate the distance $d _ { q } ( q _ { i } , q _ { j } )$ from the projections $( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } )$ . " + ], + "image_footnote": [], + "bbox": [ + 555, + 88, + 779, + 257 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 452, + 825, + 494 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 501, + 825, + 654 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 659, + 825, + 728 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 734, + 825, + 819 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "60 2 Method ", + "text_level": 1, + "bbox": [ + 148, + 847, + 271, + 863 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "61 Our approach relies on two observations (Figure 2), yielding two steps (Figure 3). First, the more similar two projections 62 $( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } )$ , the more likely they originated from two particles that adopted close ", + "bbox": [ + 148, + 883, + 823, + 912 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/3f3b0db6773005f59b0f2a35d0d58aab70fa3e035961632f38d44e661423dc1b.jpg", + "image_caption": [ + "Figure 3: Our method consists of two steps. First, we estimate distances between pairs of projections. Second, we recover the orientation of each projection from these distances. " + ], + "image_footnote": [], + "bbox": [ + 176, + 85, + 825, + 229 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "63 orientations $( q _ { i } , q _ { j } )$ in the ice prior to imaging;1 this observation guides a number of applications in \n64 the field [2]. Hence, we aim to estimate distances between orientations $d _ { q } ( q _ { i } , q _ { j } )$ from the projections \n65 as $\\widehat { d } _ { p } ( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } )$ , which we discuss in $\\ S 2 . 2$ . Second, an orientation $q$ is constrained by the distances \n66 between itself and the other orientations $\\{ d ( q , q _ { j } ) \\}$ . Hence, we aim to recover orientations $\\left\\{ { \\widehat { q } } _ { k } \\right\\}$ such \n67 that the induced distances $\\{ d _ { q } ( \\widehat { q } _ { i } , \\widehat { q } _ { j } ) \\}$ are close to the estimated distances $\\{ \\widehat { d _ { p } } ( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } ) \\}$ , which we \n68 discuss in $\\ S 2 . 3$ b b. All in all, from a set of projections $\\left\\{ \\mathbf { p } _ { k } \\right\\}$ , we aim to recover their orientations $\\left\\{ { \\widehat { q _ { k } } } \\right\\}$ \n69 such that $d _ { q } ( \\widehat { q _ { i } } , \\widehat { q _ { j } } ) \\approx \\widehat { d _ { p } } ( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } ) \\approx d _ { q } ( q _ { i } , q _ { j } )$ , with equality if $\\widehat { d } _ { p }$ and $\\left\\{ { \\widehat { q } } _ { k } \\right\\}$ are perfectly estimated. \n70 Our approach is similar to [31]. While the authors reconstruct 2D images from 1D projections, they \n71 rely on the same two-step approach: they (i) estimate distances as $\\hat { \\hat { d _ { p } } } ( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } ) = \\vert \\vert \\mathbf { p } _ { i } - \\mathbf { p } _ { j } \\vert \\vert _ { 2 }$ then \n72 (ii) recover the orientations by spectrally embedding that distance graph. The Euclidean distance is \n73 however not robust to perturbations: for example, two projections that only differ by a shift $\\bf { S _ { t } }$ of one \n74 pixel would be considered far apart while their orientations are the same. They noted that issue and \n75 we observed it too (Appendix E). To circumvent this, we propose to learn $\\widehat { d } _ { p }$ from examples $( \\ S 2 . 2 )$ . ", + "bbox": [ + 145, + 287, + 825, + 398 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 401, + 825, + 492 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "76 2.1 Representation of orientations with quaternions ", + "text_level": 1, + "bbox": [ + 150, + 506, + 545, + 521 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "77 The orientation of a 3D particle with respect to the microscope’s detector plane is a rotation relative to \n78 a reference orientation (Figure 1). The group of all 3D rotations under composition is identified with \n79 SO(3), the group of $3 \\times 3$ orthogonal matrices with determinant 1 under matrix multiplication. A \n80 rotation matrix $\\mathbf { R } _ { \\theta } \\in \\mathbf { S O } ( 3 )$ can be decomposed as a product of ${ \\binom { 3 } { 2 } } = 3$ independent rotations, for \n81 example as $\\mathbf { R } _ { \\theta } = \\mathbf { R } _ { \\theta _ { 3 } } \\mathbf { R } _ { \\theta _ { 2 } } \\mathbf { R } _ { \\theta _ { 1 } }$ , where $\\pmb { \\theta } = ( \\theta _ { 3 } , \\theta _ { 2 } , \\theta _ { 1 } ) \\in [ 0 , 2 \\pi [ \\times \\overline { { [ 0 , \\pi ] } } \\times [ 0 , 2 \\pi [$ [ are the (extrinsic \n82 and proper) Euler angles in the $Z Y Z$ convention (a common parameterization in cryo-EM) [32]. \n83 While Euler angles are a concise representation of orientation (3 numbers for 3 degrees of freedom), \n84 they suffer from a topological constraint—there is no covering map from the 3-torus to SO(3)— \n85 which manifests itself in the gimbal lock, the loss of one degree of freedom when $\\theta _ { 2 } = 0$ . This makes \n86 their optimization by gradient descent $( \\ S 2 . 3 )$ problematic. On the other hand, optimizing rotation \n87 matrices (made of 9 numbers) would require computationally costly constraints (orthogonality and \n88 determinant 1) to reduce the degrees of freedom to 3. Moreover, the distance between orientations \n89 cannot be directly computed from Euler angles and is costly (30 multiplications) to compute from \n90 rotation matrices [33]. We solve both problems by representing orientations with unit quaternions. \n91 Quaternions $q \\in \\mathbb { H }$ are an extension of complex numbers2 of the form $q = a + b i + c j + d \\pmb { k }$ where \n92 $a , b , c , d \\in \\mathbb { R }$ . Unit quaternions $q \\in \\mathbb { S } ^ { 3 }$ , where $\\mathbb { S } ^ { 3 } = \\{ q \\in \\mathbb { H } : | q | \\lceil = 1 \\}$ is the 3-sphere (with \n93 the additional group structure inherited from quaternion multiplication), concisely and elegantly \n94 represent a rotation of angle $\\theta$ about axis $( x _ { 1 } , x _ { 2 } , x _ { 3 } )$ as $q = \\cos ( \\bar { \\theta } / 2 ) + x _ { 1 } \\sin ( \\theta / 2 ) i \\dot { + } x _ { 2 } \\sin ( \\bar { \\theta / 2 } ) \\dot { { \\bf j } _ { + } }$ \n95 $\\bar { x _ { 3 } } \\sin ( \\theta / 2 ) k$ . They parameterize rotation matrices as ", + "bbox": [ + 147, + 531, + 826, + 618 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 625, + 826, + 736 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 741, + 826, + 810 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d63935fd958375da914f45efcd96dea730db939e5e34835b94b0578b08f5ed21.jpg", + "text": "$$\n\\mathbf { R } _ { q } = \\left( \\begin{array} { c c c } { a ^ { 2 } + b ^ { 2 } - c ^ { 2 } - d ^ { 2 } } & { 2 b c - 2 a d } & { 2 b d + 2 a c } \\\\ { 2 b c + 2 a d } & { a ^ { 2 } - b ^ { 2 } + c ^ { 2 } - d ^ { 2 } } & { 2 c d - 2 a b } \\\\ { 2 b d - 2 a c } & { 2 c d + 2 a b } & { a ^ { 2 } - b ^ { 2 } - c ^ { 2 } + d ^ { 2 } } \\end{array} \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 259, + 814, + 736, + 864 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/a4869dab607e0b19aefdd984dbad627ea589878c7313f5aba9d2ea397257420b.jpg", + "image_caption": [ + "Figure 4: Distance learning. We are looking for a distance $\\widehat { d } _ { p }$ between projections that is an accurate estimator of the distance $d _ { q }$ between their orientations. We propose to parameterize $\\widehat { d } _ { p }$ as a Siamese neural network (SNN), trained on a synthetic dataset of projections with associated orientation. " + ], + "image_footnote": [], + "bbox": [ + 174, + 88, + 826, + 205 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "96 Note that $\\mathbb { S } ^ { 3 } \\to { \\bf S O } ( 3 )$ is a two-to-one mapping (a double cover) as $q$ and $- q$ represent the same \n97 orientation. Unlike Euler angles, ${ \\mathbb S } ^ { 3 }$ is isomorphic to the universal cover of $\\mathbf { S O } ( 3 )$ . Hence, the \n98 distance between two orientations, i.e., the length of the geodesic between them on $\\mathbf { \\bar { S } O ( 3 ) }$ , is ", + "bbox": [ + 147, + 286, + 825, + 329 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1563738b4921a503b6a71279eff0d86923d80e443f70db04d5163f09a6a14d3b.jpg", + "text": "$$\n\\begin{array} { c } { d _ { q } : \\mathbb { S } ^ { 3 } \\times \\mathbb { S } ^ { 3 } \\to [ 0 , \\pi ] , } \\\\ { d _ { q } ( q _ { i } , q _ { j } ) = 2 \\operatorname { a r c c o s } \\left( | \\langle q _ { i } , q _ { j } \\rangle | \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 388, + 335, + 609, + 375 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where 99 $\\langle \\cdot , \\cdot \\rangle$ is the inner product, and the absolute value $\\left. \\cdot \\right.$ ensures that $d _ { q } ( q _ { i } , q _ { j } ) = d _ { q } ( q _ { i } , - q _ { j } )$ . The distance 00 $d _ { q } ( q _ { i } , q _ { j } )$ corresponds to the magnitude of the rotation $\\mathbf { R } _ { * }$ such that ${ \\bf R } _ { q _ { i } } = { \\bf R } _ { * } { \\bf R } _ { q _ { j } }$ [33]. ", + "bbox": [ + 153, + 380, + 825, + 410 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 Distance learning ", + "text_level": 1, + "bbox": [ + 173, + 425, + 334, + 440 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "102 We aim to estimate a function $\\widehat { d } _ { p }$ such that $\\widehat { d } _ { p } ( { \\bf p } _ { i } , { \\bf p } _ { j } ) \\approx d _ { q } ( q _ { i } , q _ { j } )$ . While we could in principle \n103 design $\\widehat { d } _ { p }$ , that would be intricate—if not impossible—partly because the invariants are difficult \n104 to specify. We instead opt to learn $\\widehat { d } _ { p }$ , capitalizing on (i) the powerful function approximation \n105 capabilities of neural networks, and (ii) the possibility to generate realistic datasets supported by the \n106 availability of numerous 3D atomic models3 and our ability to model the cryo-EM imaging procedure. ", + "bbox": [ + 140, + 450, + 826, + 531 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "From a training dataset 107 $\\left\\{ { \\bf p } _ { i } , q _ { i } \\right\\} _ { i = 1 } ^ { P }$ , we learn the projection distance ", + "bbox": [ + 143, + 536, + 622, + 555 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c8b5fbe87899589482c182ea2adb267fb557ff7a5dfaeea85cdffd3af0caacbe.jpg", + "text": "$$\n\\widehat { d } _ { p } = \\mathop { \\mathrm { a r g } } \\underset { d _ { p } } { \\mathrm { a r g } } \\mathrm { m i n } L _ { \\mathrm { D E } } , \\quad \\mathrm { w h e r e } \\quad L _ { \\mathrm { D E } } = \\sum _ { i , j } \\left| d _ { p } \\big ( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } \\big ) - d _ { q } \\big ( q _ { i } , q _ { j } \\big ) \\right| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 271, + 560, + 725, + 597 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "108 is the loss and $d _ { q }$ is defined in (2). The $d _ { p }$ is parameterized as the Siamese neural network (SNN) [34] ", + "bbox": [ + 140, + 602, + 826, + 618 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f2b8d1548acf18714d657060c7aa1c07aefac4fdc9bb71474e2c595089c9066e.jpg", + "text": "$$\nd _ { p } ( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } ) = d _ { f } ( \\mathcal { G } _ { w } ( \\mathbf { p } _ { i } ) , \\mathcal { G } _ { w } ( \\mathbf { p } _ { j } ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 625, + 614, + 642 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "109 where $\\mathcal { G } _ { w }$ is a convolutional neural network with weights $w$ that is trained to extract the most relevant \n110 features $\\mathbf { f } _ { i } \\in \\mathbb { R } ^ { n _ { f } }$ from a projection $\\mathbf { p } _ { i }$ . SNNs, also termed “twin networks”, are commonly used in \n111 the field of deep metric learning to learn similarity functions [35]. We set the feature space distance \n112 $d _ { f }$ as the cosine distance to facilitate the learning of a $\\widehat { d } _ { p }$ that respects the elliptic geometry of $\\mathbb { S } ^ { 3 }$ \n113 (Appendix F). Figure 4 illustrates the proposed learning paradigm. \n114 As evaluating a sum over $P ^ { 2 }$ pairs is computationally intractable for cryo-EM datasets with typically \n115 $P$ in the order of $1 0 ^ { 5 }$ projections, we sample the sum and minimize (3) with stochastic gradient \n116 descent (SGD) over small batches of pairs. The weights $w$ are updated by back-propagation. \n117 The architecture of $\\mathcal { G } _ { w }$ is described in Appendix G. When designing the architecture, we constrain \n118 the functional space from which the trained $\\mathcal { G } _ { w }$ is drawn and express our prior expert knowledge. For \n119 example, we realize shift invariance, i.e., a guarantee that a shift $\\bf { S _ { t } }$ does not change our estimated \n120 distances and orientations, with a fully convolutional architecture. Size invariance, i.e., taking \n121 projections $\\mathbf { p }$ of varying sizes $n _ { p }$ while yielding a representation f of a fixed size $n _ { f }$ , is realized by a \n122 final average pooling layer. As we do not (yet) know how to realize an invariance to noise or PSF, we \n123 resort to data augmentation, i.e., training on perturbed projections. In $\\ S 3 . 4$ , we show that a built-in \n124 invariance (shift) is far preferable to one learned through augmentation (noise). Finally, as projections ", + "bbox": [ + 142, + 648, + 825, + 722 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 143, + 727, + 825, + 770 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 775, + 825, + 887 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "125 are made by integrating through the 3D volume, projections from opposed directions are mirrors of each other.4126 That is another kind of physical knowledge that should ideally be built into our method. ", + "bbox": [ + 148, + 90, + 826, + 119 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "127 One could hope to train $\\mathcal { G } _ { w }$ to directly map projections to orientations as ${ \\widehat { q _ { i } } } = \\mathbf { f } _ { i } = { \\mathcal { G } } _ { w } ( \\mathbf { p } _ { i } )$ . While \n128 that would avoid the orientation recovery step, a space of ${ n } _ { f } = 4$ bdimensions does not have room for \n129 $\\mathcal { G } _ { w }$ to represent the other factors of variation in $\\mathbf { p }$ , such as different noise levels, PSFs, or proteins. \n130 We tested that hypothesis in Appendix F. ", + "bbox": [ + 142, + 126, + 826, + 181 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.3 Orientation recovery ", + "text_level": 1, + "bbox": [ + 169, + 202, + 356, + 217 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "132 The task of recovering points based on their relative distances has been extensively studied. Many \n133 methods aim at mapping high-dimensional data onto a lower-dimensional space while preserving \n134 distances, primarily for dimensionality reduction and data visualization. Well-known examples \n135 include MDS [36], Isomap [37], LLE [38], Laplacian eigenmaps [39], t-SNE [40], and UMAP [41]. \n136 The embedding of distance matrices in Euclidean space (given by their eigenvectors) is especially \n137 well-described. In particular, the framework of Euclidean distance matrices (EDMs) [42] provides \n138 theoretical guarantees on the recovery of points from distances. \n139 We however aim to embed the orientations $q$ in $\\mathbb { S } ^ { 3 }$ (§2.1), a setting for which we are unaware of any \n140 theoretical characterization (e.g., on the shape of the loss function or its behavior when distances are \n141 missing or noisy). The fact that ${ \\mathbb S } ^ { 3 }$ is locally Euclidean does however offer some hope. Indeed, despite \n142 the non-convexity and the lack of theoretical guarantees, we are able to appropriately minimize our \n143 loss function, as we experimentally demonstrate in Appendix D. ", + "bbox": [ + 140, + 227, + 825, + 325 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 330, + 825, + 401 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We recover the orientations of a set of projections 144 $\\left\\{ { \\bf p } _ { k } \\right\\} _ { k = 1 } ^ { P }$ through ", + "bbox": [ + 143, + 406, + 620, + 428 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/dc31193ce98560699aff637469cacd0caf26d003a599eb96d4d087d134693985.jpg", + "text": "$$\n\\left\\{ \\widehat { q } _ { k } \\right\\} _ { k = 1 } ^ { P } = \\underset { \\left\\{ q _ { k } \\in \\mathbb { S } ^ { 3 } \\right\\} } { \\arg \\operatorname* { m i n } } L _ { \\mathrm { O R } } , \\quad \\mathrm { w h e r e } \\quad L _ { \\mathrm { O R } } = \\sum _ { i , j } \\left| \\widehat { d } _ { p } \\left( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } \\right) - d _ { q } \\left( q _ { i } , q _ { j } \\right) \\right| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 250, + 438, + 746, + 477 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "145 is the loss and $\\widehat { d } _ { p }$ is the estimator trained in (3). Note that the sole difference with (3) is that the \n146 minimization is performed over the orientations $q$ rather than the distance $d _ { p }$ . Here again, we sample \n147 the sum in practice and minimize (4) with mini-batch SGD. Sampling the sum amounts to building a \n148 sparse (instead of complete) distance graph before embedding, a common strategy. ", + "bbox": [ + 140, + 487, + 825, + 545 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "149 2.4 Evaluation ", + "text_level": 1, + "bbox": [ + 143, + 565, + 289, + 579 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "150 151 Unfortunately, we cannand the true orientations y take the difference between the recovered orientations as orientations are rotations up to an arbitrary reference or $\\{ \\widehat { q _ { k } } \\} _ { k = 1 } ^ { P }$ \n$\\{ q _ { k } \\} _ { k = 1 } ^ { P }$ \n153 Any global rotation or reflection of the recovered orientations is as valid as any other, i.e., $d _ { q } ( q _ { i } , q _ { j } ) =$ \n154 $d _ { q } ( \\mathbf { T } q _ { i } , \\mathbf { T } q _ { j } ) \\ \\forall \\mathbf { T } \\in \\mathbf { O } ( 4 )$ , where $\\mathbf { O } ( 4 )$ is the group of $4 \\times 4$ orthogonal matrices. Hence, we align \n155 the sets of orientations and compute the mean orientation recovery error as ", + "bbox": [ + 140, + 590, + 826, + 676 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/757a75a263f34ed731a98c003349a1a328120404e31f5e51cda68adfbad3b9e8.jpg", + "text": "$$\nE _ { \\mathrm { O R } } = \\operatorname* { m i n } _ { \\mathbf { T } \\in \\mathbf { O } ( 4 ) } \\frac { 1 } { P } \\sum _ { i = 1 } ^ { P } \\left| d _ { q } \\left( q _ { i } , \\mathbf { T } \\widehat { q _ { i } } \\right) \\right| .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 685, + 620, + 729 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We implement 156 $\\mathbf { T }$ as a product of ${ \\binom { 4 } { 2 } } = 6$ independent rotations and an optional reflection: ", + "bbox": [ + 137, + 739, + 763, + 757 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/69eff99069d1d3c687db80db77f4abeedd21ee9ae19c88d0956bcf6808b93592.jpg", + "text": "$$\n\\mathbf { T } = \\left[ \\begin{array} { l l } { m } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { I } } \\end{array} \\right] \\prod _ { \\substack { 1 \\leq i < j \\leq 4 } } \\mathbf { T } _ { \\theta _ { i j } } , \\quad m \\in \\{ - 1 , 1 \\} , \\ \\theta _ { i j } \\in [ 0 , 2 \\pi [ ,\n$$", + "text_format": "latex", + "bbox": [ + 300, + 766, + 694, + 808 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where 157 $\\mathbf { T } _ { \\theta _ { i j } } \\in \\mathbf { S O } ( 4 )$ is a rotation by angle $\\theta _ { i j }$ on the $( x _ { i } , x _ { j } )$ plane. ", + "bbox": [ + 143, + 816, + 620, + 833 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "158 In practice, we again minimize (5) with mini-batch SGD. Because $\\mathbf { O } ( 4 )$ is disconnected, we optimize \n159 the 6 angles separately for $m = 1$ (proper rotations) and $m = - 1$ (improper rotations). Figure 15 \n160 shows an alignment to $E _ { \\mathrm { O R } } = 0$ after a perfect recovery. \n162 We first evaluated whether orientation recovery through (4) was feasible assuming perfect distances, \n163 and how it was affected by errors in the distances (§3.2). We then learned to estimate the distances \n164 through (3), and evaluated the accuracy of this procedure (§3.3) and its robustness to perturbations of \n165 the projections (§3.4). Finally, we ran the whole machinery on a synthetic dataset to assess how well \n166 orientations could be recovered from estimated distances (§3.5). ", + "bbox": [ + 142, + 838, + 823, + 882 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 121, + 825, + 190 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 Experimental conditions ", + "text_level": 1, + "bbox": [ + 169, + 208, + 383, + 223 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "168 Density maps. We considered two proteins (Figure 10): the $\\beta$ -galactosidase, a protein with a \n169 dihedral (D2) symmetry, and the lambda excision HJ intermediate (HJI), an asymmetric protein \n170 with local cyclic (C1) symmetry. Their deposited PDB atomic models are 5a1a [43] and $5 \\mathrm { j } 0 \\mathrm { n }$ [44], \n171 respectively. From these atomic models, we generated the density maps in Chimera [45] by fitting the \n172 models with a $1 \\mathring \\mathrm { A }$ map for 5a1a and a $3 . 6 7 \\mathring \\mathrm { A }$ map for $5 \\mathrm { j } 0 \\mathrm { n }$ ; this gave us a volume of $1 1 0 \\times 1 5 5 \\times 1 9 9$ \n173 voxels for 5a1a and one of $6 9 \\times 5 7 \\times 7 5$ voxels for $5 \\mathrm { j } 0 \\mathrm { n }$ . \n174 Protein symmetries. Symmetries are problematic when learning distances: two projections can \n175 be identical while not originating from the same orientation, which breaks an axiom of distance \n176 functions (identity of indiscernibles). Figure 16b illustrates this problem. To capture only one of four \n177 identical projections of 5a1a, we restricted directions to $( \\theta _ { 2 } , \\theta _ { 1 } ) \\in [ 0 , \\pi [ \\times [ 0 , \\frac { \\pi } { 2 } [$ (a quarter of the \n178 sphere, illustrated in Figure 12a) for that protein. This treatment of symmetries is incomplete5 but \n179 sufficient for a proof-of-concept. \n180 Projections. Using the ASTRA projector [46], we generated $P = 5 , 0 0 0$ synthetic projections \n181 of $2 7 5 \\times 2 7 5$ pixels (downsampled to $1 1 6 \\times 1 1 6 )$ for 5a1a and $1 1 6 \\times 1 1 6$ pixels for $5 { \\dot { \\jmath } } 0 \\mathbf { n }$ , taken \n182 from uniformly sampled orientations. 6 We then perturbed the measurements with different levels of \n183 additive Gaussian noise [47, 48] and off-centering shifts. Figure 11 displays some samples. ", + "bbox": [ + 142, + 233, + 825, + 319 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 334, + 825, + 420 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 435, + 825, + 492 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Datasets. For each protein, we split the projections into training, validation, and test subsets, and created disjoint pairs of projections from each (Table 1). The training and validation sets were used to train and evaluate the SNN, while the test set was used to evaluate orientation recovery given a trained SNN. Sampling orientations (mostly) uniformly induces a distribution of distances that is skewed towards larger distances (shown in Figure 12b). As this would skew $L _ { \\mathrm { D E } }$ and bias $\\widehat { d } _ { p }$ , we further sampled $1 \\%$ of the training and validation pairs to make the distribution of distances uniform—for $\\widehat { d } _ { p }$ to be uniformly accurate over the whole $[ 0 , \\pi ]$ range of distances (see Appendix B for further illustrations). While 1, 650 projections were enough to perfectly reconstruct the density maps (as shown in Figures 9e and 9j), our method is not limited by the number of projections as optimization is done per batch. Optimization settings are described in Appendix C. ", + "bbox": [ + 166, + 507, + 825, + 651 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2 Sensitivity of orientation recovery to errors in distance estimation ", + "text_level": 1, + "bbox": [ + 171, + 669, + 666, + 683 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We first evaluated the feasibility of orientation recovery assuming that the exact distances were known. \nThe method successfully recovers the orientation of every projection in this case (see Appendix D). ", + "bbox": [ + 165, + 694, + 820, + 723 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To evaluate the robustness of (4), we perturbed the distances prior to recovery with an error sampled from a Gaussian distribution with mean 0 and variances $\\sigma ^ { \\hat { 2 } } \\in [ 0 . 0 , 0 . 8 ]$ . Figure 5 shows that the recovery error $E _ { \\mathrm { O R } }$ is a monotonic function of the error in distances: from $E _ { \\mathrm { O R } } = 0$ with exact distances to $E _ { 0 \\mathrm { R } } \\approx 0 . 2$ radians $( \\approx 1 1 . 5 ^ { \\circ } )$ for $\\sigma ^ { 2 } = 0 . 8$ . ", + "bbox": [ + 173, + 728, + 825, + 785 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "These results demonstrate that the performance of orientation recovery (4) depends on the quality of the estimated distances, which advocates for a proper and extensive training of the SNN. Moreover, we observe that $L _ { \\mathrm { O R } }$ is a reliable proxy for $E _ { \\mathrm { O R } }$ , allowing us to assess recovery performance in the absence of ground-truth orientations (i.e., when recovering the orientations of real projections). ", + "bbox": [ + 174, + 791, + 825, + 847 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/14ffbcfa56c8f9cef8fdac0640d35b581d88e1784d431ca1bc818a772babfcfc.jpg", + "table_caption": [ + "Table 1: Split of $P = 5 , 0 0 0$ projections in training, validation, and test subsets. " + ], + "table_footnote": [], + "table_body": "
DatasetPp2Used pairs
Training2,512 (50%)6,310,14463,101
Validation838 (17%)702,2447,022
Test1,650 (33%)2,722,5002,722,500
", + "bbox": [ + 173, + 136, + 485, + 198 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/2aaf4e1ffebb6061b0d70680344575ada494cd691329db794acfa53f635c55b1.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 529, + 87, + 825, + 183 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/8cec7c08a80f0bbcb089809f087d6f046a01157d520bd8c120d20494ab91474b.jpg", + "image_caption": [ + "Figure 5: Orientation recovery from perturbed distances on $5 { \\dot { \\jmath } } 0 \\mathbf { n }$ (left) and 5a1a (right). " + ], + "image_footnote": [], + "bbox": [ + 173, + 238, + 488, + 319 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/705437b5f98dff2a7bfcb3af8a130419f1d633b057aa6fcecdae2727f00411f2.jpg", + "image_caption": [ + "Figure 6: Distance learning. " + ], + "image_footnote": [], + "bbox": [ + 542, + 203, + 812, + 297 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "(b) Relationship between $\\widehat { d _ { p } }$ and $d _ { q }$ on 1, 000 pairs from the test sets of $5 { \\dot { \\jmath } } 0 \\mathbf { n }$ (left) and 5a1a (right). ", + "bbox": [ + 529, + 304, + 825, + 330 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "205 3.3 Learning to estimate distances ", + "text_level": 1, + "bbox": [ + 150, + 397, + 423, + 411 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "06 We evaluated the ability of the SNN to learn to approximate the orientation distance $d _ { q }$ . For \n7 comparison, we evaluated a baseline, the Euclidean distance $\\widehat { d } _ { p } ( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } ) = \\| \\mathbf { p } _ { i } , \\mathbf { p } _ { j } \\| _ { 2 }$ , in Appendix $\\mathrm { E }$ \n8 Figure 6a shows the convergence of $L _ { \\mathrm { D E } }$ , reached in about 50 epochs. Figure 6b shows the relationship \n09 between the distance $\\widehat { d } _ { p }$ estimated from projections and the true distance $d _ { q }$ . The outliers for 5a1a \n10 are explained by our incomplete treatment of its symmetry. While our learned distance function \nis a much better estimator than the Euclidean distance—compare Figure 6b with Figure 16—they \n2 share one characteristic: both plateau and underestimate the largest distances. We did attenuate \nthe phenomenon by sampling training distances uniformly (see $\\ S 3 . 1 \\ r ,$ ), and the issue is much less \n4 severe than with the Euclidean distance. An alternative could be to only rely on smaller distances for \n15 recovery. That would however require the addition of a spreading term in (4) to prevent the recovered \n16 orientations to collapse. \n217 These results confirm that a SNN is able to estimate differences in orientations from projections alone, \n218 even though much has yet to be gained from improving upon the rather primitive SNN architecture \n219 we are currently using. The use of additional training data should help further diminish overfitting. ", + "bbox": [ + 155, + 429, + 825, + 594 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 601, + 826, + 643 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "220 3.4 Sensitivity of distance learning to perturbations in the projections ", + "text_level": 1, + "bbox": [ + 147, + 678, + 666, + 693 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "221 We first demonstrated that the learning of distances is insensible to off-centering shifts (Figure 7a), \n222 which is expected given that shift invariance is built in our SNN (see $\\ S 2 . 2 \\AA ,$ ). ", + "bbox": [ + 151, + 710, + 828, + 739 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As we cannot—or do not yet know how to—build noise invariance in the SNN architecture, we trained the SNN on noisy projections and evaluated whether it could learn to treat noise as an irrelevant information. Figure 7b shows $E _ { \\mathrm { O R } } \\approx 0 . 1 6$ radians $( \\approx 9 ^ { \\circ } )$ for noiseless projections and $E _ { \\mathrm { O R } } \\approx 0 . 4 2$ radians $( \\approx 2 4 ^ { \\circ } )$ for a more realistic noise variance of $\\sigma ^ { 2 } = 1 6$ (with signal-to-noise ratio of $- 1 2 \\ \\mathrm { d B }$ ). Whereas a naive distance function (e.g., an Euclidean distance) would be extremely sensitive to noise, the SNN mostly learned to discard it. Moreover, the observed overfitting indicates that more training data should further decrease the sensitivity of the SNN to noise. ", + "bbox": [ + 158, + 746, + 825, + 842 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "230 Note that we did not evaluate sensitivity to the PSF at this stage but expect a similar behavior. ", + "bbox": [ + 150, + 848, + 785, + 863 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "231 Here again (§3.2), we observed that (i) the estimation of more accurate distances (a smaller $L _ { \\mathrm { D E } } ,$ ) \n232 leads to the recovery of more accurate orientations (a smaller $L _ { \\mathrm { O R } }$ and $E _ { \\mathrm { O R } } \\mathrm { , }$ ), and that (ii) an higher \n233 recovery loss $L _ { \\mathrm { O R } }$ induces an higher error $E _ { \\mathrm { O R } }$ . ", + "bbox": [ + 142, + 869, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/6f70f3284125a2c36d4a356143a7dcfa7b58837a136ef24c27fd3c69e4865116.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 173, + 88, + 482, + 185 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(a) Learning from shifted projections $\\{ \\mathbf { S } _ { \\mathbf { t } _ { i } } \\mathbf { P } _ { \\pmb { \\theta } _ { i } } \\mathbf { x } \\}$ , with shifts $t _ { i _ { 1 } }$ and $t _ { i _ { 2 } }$ sampled from a triangular distribution with mean 0 and of increasing limits. ", + "bbox": [ + 173, + 190, + 482, + 229 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/fa385fb38fc1189c00191e90224e286cf12a4a8e9dac8951664eec3eb9e6ac1a.jpg", + "image_caption": [ + "(b) Learning from noisy projections $\\{ \\mathbf { P } _ { \\pmb { \\theta } _ { i } } \\mathbf { x } + \\mathbf { n } \\}$ , with white noise $\\mathbf { n } \\sim { \\mathcal { N } } ( 0 , \\sigma ^ { 2 } \\mathbf { I } )$ of increasing variance $\\sigma ^ { 2 }$ . " + ], + "image_footnote": [], + "bbox": [ + 516, + 89, + 825, + 185 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/0264fa45e6883ef7433b143093a4bdda85e2307012d7e4a9d89d7791417030a3.jpg", + "image_caption": [ + "Figure 7: Sensitivity of distance learning to perturbations in the projections of $5 \\mathrm { j } 0 \\mathrm { n }$ . The box plots show the distance learning loss $L _ { \\mathrm { D E } }$ (the distribution is taken over epochs). Boxes show the orientation recovery loss $L _ { \\mathrm { O R } }$ and error $E _ { \\mathrm { O R } }$ . ", + "Figure 8: Distance learning and orientation recovery from estimated distances. The green and orange boxes show $L _ { \\mathrm { D E } }$ (3) on the training and validation sets. The blue curve shows the evolution of the recovery loss until convergence, with the minimum $L _ { \\mathrm { O R } }$ (4) highlighted. The red histogram shows the errors in the recovered orientations $\\{ d _ { q } ( q _ { i } , \\mathbf { T } \\widehat { q _ { i } } ) \\}$ , with the mean $E _ { \\mathrm { O R } }$ (5) highlighted. " + ], + "image_footnote": [], + "bbox": [ + 173, + 295, + 825, + 396 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "234 3.5 Orientation recovery and reconstruction of density maps ", + "text_level": 1, + "bbox": [ + 148, + 488, + 606, + 503 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As a proof-of-concept, we attempted to solve the full inverse problem posed by (1), i.e., to reconstruct the density maps $\\widehat { \\mathbf { x } }$ from sets of projections $\\{ { \\bf { p } } _ { i } \\}$ and their orientations $\\left\\{ { \\widehat { q } } _ { i } \\right\\}$ recovered through the b bproposed method. It is worth noting that, at this stage of development, we only trained the SNN on projections originating from the protein we were attempting to reconstruct. In addition, reconstruction was performed with a direct reconstruction algorithm (ASTRA’s GPU implementation of the CGLS algorithm) rather than with a robuster iterative method. This is a specific experimental case that only partially shines light on the applicability of the method in real situations; this is discussed in $\\ S 4$ . ", + "bbox": [ + 173, + 515, + 825, + 612 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 8a shows the recovery of orientations from distances that were estimated from noiseless projections of $5 \\mathrm { j } 0 \\mathrm { n }$ . A mean error of $E _ { 0 \\mathrm { R } } \\approx 0 . 2 0$ radians $( \\approx 1 1 ^ { \\circ } )$ in the recovered orientations led to a reconstruction with a resolution of $1 2 . 2 \\mathring \\mathrm { A }$ at a Fourier shell coefficient (FSC) of 0.5, shown in Figure ${ 9 \\mathrm { c } }$ . As predicted by our other experiments, corrupting the projections with noise ( $\\sigma ^ { 2 } = 1 6$ ) negatively impacts the quality of the recovered orientations (Figure 8b); the obtained mean error is then $E _ { 0 \\mathrm { R } } \\approx 0 . 2 5$ radians $( \\approx 1 4 ^ { \\circ } )$ ). Unsurprisingly, this leads to a reconstruction with a lower resolution of $1 5 . 2 \\mathring \\mathrm { A }$ , shown in Figure 9d. (Note that reconstruction was here obtained from the noiseless projections, the goal being to evaluate only the impact of orientation mis-estimation.) ", + "bbox": [ + 171, + 618, + 825, + 733 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Finally, Figures 8c,d show the recovery of orientations from noiseless and noisy projections of 5a1a. A mean error of $E _ { \\mathrm { O R } } \\approx 0 . 1 3$ radians $( \\approx 7 ^ { \\circ } )$ in both cases led to reconstructions with resolutions of $8 . 0 \\mathring \\mathrm { A }$ and $9 . 6 \\mathring \\mathrm { A }$ , shown in Figures 9h,i. Distance estimation, orientation recovery, and reconstruction performed better on 5a1a than $5 \\mathrm { j } 0 \\mathrm { n }$ because its ground-truth density is of higher resolution. ", + "bbox": [ + 165, + 738, + 825, + 794 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "These results tend to indicate that a reasonable first structure can be reconstructed from projections whose orientations have been recovered through our method. ", + "bbox": [ + 165, + 800, + 823, + 829 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 Discussion ", + "text_level": 1, + "bbox": [ + 169, + 849, + 294, + 867 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "257 In this work, we explored the use of distance learning between pairs of 2D cryo-EM projections \n258 from a 3D protein structure to infer the unknown orientation at which each projection was imaged \n259 from. Our two-step method relies on the estimation of pairwise distances between unseen projections, \n260 followed by the recovery of the orientations from these distances. ", + "bbox": [ + 147, + 883, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/be11a1e7a160d661e763531e4c3e3700d0c00e6a7b41db7a15cda69b0a0bdc59.jpg", + "image_caption": [ + "Figure 9: Density maps $\\widehat { \\mathbf { x } }$ reconstructed from (a,f) ground-truth orientations, (b,g) random orientations, b(c,h) orientations recovered from noiseless projections, and (d,i) orientations recovered from noisy projections. The Fourier shell correlation (FSC) curves in (e,j) indicate the resolutions of the densities (w.r.t. ground-truth densities, shown in Figures 10b,d). " + ], + "image_footnote": [], + "bbox": [ + 174, + 85, + 826, + 315 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 407, + 823, + 436 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The method has been evaluated on synthetic datasets for two different proteins. The results provide key insights on the viability of the proposed scheme. First, they demonstrate that a SNN can learn a distance function between projections that estimates the difference in their orientation $( \\ S 3 . 3 )$ and that is invariant to shifts and robust to increasing levels of noise (§3.4)—an important condition in cryo-EM. Second, they demonstrate that an accurate estimation of distances leads to an accurate recovery of orientations $( \\ S 3 . 2 , \\ \\ S 3 . 4 )$ . Finally, our method was able to recover orientations with an error of 0.12 to 0.25 radians (7 to $1 4 ^ { \\circ }$ )—leading to an initial volume with a resolution of 8 to $1 5 \\mathring \\mathrm { A }$ (§3.5). In summary, the more accurate the estimated distances, the more precise the recovered orientations, and, ultimately, the higher-resolution the reconstructed volume. ", + "bbox": [ + 171, + 443, + 825, + 569 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "270 While the method is not yet ready to be deployed in practice, we believe that a series of developments \n271 could make it relevant for single-particle cryo-EM reconstruction. 7 As previously discussed, the \n272 results underline the importance of learning an accurate distance estimator. In this regard, the \n273 performance of the SNN could be improved. First, the architecture of the twin convolutional neural \n274 networks should be expanded and tuned. Second, training could be improved, perhaps by providing \n275 more supervision by separately predicting the differences in direction $( \\theta _ { 2 } , \\theta _ { 1 } )$ and in-plane angle $\\theta _ { 3 }$ . ", + "bbox": [ + 142, + 574, + 825, + 659 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Importantly, the SNN would be better trained on a more diverse cryo-EM dataset. Indeed, its success as a faithful estimator eventually relies on our capacity to generate a synthetic training dataset whose data distribution is diverse enough to cover that of unseen projection datasets. Such realistic cryo-EM projections could be generated by relying on a more expressive formulation of the cryo-EM physics and taking advantage of the thousands of atomic models available in the PDB. In particular, a necessary extension will be to include the effects of the PSF and to evaluate its impact. ", + "bbox": [ + 171, + 664, + 825, + 747 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "A final phase of tests before deploying the method on real cryo-EM measurements will be to extensively test the method on “unseen proteins”, i.e., proteins whose simulated projections have never been seen by the SNN. In this regard, an interesting aspect of our method is that the twin networks within the SNN intrinsically predict the relationship between projections, allowing the SNN as a whole to abstract the particular volume. Learning should benefit from the profound structural similarity shared by proteins—after all, they are all derived from the same 21 building blocks. ", + "bbox": [ + 174, + 753, + 825, + 838 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Training our 4.5M parameter model (see Appendices G and C) has the following negative environmental impact: it consumes 13 kWh of energy, which produces 6.36 lbs of $\\mathrm { C O _ { 2 } }$ on average [49]. ", + "bbox": [ + 160, + 843, + 825, + 872 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "91 [1] J. Dubochet, M. Adrian, J.-J. Chang, J.-C. Homo, J. Lepault, A. W. McDowall, and P. 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Available: http://dl.acm.org/citation.cfm?doid $\\mathbf { \\Psi } =$ 2487575.2488200 ", + "bbox": [ + 156, + 93, + 828, + 915 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 161, + 92, + 826, + 698 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "34 Checklist ", + "text_level": 1, + "bbox": [ + 151, + 713, + 254, + 729 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 741, + 339, + 755 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] \n(b) Did you describe the limitations of your work? [Yes] See $\\ S 4$ . \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] We didn’t identify any potential risk for improving protein imaging. Moreover, our work only addresses a small step in a huge pipeline. We however mentioned the environmental impact of training our model (see $\\ S 4$ ). \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 238, + 760, + 825, + 893 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 215, + 897, + 493, + 911 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] ", + "bbox": [ + 236, + 90, + 738, + 122 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 214, + 127, + 393, + 141 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We include a URL in the Abstract to a git repository that includes code, data, and instructions to reproduce our results. Moreover, notebooks and an interactive website are provided to further play with the method. \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See $\\ S 3 . 1$ (including Table 1) for the preparation of data and how they were split. See Appendix C for the hyperparameters. \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] When there was variance, e.g., on Figure 7 and Figure 17b. \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix C. ", + "bbox": [ + 238, + 145, + 825, + 332 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 218, + 337, + 823, + 352 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] We used proteins from the publicly available Protein Data Bank (PDB) and cited the ones we used, see $\\ S 3 . 1$ . We also used and cited the ASTRA toolbox in the same section. \n(b) Did you mention the license of the assets? [Yes] The license of our code is mentioned in the README.md and included in a LICENSE.txt file in our git repository. PDB data are free of all copyright restrictions and made fully and freely available for both non-commercial and commercial use. \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] As a URL in the Abstract. \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] Our data are proteins. \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] Our data are proteins. ", + "bbox": [ + 238, + 356, + 825, + 545 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 214, + 550, + 705, + 564 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? 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The use of electron beams to image ice-embedded samples has", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 530, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 89, + 532, + 99, + 542 + ], + "score": 1.0, + "content": "21", + "type": "text" + }, + { + "bbox": [ + 105, + 530, + 506, + 542 + ], + "score": 1.0, + "content": "permitted the recovery of 3D bio-structures at unprecedented resolution. 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The challenge", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 701, + 490, + 716 + ], + "spans": [ + { + "bbox": [ + 89, + 704, + 100, + 713 + ], + "score": 1.0, + "content": "34", + "type": "text" + }, + { + "bbox": [ + 104, + 701, + 192, + 716 + ], + "score": 1.0, + "content": "is then to reconstruct", + "type": "text" + }, + { + "bbox": [ + 192, + 704, + 200, + 712 + ], + "score": 0.53, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 701, + 301, + 716 + ], + "score": 1.0, + "content": "from a set of projections", + "type": "text" + }, + { + "bbox": [ + 301, + 701, + 335, + 714 + ], + "score": 0.93, + "content": "\\{ { \\bf p } _ { i } \\} _ { i = 1 } ^ { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 701, + 490, + 716 + ], + "score": 1.0, + "content": "acquired along unknown orientations.", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 88, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "35 A popular approach is to alternatively refine the 3D structure and estimated orientations [8, 9, 10, 11,", + "type": "text", + "cross_page": true + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 90, + 372, + 100, + 381 + ], + "score": 1.0, + "content": "36", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "12, 13]. Yet, the outcome of these iterative-refinement procedures is often predicated on the quality", + "type": "text", + "cross_page": true + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 380, + 494, + 393 + ], + "spans": [ + { + "bbox": [ + 88, + 380, + 494, + 393 + ], + "score": 1.0, + "content": "37 of the initial reconstruction, or, equivalently, on the initial estimation of the orientations [14, 15].", + "type": "text", + "cross_page": true + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 88, + 399, + 99, + 409 + ], + "score": 1.0, + "content": "38", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "Several methods have been designed to produce a first rough ab initio structure for the refinement", + "type": "text", + "cross_page": true + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 88, + 410, + 99, + 420 + ], + "score": 1.0, + "content": "39", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "procedure [16]. 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They parameterize rotation matrices as", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 645, + 451, + 685 + ], + "lines": [ + { + "bbox": [ + 159, + 645, + 451, + 685 + ], + "spans": [ + { + "bbox": [ + 159, + 645, + 451, + 685 + ], + "score": 0.91, + "content": "\\mathbf { R } _ { q } = \\left( \\begin{array} { c c c } { a ^ { 2 } + b ^ { 2 } - c ^ { 2 } - d ^ { 2 } } & { 2 b c - 2 a d } & { 2 b d + 2 a c } \\\\ { 2 b c + 2 a d } & { a ^ { 2 } - b ^ { 2 } + c ^ { 2 } - d ^ { 2 } } & { 2 c d - 2 a b } \\\\ { 2 b d - 2 a c } & { 2 c d + 2 a b } & { a ^ { 2 } - b ^ { 2 } - c ^ { 2 } + d ^ { 2 } } \\end{array} \\right) .", + "type": "interline_equation", + "image_path": "d63935fd958375da914f45efcd96dea730db939e5e34835b94b0578b08f5ed21.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 159, + 645, + 451, + 658.3333333333334 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 159, + 658.3333333333334, + 451, + 671.6666666666667 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 159, + 671.6666666666667, + 451, + 685.0000000000001 + ], + "spans": [], + "index": 40 + } + ] + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 690, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 119, + 689, + 304, + 702 + ], + "spans": [ + { + "bbox": [ + 119, + 689, + 304, + 702 + ], + "score": 1.0, + "content": "1Up to protein symmetries, which we discuss later.", + "type": "text" + } + ] + }, + { + "bbox": [ + 117, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 117, + 699, + 168, + 714 + ], + "score": 1.0, + "content": "2The algebra", + "type": "text" + }, + { + "bbox": [ + 168, + 702, + 176, + 711 + ], + "score": 0.73, + "content": "\\mathbb { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 699, + 340, + 714 + ], + "score": 1.0, + "content": "is similar to the algebra of complex numbers", + "type": "text" + }, + { + "bbox": [ + 340, + 703, + 348, + 710 + ], + "score": 0.81, + "content": "\\mathbb { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 699, + 506, + 714 + ], + "score": 1.0, + "content": ", with the exception of multiplication being", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 712, + 174, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 712, + 174, + 722 + ], + "score": 1.0, + "content": "non-commutative.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 68, + 505, + 182 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 68, + 505, + 182 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 68, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 108, + 68, + 505, + 182 + ], + "score": 0.95, + "type": "image", + "image_path": "3f3b0db6773005f59b0f2a35d0d58aab70fa3e035961632f38d44e661423dc1b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 68, + 505, + 106.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 106.0, + 505, + 144.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 144.0, + 505, + 182.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 187, + 505, + 210 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 186, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 506, + 201 + ], + "score": 1.0, + "content": "Figure 3: Our method consists of two steps. First, we estimate distances between pairs of projections.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 198, + 408, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 408, + 211 + ], + "score": 1.0, + "content": "Second, we recover the orientation of each projection from these distances.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "index", + "bbox": [ + 89, + 228, + 505, + 316 + ], + "lines": [ + { + "bbox": [ + 89, + 228, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 89, + 231, + 100, + 240 + ], + "score": 1.0, + "content": "63", + "type": "text" + }, + { + "bbox": [ + 104, + 228, + 156, + 243 + ], + "score": 1.0, + "content": "orientations", + "type": "text" + }, + { + "bbox": [ + 156, + 229, + 185, + 241 + ], + "score": 0.92, + "content": "( q _ { i } , q _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 228, + 506, + 243 + ], + "score": 1.0, + "content": "in the ice prior to imaging;1 this observation guides a number of applications in", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 238, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 88, + 242, + 99, + 252 + ], + "score": 1.0, + "content": "64", + "type": "text" + }, + { + "bbox": [ + 105, + 238, + 383, + 254 + ], + "score": 1.0, + "content": "the field [2]. Hence, we aim to estimate distances between orientations", + "type": "text" + }, + { + "bbox": [ + 384, + 240, + 423, + 253 + ], + "score": 0.94, + "content": "d _ { q } ( q _ { i } , q _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 238, + 505, + 254 + ], + "score": 1.0, + "content": "from the projections", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 251, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 88, + 254, + 100, + 266 + ], + "score": 1.0, + "content": "65", + "type": "text" + }, + { + "bbox": [ + 105, + 253, + 118, + 267 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 251, + 161, + 266 + ], + "score": 0.93, + "content": "\\widehat { d } _ { p } ( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 253, + 250, + 267 + ], + "score": 1.0, + "content": ", which we discuss in", + "type": "text" + }, + { + "bbox": [ + 251, + 254, + 270, + 265 + ], + "score": 0.87, + "content": "\\ S 2 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 253, + 370, + 267 + ], + "score": 1.0, + "content": ". Second, an orientation", + "type": "text" + }, + { + "bbox": [ + 370, + 256, + 376, + 266 + ], + "score": 0.78, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 253, + 505, + 267 + ], + "score": 1.0, + "content": "is constrained by the distances", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 264, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 88, + 267, + 99, + 276 + ], + "score": 1.0, + "content": "66", + "type": "text" + }, + { + "bbox": [ + 106, + 264, + 265, + 278 + ], + "score": 1.0, + "content": "between itself and the other orientations", + "type": "text" + }, + { + "bbox": [ + 266, + 264, + 307, + 277 + ], + "score": 0.94, + "content": "\\{ d ( q , q _ { j } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 264, + 463, + 278 + ], + "score": 1.0, + "content": ". Hence, we aim to recover orientations", + "type": "text" + }, + { + "bbox": [ + 464, + 264, + 483, + 277 + ], + "score": 0.91, + "content": "\\left\\{ { \\widehat { q } } _ { k } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 264, + 505, + 278 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 276, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 89, + 280, + 100, + 290 + ], + "score": 1.0, + "content": "67", + "type": "text" + }, + { + "bbox": [ + 105, + 277, + 213, + 292 + ], + "score": 1.0, + "content": "that the induced distances", + "type": "text" + }, + { + "bbox": [ + 213, + 278, + 262, + 291 + ], + "score": 0.94, + "content": "\\{ d _ { q } ( \\widehat { q } _ { i } , \\widehat { q } _ { j } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 277, + 407, + 292 + ], + "score": 1.0, + "content": "are close to the estimated distances", + "type": "text" + }, + { + "bbox": [ + 407, + 276, + 460, + 291 + ], + "score": 0.93, + "content": "\\{ \\widehat { d _ { p } } ( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 277, + 506, + 292 + ], + "score": 1.0, + "content": ", which we", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 289, + 504, + 302 + ], + "spans": [ + { + "bbox": [ + 88, + 291, + 100, + 301 + ], + "score": 1.0, + "content": "68", + "type": "text" + }, + { + "bbox": [ + 105, + 289, + 148, + 302 + ], + "score": 1.0, + "content": "discuss in", + "type": "text" + }, + { + "bbox": [ + 148, + 289, + 167, + 300 + ], + "score": 0.84, + "content": "\\ S 2 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 289, + 312, + 302 + ], + "score": 1.0, + "content": "b b. All in all, from a set of projections", + "type": "text" + }, + { + "bbox": [ + 313, + 289, + 334, + 302 + ], + "score": 0.92, + "content": "\\left\\{ \\mathbf { p } _ { k } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 289, + 483, + 302 + ], + "score": 1.0, + "content": ", we aim to recover their orientations", + "type": "text" + }, + { + "bbox": [ + 484, + 289, + 504, + 302 + ], + "score": 0.91, + "content": "\\left\\{ { \\widehat { q _ { k } } } \\right\\}", + "type": "inline_equation" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 300, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 89, + 304, + 99, + 314 + ], + "score": 1.0, + "content": "69", + "type": "text" + }, + { + "bbox": [ + 105, + 301, + 145, + 316 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 145, + 300, + 291, + 315 + ], + "score": 0.93, + "content": "d _ { q } ( \\widehat { q _ { i } } , \\widehat { q _ { j } } ) \\approx \\widehat { d _ { p } } ( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } ) \\approx d _ { q } ( q _ { i } , q _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 301, + 358, + 316 + ], + "score": 1.0, + "content": ", with equality if", + "type": "text" + }, + { + "bbox": [ + 358, + 300, + 369, + 315 + ], + "score": 0.9, + "content": "\\widehat { d } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 301, + 387, + 316 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 388, + 302, + 408, + 315 + ], + "score": 0.91, + "content": "\\left\\{ { \\widehat { q } } _ { k } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 301, + 505, + 316 + ], + "score": 1.0, + "content": "are perfectly estimated.", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 317, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 89, + 321, + 100, + 331 + ], + "score": 1.0, + "content": "70", + "type": "text" + }, + { + "bbox": [ + 104, + 317, + 506, + 333 + ], + "score": 1.0, + "content": "Our approach is similar to [31]. While the authors reconstruct 2D images from 1D projections, they", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 330, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 89, + 333, + 99, + 344 + ], + "score": 1.0, + "content": "71", + "type": "text" + }, + { + "bbox": [ + 105, + 331, + 379, + 345 + ], + "score": 1.0, + "content": "rely on the same two-step approach: they (i) estimate distances as", + "type": "text" + }, + { + "bbox": [ + 379, + 330, + 483, + 344 + ], + "score": 0.92, + "content": "\\hat { \\hat { d _ { p } } } ( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } ) = \\vert \\vert \\mathbf { p } _ { i } - \\mathbf { p } _ { j } \\vert \\vert _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 331, + 505, + 345 + ], + "score": 1.0, + "content": "then", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 341, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 88, + 344, + 101, + 355 + ], + "score": 1.0, + "content": "72", + "type": "text" + }, + { + "bbox": [ + 104, + 341, + 506, + 357 + ], + "score": 1.0, + "content": "(ii) recover the orientations by spectrally embedding that distance graph. 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They noted that issue and", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 375, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 89, + 380, + 100, + 389 + ], + "score": 1.0, + "content": "75", + "type": "text" + }, + { + "bbox": [ + 105, + 378, + 402, + 390 + ], + "score": 1.0, + "content": "we observed it too (Appendix E). 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Well-known examples", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 212, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 86, + 216, + 100, + 225 + ], + "score": 1.0, + "content": "135", + "type": "text" + }, + { + "bbox": [ + 104, + 212, + 506, + 227 + ], + "score": 1.0, + "content": "include MDS [36], Isomap [37], LLE [38], Laplacian eigenmaps [39], t-SNE [40], and UMAP [41].", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 224, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 86, + 226, + 100, + 236 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 106, + 224, + 505, + 237 + ], + "score": 1.0, + "content": "The embedding of distance matrices in Euclidean space (given by their eigenvectors) is especially", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 86, + 237, + 99, + 247 + ], + "score": 1.0, + "content": "137", + "type": "text" + }, + { + "bbox": [ + 106, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "well-described. 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The fact that", + "type": "text" + }, + { + "bbox": [ + 230, + 284, + 241, + 294 + ], + "score": 0.88, + "content": "{ \\mathbb S } ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 283, + 506, + 298 + ], + "score": 1.0, + "content": "is locally Euclidean does however offer some hope. 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Note that the sole difference with (3) is that the", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 86, + 401, + 100, + 411 + ], + "score": 1.0, + "content": "146", + "type": "text" + }, + { + "bbox": [ + 105, + 398, + 296, + 412 + ], + "score": 1.0, + "content": "minimization is performed over the orientations", + "type": "text" + }, + { + "bbox": [ + 297, + 401, + 303, + 410 + ], + "score": 0.83, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 398, + 397, + 412 + ], + "score": 1.0, + "content": "rather than the distance", + "type": "text" + }, + { + "bbox": [ + 398, + 399, + 408, + 411 + ], + "score": 0.87, + "content": "d _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 398, + 505, + 412 + ], + "score": 1.0, + "content": ". Here again, we sample", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 408, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 86, + 412, + 100, + 421 + ], + "score": 1.0, + "content": "147", + "type": "text" + }, + { + "bbox": [ + 105, + 408, + 506, + 424 + ], + "score": 1.0, + "content": "the sum in practice and minimize (4) with mini-batch SGD. Sampling the sum amounts to building a", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 419, + 439, + 435 + ], + "spans": [ + { + "bbox": [ + 86, + 423, + 100, + 432 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 105, + 419, + 439, + 435 + ], + "score": 1.0, + "content": "sparse (instead of complete) distance graph before embedding, a common strategy.", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + } + ], + "index": 24.5, + "bbox_fs": [ + 86, + 385, + 506, + 435 + ] + }, + { + "type": "title", + "bbox": [ + 88, + 448, + 177, + 459 + ], + "lines": [ + { + "bbox": [ + 85, + 447, + 178, + 460 + ], + "spans": [ + { + "bbox": [ + 85, + 447, + 178, + 460 + ], + "score": 1.0, + "content": "149 2.4 Evaluation", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "index", + "bbox": [ + 86, + 468, + 506, + 536 + ], + "lines": [ + { + "bbox": [ + 85, + 468, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 85, + 470, + 100, + 504 + ], + "score": 1.0, + "content": "150 151", + "type": "text" + }, + { + "bbox": [ + 101, + 468, + 204, + 513 + ], + "score": 1.0, + "content": "Unfortunately, we cannand the true orientations", + "type": "text" + }, + { + "bbox": [ + 239, + 468, + 468, + 513 + ], + "score": 1.0, + "content": "y take the difference between the recovered orientations as orientations are rotations up to an arbitrary reference or", + "type": "text" + }, + { + "bbox": [ + 469, + 479, + 504, + 492 + ], + "score": 0.91, + "content": "\\{ \\widehat { q _ { k } } \\} _ { k = 1 } ^ { P }", + "type": "inline_equation" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 204, + 491, + 239, + 504 + ], + "spans": [ + { + "bbox": [ + 204, + 491, + 239, + 504 + ], + "score": 0.92, + "content": "\\{ q _ { k } \\} _ { k = 1 } ^ { P }", + "type": "inline_equation" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 85, + 504, + 100, + 514 + ], + "score": 1.0, + "content": "153", + "type": "text" + }, + { + "bbox": [ + 105, + 502, + 454, + 516 + ], + "score": 1.0, + "content": "Any global rotation or reflection of the recovered orientations is as valid as any other, i.e.,", + "type": "text" + }, + { + "bbox": [ + 454, + 503, + 506, + 515 + ], + "score": 0.91, + "content": "d _ { q } ( q _ { i } , q _ { j } ) =", + "type": "inline_equation" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 513, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 85, + 515, + 100, + 525 + ], + "score": 1.0, + "content": "154", + "type": "text" + }, + { + "bbox": [ + 107, + 514, + 213, + 526 + ], + "score": 0.91, + "content": "d _ { q } ( \\mathbf { T } q _ { i } , \\mathbf { T } q _ { j } ) \\ \\forall \\mathbf { T } \\in \\mathbf { O } ( 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 513, + 244, + 527 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 244, + 514, + 266, + 525 + ], + "score": 0.82, + "content": "\\mathbf { O } ( 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 513, + 328, + 527 + ], + "score": 1.0, + "content": "is the group of", + "type": "text" + }, + { + "bbox": [ + 328, + 514, + 352, + 524 + ], + "score": 0.9, + "content": "4 \\times 4", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 513, + 505, + 527 + ], + "score": 1.0, + "content": "orthogonal matrices. 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Because", + "type": "text" + }, + { + "bbox": [ + 366, + 665, + 389, + 677 + ], + "score": 0.74, + "content": "\\mathbf { O } ( 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 664, + 505, + 678 + ], + "score": 1.0, + "content": "is disconnected, we optimize", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 87, + 678, + 99, + 687 + ], + "score": 1.0, + "content": "159", + "type": "text" + }, + { + "bbox": [ + 104, + 676, + 216, + 689 + ], + "score": 1.0, + "content": "the 6 angles separately for", + "type": "text" + }, + { + "bbox": [ + 216, + 677, + 245, + 686 + ], + "score": 0.89, + "content": "m = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 676, + 337, + 689 + ], + "score": 1.0, + "content": "(proper rotations) and", + "type": "text" + }, + { + "bbox": [ + 338, + 677, + 374, + 687 + ], + "score": 0.91, + "content": "m = - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "(improper rotations). Figure 15", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 686, + 333, + 701 + ], + "spans": [ + { + "bbox": [ + 87, + 690, + 100, + 698 + ], + "score": 1.0, + "content": "160", + "type": "text" + }, + { + "bbox": [ + 104, + 686, + 198, + 701 + ], + "score": 1.0, + "content": "shows an alignment to", + "type": "text" + }, + { + "bbox": [ + 198, + 687, + 235, + 698 + ], + "score": 0.91, + "content": "E _ { \\mathrm { O R } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 686, + 333, + 701 + ], + "score": 1.0, + "content": "after a perfect recovery.", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 96, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 86, + 99, + 100, + 108 + ], + "score": 1.0, + "content": "162", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 96, + 506, + 108 + ], + "score": 1.0, + "content": "We first evaluated whether orientation recovery through (4) was feasible assuming perfect distances,", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 108, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 86, + 109, + 100, + 119 + ], + "score": 1.0, + "content": "163", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 108, + 505, + 119 + ], + "score": 1.0, + "content": "and how it was affected by errors in the distances (§3.2). We then learned to estimate the distances", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 118, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 86, + 120, + 100, + 129 + ], + "score": 1.0, + "content": "164", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 118, + 505, + 130 + ], + "score": 1.0, + "content": "through (3), and evaluated the accuracy of this procedure (§3.3) and its robustness to perturbations of", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 128, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 86, + 131, + 100, + 141 + ], + "score": 1.0, + "content": "165", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 128, + 506, + 142 + ], + "score": 1.0, + "content": "the projections (§3.4). Finally, we ran the whole machinery on a synthetic dataset to assess how well", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 140, + 365, + 153 + ], + "spans": [ + { + "bbox": [ + 86, + 142, + 100, + 151 + ], + "score": 1.0, + "content": "166", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 140, + 365, + 153 + ], + "score": 1.0, + "content": "orientations could be recovered from estimated distances (§3.5).", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + } + ], + "index": 40, + "bbox_fs": [ + 87, + 664, + 505, + 701 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 87, + 96, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 86, + 96, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 86, + 99, + 100, + 108 + ], + "score": 1.0, + "content": "162", + "type": "text" + }, + { + "bbox": [ + 105, + 96, + 506, + 108 + ], + "score": 1.0, + "content": "We first evaluated whether orientation recovery through (4) was feasible assuming perfect distances,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 86, + 108, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 86, + 109, + 100, + 119 + ], + "score": 1.0, + "content": "163", + "type": "text" + }, + { + "bbox": [ + 106, + 108, + 505, + 119 + ], + "score": 1.0, + "content": "and how it was affected by errors in the distances (§3.2). We then learned to estimate the distances", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 86, + 118, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 86, + 120, + 100, + 129 + ], + "score": 1.0, + "content": "164", + "type": "text" + }, + { + "bbox": [ + 106, + 118, + 505, + 130 + ], + "score": 1.0, + "content": "through (3), and evaluated the accuracy of this procedure (§3.3) and its robustness to perturbations of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 86, + 128, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 86, + 131, + 100, + 141 + ], + "score": 1.0, + "content": "165", + "type": "text" + }, + { + "bbox": [ + 106, + 128, + 506, + 142 + ], + "score": 1.0, + "content": "the projections (§3.4). Finally, we ran the whole machinery on a synthetic dataset to assess how well", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 86, + 140, + 365, + 153 + ], + "spans": [ + { + "bbox": [ + 86, + 142, + 100, + 151 + ], + "score": 1.0, + "content": "166", + "type": "text" + }, + { + "bbox": [ + 105, + 140, + 365, + 153 + ], + "score": 1.0, + "content": "orientations could be recovered from estimated distances (§3.5).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 104, + 165, + 235, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 236, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 236, + 179 + ], + "score": 1.0, + "content": "3.1 Experimental conditions", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 87, + 185, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 86, + 186, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 86, + 187, + 100, + 197 + ], + "score": 1.0, + "content": "168", + "type": "text" + }, + { + "bbox": [ + 106, + 186, + 366, + 197 + ], + "score": 1.0, + "content": "Density maps. We considered two proteins (Figure 10): the", + "type": "text" + }, + { + "bbox": [ + 366, + 186, + 374, + 197 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 186, + 506, + 197 + ], + "score": 1.0, + "content": "-galactosidase, a protein with a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 86, + 196, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 86, + 198, + 100, + 208 + ], + "score": 1.0, + "content": "169", + "type": "text" + }, + { + "bbox": [ + 105, + 196, + 506, + 209 + ], + "score": 1.0, + "content": "dihedral (D2) symmetry, and the lambda excision HJ intermediate (HJI), an asymmetric protein", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 86, + 207, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 86, + 210, + 99, + 218 + ], + "score": 1.0, + "content": "170", + "type": "text" + }, + { + "bbox": [ + 105, + 207, + 461, + 220 + ], + "score": 1.0, + "content": "with local cyclic (C1) symmetry. Their deposited PDB atomic models are 5a1a [43] and", + "type": "text" + }, + { + "bbox": [ + 461, + 208, + 484, + 219 + ], + "score": 0.72, + "content": "5 \\mathrm { j } 0 \\mathrm { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 207, + 506, + 220 + ], + "score": 1.0, + "content": "[44],", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 86, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 86, + 221, + 99, + 230 + ], + "score": 1.0, + "content": "171", + "type": "text" + }, + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "respectively. From these atomic models, we generated the density maps in Chimera [45] by fitting the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 229, + 504, + 243 + ], + "spans": [ + { + "bbox": [ + 86, + 232, + 99, + 242 + ], + "score": 1.0, + "content": "172", + "type": "text" + }, + { + "bbox": [ + 106, + 230, + 163, + 243 + ], + "score": 1.0, + "content": "models with a", + "type": "text" + }, + { + "bbox": [ + 164, + 229, + 177, + 241 + ], + "score": 0.39, + "content": "1 \\mathring \\mathrm { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 230, + 256, + 243 + ], + "score": 1.0, + "content": "map for 5a1a and a", + "type": "text" + }, + { + "bbox": [ + 256, + 229, + 282, + 241 + ], + "score": 0.74, + "content": "3 . 6 7 \\mathring \\mathrm { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 230, + 315, + 243 + ], + "score": 1.0, + "content": "map for", + "type": "text" + }, + { + "bbox": [ + 315, + 231, + 337, + 242 + ], + "score": 0.75, + "content": "5 \\mathrm { j } 0 \\mathrm { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 230, + 437, + 243 + ], + "score": 1.0, + "content": "; this gave us a volume of", + "type": "text" + }, + { + "bbox": [ + 438, + 230, + 504, + 241 + ], + "score": 0.91, + "content": "1 1 0 \\times 1 5 5 \\times 1 9 9", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 241, + 341, + 253 + ], + "spans": [ + { + "bbox": [ + 86, + 243, + 100, + 253 + ], + "score": 1.0, + "content": "173", + "type": "text" + }, + { + "bbox": [ + 106, + 241, + 217, + 253 + ], + "score": 1.0, + "content": "voxels for 5a1a and one of", + "type": "text" + }, + { + "bbox": [ + 217, + 242, + 272, + 252 + ], + "score": 0.9, + "content": "6 9 \\times 5 7 \\times 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 241, + 315, + 253 + ], + "score": 1.0, + "content": "voxels for", + "type": "text" + }, + { + "bbox": [ + 316, + 243, + 338, + 253 + ], + "score": 0.75, + "content": "5 \\mathrm { j } 0 \\mathrm { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 241, + 341, + 253 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 87, + 265, + 505, + 333 + ], + "lines": [ + { + "bbox": [ + 87, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 87, + 268, + 99, + 276 + ], + "score": 1.0, + "content": "174", + "type": "text" + }, + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "Protein symmetries. Symmetries are problematic when learning distances: two projections can", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 86, + 278, + 100, + 288 + ], + "score": 1.0, + "content": "175", + "type": "text" + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "be identical while not originating from the same orientation, which breaks an axiom of distance", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 86, + 288, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 86, + 289, + 100, + 299 + ], + "score": 1.0, + "content": "176", + "type": "text" + }, + { + "bbox": [ + 106, + 288, + 505, + 299 + ], + "score": 1.0, + "content": "functions (identity of indiscernibles). Figure 16b illustrates this problem. To capture only one of four", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 86, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 86, + 300, + 99, + 310 + ], + "score": 1.0, + "content": "177", + "type": "text" + }, + { + "bbox": [ + 105, + 298, + 337, + 311 + ], + "score": 1.0, + "content": "identical projections of 5a1a, we restricted directions to", + "type": "text" + }, + { + "bbox": [ + 338, + 298, + 437, + 311 + ], + "score": 0.93, + "content": "( \\theta _ { 2 } , \\theta _ { 1 } ) \\in [ 0 , \\pi [ \\times [ 0 , \\frac { \\pi } { 2 } [", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "(a quarter of the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 311, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 86, + 312, + 100, + 323 + ], + "score": 1.0, + "content": "178", + "type": "text" + }, + { + "bbox": [ + 105, + 311, + 506, + 323 + ], + "score": 1.0, + "content": "sphere, illustrated in Figure 12a) for that protein. This treatment of symmetries is incomplete5 but", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 322, + 239, + 334 + ], + "spans": [ + { + "bbox": [ + 86, + 324, + 100, + 333 + ], + "score": 1.0, + "content": "179", + "type": "text" + }, + { + "bbox": [ + 105, + 322, + 239, + 334 + ], + "score": 1.0, + "content": "sufficient for a proof-of-concept.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 87, + 345, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 86, + 345, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 86, + 348, + 99, + 357 + ], + "score": 1.0, + "content": "180", + "type": "text" + }, + { + "bbox": [ + 105, + 345, + 366, + 359 + ], + "score": 1.0, + "content": "Projections. Using the ASTRA projector [46], we generated", + "type": "text" + }, + { + "bbox": [ + 367, + 346, + 416, + 357 + ], + "score": 0.86, + "content": "P = 5 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 345, + 506, + 359 + ], + "score": 1.0, + "content": "synthetic projections", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 86, + 358, + 99, + 368 + ], + "score": 1.0, + "content": "181", + "type": "text" + }, + { + "bbox": [ + 105, + 356, + 118, + 369 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 357, + 162, + 367 + ], + "score": 0.89, + "content": "2 7 5 \\times 2 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 356, + 263, + 369 + ], + "score": 1.0, + "content": "pixels (downsampled to", + "type": "text" + }, + { + "bbox": [ + 263, + 357, + 308, + 367 + ], + "score": 0.86, + "content": "1 1 6 \\times 1 1 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 356, + 367, + 369 + ], + "score": 1.0, + "content": "for 5a1a and", + "type": "text" + }, + { + "bbox": [ + 367, + 357, + 411, + 367 + ], + "score": 0.88, + "content": "1 1 6 \\times 1 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 356, + 454, + 369 + ], + "score": 1.0, + "content": "pixels for", + "type": "text" + }, + { + "bbox": [ + 455, + 357, + 477, + 368 + ], + "score": 0.73, + "content": "5 { \\dot { \\jmath } } 0 \\mathbf { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 356, + 505, + 369 + ], + "score": 1.0, + "content": ", taken", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 86, + 370, + 100, + 379 + ], + "score": 1.0, + "content": "182", + "type": "text" + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "from uniformly sampled orientations. 6 We then perturbed the measurements with different levels of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 377, + 472, + 392 + ], + "spans": [ + { + "bbox": [ + 86, + 380, + 100, + 390 + ], + "score": 1.0, + "content": "183", + "type": "text" + }, + { + "bbox": [ + 104, + 377, + 472, + 392 + ], + "score": 1.0, + "content": "additive Gaussian noise [47, 48] and off-centering shifts. Figure 11 displays some samples.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 102, + 402, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "Datasets. For each protein, we split the projections into training, validation, and test subsets, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "created disjoint pairs of projections from each (Table 1). The training and validation sets were used to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "train and evaluate the SNN, while the test set was used to evaluate orientation recovery given a trained", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "SNN. Sampling orientations (mostly) uniformly induces a distribution of distances that is skewed", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 446, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 389, + 460 + ], + "score": 1.0, + "content": "towards larger distances (shown in Figure 12b). As this would skew", + "type": "text" + }, + { + "bbox": [ + 389, + 448, + 407, + 459 + ], + "score": 0.89, + "content": "L _ { \\mathrm { D E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 448, + 444, + 460 + ], + "score": 1.0, + "content": "and bias", + "type": "text" + }, + { + "bbox": [ + 445, + 446, + 456, + 461 + ], + "score": 0.89, + "content": "\\widehat { d } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 448, + 505, + 460 + ], + "score": 1.0, + "content": ", we further", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 142, + 472 + ], + "score": 1.0, + "content": "sampled", + "type": "text" + }, + { + "bbox": [ + 142, + 459, + 157, + 470 + ], + "score": 0.88, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "of the training and validation pairs to make the distribution of distances uniform—for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 469, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 118, + 484 + ], + "score": 0.89, + "content": "\\widehat { d } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 472, + 287, + 485 + ], + "score": 1.0, + "content": "to be uniformly accurate over the whole", + "type": "text" + }, + { + "bbox": [ + 288, + 472, + 309, + 484 + ], + "score": 0.91, + "content": "[ 0 , \\pi ]", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "range of distances (see Appendix B for further", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "score": 1.0, + "content": "illustrations). While 1, 650 projections were enough to perfectly reconstruct the density maps (as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 506 + ], + "score": 1.0, + "content": "shown in Figures 9e and 9j), our method is not limited by the number of projections as optimization", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 505, + 387, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 387, + 518 + ], + "score": 1.0, + "content": "is done per batch. Optimization settings are described in Appendix C.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 105, + 530, + 408, + 541 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 410, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 410, + 544 + ], + "score": 1.0, + "content": "3.2 Sensitivity of orientation recovery to errors in distance estimation", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 101, + 550, + 502, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "We first evaluated the feasibility of orientation recovery assuming that the exact distances were known.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 560, + 504, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 504, + 574 + ], + "score": 1.0, + "content": "The method successfully recovers the orientation of every projection in this case (see Appendix D).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 577, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 104, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "To evaluate the robustness of (4), we perturbed the distances prior to recovery with an error sampled", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 339, + 600 + ], + "score": 1.0, + "content": "from a Gaussian distribution with mean 0 and variances", + "type": "text" + }, + { + "bbox": [ + 339, + 588, + 400, + 601 + ], + "score": 0.91, + "content": "\\sigma ^ { \\hat { 2 } } \\in [ 0 . 0 , 0 . 8 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 588, + 505, + 600 + ], + "score": 1.0, + "content": ". Figure 5 shows that the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 600, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 168, + 611 + ], + "score": 1.0, + "content": "recovery error", + "type": "text" + }, + { + "bbox": [ + 168, + 600, + 187, + 610 + ], + "score": 0.88, + "content": "E _ { \\mathrm { O R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 600, + 419, + 611 + ], + "score": 1.0, + "content": "is a monotonic function of the error in distances: from", + "type": "text" + }, + { + "bbox": [ + 419, + 600, + 458, + 610 + ], + "score": 0.91, + "content": "E _ { \\mathrm { O R } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 600, + 506, + 611 + ], + "score": 1.0, + "content": "with exact", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 610, + 330, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 155, + 622 + ], + "score": 1.0, + "content": "distances to", + "type": "text" + }, + { + "bbox": [ + 156, + 611, + 201, + 622 + ], + "score": 0.9, + "content": "E _ { 0 \\mathrm { R } } \\approx 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 610, + 234, + 622 + ], + "score": 1.0, + "content": "radians", + "type": "text" + }, + { + "bbox": [ + 234, + 610, + 272, + 621 + ], + "score": 0.86, + "content": "( \\approx 1 1 . 5 ^ { \\circ } )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 610, + 288, + 622 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 289, + 610, + 326, + 621 + ], + "score": 0.91, + "content": "\\sigma ^ { 2 } = 0 . 8", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 610, + 330, + 622 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "These results demonstrate that the performance of orientation recovery (4) depends on the quality of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 507, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 507, + 650 + ], + "score": 1.0, + "content": "the estimated distances, which advocates for a proper and extensive training of the SNN. Moreover,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 171, + 662 + ], + "score": 1.0, + "content": "we observe that", + "type": "text" + }, + { + "bbox": [ + 172, + 649, + 190, + 660 + ], + "score": 0.89, + "content": "L _ { \\mathrm { O R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 649, + 280, + 662 + ], + "score": 1.0, + "content": "is a reliable proxy for", + "type": "text" + }, + { + "bbox": [ + 280, + 649, + 299, + 660 + ], + "score": 0.89, + "content": "E _ { \\mathrm { O R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 649, + 505, + 662 + ], + "score": 1.0, + "content": ", allowing us to assess recovery performance in the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 488, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 488, + 673 + ], + "score": 1.0, + "content": "absence of ground-truth orientations (i.e., when recovering the orientations of real projections).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 681, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 678, + 456, + 695 + ], + "spans": [ + { + "bbox": [ + 118, + 678, + 456, + 695 + ], + "score": 1.0, + "content": "5The remaining issue is that one of four distances is arbitrarily chosen per pair of projections.", + "type": "text" + } + ] + }, + { + "bbox": [ + 117, + 689, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 117, + 689, + 199, + 704 + ], + "score": 1.0, + "content": "6Orientations used in", + "type": "text" + }, + { + "bbox": [ + 200, + 692, + 217, + 702 + ], + "score": 0.84, + "content": "\\ S 3 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 689, + 273, + 704 + ], + "score": 1.0, + "content": "(Figure 5) and", + "type": "text" + }, + { + "bbox": [ + 274, + 692, + 291, + 702 + ], + "score": 0.84, + "content": "\\ S 3 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 689, + 505, + 704 + ], + "score": 1.0, + "content": "(Figure 7) were actually obtained by uniformly sampling", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 701, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 166, + 713 + ], + "score": 1.0, + "content": "the Euler angles", + "type": "text" + }, + { + "bbox": [ + 167, + 702, + 173, + 711 + ], + "score": 0.61, + "content": "\\pmb { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 701, + 229, + 713 + ], + "score": 1.0, + "content": ", constrained to", + "type": "text" + }, + { + "bbox": [ + 229, + 702, + 371, + 713 + ], + "score": 0.9, + "content": "\\begin{array}{c} \\begin{array} { r } { ( \\theta _ { 3 } , \\theta _ { 2 } , \\theta _ { 1 } ) \\in [ 0 , 2 \\pi [ \\times [ 0 , \\frac { \\pi } { 2 } [ \\times [ 0 , 2 \\pi [ } \\end{array} ] ] \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 701, + 385, + 713 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 385, + 702, + 405, + 712 + ], + "score": 0.8, + "content": "5 { \\dot { \\jmath } } 0 \\mathbf { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 701, + 505, + 713 + ], + "score": 1.0, + "content": ". Our conclusions would be", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 711, + 359, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 359, + 723 + ], + "score": 1.0, + "content": "identical if orientations were uniformly sampled from SO(3) instead.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 87, + 71, + 191, + 85 + ], + "lines": [ + { + "bbox": [ + 84, + 69, + 193, + 88 + ], + "spans": [ + { + "bbox": [ + 84, + 69, + 193, + 88 + ], + "score": 1.0, + "content": "161 3 Experiments", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 87, + 96, + 505, + 151 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 86, + 96, + 506, + 153 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 104, + 165, + 235, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 163, + 236, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 236, + 179 + ], + "score": 1.0, + "content": "3.1 Experimental conditions", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "index", + "bbox": [ + 87, + 185, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 86, + 186, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 86, + 187, + 100, + 197 + ], + "score": 1.0, + "content": "168", + "type": "text" + }, + { + "bbox": [ + 106, + 186, + 366, + 197 + ], + "score": 1.0, + "content": "Density maps. We considered two proteins (Figure 10): the", + "type": "text" + }, + { + "bbox": [ + 366, + 186, + 374, + 197 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 186, + 506, + 197 + ], + "score": 1.0, + "content": "-galactosidase, a protein with a", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 196, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 86, + 198, + 100, + 208 + ], + "score": 1.0, + "content": "169", + "type": "text" + }, + { + "bbox": [ + 105, + 196, + 506, + 209 + ], + "score": 1.0, + "content": "dihedral (D2) symmetry, and the lambda excision HJ intermediate (HJI), an asymmetric protein", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 207, + 506, + 220 + ], + "spans": [ + { + "bbox": [ + 86, + 210, + 99, + 218 + ], + "score": 1.0, + "content": "170", + "type": "text" + }, + { + "bbox": [ + 105, + 207, + 461, + 220 + ], + "score": 1.0, + "content": "with local cyclic (C1) symmetry. Their deposited PDB atomic models are 5a1a [43] and", + "type": "text" + }, + { + "bbox": [ + 461, + 208, + 484, + 219 + ], + "score": 0.72, + "content": "5 \\mathrm { j } 0 \\mathrm { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 207, + 506, + 220 + ], + "score": 1.0, + "content": "[44],", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 86, + 221, + 99, + 230 + ], + "score": 1.0, + "content": "171", + "type": "text" + }, + { + "bbox": [ + 106, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "respectively. From these atomic models, we generated the density maps in Chimera [45] by fitting the", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 229, + 504, + 243 + ], + "spans": [ + { + "bbox": [ + 86, + 232, + 99, + 242 + ], + "score": 1.0, + "content": "172", + "type": "text" + }, + { + "bbox": [ + 106, + 230, + 163, + 243 + ], + "score": 1.0, + "content": "models with a", + "type": "text" + }, + { + "bbox": [ + 164, + 229, + 177, + 241 + ], + "score": 0.39, + "content": "1 \\mathring \\mathrm { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 230, + 256, + 243 + ], + "score": 1.0, + "content": "map for 5a1a and a", + "type": "text" + }, + { + "bbox": [ + 256, + 229, + 282, + 241 + ], + "score": 0.74, + "content": "3 . 6 7 \\mathring \\mathrm { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 230, + 315, + 243 + ], + "score": 1.0, + "content": "map for", + "type": "text" + }, + { + "bbox": [ + 315, + 231, + 337, + 242 + ], + "score": 0.75, + "content": "5 \\mathrm { j } 0 \\mathrm { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 230, + 437, + 243 + ], + "score": 1.0, + "content": "; this gave us a volume of", + "type": "text" + }, + { + "bbox": [ + 438, + 230, + 504, + 241 + ], + "score": 0.91, + "content": "1 1 0 \\times 1 5 5 \\times 1 9 9", + "type": "inline_equation" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 241, + 341, + 253 + ], + "spans": [ + { + "bbox": [ + 86, + 243, + 100, + 253 + ], + "score": 1.0, + "content": "173", + "type": "text" + }, + { + "bbox": [ + 106, + 241, + 217, + 253 + ], + "score": 1.0, + "content": "voxels for 5a1a and one of", + "type": "text" + }, + { + "bbox": [ + 217, + 242, + 272, + 252 + ], + "score": 0.9, + "content": "6 9 \\times 5 7 \\times 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 241, + 315, + 253 + ], + "score": 1.0, + "content": "voxels for", + "type": "text" + }, + { + "bbox": [ + 316, + 243, + 338, + 253 + ], + "score": 0.75, + "content": "5 \\mathrm { j } 0 \\mathrm { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 241, + 341, + 253 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 87, + 268, + 99, + 276 + ], + "score": 1.0, + "content": "174", + "type": "text" + }, + { + "bbox": [ + 105, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "Protein symmetries. Symmetries are problematic when learning distances: two projections can", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 86, + 278, + 100, + 288 + ], + "score": 1.0, + "content": "175", + "type": "text" + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "be identical while not originating from the same orientation, which breaks an axiom of distance", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 288, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 86, + 289, + 100, + 299 + ], + "score": 1.0, + "content": "176", + "type": "text" + }, + { + "bbox": [ + 106, + 288, + 505, + 299 + ], + "score": 1.0, + "content": "functions (identity of indiscernibles). Figure 16b illustrates this problem. To capture only one of four", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 86, + 300, + 99, + 310 + ], + "score": 1.0, + "content": "177", + "type": "text" + }, + { + "bbox": [ + 105, + 298, + 337, + 311 + ], + "score": 1.0, + "content": "identical projections of 5a1a, we restricted directions to", + "type": "text" + }, + { + "bbox": [ + 338, + 298, + 437, + 311 + ], + "score": 0.93, + "content": "( \\theta _ { 2 } , \\theta _ { 1 } ) \\in [ 0 , \\pi [ \\times [ 0 , \\frac { \\pi } { 2 } [", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "(a quarter of the", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 311, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 86, + 312, + 100, + 323 + ], + "score": 1.0, + "content": "178", + "type": "text" + }, + { + "bbox": [ + 105, + 311, + 506, + 323 + ], + "score": 1.0, + "content": "sphere, illustrated in Figure 12a) for that protein. This treatment of symmetries is incomplete5 but", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 322, + 239, + 334 + ], + "spans": [ + { + "bbox": [ + 86, + 324, + 100, + 333 + ], + "score": 1.0, + "content": "179", + "type": "text" + }, + { + "bbox": [ + 105, + 322, + 239, + 334 + ], + "score": 1.0, + "content": "sufficient for a proof-of-concept.", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 345, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 86, + 348, + 99, + 357 + ], + "score": 1.0, + "content": "180", + "type": "text" + }, + { + "bbox": [ + 105, + 345, + 366, + 359 + ], + "score": 1.0, + "content": "Projections. Using the ASTRA projector [46], we generated", + "type": "text" + }, + { + "bbox": [ + 367, + 346, + 416, + 357 + ], + "score": 0.86, + "content": "P = 5 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 345, + 506, + 359 + ], + "score": 1.0, + "content": "synthetic projections", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 86, + 358, + 99, + 368 + ], + "score": 1.0, + "content": "181", + "type": "text" + }, + { + "bbox": [ + 105, + 356, + 118, + 369 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 357, + 162, + 367 + ], + "score": 0.89, + "content": "2 7 5 \\times 2 7 5", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 356, + 263, + 369 + ], + "score": 1.0, + "content": "pixels (downsampled to", + "type": "text" + }, + { + "bbox": [ + 263, + 357, + 308, + 367 + ], + "score": 0.86, + "content": "1 1 6 \\times 1 1 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 356, + 367, + 369 + ], + "score": 1.0, + "content": "for 5a1a and", + "type": "text" + }, + { + "bbox": [ + 367, + 357, + 411, + 367 + ], + "score": 0.88, + "content": "1 1 6 \\times 1 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 356, + 454, + 369 + ], + "score": 1.0, + "content": "pixels for", + "type": "text" + }, + { + "bbox": [ + 455, + 357, + 477, + 368 + ], + "score": 0.73, + "content": "5 { \\dot { \\jmath } } 0 \\mathbf { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 356, + 505, + 369 + ], + "score": 1.0, + "content": ", taken", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 86, + 370, + 100, + 379 + ], + "score": 1.0, + "content": "182", + "type": "text" + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "from uniformly sampled orientations. 6 We then perturbed the measurements with different levels of", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 377, + 472, + 392 + ], + "spans": [ + { + "bbox": [ + 86, + 380, + 100, + 390 + ], + "score": 1.0, + "content": "183", + "type": "text" + }, + { + "bbox": [ + 104, + 377, + 472, + 392 + ], + "score": 1.0, + "content": "additive Gaussian noise [47, 48] and off-centering shifts. Figure 11 displays some samples.", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + } + ], + "index": 8.5, + "bbox_fs": [ + 86, + 186, + 506, + 253 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 265, + 505, + 333 + ], + "lines": [], + "index": 14.5, + "bbox_fs": [ + 86, + 264, + 506, + 334 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 87, + 345, + 505, + 390 + ], + "lines": [], + "index": 19.5, + "bbox_fs": [ + 86, + 345, + 506, + 392 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 102, + 402, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "Datasets. For each protein, we split the projections into training, validation, and test subsets, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "created disjoint pairs of projections from each (Table 1). The training and validation sets were used to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "train and evaluate the SNN, while the test set was used to evaluate orientation recovery given a trained", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "SNN. 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As this would skew", + "type": "text" + }, + { + "bbox": [ + 389, + 448, + 407, + 459 + ], + "score": 0.89, + "content": "L _ { \\mathrm { D E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 448, + 444, + 460 + ], + "score": 1.0, + "content": "and bias", + "type": "text" + }, + { + "bbox": [ + 445, + 446, + 456, + 461 + ], + "score": 0.89, + "content": "\\widehat { d } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 448, + 505, + 460 + ], + "score": 1.0, + "content": ", we further", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 142, + 472 + ], + "score": 1.0, + "content": "sampled", + "type": "text" + }, + { + "bbox": [ + 142, + 459, + 157, + 470 + ], + "score": 0.88, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "of the training and validation pairs to make the distribution of distances uniform—for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 469, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 118, + 484 + ], + "score": 0.89, + "content": "\\widehat { d } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 472, + 287, + 485 + ], + "score": 1.0, + "content": "to be uniformly accurate over the whole", + "type": "text" + }, + { + "bbox": [ + 288, + 472, + 309, + 484 + ], + "score": 0.91, + "content": "[ 0 , \\pi ]", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "range of distances (see Appendix B for further", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "score": 1.0, + "content": "illustrations). 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For", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 94, + 352, + 503, + 367 + ], + "spans": [ + { + "bbox": [ + 94, + 358, + 99, + 365 + ], + "score": 1.0, + "content": "7", + "type": "text" + }, + { + "bbox": [ + 104, + 354, + 345, + 367 + ], + "score": 1.0, + "content": "comparison, we evaluated a baseline, the Euclidean distance", + "type": "text" + }, + { + "bbox": [ + 345, + 352, + 441, + 367 + ], + "score": 0.93, + "content": "\\widehat { d } _ { p } ( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } ) = \\| \\mathbf { p } _ { i } , \\mathbf { p } _ { j } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 354, + 495, + 367 + ], + "score": 1.0, + "content": ", in Appendix", + "type": "text" + }, + { + "bbox": [ + 496, + 354, + 503, + 364 + ], + "score": 0.25, + "content": "\\mathrm { E }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 95, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 95, + 375, + 98, + 380 + ], + "score": 1.0, + "content": "8", + "type": "text" + }, + { + "bbox": [ + 104, + 370, + 246, + 383 + ], + "score": 1.0, + "content": "Figure 6a shows the convergence of", + "type": "text" + }, + { + "bbox": [ + 246, + 371, + 263, + 381 + ], + "score": 0.89, + "content": "L _ { \\mathrm { D E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 370, + 505, + 383 + ], + "score": 1.0, + "content": ", reached in about 50 epochs. Figure 6b shows the relationship", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 93, + 381, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 93, + 387, + 99, + 394 + ], + "score": 1.0, + "content": "09", + "type": "text" + }, + { + "bbox": [ + 104, + 383, + 192, + 396 + ], + "score": 1.0, + "content": "between the distance", + "type": "text" + }, + { + "bbox": [ + 193, + 381, + 204, + 396 + ], + "score": 0.89, + "content": "\\widehat { d } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 383, + 401, + 396 + ], + "score": 1.0, + "content": "estimated from projections and the true distance", + "type": "text" + }, + { + "bbox": [ + 401, + 384, + 412, + 396 + ], + "score": 0.87, + "content": "d _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 383, + 506, + 396 + ], + "score": 1.0, + "content": ". The outliers for 5a1a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 93, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 93, + 399, + 99, + 405 + ], + "score": 1.0, + "content": "10", + "type": "text" + }, + { + "bbox": [ + 104, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "are explained by our incomplete treatment of its symmetry. While our learned distance function", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "is a much better estimator than the Euclidean distance—compare Figure 6b with Figure 16—they", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 94, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 94, + 419, + 98, + 426 + ], + "score": 1.0, + "content": "2", + "type": "text" + }, + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "share one characteristic: both plateau and underestimate the largest distances. We did attenuate", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 369, + 439 + ], + "score": 1.0, + "content": "the phenomenon by sampling training distances uniformly (see", + "type": "text" + }, + { + "bbox": [ + 369, + 428, + 389, + 439 + ], + "score": 0.74, + "content": "\\ S 3 . 1 \\ r ,", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 428, + 505, + 439 + ], + "score": 1.0, + "content": "), and the issue is much less", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 95, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 95, + 443, + 98, + 447 + ], + "score": 1.0, + "content": "4", + "type": "text" + }, + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "severe than with the Euclidean distance. An alternative could be to only rely on smaller distances for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 93, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 93, + 452, + 99, + 459 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "score": 1.0, + "content": "recovery. That would however require the addition of a spreading term in (4) to prevent the recovered", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 93, + 460, + 204, + 473 + ], + "spans": [ + { + "bbox": [ + 93, + 463, + 99, + 470 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 105, + 460, + 204, + 473 + ], + "score": 1.0, + "content": "orientations to collapse.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 90, + 476, + 506, + 510 + ], + "lines": [ + { + "bbox": [ + 88, + 475, + 507, + 489 + ], + "spans": [ + { + "bbox": [ + 88, + 479, + 99, + 487 + ], + "score": 1.0, + "content": "217", + "type": "text" + }, + { + "bbox": [ + 104, + 475, + 507, + 489 + ], + "score": 1.0, + "content": "These results confirm that a SNN is able to estimate differences in orientations from projections alone,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 88, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 88, + 490, + 100, + 498 + ], + "score": 1.0, + "content": "218", + "type": "text" + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "even though much has yet to be gained from improving upon the rather primitive SNN architecture", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 88, + 498, + 502, + 511 + ], + "spans": [ + { + "bbox": [ + 88, + 501, + 100, + 510 + ], + "score": 1.0, + "content": "219", + "type": "text" + }, + { + "bbox": [ + 104, + 498, + 502, + 511 + ], + "score": 1.0, + "content": "we are currently using. The use of additional training data should help further diminish overfitting.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 90, + 537, + 408, + 549 + ], + "lines": [ + { + "bbox": [ + 87, + 537, + 410, + 552 + ], + "spans": [ + { + "bbox": [ + 87, + 537, + 410, + 552 + ], + "score": 1.0, + "content": "220 3.4 Sensitivity of distance learning to perturbations in the projections", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 93, + 563, + 507, + 586 + ], + "lines": [ + { + "bbox": [ + 89, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 89, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "221 We first demonstrated that the learning of distances is insensible to off-centering shifts (Figure 7a),", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 89, + 574, + 408, + 587 + ], + "spans": [ + { + "bbox": [ + 89, + 574, + 381, + 587 + ], + "score": 1.0, + "content": "222 which is expected given that shift invariance is built in our SNN (see", + "type": "text" + }, + { + "bbox": [ + 382, + 575, + 402, + 586 + ], + "score": 0.77, + "content": "\\ S 2 . 2 \\AA ,", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 574, + 408, + 587 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 97, + 591, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 506, + 603 + ], + "score": 1.0, + "content": "As we cannot—or do not yet know how to—build noise invariance in the SNN architecture, we trained", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 602, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 614 + ], + "score": 1.0, + "content": "the SNN on noisy projections and evaluated whether it could learn to treat noise as an irrelevant", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 227, + 624 + ], + "score": 1.0, + "content": "information. Figure 7b shows", + "type": "text" + }, + { + "bbox": [ + 227, + 613, + 277, + 624 + ], + "score": 0.91, + "content": "E _ { \\mathrm { O R } } \\approx 0 . 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 612, + 309, + 624 + ], + "score": 1.0, + "content": "radians", + "type": "text" + }, + { + "bbox": [ + 310, + 613, + 336, + 623 + ], + "score": 0.85, + "content": "( \\approx 9 ^ { \\circ } )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 612, + 454, + 624 + ], + "score": 1.0, + "content": "for noiseless projections and", + "type": "text" + }, + { + "bbox": [ + 454, + 613, + 505, + 623 + ], + "score": 0.89, + "content": "E _ { \\mathrm { O R } } \\approx 0 . 4 2", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 138, + 636 + ], + "score": 1.0, + "content": "radians", + "type": "text" + }, + { + "bbox": [ + 138, + 624, + 169, + 634 + ], + "score": 0.84, + "content": "( \\approx 2 4 ^ { \\circ } )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 623, + 318, + 636 + ], + "score": 1.0, + "content": "for a more realistic noise variance of", + "type": "text" + }, + { + "bbox": [ + 319, + 623, + 354, + 633 + ], + "score": 0.91, + "content": "\\sigma ^ { 2 } = 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 623, + 471, + 636 + ], + "score": 1.0, + "content": "(with signal-to-noise ratio of", + "type": "text" + }, + { + "bbox": [ + 471, + 624, + 500, + 634 + ], + "score": 0.29, + "content": "- 1 2 \\ \\mathrm { d B }", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 623, + 506, + 636 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "Whereas a naive distance function (e.g., an Euclidean distance) would be extremely sensitive to noise,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "the SNN mostly learned to discard it. Moreover, the observed overfitting indicates that more training", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 655, + 363, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 363, + 669 + ], + "score": 1.0, + "content": "data should further decrease the sensitivity of the SNN to noise.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 92, + 672, + 481, + 684 + ], + "lines": [ + { + "bbox": [ + 89, + 671, + 482, + 685 + ], + "spans": [ + { + "bbox": [ + 89, + 671, + 482, + 685 + ], + "score": 1.0, + "content": "230 Note that we did not evaluate sensitivity to the PSF at this stage but expect a similar behavior.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 87, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 86, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 86, + 691, + 99, + 700 + ], + "score": 1.0, + "content": "231", + "type": "text" + }, + { + "bbox": [ + 103, + 687, + 483, + 702 + ], + "score": 1.0, + "content": "Here again (§3.2), we observed that (i) the estimation of more accurate distances (a smaller", + "type": "text" + }, + { + "bbox": [ + 483, + 689, + 502, + 700 + ], + "score": 0.83, + "content": "L _ { \\mathrm { D E } } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 687, + 506, + 702 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 86, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 86, + 702, + 100, + 711 + ], + "score": 1.0, + "content": "232", + "type": "text" + }, + { + "bbox": [ + 105, + 699, + 353, + 712 + ], + "score": 1.0, + "content": "leads to the recovery of more accurate orientations (a smaller", + "type": "text" + }, + { + "bbox": [ + 354, + 700, + 372, + 711 + ], + "score": 0.89, + "content": "L _ { \\mathrm { O R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 699, + 390, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 390, + 700, + 410, + 711 + ], + "score": 0.88, + "content": "E _ { \\mathrm { O R } } \\mathrm { , }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "), and that (ii) an higher", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 87, + 710, + 299, + 724 + ], + "spans": [ + { + "bbox": [ + 87, + 713, + 99, + 722 + ], + "score": 1.0, + "content": "233", + "type": "text" + }, + { + "bbox": [ + 103, + 710, + 161, + 724 + ], + "score": 1.0, + "content": "recovery loss", + "type": "text" + }, + { + "bbox": [ + 162, + 711, + 180, + 722 + ], + "score": 0.88, + "content": "L _ { \\mathrm { O R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 710, + 276, + 724 + ], + "score": 1.0, + "content": "induces an higher error", + "type": "text" + }, + { + "bbox": [ + 276, + 711, + 294, + 722 + ], + "score": 0.9, + "content": "E _ { \\mathrm { O R } }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 710, + 299, + 724 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 108, + 297, + 157 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 79, + 299, + 102 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 79, + 299, + 91 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 173, + 91 + ], + "score": 1.0, + "content": "Table 1: Split of", + "type": "text" + }, + { + "bbox": [ + 173, + 79, + 220, + 91 + ], + "score": 0.48, + "content": "P = 5 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 79, + 299, + 91 + ], + "score": 1.0, + "content": "projections in train-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 235, + 102 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 235, + 102 + ], + "score": 1.0, + "content": "ing, validation, and test subsets.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 108, + 297, + 157 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 108, + 297, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 108, + 297, + 157 + ], + "score": 0.966, + "html": "
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distances", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "index", + "bbox": [ + 95, + 340, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 93, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 93, + 344, + 99, + 351 + ], + "score": 1.0, + "content": "06", + "type": "text" + }, + { + "bbox": [ + 104, + 340, + 469, + 353 + ], + "score": 1.0, + "content": "We evaluated the ability of the SNN to learn to approximate the orientation distance", + "type": "text" + }, + { + "bbox": [ + 469, + 341, + 480, + 353 + ], + "score": 0.87, + "content": "d _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 340, + 506, + 353 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 94, + 352, + 503, + 367 + ], + "spans": [ + { + "bbox": [ + 94, + 358, + 99, + 365 + ], + "score": 1.0, + "content": "7", + "type": "text" + }, + { + "bbox": [ + 104, + 354, + 345, + 367 + ], + "score": 1.0, + "content": "comparison, we evaluated a baseline, the Euclidean distance", + "type": "text" + }, + { + "bbox": [ + 345, + 352, + 441, + 367 + ], + "score": 0.93, + "content": "\\widehat { d } _ { p } ( \\mathbf { p } _ { i } , \\mathbf { p } _ { j } ) = \\| \\mathbf { p } _ { i } , \\mathbf { p } _ { j } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 354, + 495, + 367 + ], + "score": 1.0, + "content": ", in Appendix", + "type": "text" + }, + { + "bbox": [ + 496, + 354, + 503, + 364 + ], + "score": 0.25, + "content": "\\mathrm { E }", + "type": "inline_equation" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 95, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 95, + 375, + 98, + 380 + ], + "score": 1.0, + "content": "8", + "type": "text" + }, + { + "bbox": [ + 104, + 370, + 246, + 383 + ], + "score": 1.0, + "content": "Figure 6a shows the convergence of", + "type": "text" + }, + { + "bbox": [ + 246, + 371, + 263, + 381 + ], + "score": 0.89, + "content": "L _ { \\mathrm { D E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 370, + 505, + 383 + ], + "score": 1.0, + "content": ", reached in about 50 epochs. Figure 6b shows the relationship", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 381, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 93, + 387, + 99, + 394 + ], + "score": 1.0, + "content": "09", + "type": "text" + }, + { + "bbox": [ + 104, + 383, + 192, + 396 + ], + "score": 1.0, + "content": "between the distance", + "type": "text" + }, + { + "bbox": [ + 193, + 381, + 204, + 396 + ], + "score": 0.89, + "content": "\\widehat { d } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 383, + 401, + 396 + ], + "score": 1.0, + "content": "estimated from projections and the true distance", + "type": "text" + }, + { + "bbox": [ + 401, + 384, + 412, + 396 + ], + "score": 0.87, + "content": "d _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 383, + 506, + 396 + ], + "score": 1.0, + "content": ". The outliers for 5a1a", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 93, + 399, + 99, + 405 + ], + "score": 1.0, + "content": "10", + "type": "text" + }, + { + "bbox": [ + 104, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "are explained by our incomplete treatment of its symmetry. While our learned distance function", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 104, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "is a much better estimator than the Euclidean distance—compare Figure 6b with Figure 16—they", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 94, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 94, + 419, + 98, + 426 + ], + "score": 1.0, + "content": "2", + "type": "text" + }, + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "share one characteristic: both plateau and underestimate the largest distances. We did attenuate", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 369, + 439 + ], + "score": 1.0, + "content": "the phenomenon by sampling training distances uniformly (see", + "type": "text" + }, + { + "bbox": [ + 369, + 428, + 389, + 439 + ], + "score": 0.74, + "content": "\\ S 3 . 1 \\ r ,", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 428, + 505, + 439 + ], + "score": 1.0, + "content": "), and the issue is much less", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 95, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 95, + 443, + 98, + 447 + ], + "score": 1.0, + "content": "4", + "type": "text" + }, + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "severe than with the Euclidean distance. An alternative could be to only rely on smaller distances for", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 93, + 452, + 99, + 459 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "score": 1.0, + "content": "recovery. That would however require the addition of a spreading term in (4) to prevent the recovered", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 460, + 204, + 473 + ], + "spans": [ + { + "bbox": [ + 93, + 463, + 99, + 470 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 105, + 460, + 204, + 473 + ], + "score": 1.0, + "content": "orientations to collapse.", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 475, + 507, + 489 + ], + "spans": [ + { + "bbox": [ + 88, + 479, + 99, + 487 + ], + "score": 1.0, + "content": "217", + "type": "text" + }, + { + "bbox": [ + 104, + 475, + 507, + 489 + ], + "score": 1.0, + "content": "These results confirm that a SNN is able to estimate differences in orientations from projections alone,", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 88, + 490, + 100, + 498 + ], + "score": 1.0, + "content": "218", + "type": "text" + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "even though much has yet to be gained from improving upon the rather primitive SNN architecture", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 498, + 502, + 511 + ], + "spans": [ + { + "bbox": [ + 88, + 501, + 100, + 510 + ], + "score": 1.0, + "content": "219", + "type": "text" + }, + { + "bbox": [ + 104, + 498, + 502, + 511 + ], + "score": 1.0, + "content": "we are currently using. The use of additional training data should help further diminish overfitting.", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + } + ], + "index": 30, + "bbox_fs": [ + 93, + 340, + 506, + 473 + ] + }, + { + "type": "index", + "bbox": [ + 90, + 476, + 506, + 510 + ], + "lines": [], + "index": 37, + "bbox_fs": [ + 88, + 475, + 507, + 511 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 90, + 537, + 408, + 549 + ], + "lines": [ + { + "bbox": [ + 87, + 537, + 410, + 552 + ], + "spans": [ + { + "bbox": [ + 87, + 537, + 410, + 552 + ], + "score": 1.0, + "content": "220 3.4 Sensitivity of distance learning to perturbations in the projections", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "index", + "bbox": [ + 93, + 563, + 507, + 586 + ], + "lines": [ + { + "bbox": [ + 89, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 89, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "221 We first demonstrated that the learning of distances is insensible to off-centering shifts (Figure 7a),", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 574, + 408, + 587 + ], + "spans": [ + { + "bbox": [ + 89, + 574, + 381, + 587 + ], + "score": 1.0, + "content": "222 which is expected given that shift invariance is built in our SNN (see", + "type": "text" + }, + { + "bbox": [ + 382, + 575, + 402, + 586 + ], + "score": 0.77, + "content": "\\ S 2 . 2 \\AA ,", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 574, + 408, + 587 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + } + ], + "index": 40.5, + "bbox_fs": [ + 89, + 563, + 506, + 587 + ] + }, + { + "type": "text", + "bbox": [ + 97, + 591, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 506, + 603 + ], + "score": 1.0, + "content": "As we cannot—or do not yet know how to—build noise invariance in the SNN architecture, we trained", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 602, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 614 + ], + "score": 1.0, + "content": "the SNN on noisy projections and evaluated whether it could learn to treat noise as an irrelevant", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 227, + 624 + ], + "score": 1.0, + "content": "information. Figure 7b shows", + "type": "text" + }, + { + "bbox": [ + 227, + 613, + 277, + 624 + ], + "score": 0.91, + "content": "E _ { \\mathrm { O R } } \\approx 0 . 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 612, + 309, + 624 + ], + "score": 1.0, + "content": "radians", + "type": "text" + }, + { + "bbox": [ + 310, + 613, + 336, + 623 + ], + "score": 0.85, + "content": "( \\approx 9 ^ { \\circ } )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 612, + 454, + 624 + ], + "score": 1.0, + "content": "for noiseless projections and", + "type": "text" + }, + { + "bbox": [ + 454, + 613, + 505, + 623 + ], + "score": 0.89, + "content": "E _ { \\mathrm { O R } } \\approx 0 . 4 2", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 138, + 636 + ], + "score": 1.0, + "content": "radians", + "type": "text" + }, + { + "bbox": [ + 138, + 624, + 169, + 634 + ], + "score": 0.84, + "content": "( \\approx 2 4 ^ { \\circ } )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 623, + 318, + 636 + ], + "score": 1.0, + "content": "for a more realistic noise variance of", + "type": "text" + }, + { + "bbox": [ + 319, + 623, + 354, + 633 + ], + "score": 0.91, + "content": "\\sigma ^ { 2 } = 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 623, + 471, + 636 + ], + "score": 1.0, + "content": "(with signal-to-noise ratio of", + "type": "text" + }, + { + "bbox": [ + 471, + 624, + 500, + 634 + ], + "score": 0.29, + "content": "- 1 2 \\ \\mathrm { d B }", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 623, + 506, + 636 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "Whereas a naive distance function (e.g., an Euclidean distance) would be extremely sensitive to noise,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "the SNN mostly learned to discard it. 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It is worth noting that, at this stage of development, we only trained the SNN on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 104, + 441, + 506, + 453 + ], + "score": 1.0, + "content": "projections originating from the protein we were attempting to reconstruct. In addition, reconstruction", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "was performed with a direct reconstruction algorithm (ASTRA’s GPU implementation of the CGLS", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "algorithm) rather than with a robuster iterative method. This is a specific experimental case that only", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 474, + 490, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 475, + 486 + ], + "score": 1.0, + "content": "partially shines light on the applicability of the method in real situations; this is discussed in", + "type": "text" + }, + { + "bbox": [ + 476, + 474, + 487, + 484 + ], + "score": 0.8, + "content": "\\ S 4", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 474, + 490, + 486 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 407, + 506, + 486 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 490, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 503 + ], + "score": 1.0, + "content": "Figure 8a shows the recovery of orientations from distances that were estimated from noiseless", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 501, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 164, + 512 + ], + "score": 1.0, + "content": "projections of", + "type": "text" + }, + { + "bbox": [ + 165, + 501, + 186, + 513 + ], + "score": 0.72, + "content": "5 \\mathrm { j } 0 \\mathrm { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 501, + 258, + 512 + ], + "score": 1.0, + "content": ". A mean error of", + "type": "text" + }, + { + "bbox": [ + 258, + 501, + 308, + 511 + ], + "score": 0.9, + "content": "E _ { 0 \\mathrm { R } } \\approx 0 . 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 501, + 340, + 512 + ], + "score": 1.0, + "content": "radians", + "type": "text" + }, + { + "bbox": [ + 341, + 501, + 372, + 512 + ], + "score": 0.83, + "content": "( \\approx 1 1 ^ { \\circ } )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 501, + 506, + 512 + ], + "score": 1.0, + "content": "in the recovered orientations led", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 267, + 525 + ], + "score": 1.0, + "content": "to a reconstruction with a resolution of", + "type": "text" + }, + { + "bbox": [ + 267, + 512, + 293, + 524 + ], + "score": 0.8, + "content": "1 2 . 2 \\mathring \\mathrm { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "at a Fourier shell coefficient (FSC) of 0.5, shown in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 135, + 537 + ], + "score": 1.0, + "content": "Figure", + "type": "text" + }, + { + "bbox": [ + 135, + 524, + 146, + 534 + ], + "score": 0.32, + "content": "{ 9 \\mathrm { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 523, + 466, + 537 + ], + "score": 1.0, + "content": ". As predicted by our other experiments, corrupting the projections with noise (", + "type": "text" + }, + { + "bbox": [ + 466, + 523, + 502, + 534 + ], + "score": 0.86, + "content": "\\sigma ^ { 2 } = 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 523, + 506, + 537 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "score": 1.0, + "content": "negatively impacts the quality of the recovered orientations (Figure 8b); the obtained mean error", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 545, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 104, + 545, + 137, + 559 + ], + "score": 1.0, + "content": "is then", + "type": "text" + }, + { + "bbox": [ + 137, + 546, + 189, + 557 + ], + "score": 0.89, + "content": "E _ { 0 \\mathrm { R } } \\approx 0 . 2 5", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 545, + 224, + 559 + ], + "score": 1.0, + "content": "radians", + "type": "text" + }, + { + "bbox": [ + 225, + 546, + 255, + 557 + ], + "score": 0.8, + "content": "( \\approx 1 4 ^ { \\circ } )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 545, + 506, + 559 + ], + "score": 1.0, + "content": "). 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The results provide", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "key insights on the viability of the proposed scheme. 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Finally, our method was able to recover orientations with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 259, + 429 + ], + "score": 1.0, + "content": "an error of 0.12 to 0.25 radians (7 to", + "type": "text" + }, + { + "bbox": [ + 260, + 417, + 275, + 427 + ], + "score": 0.81, + "content": "1 4 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 416, + 506, + 429 + ], + "score": 1.0, + "content": ")—leading to an initial volume with a resolution of 8 to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 125, + 439 + ], + "score": 0.39, + "content": "1 5 \\mathring \\mathrm { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "(§3.5). In summary, the more accurate the estimated distances, the more precise the recovered", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 440, + 413, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 413, + 452 + ], + "score": 1.0, + "content": "orientations, and, ultimately, the higher-resolution the reconstructed volume.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 87, + 455, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 85, + 456, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 85, + 457, + 100, + 467 + ], + "score": 1.0, + "content": "270", + "type": "text" + }, + { + "bbox": [ + 106, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "While the method is not yet ready to be deployed in practice, we believe that a series of developments", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 85, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 85, + 468, + 99, + 478 + ], + "score": 1.0, + "content": "271", + "type": "text" + }, + { + "bbox": [ + 106, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "could make it relevant for single-particle cryo-EM reconstruction. 7 As previously discussed, the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 85, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 85, + 479, + 99, + 489 + ], + "score": 1.0, + "content": "272", + "type": "text" + }, + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "results underline the importance of learning an accurate distance estimator. 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The results provide", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "key insights on the viability of the proposed scheme. 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Finally, our method was able to recover orientations with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 259, + 429 + ], + "score": 1.0, + "content": "an error of 0.12 to 0.25 radians (7 to", + "type": "text" + }, + { + "bbox": [ + 260, + 417, + 275, + 427 + ], + "score": 0.81, + "content": "1 4 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 416, + 506, + 429 + ], + "score": 1.0, + "content": ")—leading to an initial volume with a resolution of 8 to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 125, + 439 + ], + "score": 0.39, + "content": "1 5 \\mathring \\mathrm { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "(§3.5). 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In this regard, the", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 86, + 491, + 99, + 499 + ], + "score": 1.0, + "content": "273", + "type": "text" + }, + { + "bbox": [ + 105, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "performance of the SNN could be improved. First, the architecture of the twin convolutional neural", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 498, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 86, + 502, + 99, + 510 + ], + "score": 1.0, + "content": "274", + "type": "text" + }, + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "score": 1.0, + "content": "networks should be expanded and tuned. 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Nie,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 127, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 127, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "T. Phillips, and E. Davydov, “Ad click prediction: a view from the trenches,” in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 126, + 514, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 126, + 514, + 506, + 531 + ], + "score": 1.0, + "content": "Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 126, + 525, + 507, + 541 + ], + "spans": [ + { + "bbox": [ + 126, + 525, + 507, + 541 + ], + "score": 1.0, + "content": "and data mining - KDD ’13. ACM Press, 2013, p. 1222. [Online]. Available:", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 126, + 538, + 348, + 551 + ], + "spans": [ + { + "bbox": [ + 126, + 538, + 267, + 551 + ], + "score": 1.0, + "content": "http://dl.acm.org/citation.cfm?doid", + "type": "text" + }, + { + "bbox": [ + 267, + 539, + 274, + 547 + ], + "score": 0.38, + "content": "\\mathbf { \\Psi } =", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 538, + 348, + 551 + ], + "score": 1.0, + "content": "2487575.2488200", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 93, + 565, + 156, + 578 + ], + "lines": [ + { + "bbox": [ + 92, + 564, + 157, + 580 + ], + "spans": [ + { + "bbox": [ + 92, + 569, + 100, + 577 + ], + "score": 1.0, + "content": "34", + "type": "text" + }, + { + "bbox": [ + 105, + 564, + 157, + 580 + ], + "score": 1.0, + "content": "Checklist", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 131, + 587, + 208, + 598 + ], + "lines": [ + { + "bbox": [ + 129, + 585, + 210, + 600 + ], + "spans": [ + { + "bbox": [ + 129, + 585, + 210, + 600 + ], + "score": 1.0, + "content": "1. For all authors...", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 146, + 602, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 146, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 146, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 162, + 613, + 288, + 625 + ], + "spans": [ + { + "bbox": [ + 162, + 613, + 288, + 625 + ], + "score": 1.0, + "content": "contributions and scope? [Yes]", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 144, + 625, + 408, + 639 + ], + "spans": [ + { + "bbox": [ + 144, + 625, + 392, + 639 + ], + "score": 1.0, + "content": "(b) Did you describe the limitations of your work? [Yes] See", + "type": "text" + }, + { + "bbox": [ + 393, + 626, + 404, + 637 + ], + "score": 0.64, + "content": "\\ S 4", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 625, + 408, + 639 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 145, + 637, + 507, + 653 + ], + "spans": [ + { + "bbox": [ + 145, + 637, + 507, + 653 + ], + "score": 1.0, + "content": "(c) Did you discuss any potential negative societal impacts of your work? [Yes] We didn’t", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 162, + 650, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 162, + 650, + 505, + 662 + ], + "score": 1.0, + "content": "identify any potential risk for improving protein imaging. Moreover, our work only", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 161, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 161, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "addresses a small step in a huge pipeline. We however mentioned the environmental", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 162, + 672, + 315, + 684 + ], + "spans": [ + { + "bbox": [ + 162, + 672, + 298, + 684 + ], + "score": 1.0, + "content": "impact of training our model (see", + "type": "text" + }, + { + "bbox": [ + 298, + 672, + 309, + 683 + ], + "score": 0.7, + "content": "\\ S 4", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 672, + 315, + 684 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 145, + 684, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 145, + 684, + 505, + 698 + ], + "score": 1.0, + "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 161, + 694, + 214, + 708 + ], + "spans": [ + { + "bbox": [ + 161, + 694, + 214, + 708 + ], + "score": 1.0, + "content": "them? 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For all authors...", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40, + "bbox_fs": [ + 129, + 585, + 210, + 600 + ] + }, + { + "type": "list", + "bbox": [ + 146, + 602, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 146, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 146, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 613, + 288, + 625 + ], + "spans": [ + { + "bbox": [ + 162, + 613, + 288, + 625 + ], + "score": 1.0, + "content": "contributions and scope? [Yes]", + "type": "text" + } + ], + "index": 42, + "is_list_end_line": true + }, + { + "bbox": [ + 144, + 625, + 408, + 639 + ], + "spans": [ + { + "bbox": [ + 144, + 625, + 392, + 639 + ], + "score": 1.0, + "content": "(b) Did you describe the limitations of your work? 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[N/A]", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 145, + 85, + 424, + 99 + ], + "spans": [ + { + "bbox": [ + 145, + 85, + 424, + 99 + ], + "score": 1.0, + "content": "(b) Did you include complete proofs of all theoretical results? [N/A]", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 131, + 101, + 241, + 112 + ], + "lines": [ + { + "bbox": [ + 128, + 99, + 243, + 115 + ], + "spans": [ + { + "bbox": [ + 128, + 99, + 243, + 115 + ], + "score": 1.0, + "content": "3. 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[Yes] We include", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 161, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 161, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "a URL in the Abstract to a git repository that includes code, data, and instructions to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 162, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 162, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "reproduce our results. 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DatasetPp2Used pairs
Training2,512 (50%)6,310,14463,101
Validation838 (17%)702,2447,022
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+ "category_id": 15, + "poly": [ + 362.0, + 737.0, + 1407.0, + 737.0, + 1407.0, + 785.0, + 362.0, + 785.0 + ], + "score": 1.0, + "text": "" + } + ], + "page_info": { + "page_no": 12, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/nkap3LV7t7O/nkap3LV7t7O.md b/parse/train/nkap3LV7t7O/nkap3LV7t7O.md new file mode 100644 index 0000000000000000000000000000000000000000..fbfc2a4670ed43688c9aab1b774440ffed1e6580 --- /dev/null +++ b/parse/train/nkap3LV7t7O/nkap3LV7t7O.md @@ -0,0 +1,349 @@ +# SIMPLE AND EFFECTIVE VAE TRAINING WITH CALIBRATED DECODERS + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Variational autoencoders (VAEs) provide an effective and simple method for modeling complex distributions. However, training VAEs often requires considerable hyperparameter tuning to determine the optimal amount of information retained by the latent variable. We study the impact of calibrated decoders, which learn the uncertainty of the decoding distribution and can determine this amount of information automatically, on the VAE performance. While many methods for learning calibrated decoders have been proposed, many of the recent papers that employ VAEs rely on heuristic hyperparameters and ad-hoc modifications instead. We perform the first comprehensive comparative analysis of calibrated decoder and provide recommendations for simple and effective VAE training. Our analysis covers a range of datasets and several single-image and sequential VAE models. We further propose a simple but novel modification to the commonly used Gaussian decoder, which computes the prediction variance analytically. We observe empirically that using heuristic modifications is not necessary with our method. + +# 1 INTRODUCTION + +Deep density models based on the variational autoencoder (VAE) (Kingma & Welling, 2014; Rezende et al., 2014) have found ubiquitous use in probabilistic modeling and representation learning as they are both conceptually simple and are able to scale to very complex distributions and large datasets. These VAE techniques are used for tasks such as future frame prediction (Castrejon et al., 2019), image segmentation (Kohl et al., 2018), generating speech (Chung et al., 2015) and music (Dhariwal et al., 2020), as well as model-based reinforcement learning (Hafner et al., 2019a). However, in practice, many of these approaches require careful manual tuning of the balance between two terms that correspond to distortion and rate from information theory (Alemi et al., 2017). This balance trades off fidelity of reconstruction and quality of samples from the model: a model with low rate would not contain enough information to reconstruct the data, while allowing the model to have high rate might lead to unrealistic samples from the prior as the KL-divergence constraint becomes weaker (Alemi et al., 2017; Higgins et al., 2017). While a proper variational lower bound does not expose any free parameters to control this tradeoff, many prior works heuristically introduce a weight on the prior KL-divergence term, often denoted $\beta$ . Usually, $\beta$ needs to be tuned for every dataset and model variant as a hyperparameter, which slows down development and can lead to poor performance as finding the optimal value is often prohibitively computationally expensive. Moreover, using $\beta \neq 1$ precludes the appealing interpretation of the VAE objective as a bound on the data likelihood, and is undesirable for applications like density modeling. + +While many architectures for calibrating decoders have been proposed in the literature (Kingma & Welling, 2014; Kingma et al., 2016; Dai & Wipf, 2019), more applied work typically employs VAEs with uncalibrated decoding distributions, such as Gaussian distributions without a learned variance, where the decoder only outputs the mean parameter (Castrejon et al., 2019; Denton & Fergus, 2018; Lee et al., 2019; Babaeizadeh et al., 2018; Lee et al., 2018; Hafner et al., 2019b; Pong et al., 2019; Zhu et al., 2017; Pavlakos et al., 2019), or uses other ad-hoc modifications to the objective (Sohn et al., 2015; Henaff et al., 2019). Indeed, it is well known that attempting to learn the variance in a Gaussian decoder may lead to numerical instability (Rezende & Viola, 2018; Dai & Wipf, 2019), and na¨ıve approaches often lead to poor results. As a result, it remains unclear whether practical empirical performance of VAEs actually benefits from calibrated decoders or not. + +To rectify this, our first contribution is a comparative analysis of various calibrated decoder architectures and practical recommendations for simple and effective VAE training. We find that, while na¨ıve calibrated decoders often lead to worse results, a careful choice of the decoder distribution can work very well, and removes the need to tune the additional parameter $\beta$ . Indeed, we note that the entropy of the decoding distribution controls the mutual information $I ( x ; z )$ . Calibrated decoders allow the model to control $I ( x ; z )$ automatically, instead of relying on manual tuning. Our second contribution is a simple but novel technique for optimizing the decoder variance analytically, without requiring the decoder network to produce it as an additional output. We call the resulting approach to learning the Gaussian variance the $\sigma$ -VAE. In our experiments, the $\sigma$ -VAE outperforms the alternative of learning the variance through gradient descent, while being simpler to implement and extend. We validate our results on several VAE and sequence VAE models and a range of image and video datasets. + +# 2 RELATED WORK + +Prior work on variational autoencoders has studied a number of different decoder parameterizations. Kingma & Welling (2014); Rezende et al. (2014) use the Bernoulli distribution for the binary MNIST data and Kingma & Welling (2014) use Gaussian distributions with learned variance parameter for grayscale images. However, modeling images with continuous distributions is prone to instability as the variance can converge to zero (Rezende & Viola, 2018; Mattei & Frellsen, 2018; Dai & Wipf, 2019). Some work has attempted to rectify this problem by using dequantization (Gregor et al., 2016), which is theoretically appealing as it is tightly related to the log-likelihood of the original discrete data (Theis et al., 2016), optimizing the variance in a two-stage procedure (Arvanitidis et al., 2017), or training a post-hoc prior (Ghosh et al., 2019). Takahashi et al. (2018); Barron (2019) proposed more expressive distributions. Additionally, different choices for representing such variance exist, including diagonal covariance (Kingma & Welling, 2014; Sønderby et al., 2016; Rolfe, 2016), or a single shared parameter (Kingma et al., 2016; Dai & Wipf, 2019; Edwards & Storkey, 2016; Rezende & Viola, 2018). We analyze these and notice that learning a single variance parameter shared across images leads to stable training and good performance, without the use of dequantization or even clipping the variance, although these techniques can be used with our decoders; and further improve the estimation of this variance with an analytic solution. + +Early work on discrete VAE decoders for color images modeled them with the Bernoulli distribution, treating the color intensities as probabilities (Gregor et al., 2015). Further work has explored various parameterizations based on discretized continuous distributions, such as discretized logistic (Kingma et al., 2016). More recent work has improved expressivity of the decoder with a mixture of discretized logistics (Chen et al., 2016; Maaløe et al., 2019). However, these models also employ powerful autoregressive decoders (Chen et al., 2016; Gulrajani et al., 2016; Maaløe et al., 2019), and the latent variables in these models may not represent all of the significant factors of variation in the data, as some factors can instead be modeled internally by the autoregressive decoder (Alemi et al., 2017).1 + +While many calibrated decoders have been proposed, outside the core generative modeling community uncalibrated decoders are ubiquitous. They are used in work on video prediction (Denton & Fergus, 2018; Castrejon et al., 2019; Lee et al., 2018; Babaeizadeh et al., 2018), image segmentation (Kohl et al., 2018), image-to-image translation (Zhu et al., 2017), 3D human pose (Pavlakos et al., 2019), as well as model-based reinforcement learning (Henaff et al., 2019; Hafner et al., 2019b;a), and representation learning (Lee et al., 2019; Watter et al., 2015; Pong et al., 2019). Most of these works utilize the heuristic hyperparameter $\beta$ instead, which is undesirable both as the resulting objective is no longer a bound on the likelihood, and as $\beta$ usually requires extensive tuning. In this work, we analyze the common pitfalls of using calibrated decoders that may have prevented the practitioners from using them, propose a simple and effective analytic way of learning such calibrated distribution, and provide a comprehensive experimental evaluation of different decoding distributions. + +Alternative discussions of the hyperparameter $\beta$ are presented by Zhao et al. (2017); Higgins et al. (2017); Alemi et al. (2017); Achille & Soatto (2018), who show that it controls the amount of information in the latent variable, $I ( x ; z )$ . Peng et al. (2018); Rezende & Viola (2018) further discuss constrained optimization objectives for VAEs, which also yield a similar hyperparameter. Here, we focus on $\beta$ -VAEs with Gaussian decoders with constant variance, as commonly used in recent work, and show that the hyperparameter $\beta$ can be incorporated in the decoding likelihood for these models. + +# 3 ANALYSING DECODING DISTRIBUTIONS + +The generative model of a VAE (Kingma & Welling, 2014; Rezende et al., 2014) with parameters $\theta$ is specified with a prior distribution over the latent variable $p _ { \theta } ( z )$ , commonly unit Gaussian, and a decoding distribution $p _ { \theta } ( x | z )$ , which for color images is commonly a conditional Gaussian parameterized with a neural network. We would like to fit this generative model to a given dataset by maximizing the evidence lower bound (ELBO (Neal & Hinton, 1998; Jordan et al., 1999; Kingma $\&$ Welling, 2014; Rezende et al., 2014)), which uses an approximate posterior distribution $q _ { \phi } ( z | x )$ , also commonly a conditional Gaussian specified with a neural network. In this work, we focus on the form of the decoding distribution $p _ { \theta } ( x | z )$ . To achieve the best results, we want a decoding distribution that represents the required probability $p ( x | z )$ accurately In this section, we will review and analyze various choices of decoding distributions that enable better decoder calibration, including expressive decoding distributions that can represent both the prediction of the image and the uncertainty about such prediction, or even multimodal predictions. + +# 3.1 GAUSSIAN DECODERS + +We first analyse the commonly used Gaussian decoders. We note that the commonly used MSE reconstruction loss between the reconstruction $\hat { x }$ and ground truth data $x$ is equivalent to the negative log-likelihood objective with a Gaussian decoding distribution with constant variance: + +$$ +- \ln p ( x | z ) = { \frac { 1 } { 2 } } | | { \hat { x } } - x | | ^ { 2 } + D \ln { \sqrt { 2 \pi } } = { \frac { 1 } { 2 } } | | { \hat { x } } - x | | ^ { 2 } + c = { \frac { D } { 2 } } \mathbf { M } \mathbf { S } \mathbf { E } ( { \hat { x } } , x ) + c , +$$ + +where $p ( x | z ) \sim \mathcal { N } ( \hat { x } , I )$ , the prediction $\hat { x }$ is produced with a neural network $\hat { x } = \mu _ { \theta } ( z )$ , and $D$ is the dimensionality of $x$ . + +This demonstrates a drawback of methods that rely simply on the MSE loss (Castrejon et al., 2019; Denton & Fergus, 2018; Lee et al., 2019; Hafner et al., 2019b; Pong et al., 2019; Zhu et al., 2017; Henaff et al., 2019), as it is equivalent to assuming a particular, constant variance of the Gaussian decoding distribution. By learning this variance, we can achieve much better performance due to better calibration of the decoder. There are several ways in which we can specify this variance. An expressive way to specify the variance is to specify a diagonal covariance matrix for the image, with one value per pixel (Kingma & Welling, 2014; Sønderby et al., 2016; Rolfe, 2016). This can be done, for example, by letting a neural network $\sigma _ { \theta }$ output the diagonal entries of the covariance matrix given a latent sample $z$ : + +$$ +p _ { \theta } ( x | z ) \sim \mathcal { N } \left( \mu _ { \theta } ( z ) , \sigma _ { \theta } ( z ) ^ { 2 } \right) . +$$ + +This parameterization of the decoding distribution outputs one variance value per each pixel and channel. While powerful, we observe in Section 5.3 that this approach attains suboptimal performance, and is moreover prone to numerical instability. Instead, we will find experimentally that a simpler parameterization, in which the covariance matrix is specified with a single shared (Kingma et al., 2016; Dai & Wipf, 2019; Edwards & Storkey, 2016; Rezende & Viola, 2018) parameter $\sigma$ as $\Sigma = \sigma I$ often works better in practice: + +$$ +p _ { \theta , \sigma } ( x | z ) \sim \mathcal { N } \left( \mu _ { \theta } ( z ) , \sigma ^ { 2 } I \right) . +$$ + +The parameter $\sigma$ can be optimized together with parameters of the neural network $\theta$ with gradient descent. Of particular interest is the interpretation of this parameter. Writing out the expression for the decoding likelihood, we obtain + +$$ +- \ln p ( x | z ) = { \frac { 1 } { 2 \sigma ^ { 2 } } } | | { \hat { x } } - x | | ^ { 2 } + D \ln \sigma { \sqrt { 2 \pi } } = { \frac { 1 } { 2 \sigma ^ { 2 } } } | | { \hat { x } } - x | | ^ { 2 } + D \ln \sigma + c = D \ln \sigma + { \frac { D } { 2 \sigma ^ { 2 } } } \mathrm { M S E } ( { \hat { x } } , x ) + c . +$$ + +The full objective of the resulting Gaussian $\sigma$ -VAE is: + +$$ +\mathcal { L } _ { \boldsymbol { \theta } , \boldsymbol { \phi } , \boldsymbol { \sigma } } = D \ln \sigma + \frac { D } { 2 \sigma ^ { 2 } } M S E ( \hat { x } , { x } ) + D _ { K L } ( q ( \boldsymbol { z } | { x } ) | | p ( \boldsymbol { z } ) ) . +$$ + +Note that $\sigma$ may be viewed as a weighting parameter between the MSE reconstruction term and the KL-divergence term in the objective. Moreover, this objective explicitly specifies how to select the optimal variance: the variance should be selected to minimize the (weighted) MSE loss while also minimizing the logarithm of the variance. + +Decoder Calibration It is important that the decoder distribution be calibrated in the statistical sense, that is, the predicted probabilities should correspond to the frequencies of seeing a particular value of $x$ given that prediction (DeGroot & Fienberg, 1983; Dawid, 1982). The calibration of a neural network can be usually improved by estimating the uncertainty of that prediction (Guo et al., 2017), such as the variance of a Gaussian (Kendall & Gal, 2017). Since the naive MSE loss assumes a constant variance, it does not effectively represent the uncertainty of the prediction, and is often poorly calibrated. Instead, learning the variance as in Eq. 3 leads to better uncertainty estimation and better calibration. In Sec 5.1, we show that learning a good estimate of this uncertainty is crucial for the quality of the VAE generations. + +Connection to $\beta$ -VAE. The $\beta$ -VAE objective (Higgins et al., 2017) for a Gaussian decoder with unit variance is: + +$$ +\mathcal { L } ^ { \beta } = \frac { D } { 2 } M S E ( \hat { x } , x ) + \beta D _ { K L } ( q ( z | x ) | | p ( z ) ) . +$$ + +We see that it can be interpreted as a particular case of the objective (3), where the variance is constant and the term $D \ln \sigma$ can be ignored during optimization. The $\beta$ -VAE objective is then equivalent to a $\sigma$ -VAE with a constant variance $\sigma = \sqrt { \beta / 2 }$ (for a particular learning rate setting). In recent work (Zhu et al., 2017; Denton $\&$ Fergus, 2018; Lee et al., 2019), $\beta$ -VAE models are often used in this exact regime. By tuning the $\beta$ term, practitioners are able to tune the variance of the decoder, manually producing a more calibrated decoder. However, by re-interpreting the $\beta$ -VAE objective as a special case of the VAE and introducing the missing $D \ln \sigma$ term, we can both obtain a valid evidence lower bound, and remove the need to manually select $\beta$ . Instead, the variance $\sigma$ can instead simply be learned end-to-end, reducing the need for hyperparameter tuning. + +An alternative discussion of this connection in the context of linear VAEs is also presented by Lucas et al. (2019). While the $\beta$ term is not necessary for good performance if the decoder is calibrated, it can still be employed if desired, such as when the aim is to attain better disentanglement (Higgins et al., 2017) or a particular rate-distortion tradeoff (Alemi et al., 2017). However, we found that with calibrated decoders, the best sample quality is obtained when $\beta = 1$ . + +Loss implementation details. For the correct evidence lower bound computation, it is necessary to add the values of the MSE loss and the KL divergence across the dimensions. We observe that common implementations of these losses (Denton & Fergus, 2018; Abadi et al., 2016; Paszke et al., 2019) use averaging instead, which will lead to poor results if the number of image dimensions is significantly different from the number of the latent dimensions. While this can be conveniently ignored in the $\beta$ -VAE regime, where the balance term is tuned manually anyway, for the $\sigma$ -VAE it is essential to compute the objective value correctly. + +Variance implementation details. Since the variance is non-negative, we parameterize it logarithmically as $\sigma ^ { \hat { 2 } } = e ^ { 2 \lambda }$ , where $\lambda$ is the logarithm of the standard deviation. For some models, such as per-pixel variance decoders, we observed that it is necessary to restrict the variance range for numerical stability. We do so by using the soft clipping operations proposed by Chua et al. (2018): + +$$ +\lambda : = \lambda _ { \mathrm { m a x } } - \mathrm { s o f t p l u s } ( \lambda _ { \mathrm { m a x } } - \lambda ) ; \qquad \lambda : = \lambda _ { \mathrm { m i n } } + \mathrm { s o f t p l u s } ( \lambda - \lambda _ { \mathrm { m i n } } ) . +$$ + +We observe that setting $\lambda _ { \operatorname* { m i n } } = - 6$ to lower bound the standard deviation to be at least half of the distance between allowed color values works well in practice. We also observe that this clipping is unnecessary when learning a shared $\sigma$ value. + +# 3.2 DISCRETE DECODERS + +It is possible to use discrete decoding distributions to generate images, as color values are commonly restricted to a fixed set of integer pixel intensities (e.g. 0..255). Indeed, for discrete color values, discrete distributions are arguably more appropriate. In the most general case, a discrete decoding distribution factorized per each pixel and channel would be specified by a probability mass vector $\hat { x }$ with 256 entries, one per each possible intensity value, similarly to a per-pixel classifier of the intensity value. We can implement it with a soft-max layer, yielding the following log-likelihood loss (sometimes called the cross-entropy loss) for a true pixel with intensity $i$ : + +$$ +- \ln { p ( x | z ) } = - \ln { \frac { \exp ( \hat { x } _ { i } ) } { \sum _ { j } \exp ( \hat { x } _ { j } ) } } , +$$ + +![](images/a72014d3eeef3fb862e1ae0c45aecac5611320d625332b68e1f68646a77844f0.jpg) +Figure 1: Different types of calibrated decoders for Gaussian VAE, model parameters are denoted with enclosing squares. Left: both the mean $\mu$ and the variance $\sigma$ are output by a neural network with parameters $\theta$ . Center: $\sigma$ -VAE with shared variance, the mean is output by a neural network with parameters $\theta$ , but the variance it iself a global parameter. Right: the proposed optimal $\sigma$ -VAE, the mean is output by a neural network with parameters $\theta$ , and the variance is computed analytically from the training data $D$ . + +We will evaluate these and further choices of discrete decoders, described in Appendix D. We recommend choosing the decoder distribution that best suits the structure of the data, such as discrete decoders for discrete data and continuous decoders for continuous data. + +# 4 OPTIMAL VARIANCE ESTIMATION FOR CALIBRATED GAUSSIAN DECODERS + +In this section, we propose a simple but novel analytic way of obtaining a calibrated decoder for continuous distributions that further improves performance. The Gaussian decoders with learned variance described in Section 3.1 are calibrated and work better than na¨ıve unit variance decoders. However, for $\sigma$ -VAE optimized with gradient descent or Adam (Kingma & Ba, 2015), we observe that careful learning rate tuning can yield significantly better performance, which is in line with prior work that reported poor performance of gradient descent for optimizing Gaussian distributions (Amari, 1998; Peters & Schaal, 2008). A smaller learning rate often produces better performance, but slows down the training, as the likelihood values $p ( x | z )$ will be very suboptimal in the beginning. Instead, here we propose an analytic solution for the value of $\sigma$ , which computes it analytically and does not require gradient descent. + +The maximum likelihood estimate of the variance given a known mean is the average squared distance from the mean: + +$$ +\boldsymbol { \sigma } ^ { * } = \mathop { \arg \operatorname* { m a x } } _ { \boldsymbol { \sigma } } \mathcal { N } ( \boldsymbol { x } | \mu , \boldsymbol { \sigma } ^ { 2 } I ) = \mathbf { M } \mathbf { S } \mathbf { E } ( \boldsymbol { x } , \mu ) , +$$ + +where $\begin{array} { r } { \mathbf { M S E } ( x , \mu ) = \frac { 1 } { D } \sum _ { i } ( x _ { i } - \mu _ { i } ) ^ { 2 } } \end{array}$ . Eq. 5 can be easily shown using manual differentiation, and is a generalization of the fact that the MLE estimate of the variance is the sample variance. + +The optimal variance for the decoder distribution under the maximum likelihood criterion is then simply the average MSE loss over the data and the encoder distribution. We leverage this to create an optimal analytic solution for the variance. In the batch setting, the optimal variance would be simply the MSE loss, and can be updated after every gradient update for the other parameters of the decoder. In the mini-batch setting, we use a batchwise estimate of the variance computed for the current minibatch. We analyze these approximations in Appendix C. At test time, a running average of the variance over the training data is used. This method, which we call optimal $\sigma { - } V A E$ , allows us to learn very efficiently as we use the optimal variance estimate at every training step. It is also easier to implement, as no separate optimizer for the variance parameter is needed. If the variance is not needed at test time, it can also be simply discarded after training. + +Per-image optimal $\sigma$ -VAE. Optimal $\sigma$ -VAE uses a single variance value shared across all data points. However, the optimal $\sigma$ -VAE also allows more powerful variance estimates, such as learning a variance value per each pixel, or even a variance value per each image, the difference in implementation simply being the dimensions across which the averaging in Equation 5 operates. This approach can be interpreted as variational variance prediction in the framework of Stirn & Knowles (2020). + +![](images/98cd3e3951801ba5ef1cb57dc92c0876982e83c4ca8bd2fc5f54c56668a9a4aa.jpg) +Figure 2: Images or videos (bottom right) sampled from the proposed optimal $\sigma$ -VAE and a unit variance Gaussian VAE models. The Gaussian VAE does not have a means to control the expressivity of the latent variable and produces suboptimal, blurry samples. The $\sigma$ -VAE controls the expressivity by learning a calibrated decoder, and produces higher quality sequences on all datasets. + +We now provide an empirical analysis of different decoding distributions, and validate the benefits of our $\sigma$ -VAE approach. We use a small convolutional VAE model on SVHN (Netzer et al., 2011), a larger hierarchical HVAE model (Maaløe et al., 2019) on the CelebA (Liu et al., 2015) and CIFAR (Krizhevsky et al., 2009) datasets, and a sequence VAE model called SVG (Denton & Fergus, 2018) on the BAIR Pushing dataset (Finn & Levine, 2017). We evaluate the ELBO values as well as visual quality measured by the Frechet Inception Distance (FID, Heusel ´ et al. (2017)). Images are $2 8 \times 2 8$ for SVHN and $3 2 \times 3 2$ for CelebA and CIFAR, while video experiments were performed on $6 4 \times 6 4$ frames + +following Denton & Fergus (2018). We do not use KL annealing as it did not improve the results in our experiments. Further experimental details are in App. B. + +Table 1: Analysis of learned variance on SVHN. The parameter $\beta$ is tuned manually in $\beta$ -VAE and learned in $\sigma$ -VAE. $\sigma$ -VAE achieves better performance, while the value of $\beta$ (implicitly defined via the decoder variance) automatically converges close the value found by manual tuning. +5.1 DO CALIBRATED DECODERS BALANCE THE VAE OBJECTIVE WITHOUT TUNING $\beta$ ? + +
β-logp↓FID↓
β-VAE0.001<21.4344.54
β-VAE0.01<-318627.93
β-VAE0.1<-122328.3
β-VAE1<138170.39
β-VAE10<4056219.3
g-VAE0.006< -333322.25
+ +As detailed in Section 3.1, a $\beta$ -VAE with a unit variance Gaussian decoder commonly used in prior work is equivalent to a $\sigma$ -VAE with constant, manually tuned variance. There is a simple relationship between beta and the variance: $\sigma = \sqrt { \beta / 2 }$ . To compare the variance that the $\sigma$ -VAE learns to the manually tuned variance in the case of the $\beta$ -VAE, we compare the ELBO values and the corresponding values of $\beta$ in Table 1. We find that learning the variance produces similar values of $\beta$ to the manually tuned values in the $\beta$ -VAE case, indicating that the $\sigma$ -VAE is able to learn the balance between the two objective terms in a single training run, without hyperparameter tuning. Moreover, the $\sigma$ -VAE outperforms the best $\beta$ + +![](images/ef9043d4e0936ecac83fd6dc61954fcd336bdf4307d4474053b1d64186d93d70.jpg) +Figure 3: Analysis of learned variance on SVHN. The parameter $\beta$ is tuned manually in $\beta$ -VAE and learned in $\sigma$ -VAE. Higher values of $\beta$ cause the images to lose detail, while lower values of $\beta$ might make samples unrealistic. The proposed optimal $\sigma$ -VAE is able to learn the balance end-to-end, here converging to an equivalent of $\beta$ -VAE with $\beta = 0 . 0 0 6$ . + +-VAE run. This is because end-to-end learning produces better estimates of the variance than is possible with manual search, improving the likelihood (as measured by the lower bound) and the visual quality. Figure 3 shows the qualitative results from this experiment. + +We further validate our results on both single-image and sequential VAE models on a range of datasets in Table 2 and Figure 2. Single-sample ELBO values are reported, and ELBO values on discretized data are reported for discrete distributions. We see that learning a shared variance in a Gaussian decoders (shared $\sigma$ -VAE) outperforms the na¨ıve unit variance decoder (Gaussian VAE) as well as tuning the $\beta$ constant for the Gaussian VAE manually. We also see that calibrated discrete decoders, such as full categorical distribution or mixture of discretized logistics, perform better than the na¨ıve Gaussian VAE. Using Bernoulli distribution by treating the color intensities as probabilities (Gregor et al., 2015; Watter et al., 2015) performs poorly. Our results further improve upon the sequence VAE method of Denton & Fergus (2018), which uses a unit variance Gaussian with the $\beta$ -VAE objective. + +# 5.2 HOW DOES LEARNING CALIBRATED DECODERS IMPACT THE LATENT VARIABLE INFORMATION CONTENT? + +We saw above that calibrated decoders result in higher log-likelihood bounds. Are calibrated decoders also beneficial for representation learning? We evaluate the mutual information $I _ { e } ( x ; z )$ between the data $p _ { d } ( x )$ and encoder samples $q ( z | x )$ , as well as the mismatch between the prior $p ( z )$ and the marginal encoder distribution $m ( z ) = E _ { p _ { d } ( x ) } q ( z | x )$ , measured by the marginal KL $D _ { K L } ( m ( z ) | | p ( z ) )$ . These terms are related to the rate term of the VAE objective as follows (Alemi et al., 2017): + +$$ +\begin{array} { r l } & { E _ { p _ { d } ( x ) } \left[ D _ { K L } ( q ( z | x ) | | p ( z ) ) \right] = E _ { p _ { d } ( x ) } \left[ D _ { K L } ( q ( z | x ) | | m ( z ) ) \right] + D _ { K L } ( m ( z ) | | p ( z ) ) } \\ & { \qquad = I _ { e } ( x ; z ) + D _ { K L } ( m ( z ) | | p ( z ) ) . } \end{array} +$$ + +That is, the rate term decomposes into the true mutual information and the marginal KL term. We want to learn expressive latent variables with high mutual information. However, doing so by tuning the $\beta$ value relaxes the constraint that the encoder and the prior distributions match, and leads to degraded quality of samples from the prior, which creates a trade-off between expressive representations and ability to generate good samples. To compare the $\beta$ -VAE and $\sigma$ - VAE in terms of these quantities, we estimate the marginal KL term via Monte Carlo sampling, as proposed by Rosca et al. (2018), and plot the results in Figure 4. As expected, we see that lower $\beta$ values lead to higher mutual information. However, after a certain point, lower values of $\beta$ also cause a significant mismatch between the marginal and the prior distributions. By calculating the “effective” $\beta$ for the $\sigma$ -VAE, as per Section 4, we can see that the $\sigma$ -VAE captures an inflection point in the $D _ { K L } ( m ( z ) | | p ( z ) )$ term, learning a representation with the highest possible MI, but without degrading sample quality. This explains the high visual quality of the optimal $\sigma$ -VAE samples: since the marginal and the prior distributions match, the samples from + +![](images/87d2b8a25eab2ea0c060438ee47de05935bf6fbb4219ca406dd0f69d64835ff7.jpg) +Figure 4: Comparison of $\beta$ -VAE and $\sigma$ -VAE on SVHN in terms of mutual information $I _ { e } ( x ; z )$ and marginal KL divergence $K L ( m ( z ) | | p ( z ) )$ (see Sec. 5.2). $I _ { e } ( x ; z )$ increases with lower $\beta$ , yielding expressive representations and better reconstruction. However, after a certain point, lowering $\beta$ leads to a rapid increase in the marginal $\mathrm { K L }$ , yielding poor samples from the prior. The $\sigma$ -VAE is able to automatically find the inflection point after which the marginal KL begins to increase, capturing as much information as possible while still producing good samples. + +the prior look similar to reconstructions, while for a $\beta$ -VAE with low $\beta$ , the samples from the prior are poor. We see that, in contrast to the $\beta$ -VAE, where the mutual information is controlled by a hyperparameter, the $\sigma$ -VAE can adjust the appropriate amount of information automatically and is able to find the setting that produces both informative latents and high quality samples. + +An alternative discussion of tuning $\beta$ is presented by Alemi et al. (2017), who show that $\beta$ controls the rate-distortion trade-off. Here, we show that the crucial trade-off also controlled by $\beta$ is the trade-off between two components of the rate itself, which control expressivity of representations and the match between the variational and the prior distributions, respectively. + +Table 2: Generative modeling performance of the proposed $\sigma$ -VAE on different models and datasets. For SVG, we compare with the original method (Denton & Fergus, 2018), which uses $\beta$ -VAE. We see that uncalibrated decoders such as mean-only Gaussian perform poorly. $\beta$ -VAE allows to calibrate the decoder but needs careful hyperparameter tuning. Calibrated decoders such as categorical or $\sigma$ -VAE perform best. [1] Gregor et al. (2015), [2] Takahashi et al. (2018), [3] Higgins et al. (2017). + +
CelebA HVAESVHN VAECIFAR HVAEBAIR SVG
-logp↓FID↓-logp↓FID↓-logp↓FID↓-logp↓FID↓
Bernoulli VAE[1]177.643.26284.5122.6
Categorical VAE<635971.5<917946.13<7179101.7N/AN/A
Bitwise-categorical VAE<906766.61<1080033.84<939091.2<4874446.13
Logistic mixture VAE<793265.3<908543.19<8443143.1<4061642.94
Gaussian VAE<7173186.5<2184112.5<7186293.7<-1037935.64
Per-pixel g-VAE<-7814159.3<2184114.7<-7222131<-1405141.98
Student-t VAE [2]<-840171.06<-365970.4<-7419123.6■
β-VAE [3]<-271361.6<-318627.93<-331103<-1347234.64
Shared g-VAE<-637460.7<-334922.25<-5435116.1<-1397434.24
Optimal g-VAE<-844660.3< （-333327.25<-5677101.4<-1417334.13
Opt. per-image g-VAE66.0126.28104.033.21
+ +# 5.3 WHAT ARE THE COMMON CHALLENGES IN LEARNING THE VARIANCE THAT PREVENTPRACTITIONERS FROM USING IT, AND HOW TO RECTIFY THEM? + +If learning the decoder variance improves generation, why are learned variances not used more often? In this section, we discuss how the na¨ıve approach to learning variances, where the decoder outputs a variance for each pixel along with the mean, leads to poor results. First, we find that this method often diverges very quickly due to numerical instability, as the network is able to predict certain pixels with very high certainty, leading to degenerate variances. In contrast, learning a shared variance is always numerically stable in our experiments. We can rectify this numerical instability by bounding the output variance (Section 3.1). However, even with bounded variance, we observe that learning per-pixel variances leads to poor results in Table 2. While the per-pixel variance achieves a good ELBO value, it produces very poor samples, as measured by FID and visual inspection. + +We see that the specific form of learned variance: a shared variance, a per-image variance, or a per-pixel variance, can lead to very different performance in practice. We hypothesize the per-pixel decoder performs poorly as it incentivizes the model to focus on particular pixels that can be predicted well, instead of focusing equally on all parts of the image. This is consistent with prior work on denoising diffusion models which noted that likelihood-based models place too much focus on imperceptible details, which leads to deteriorated results (Ho et al., 2020). The shared and per-image variance models mitigate this issue at the cost of introducing more bias, and work better in practice. + +# 5.4 CAN AN ANALYTIC SOLUTION FOR OPTIMAL VARIANCE FURTHER IMPROVE LEARNING? + +We evaluate the optimal $\sigma$ -VAE which uses an analytic solution for the variance (Section 4). Table 2 shows that it achieves superior results in terms of log-likelihood. We also note that the optimal $\sigma$ -VAE converges to a good variance estimate instantaneously, which speeds up learning (highlighted in Figure 9 in the Appendix). In addition, we evaluate the per-image optimal $\sigma$ -VAE, in which a single variance is computed per image. This model achieves significantly higher visual quality. While producing this per-image variance with a neural network would require additional architecture tuning, optimal $\sigma$ -VAE is extremely simple to implement (it can be implemented simply as changing the axes of summation), not requiring any new tunable parameters. + +# 6 CONCLUSION + +We presented a simple and effective method for learning calibrated decoders, as well as an evaluation of different decoding distributions with several VAE and sequential VAE models. The proposed method outperforms methods that use na¨ıve unit variance Gaussian decoders and tune a heuristic weight $\beta$ on the KL-divergence loss, as commonly done in prior work. Moreover, it does not use the heuristic weight $\beta$ , making it easier to train than this prior work. 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In IJCAI, pp. 2696–2702, 2018. + +![](images/2b3ba81e785aa592ac00900595a968588fd2c1d64f9124ae200bbd89fc7e1eeb.jpg) +Figure 5: Samples from the $\sigma$ -VAE (left) and the Gaussian VAE (right) on the SVHN dataset. The Gaussian VAE produces blurry results with muted colors, while the $\sigma$ -VAE is able to produce accurate images of digits. + +Lucas Theis, Aaron van den Oord, and Matthias Bethge. A note on the evaluation of generative ¨ models. ICLR, 2016. + +Manuel Watter, Jost Springenberg, Joschka Boedecker, and Martin Riedmiller. Embed to control: A locally linear latent dynamics model for control from raw images. In Advances in neural information processing systems, pp. 2746–2754, 2015. + +Shengjia Zhao, Jiaming Song, and Stefano Ermon. Infovae: Information maximizing variational autoencoders. arXiv preprint arXiv:1706.02262, 2017. + +Jun-Yan Zhu, Richard Zhang, Deepak Pathak, Trevor Darrell, Alexei A Efros, Oliver Wang, and Eli Shechtman. Toward multimodal image-to-image translation. In Advances in neural information processing systems, pp. 465–476, 2017. + +# A ADDITIONAL EXPERIMENTAL RESULTS + +In this section, we provide more qualitative results in Figures 7, 6, 8, 5 as well as a graph showing the convergence properties of the variance for different models in Fig. 9. In order to validate our method with a different architecture, we also report performance of different decoders with a small 5-layer convolutional architecture on the CelebA and CIFAR dataset in Table 3. We see that the ordering of the methods is consistent with this smaller architecture. + +# B EXPERIMENTAL DETAILS + +For the small convolutional network test on SVHN, the encoder has 3 convolutional layers followed by a fully connected layer, while the decoder has a fully connected layer followed by 3 convolutional layers. The $\beta$ was tuned from 100 to 0.0001 for $\beta$ -VAE. The number of channels in the convolutional layers starts with 32 and increases 2 times in every layer. The dimension of the latent variable is 20. Adam (Kingma & Ba, 2015) with learning rate of 1e-3 is used for optimization. Batch size of 128 was used and all models were trained for 10 epochs. We additionally evaluate this small convolutional network on CelebA, CIFAR, and Frey Face2 datasets in Table 3. Unit Gaussian prior and Gaussian posteriors with diagonal covariance were used. For the larger hierarchical VAE, we used the official pytorch implementation of (Maaløe et al., 2019). We use the baseline hierarchical VAE with 15 layers of latent variables, without the top-down and bottom-up connections. For the hierarchical VAE and the SVG-LP model, we use the default hyperparameters in the respective implementations. We use the standard train-val-test split for all datasets. All models were trained on a single high-end GPU. We use the official PyTorch implementation of the Inception network to compute FID. All methods are compared on the same hyperparameters. + +![](images/d38593657ca6b60d33ed0b9cdf75405e10302b180aabe801f09b58dd79319736.jpg) +Figure 6: Samples from the $\sigma$ -VAE (left) and the Gaussian VAE (right) on the CelebA dataset, images cropped to the face for clarity. The Gaussian VAE produces blurry results with indistinct face features, while the $\sigma$ -VAE is able to produce accurate images of faces. + +Table 3: Generative modeling performance of the proposed $\sigma$ -VAE on CelebA, CIFAR, and Frey Face with a smaller model. We see that uncalibrated decoders such as mean-only Gaussian perform poorly. $\beta$ -VAE allows to calibrate the decoder but needs careful hyperparameter tuning. Calibrated decoders such as categorical or $\sigma$ -VAE perform best. + +
CelebA VAECIFAR VAEFrey Face VAE
-logp↓FID↓-logp↓FID↓-logp↓FID↓
Bernoulli VAE Gregor et al. (2015)102.7165.147.7
Categorical VAE;1019550.45;10673124.1<245450.16
bitwise-categorical VAE1101956.361160499.65<317366.77
Logistic mixture VAE;1015461.81;10648100.2<256250.28
Gaussian VAE<2201144.8<1409205.8<726.480.17
β-VAE Higgins et al. (2017)<-194258.73<-1318117.9< -420.037.61
Shared g-VAE (Ours)<-193973.27< （-1830137.8< -49.7842.86
Optimal g-VAE (Ours)<-195161.27< （-183280.9< -162253.36
Opt. per-image g-VAE (Ours)53.1389.8856.07
+ +![](images/ca6fc2105698b7956e8b56b977c9c8da04a3379e8a83c6b66b469bc5cb64f51b.jpg) +Figure 7: Samples from the $\sigma$ -VAE (top) and the Gaussian VAE (bottom) on the BAIR dataset. Sampled sequences conditioned on two initial frames are shown, and the ground truth sequence is shown at the top. The Gaussian VAE produces blurry robot arm texture and the arm often disappears towards the end of the sequence, while the $\sigma$ -VAE is able to produce sequences with realistic motion and model the details of the arm texture, such as the gripper. + +![](images/37438c302fad1c82eb34bd451e5ebc5793a6d0778bfc657860781536da22714b.jpg) +Figure 8: Samples from the $\sigma$ -VAE (left) and the Gaussian VAE (right) on the challenging CIFAR dataset. The Gaussian VAE produces blurry results with muted colors, while the $\sigma$ -VAE models the distribution of shapes in the CIFAR data more faithfully. + +![](images/8469db8ea7a9237e00486921f5ccc4e7297db5670f610fba5908ade727026f74.jpg) +Figure 9: Variance convergence speed on SVHN. We see that the shared $\sigma$ -VAE which optimizes the variance with gradient descent has an initial period of convergence when the variance converges to the region of the optimal value. In contrast, $\sigma$ -VAE with analytical (optimal) variance quickly learns a good estimate of the variance, which leads to better performance. The unit variance Gaussian $\beta$ -VAE can be interpreted as having a constant variance determined by $\beta$ , shown here. Since the variance doesn’t change throughout training, it achieves suboptimal performance. + +# C EMPIRICAL ANALYZIS OF APPROXIMATIONS FOR OPTIMAL $\sigma$ -VAE + +The optimal $\sigma$ -VAE requires computing the following estimate of the variance + +$$ +\sigma ^ { * } = \underset { \sigma } { \arg \operatorname* { m a x } } \mathbb { E } _ { x \sim \mathrm { D a t a } } \mathbb { E } _ { q ( z | x ) } \left[ \ln p ( x | \mu _ { \theta } ( z ) , \sigma ^ { 2 } I ) \right] = \mathbb { E } _ { x \sim \mathrm { D a t a } } \mathbb { E } _ { q ( z | x ) } { \mathbf { M S E } } ( x , \mu _ { \theta } ( z ) ) . +$$ + +This requires computing two expectations, with respect to the data in the dataset, and with respect to the encoder distribution. We use MC sampling with one sample per data point to approximate both expectations. Inspired by common practices in VAEs, we use one sample per data point to approximate the inner expectation. On SVHN, the standard error of this approximation is $0 . 2 6 \%$ of the value of sigma. We further approximate the outer expectation with a single batch instead of the entire dataset. On SVHN, the standard error of this approximation is $2 \%$ of the value of sigma. We see that both approximations are accurate in practice. The second approximation yields a biased estimate of the evidence lower bound because the same batch is used to approximate the variance and compute the lower bound estimate. However, this bias can be corrected by using a different batch, or with a running average of the variance with an appropriate decay. This running average can also be used to reduce the variance of the estimate and to achieve convergence guarantees, but we did not find it necessary in our experiments. + +# D ALTERNATIVE DECODER CHOICES + +We describe the alternative decoders evaluated in Table 2: using the bitwise-categorical, and the logistic mixture distributions. + +Bitwise-categorical VAE While the 256-way categorical decoder described in Section 3.2 is very powerful due to the ability to specify any possible intensity distribution, it suffers from high computational and memory requirements. Because 256 values need to be kept for each pixel and channel, simply keeping this distribution in memory for one 3-channel $1 0 2 4 \times 1 0 2 4$ image would require 3 GiB of memory, compared to 0.012 GiB for the Gaussian decoder. Therefore, training deep + +Table 4: ELBO on discretized data. All distributions except categorical have scalar scale parameters. The $\sigma$ -VAE performs well on the discretized ELBO metric, performing similarly to a discrete distribution parametrized as a discretized Gaussian or discretized Logistic. Full categorical distribution attains highest likelihood due to having the most statistical power. + +
CIFAR VAE
- log pdf↓-logp↓FID↓
Categorical VAE<10673137.6
Gaussian VAE<740.5<15131212.7
Gaussianσ-VAE<-896.1<11120136.7
Disc.Gaussian g-VAE<11117136.9
Disc.Logistic g-VAE<11103136.7
+ +neural networks with this full categorical distribution is impractical for high-resolution images or videos. The bitwise-categorical VAE improves the memory complexity by defining the distribution over 256 values in a more compact way. Specifically, it defines a binary distribution over each bit in the pixel intensity value, requiring 8 values in total, one for each bit. This distribution can be thought of as a classifier that predicts the value of each bit in the image separately. In our implementation of the bitwise-categorical likelihood, we convert the image channels to binary format and use the standard binary cross-entropy loss (which reduces to binary log-likelihood since all bits in the image are deterministically either zero or one). While in our experiments the bitwise-categorical distribution did not outperform other choices, it often performs on par with our proposed method. We expect this distribution to be useful due to its generality as it is able to represent values stored in any digital format by converting them into binary. + +Logistic mixture VAE For this decoder, we adapt the discretized logistic mixture from Salimans et al. (2017). To define a discrete 256-way distribution, it divides the corresponding continuous distribution into 256 bins, where the probability mass is defined as the integral of the PDF over the corresponding bin. (Kingma et al., 2016) uses the logistic distribution discretized in this manner for the decoder. Salimans et al. (2017) suggests to make all bins except the first and the last be of equal size, whereas the first and the last bin include, respectively, the intervals $( - \infty , 0 ]$ and $\lbrack 1 , \infty )$ . Salimans et al. (2017) further suggests using a mixture of discretized logistics for improved capacity. Our implementation largely follows the one in Salimans et al. (2017), however, we note that the original implementation is not suitable for learning latent variable models, as it generates the channels autoregressively. This will cause the latent variable to lose color information since it can be represented by the autoregressive decoder. We therefore adapt the mixture of discretized logistics to the pure latent variable setup by removing the mean-adjusting coefficients from (Salimans et al., 2017). In our experiments, the logistic mixture outperformed other discrete distributions. \ No newline at end of file diff --git a/parse/train/nkap3LV7t7O/nkap3LV7t7O_content_list.json b/parse/train/nkap3LV7t7O/nkap3LV7t7O_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..91901c0475f5eb6b9b5abd5a8031a89a117242e5 --- /dev/null +++ b/parse/train/nkap3LV7t7O/nkap3LV7t7O_content_list.json @@ -0,0 +1,1847 @@ +[ + { + "type": "text", + "text": "SIMPLE AND EFFECTIVE VAE TRAINING WITH CALIBRATED DECODERS ", + "text_level": 1, + "bbox": [ + 174, + 98, + 665, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Variational autoencoders (VAEs) provide an effective and simple method for modeling complex distributions. However, training VAEs often requires considerable hyperparameter tuning to determine the optimal amount of information retained by the latent variable. We study the impact of calibrated decoders, which learn the uncertainty of the decoding distribution and can determine this amount of information automatically, on the VAE performance. While many methods for learning calibrated decoders have been proposed, many of the recent papers that employ VAEs rely on heuristic hyperparameters and ad-hoc modifications instead. We perform the first comprehensive comparative analysis of calibrated decoder and provide recommendations for simple and effective VAE training. Our analysis covers a range of datasets and several single-image and sequential VAE models. We further propose a simple but novel modification to the commonly used Gaussian decoder, which computes the prediction variance analytically. We observe empirically that using heuristic modifications is not necessary with our method. ", + "bbox": [ + 232, + 268, + 766, + 463 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 494, + 334, + 511 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep density models based on the variational autoencoder (VAE) (Kingma & Welling, 2014; Rezende et al., 2014) have found ubiquitous use in probabilistic modeling and representation learning as they are both conceptually simple and are able to scale to very complex distributions and large datasets. These VAE techniques are used for tasks such as future frame prediction (Castrejon et al., 2019), image segmentation (Kohl et al., 2018), generating speech (Chung et al., 2015) and music (Dhariwal et al., 2020), as well as model-based reinforcement learning (Hafner et al., 2019a). However, in practice, many of these approaches require careful manual tuning of the balance between two terms that correspond to distortion and rate from information theory (Alemi et al., 2017). This balance trades off fidelity of reconstruction and quality of samples from the model: a model with low rate would not contain enough information to reconstruct the data, while allowing the model to have high rate might lead to unrealistic samples from the prior as the KL-divergence constraint becomes weaker (Alemi et al., 2017; Higgins et al., 2017). While a proper variational lower bound does not expose any free parameters to control this tradeoff, many prior works heuristically introduce a weight on the prior KL-divergence term, often denoted $\\beta$ . Usually, $\\beta$ needs to be tuned for every dataset and model variant as a hyperparameter, which slows down development and can lead to poor performance as finding the optimal value is often prohibitively computationally expensive. Moreover, using $\\beta \\neq 1$ precludes the appealing interpretation of the VAE objective as a bound on the data likelihood, and is undesirable for applications like density modeling. ", + "bbox": [ + 174, + 529, + 825, + 777 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "While many architectures for calibrating decoders have been proposed in the literature (Kingma & Welling, 2014; Kingma et al., 2016; Dai & Wipf, 2019), more applied work typically employs VAEs with uncalibrated decoding distributions, such as Gaussian distributions without a learned variance, where the decoder only outputs the mean parameter (Castrejon et al., 2019; Denton & Fergus, 2018; Lee et al., 2019; Babaeizadeh et al., 2018; Lee et al., 2018; Hafner et al., 2019b; Pong et al., 2019; Zhu et al., 2017; Pavlakos et al., 2019), or uses other ad-hoc modifications to the objective (Sohn et al., 2015; Henaff et al., 2019). Indeed, it is well known that attempting to learn the variance in a Gaussian decoder may lead to numerical instability (Rezende & Viola, 2018; Dai & Wipf, 2019), and na¨ıve approaches often lead to poor results. As a result, it remains unclear whether practical empirical performance of VAEs actually benefits from calibrated decoders or not. ", + "bbox": [ + 174, + 785, + 825, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To rectify this, our first contribution is a comparative analysis of various calibrated decoder architectures and practical recommendations for simple and effective VAE training. We find that, while na¨ıve calibrated decoders often lead to worse results, a careful choice of the decoder distribution can work very well, and removes the need to tune the additional parameter $\\beta$ . Indeed, we note that the entropy of the decoding distribution controls the mutual information $I ( x ; z )$ . Calibrated decoders allow the model to control $I ( x ; z )$ automatically, instead of relying on manual tuning. Our second contribution is a simple but novel technique for optimizing the decoder variance analytically, without requiring the decoder network to produce it as an additional output. We call the resulting approach to learning the Gaussian variance the $\\sigma$ -VAE. In our experiments, the $\\sigma$ -VAE outperforms the alternative of learning the variance through gradient descent, while being simpler to implement and extend. We validate our results on several VAE and sequence VAE models and a range of image and video datasets. ", + "bbox": [ + 173, + 103, + 825, + 256 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 268, + 344, + 285 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Prior work on variational autoencoders has studied a number of different decoder parameterizations. Kingma & Welling (2014); Rezende et al. (2014) use the Bernoulli distribution for the binary MNIST data and Kingma & Welling (2014) use Gaussian distributions with learned variance parameter for grayscale images. However, modeling images with continuous distributions is prone to instability as the variance can converge to zero (Rezende & Viola, 2018; Mattei & Frellsen, 2018; Dai & Wipf, 2019). Some work has attempted to rectify this problem by using dequantization (Gregor et al., 2016), which is theoretically appealing as it is tightly related to the log-likelihood of the original discrete data (Theis et al., 2016), optimizing the variance in a two-stage procedure (Arvanitidis et al., 2017), or training a post-hoc prior (Ghosh et al., 2019). Takahashi et al. (2018); Barron (2019) proposed more expressive distributions. Additionally, different choices for representing such variance exist, including diagonal covariance (Kingma & Welling, 2014; Sønderby et al., 2016; Rolfe, 2016), or a single shared parameter (Kingma et al., 2016; Dai & Wipf, 2019; Edwards & Storkey, 2016; Rezende & Viola, 2018). We analyze these and notice that learning a single variance parameter shared across images leads to stable training and good performance, without the use of dequantization or even clipping the variance, although these techniques can be used with our decoders; and further improve the estimation of this variance with an analytic solution. ", + "bbox": [ + 173, + 297, + 826, + 520 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Early work on discrete VAE decoders for color images modeled them with the Bernoulli distribution, treating the color intensities as probabilities (Gregor et al., 2015). Further work has explored various parameterizations based on discretized continuous distributions, such as discretized logistic (Kingma et al., 2016). More recent work has improved expressivity of the decoder with a mixture of discretized logistics (Chen et al., 2016; Maaløe et al., 2019). However, these models also employ powerful autoregressive decoders (Chen et al., 2016; Gulrajani et al., 2016; Maaløe et al., 2019), and the latent variables in these models may not represent all of the significant factors of variation in the data, as some factors can instead be modeled internally by the autoregressive decoder (Alemi et al., 2017).1 ", + "bbox": [ + 174, + 526, + 825, + 638 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While many calibrated decoders have been proposed, outside the core generative modeling community uncalibrated decoders are ubiquitous. They are used in work on video prediction (Denton & Fergus, 2018; Castrejon et al., 2019; Lee et al., 2018; Babaeizadeh et al., 2018), image segmentation (Kohl et al., 2018), image-to-image translation (Zhu et al., 2017), 3D human pose (Pavlakos et al., 2019), as well as model-based reinforcement learning (Henaff et al., 2019; Hafner et al., 2019b;a), and representation learning (Lee et al., 2019; Watter et al., 2015; Pong et al., 2019). Most of these works utilize the heuristic hyperparameter $\\beta$ instead, which is undesirable both as the resulting objective is no longer a bound on the likelihood, and as $\\beta$ usually requires extensive tuning. In this work, we analyze the common pitfalls of using calibrated decoders that may have prevented the practitioners from using them, propose a simple and effective analytic way of learning such calibrated distribution, and provide a comprehensive experimental evaluation of different decoding distributions. ", + "bbox": [ + 174, + 645, + 825, + 797 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Alternative discussions of the hyperparameter $\\beta$ are presented by Zhao et al. (2017); Higgins et al. (2017); Alemi et al. (2017); Achille & Soatto (2018), who show that it controls the amount of information in the latent variable, $I ( x ; z )$ . Peng et al. (2018); Rezende & Viola (2018) further discuss constrained optimization objectives for VAEs, which also yield a similar hyperparameter. Here, we focus on $\\beta$ -VAEs with Gaussian decoders with constant variance, as commonly used in recent work, and show that the hyperparameter $\\beta$ can be incorporated in the decoding likelihood for these models. ", + "bbox": [ + 174, + 805, + 825, + 888 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 ANALYSING DECODING DISTRIBUTIONS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 542, + 118 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The generative model of a VAE (Kingma & Welling, 2014; Rezende et al., 2014) with parameters $\\theta$ is specified with a prior distribution over the latent variable $p _ { \\theta } ( z )$ , commonly unit Gaussian, and a decoding distribution $p _ { \\theta } ( x | z )$ , which for color images is commonly a conditional Gaussian parameterized with a neural network. We would like to fit this generative model to a given dataset by maximizing the evidence lower bound (ELBO (Neal & Hinton, 1998; Jordan et al., 1999; Kingma $\\&$ Welling, 2014; Rezende et al., 2014)), which uses an approximate posterior distribution $q _ { \\phi } ( z | x )$ , also commonly a conditional Gaussian specified with a neural network. In this work, we focus on the form of the decoding distribution $p _ { \\theta } ( x | z )$ . To achieve the best results, we want a decoding distribution that represents the required probability $p ( x | z )$ accurately In this section, we will review and analyze various choices of decoding distributions that enable better decoder calibration, including expressive decoding distributions that can represent both the prediction of the image and the uncertainty about such prediction, or even multimodal predictions. ", + "bbox": [ + 173, + 132, + 825, + 300 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 GAUSSIAN DECODERS ", + "text_level": 1, + "bbox": [ + 174, + 316, + 370, + 332 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We first analyse the commonly used Gaussian decoders. We note that the commonly used MSE reconstruction loss between the reconstruction $\\hat { x }$ and ground truth data $x$ is equivalent to the negative log-likelihood objective with a Gaussian decoding distribution with constant variance: ", + "bbox": [ + 174, + 343, + 825, + 386 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c236e8704d065078d3942c790a28d80b4b1fa686a12a6225196ec3965226d373.jpg", + "text": "$$\n- \\ln p ( x | z ) = { \\frac { 1 } { 2 } } | | { \\hat { x } } - x | | ^ { 2 } + D \\ln { \\sqrt { 2 \\pi } } = { \\frac { 1 } { 2 } } | | { \\hat { x } } - x | | ^ { 2 } + c = { \\frac { D } { 2 } } \\mathbf { M } \\mathbf { S } \\mathbf { E } ( { \\hat { x } } , x ) + c ,\n$$", + "text_format": "latex", + "bbox": [ + 232, + 391, + 763, + 421 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $p ( x | z ) \\sim \\mathcal { N } ( \\hat { x } , I )$ , the prediction $\\hat { x }$ is produced with a neural network $\\hat { x } = \\mu _ { \\theta } ( z )$ , and $D$ is the dimensionality of $x$ . ", + "bbox": [ + 171, + 428, + 823, + 457 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This demonstrates a drawback of methods that rely simply on the MSE loss (Castrejon et al., 2019; Denton & Fergus, 2018; Lee et al., 2019; Hafner et al., 2019b; Pong et al., 2019; Zhu et al., 2017; Henaff et al., 2019), as it is equivalent to assuming a particular, constant variance of the Gaussian decoding distribution. By learning this variance, we can achieve much better performance due to better calibration of the decoder. There are several ways in which we can specify this variance. An expressive way to specify the variance is to specify a diagonal covariance matrix for the image, with one value per pixel (Kingma & Welling, 2014; Sønderby et al., 2016; Rolfe, 2016). This can be done, for example, by letting a neural network $\\sigma _ { \\theta }$ output the diagonal entries of the covariance matrix given a latent sample $z$ : ", + "bbox": [ + 173, + 462, + 826, + 588 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/4d794c3401c07a472866baac752c1485e32fde12d60d0ce7075bfe703c077245.jpg", + "text": "$$\np _ { \\theta } ( x | z ) \\sim \\mathcal { N } \\left( \\mu _ { \\theta } ( z ) , \\sigma _ { \\theta } ( z ) ^ { 2 } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 395, + 585, + 602, + 604 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This parameterization of the decoding distribution outputs one variance value per each pixel and channel. While powerful, we observe in Section 5.3 that this approach attains suboptimal performance, and is moreover prone to numerical instability. Instead, we will find experimentally that a simpler parameterization, in which the covariance matrix is specified with a single shared (Kingma et al., 2016; Dai & Wipf, 2019; Edwards & Storkey, 2016; Rezende & Viola, 2018) parameter $\\sigma$ as $\\Sigma = \\sigma I$ often works better in practice: ", + "bbox": [ + 173, + 606, + 826, + 690 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/10ece0a186fda45ce7ab58d2efe0f07328f9eccc8c58e98e0fdec58a2f0b2351.jpg", + "text": "$$\np _ { \\theta , \\sigma } ( x | z ) \\sim \\mathcal { N } \\left( \\mu _ { \\theta } ( z ) , \\sigma ^ { 2 } I \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 398, + 695, + 599, + 715 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The parameter $\\sigma$ can be optimized together with parameters of the neural network $\\theta$ with gradient descent. Of particular interest is the interpretation of this parameter. Writing out the expression for the decoding likelihood, we obtain ", + "bbox": [ + 174, + 719, + 826, + 762 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/e9ce1b76940634dda70fb64b4f302a863d2642241f14f5fc232a274043855271.jpg", + "text": "$$\n- \\ln p ( x | z ) = { \\frac { 1 } { 2 \\sigma ^ { 2 } } } | | { \\hat { x } } - x | | ^ { 2 } + D \\ln \\sigma { \\sqrt { 2 \\pi } } = { \\frac { 1 } { 2 \\sigma ^ { 2 } } } | | { \\hat { x } } - x | | ^ { 2 } + D \\ln \\sigma + c = D \\ln \\sigma + { \\frac { D } { 2 \\sigma ^ { 2 } } } \\mathrm { M S E } ( { \\hat { x } } , x ) + c .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 768, + 833, + 799 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The full objective of the resulting Gaussian $\\sigma$ -VAE is: ", + "bbox": [ + 174, + 810, + 529, + 827 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1ff481ef555b6fcd22ba8bd4652b3a15d00fec2ca56c1d2e1aa2d3a5b9ba17c2.jpg", + "text": "$$\n\\mathcal { L } _ { \\boldsymbol { \\theta } , \\boldsymbol { \\phi } , \\boldsymbol { \\sigma } } = D \\ln \\sigma + \\frac { D } { 2 \\sigma ^ { 2 } } M S E ( \\hat { x } , { x } ) + D _ { K L } ( q ( \\boldsymbol { z } | { x } ) | | p ( \\boldsymbol { z } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 302, + 832, + 696, + 863 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Note that $\\sigma$ may be viewed as a weighting parameter between the MSE reconstruction term and the KL-divergence term in the objective. Moreover, this objective explicitly specifies how to select the optimal variance: the variance should be selected to minimize the (weighted) MSE loss while also minimizing the logarithm of the variance. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Decoder Calibration It is important that the decoder distribution be calibrated in the statistical sense, that is, the predicted probabilities should correspond to the frequencies of seeing a particular value of $x$ given that prediction (DeGroot & Fienberg, 1983; Dawid, 1982). The calibration of a neural network can be usually improved by estimating the uncertainty of that prediction (Guo et al., 2017), such as the variance of a Gaussian (Kendall & Gal, 2017). Since the naive MSE loss assumes a constant variance, it does not effectively represent the uncertainty of the prediction, and is often poorly calibrated. Instead, learning the variance as in Eq. 3 leads to better uncertainty estimation and better calibration. In Sec 5.1, we show that learning a good estimate of this uncertainty is crucial for the quality of the VAE generations. ", + "bbox": [ + 173, + 103, + 826, + 229 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Connection to $\\beta$ -VAE. The $\\beta$ -VAE objective (Higgins et al., 2017) for a Gaussian decoder with unit variance is: ", + "bbox": [ + 173, + 242, + 823, + 270 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e93051f671722b89c544495fecdd8cb58bb777c81c0a004270b3c3f43e353324.jpg", + "text": "$$\n\\mathcal { L } ^ { \\beta } = \\frac { D } { 2 } M S E ( \\hat { x } , x ) + \\beta D _ { K L } ( q ( z | x ) | | p ( z ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 267, + 651, + 297 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We see that it can be interpreted as a particular case of the objective (3), where the variance is constant and the term $D \\ln \\sigma$ can be ignored during optimization. The $\\beta$ -VAE objective is then equivalent to a $\\sigma$ -VAE with a constant variance $\\sigma = \\sqrt { \\beta / 2 }$ (for a particular learning rate setting). In recent work (Zhu et al., 2017; Denton $\\&$ Fergus, 2018; Lee et al., 2019), $\\beta$ -VAE models are often used in this exact regime. By tuning the $\\beta$ term, practitioners are able to tune the variance of the decoder, manually producing a more calibrated decoder. However, by re-interpreting the $\\beta$ -VAE objective as a special case of the VAE and introducing the missing $D \\ln \\sigma$ term, we can both obtain a valid evidence lower bound, and remove the need to manually select $\\beta$ . Instead, the variance $\\sigma$ can instead simply be learned end-to-end, reducing the need for hyperparameter tuning. ", + "bbox": [ + 173, + 297, + 825, + 425 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "An alternative discussion of this connection in the context of linear VAEs is also presented by Lucas et al. (2019). While the $\\beta$ term is not necessary for good performance if the decoder is calibrated, it can still be employed if desired, such as when the aim is to attain better disentanglement (Higgins et al., 2017) or a particular rate-distortion tradeoff (Alemi et al., 2017). However, we found that with calibrated decoders, the best sample quality is obtained when $\\beta = 1$ . ", + "bbox": [ + 173, + 430, + 825, + 501 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Loss implementation details. For the correct evidence lower bound computation, it is necessary to add the values of the MSE loss and the KL divergence across the dimensions. We observe that common implementations of these losses (Denton & Fergus, 2018; Abadi et al., 2016; Paszke et al., 2019) use averaging instead, which will lead to poor results if the number of image dimensions is significantly different from the number of the latent dimensions. While this can be conveniently ignored in the $\\beta$ -VAE regime, where the balance term is tuned manually anyway, for the $\\sigma$ -VAE it is essential to compute the objective value correctly. ", + "bbox": [ + 173, + 515, + 825, + 614 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Variance implementation details. Since the variance is non-negative, we parameterize it logarithmically as $\\sigma ^ { \\hat { 2 } } = e ^ { 2 \\lambda }$ , where $\\lambda$ is the logarithm of the standard deviation. For some models, such as per-pixel variance decoders, we observed that it is necessary to restrict the variance range for numerical stability. We do so by using the soft clipping operations proposed by Chua et al. (2018): ", + "bbox": [ + 173, + 627, + 825, + 684 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/069c18d427b4d691f0bd28e1d05bab7551b339d9e5175e7b10afc9c53905f025.jpg", + "text": "$$\n\\lambda : = \\lambda _ { \\mathrm { m a x } } - \\mathrm { s o f t p l u s } ( \\lambda _ { \\mathrm { m a x } } - \\lambda ) ; \\qquad \\lambda : = \\lambda _ { \\mathrm { m i n } } + \\mathrm { s o f t p l u s } ( \\lambda - \\lambda _ { \\mathrm { m i n } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 250, + 686, + 746, + 704 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We observe that setting $\\lambda _ { \\operatorname* { m i n } } = - 6$ to lower bound the standard deviation to be at least half of the distance between allowed color values works well in practice. We also observe that this clipping is unnecessary when learning a shared $\\sigma$ value. ", + "bbox": [ + 176, + 707, + 825, + 750 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 DISCRETE DECODERS ", + "text_level": 1, + "bbox": [ + 176, + 765, + 367, + 780 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "It is possible to use discrete decoding distributions to generate images, as color values are commonly restricted to a fixed set of integer pixel intensities (e.g. 0..255). Indeed, for discrete color values, discrete distributions are arguably more appropriate. In the most general case, a discrete decoding distribution factorized per each pixel and channel would be specified by a probability mass vector $\\hat { x }$ with 256 entries, one per each possible intensity value, similarly to a per-pixel classifier of the intensity value. We can implement it with a soft-max layer, yielding the following log-likelihood loss (sometimes called the cross-entropy loss) for a true pixel with intensity $i$ : ", + "bbox": [ + 173, + 791, + 826, + 890 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7a592e2dd0fc7267beca57f9a2ab0de92e4d4abd62ede1bb6299a81ed41d7736.jpg", + "text": "$$\n- \\ln { p ( x | z ) } = - \\ln { \\frac { \\exp ( \\hat { x } _ { i } ) } { \\sum _ { j } \\exp ( \\hat { x } _ { j } ) } } ,\n$$", + "text_format": "latex", + "bbox": [ + 387, + 892, + 607, + 929 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/a72014d3eeef3fb862e1ae0c45aecac5611320d625332b68e1f68646a77844f0.jpg", + "image_caption": [ + "Figure 1: Different types of calibrated decoders for Gaussian VAE, model parameters are denoted with enclosing squares. Left: both the mean $\\mu$ and the variance $\\sigma$ are output by a neural network with parameters $\\theta$ . Center: $\\sigma$ -VAE with shared variance, the mean is output by a neural network with parameters $\\theta$ , but the variance it iself a global parameter. Right: the proposed optimal $\\sigma$ -VAE, the mean is output by a neural network with parameters $\\theta$ , and the variance is computed analytically from the training data $D$ . " + ], + "image_footnote": [], + "bbox": [ + 178, + 82, + 816, + 217 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We will evaluate these and further choices of discrete decoders, described in Appendix D. We recommend choosing the decoder distribution that best suits the structure of the data, such as discrete decoders for discrete data and continuous decoders for continuous data. ", + "bbox": [ + 173, + 324, + 825, + 366 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 OPTIMAL VARIANCE ESTIMATION FOR CALIBRATED GAUSSIAN DECODERS ", + "text_level": 1, + "bbox": [ + 171, + 387, + 821, + 405 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, we propose a simple but novel analytic way of obtaining a calibrated decoder for continuous distributions that further improves performance. The Gaussian decoders with learned variance described in Section 3.1 are calibrated and work better than na¨ıve unit variance decoders. However, for $\\sigma$ -VAE optimized with gradient descent or Adam (Kingma & Ba, 2015), we observe that careful learning rate tuning can yield significantly better performance, which is in line with prior work that reported poor performance of gradient descent for optimizing Gaussian distributions (Amari, 1998; Peters & Schaal, 2008). A smaller learning rate often produces better performance, but slows down the training, as the likelihood values $p ( x | z )$ will be very suboptimal in the beginning. Instead, here we propose an analytic solution for the value of $\\sigma$ , which computes it analytically and does not require gradient descent. ", + "bbox": [ + 173, + 419, + 826, + 559 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The maximum likelihood estimate of the variance given a known mean is the average squared distance from the mean: ", + "bbox": [ + 176, + 565, + 821, + 593 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7a61ad70f46b4e59d8462e381b67b2ffc391be510eb34b1cd6bc7ffa80c065af.jpg", + "text": "$$\n\\boldsymbol { \\sigma } ^ { * } = \\mathop { \\arg \\operatorname* { m a x } } _ { \\boldsymbol { \\sigma } } \\mathcal { N } ( \\boldsymbol { x } | \\mu , \\boldsymbol { \\sigma } ^ { 2 } I ) = \\mathbf { M } \\mathbf { S } \\mathbf { E } ( \\boldsymbol { x } , \\mu ) ,\n$$", + "text_format": "latex", + "bbox": [ + 356, + 590, + 640, + 617 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\mathbf { M S E } ( x , \\mu ) = \\frac { 1 } { D } \\sum _ { i } ( x _ { i } - \\mu _ { i } ) ^ { 2 } } \\end{array}$ . Eq. 5 can be easily shown using manual differentiation, and is a generalization of the fact that the MLE estimate of the variance is the sample variance. ", + "bbox": [ + 174, + 621, + 826, + 651 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The optimal variance for the decoder distribution under the maximum likelihood criterion is then simply the average MSE loss over the data and the encoder distribution. We leverage this to create an optimal analytic solution for the variance. In the batch setting, the optimal variance would be simply the MSE loss, and can be updated after every gradient update for the other parameters of the decoder. In the mini-batch setting, we use a batchwise estimate of the variance computed for the current minibatch. We analyze these approximations in Appendix C. At test time, a running average of the variance over the training data is used. This method, which we call optimal $\\sigma { - } V A E$ , allows us to learn very efficiently as we use the optimal variance estimate at every training step. It is also easier to implement, as no separate optimizer for the variance parameter is needed. If the variance is not needed at test time, it can also be simply discarded after training. ", + "bbox": [ + 173, + 656, + 825, + 797 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Per-image optimal $\\sigma$ -VAE. Optimal $\\sigma$ -VAE uses a single variance value shared across all data points. However, the optimal $\\sigma$ -VAE also allows more powerful variance estimates, such as learning a variance value per each pixel, or even a variance value per each image, the difference in implementation simply being the dimensions across which the averaging in Equation 5 operates. This approach can be interpreted as variational variance prediction in the framework of Stirn & Knowles (2020). ", + "bbox": [ + 174, + 813, + 825, + 882 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/98cd3e3951801ba5ef1cb57dc92c0876982e83c4ca8bd2fc5f54c56668a9a4aa.jpg", + "image_caption": [ + "Figure 2: Images or videos (bottom right) sampled from the proposed optimal $\\sigma$ -VAE and a unit variance Gaussian VAE models. The Gaussian VAE does not have a means to control the expressivity of the latent variable and produces suboptimal, blurry samples. The $\\sigma$ -VAE controls the expressivity by learning a calibrated decoder, and produces higher quality sequences on all datasets. " + ], + "image_footnote": [], + "bbox": [ + 173, + 103, + 823, + 270 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We now provide an empirical analysis of different decoding distributions, and validate the benefits of our $\\sigma$ -VAE approach. We use a small convolutional VAE model on SVHN (Netzer et al., 2011), a larger hierarchical HVAE model (Maaløe et al., 2019) on the CelebA (Liu et al., 2015) and CIFAR (Krizhevsky et al., 2009) datasets, and a sequence VAE model called SVG (Denton & Fergus, 2018) on the BAIR Pushing dataset (Finn & Levine, 2017). We evaluate the ELBO values as well as visual quality measured by the Frechet Inception Distance (FID, Heusel ´ et al. (2017)). Images are $2 8 \\times 2 8$ for SVHN and $3 2 \\times 3 2$ for CelebA and CIFAR, while video experiments were performed on $6 4 \\times 6 4$ frames ", + "bbox": [ + 174, + 358, + 483, + 564 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "following Denton & Fergus (2018). We do not use KL annealing as it did not improve the results in our experiments. Further experimental details are in App. B. ", + "bbox": [ + 169, + 564, + 823, + 593 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/f3bc299ff2ef0fd5299d713008b8050ad20da16e0dd5a6b53e28def1ce307713.jpg", + "table_caption": [ + "Table 1: Analysis of learned variance on SVHN. The parameter $\\beta$ is tuned manually in $\\beta$ -VAE and learned in $\\sigma$ -VAE. $\\sigma$ -VAE achieves better performance, while the value of $\\beta$ (implicitly defined via the decoder variance) automatically converges close the value found by manual tuning. ", + "5.1 DO CALIBRATED DECODERS BALANCE THE VAE OBJECTIVE WITHOUT TUNING $\\beta$ ? " + ], + "table_footnote": [], + "table_body": "
β-logp↓FID↓
β-VAE0.001<21.4344.54
β-VAE0.01<-318627.93
β-VAE0.1<-122328.3
β-VAE1<138170.39
β-VAE10<4056219.3
g-VAE0.006< -333322.25
", + "bbox": [ + 529, + 453, + 784, + 558 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "As detailed in Section 3.1, a $\\beta$ -VAE with a unit variance Gaussian decoder commonly used in prior work is equivalent to a $\\sigma$ -VAE with constant, manually tuned variance. There is a simple relationship between beta and the variance: $\\sigma = \\sqrt { \\beta / 2 }$ . To compare the variance that the $\\sigma$ -VAE learns to the manually tuned variance in the case of the $\\beta$ -VAE, we compare the ELBO values and the corresponding values of $\\beta$ in Table 1. We find that learning the variance produces similar values of $\\beta$ to the manually tuned values in the $\\beta$ -VAE case, indicating that the $\\sigma$ -VAE is able to learn the balance between the two objective terms in a single training run, without hyperparameter tuning. Moreover, the $\\sigma$ -VAE outperforms the best $\\beta$ ", + "bbox": [ + 174, + 643, + 419, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/ef9043d4e0936ecac83fd6dc61954fcd336bdf4307d4474053b1d64186d93d70.jpg", + "image_caption": [ + "Figure 3: Analysis of learned variance on SVHN. The parameter $\\beta$ is tuned manually in $\\beta$ -VAE and learned in $\\sigma$ -VAE. Higher values of $\\beta$ cause the images to lose detail, while lower values of $\\beta$ might make samples unrealistic. The proposed optimal $\\sigma$ -VAE is able to learn the balance end-to-end, here converging to an equivalent of $\\beta$ -VAE with $\\beta = 0 . 0 0 6$ . " + ], + "image_footnote": [], + "bbox": [ + 434, + 643, + 821, + 820 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "-VAE run. This is because end-to-end learning produces better estimates of the variance than is possible with manual search, improving the likelihood (as measured by the lower bound) and the visual quality. Figure 3 shows the qualitative results from this experiment. ", + "bbox": [ + 171, + 910, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We further validate our results on both single-image and sequential VAE models on a range of datasets in Table 2 and Figure 2. Single-sample ELBO values are reported, and ELBO values on discretized data are reported for discrete distributions. We see that learning a shared variance in a Gaussian decoders (shared $\\sigma$ -VAE) outperforms the na¨ıve unit variance decoder (Gaussian VAE) as well as tuning the $\\beta$ constant for the Gaussian VAE manually. We also see that calibrated discrete decoders, such as full categorical distribution or mixture of discretized logistics, perform better than the na¨ıve Gaussian VAE. Using Bernoulli distribution by treating the color intensities as probabilities (Gregor et al., 2015; Watter et al., 2015) performs poorly. Our results further improve upon the sequence VAE method of Denton & Fergus (2018), which uses a unit variance Gaussian with the $\\beta$ -VAE objective. ", + "bbox": [ + 174, + 138, + 825, + 263 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 HOW DOES LEARNING CALIBRATED DECODERS IMPACT THE LATENT VARIABLE INFORMATION CONTENT? ", + "text_level": 1, + "bbox": [ + 174, + 285, + 759, + 311 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We saw above that calibrated decoders result in higher log-likelihood bounds. Are calibrated decoders also beneficial for representation learning? We evaluate the mutual information $I _ { e } ( x ; z )$ between the data $p _ { d } ( x )$ and encoder samples $q ( z | x )$ , as well as the mismatch between the prior $p ( z )$ and the marginal encoder distribution $m ( z ) = E _ { p _ { d } ( x ) } q ( z | x )$ , measured by the marginal KL $D _ { K L } ( m ( z ) | | p ( z ) )$ . These terms are related to the rate term of the VAE objective as follows (Alemi et al., 2017): ", + "bbox": [ + 173, + 325, + 825, + 410 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/63209d8bf4318842c5b54bea468d7ecd32eb42a2f122996d585ce4aa0e2ce007.jpg", + "text": "$$\n\\begin{array} { r l } & { E _ { p _ { d } ( x ) } \\left[ D _ { K L } ( q ( z | x ) | | p ( z ) ) \\right] = E _ { p _ { d } ( x ) } \\left[ D _ { K L } ( q ( z | x ) | | m ( z ) ) \\right] + D _ { K L } ( m ( z ) | | p ( z ) ) } \\\\ & { \\qquad = I _ { e } ( x ; z ) + D _ { K L } ( m ( z ) | | p ( z ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 227, + 417, + 769, + 458 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "That is, the rate term decomposes into the true mutual information and the marginal KL term. We want to learn expressive latent variables with high mutual information. However, doing so by tuning the $\\beta$ value relaxes the constraint that the encoder and the prior distributions match, and leads to degraded quality of samples from the prior, which creates a trade-off between expressive representations and ability to generate good samples. To compare the $\\beta$ -VAE and $\\sigma$ - VAE in terms of these quantities, we estimate the marginal KL term via Monte Carlo sampling, as proposed by Rosca et al. (2018), and plot the results in Figure 4. As expected, we see that lower $\\beta$ values lead to higher mutual information. However, after a certain point, lower values of $\\beta$ also cause a significant mismatch between the marginal and the prior distributions. By calculating the “effective” $\\beta$ for the $\\sigma$ -VAE, as per Section 4, we can see that the $\\sigma$ -VAE captures an inflection point in the $D _ { K L } ( m ( z ) | | p ( z ) )$ term, learning a representation with the highest possible MI, but without degrading sample quality. This explains the high visual quality of the optimal $\\sigma$ -VAE samples: since the marginal and the prior distributions match, the samples from ", + "bbox": [ + 174, + 474, + 485, + 832 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/87d2b8a25eab2ea0c060438ee47de05935bf6fbb4219ca406dd0f69d64835ff7.jpg", + "image_caption": [ + "Figure 4: Comparison of $\\beta$ -VAE and $\\sigma$ -VAE on SVHN in terms of mutual information $I _ { e } ( x ; z )$ and marginal KL divergence $K L ( m ( z ) | | p ( z ) )$ (see Sec. 5.2). $I _ { e } ( x ; z )$ increases with lower $\\beta$ , yielding expressive representations and better reconstruction. However, after a certain point, lowering $\\beta$ leads to a rapid increase in the marginal $\\mathrm { K L }$ , yielding poor samples from the prior. The $\\sigma$ -VAE is able to automatically find the inflection point after which the marginal KL begins to increase, capturing as much information as possible while still producing good samples. " + ], + "image_footnote": [], + "bbox": [ + 500, + 491, + 816, + 632 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "the prior look similar to reconstructions, while for a $\\beta$ -VAE with low $\\beta$ , the samples from the prior are poor. We see that, in contrast to the $\\beta$ -VAE, where the mutual information is controlled by a hyperparameter, the $\\sigma$ -VAE can adjust the appropriate amount of information automatically and is able to find the setting that produces both informative latents and high quality samples. ", + "bbox": [ + 176, + 833, + 825, + 888 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "An alternative discussion of tuning $\\beta$ is presented by Alemi et al. (2017), who show that $\\beta$ controls the rate-distortion trade-off. Here, we show that the crucial trade-off also controlled by $\\beta$ is the trade-off between two components of the rate itself, which control expressivity of representations and the match between the variational and the prior distributions, respectively. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/6948c0b9118b191788060badd87cc22084df42ec3840ab5d17a11e1715cb454e.jpg", + "table_caption": [ + "Table 2: Generative modeling performance of the proposed $\\sigma$ -VAE on different models and datasets. For SVG, we compare with the original method (Denton & Fergus, 2018), which uses $\\beta$ -VAE. We see that uncalibrated decoders such as mean-only Gaussian perform poorly. $\\beta$ -VAE allows to calibrate the decoder but needs careful hyperparameter tuning. Calibrated decoders such as categorical or $\\sigma$ -VAE perform best. [1] Gregor et al. (2015), [2] Takahashi et al. (2018), [3] Higgins et al. (2017). " + ], + "table_footnote": [], + "table_body": "
CelebA HVAESVHN VAECIFAR HVAEBAIR SVG
-logp↓FID↓-logp↓FID↓-logp↓FID↓-logp↓FID↓
Bernoulli VAE[1]177.643.26284.5122.6
Categorical VAE<635971.5<917946.13<7179101.7N/AN/A
Bitwise-categorical VAE<906766.61<1080033.84<939091.2<4874446.13
Logistic mixture VAE<793265.3<908543.19<8443143.1<4061642.94
Gaussian VAE<7173186.5<2184112.5<7186293.7<-1037935.64
Per-pixel g-VAE<-7814159.3<2184114.7<-7222131<-1405141.98
Student-t VAE [2]<-840171.06<-365970.4<-7419123.6■
β-VAE [3]<-271361.6<-318627.93<-331103<-1347234.64
Shared g-VAE<-637460.7<-334922.25<-5435116.1<-1397434.24
Optimal g-VAE<-844660.3< （-333327.25<-5677101.4<-1417334.13
Opt. per-image g-VAE66.0126.28104.033.21
", + "bbox": [ + 178, + 183, + 820, + 358 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 376, + 823, + 404 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.3 WHAT ARE THE COMMON CHALLENGES IN LEARNING THE VARIANCE THAT PREVENTPRACTITIONERS FROM USING IT, AND HOW TO RECTIFY THEM?", + "text_level": 1, + "bbox": [ + 176, + 421, + 803, + 448 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "If learning the decoder variance improves generation, why are learned variances not used more often? In this section, we discuss how the na¨ıve approach to learning variances, where the decoder outputs a variance for each pixel along with the mean, leads to poor results. First, we find that this method often diverges very quickly due to numerical instability, as the network is able to predict certain pixels with very high certainty, leading to degenerate variances. In contrast, learning a shared variance is always numerically stable in our experiments. We can rectify this numerical instability by bounding the output variance (Section 3.1). However, even with bounded variance, we observe that learning per-pixel variances leads to poor results in Table 2. While the per-pixel variance achieves a good ELBO value, it produces very poor samples, as measured by FID and visual inspection. ", + "bbox": [ + 174, + 459, + 825, + 585 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We see that the specific form of learned variance: a shared variance, a per-image variance, or a per-pixel variance, can lead to very different performance in practice. We hypothesize the per-pixel decoder performs poorly as it incentivizes the model to focus on particular pixels that can be predicted well, instead of focusing equally on all parts of the image. This is consistent with prior work on denoising diffusion models which noted that likelihood-based models place too much focus on imperceptible details, which leads to deteriorated results (Ho et al., 2020). The shared and per-image variance models mitigate this issue at the cost of introducing more bias, and work better in practice. ", + "bbox": [ + 174, + 593, + 825, + 690 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.4 CAN AN ANALYTIC SOLUTION FOR OPTIMAL VARIANCE FURTHER IMPROVE LEARNING? ", + "text_level": 1, + "bbox": [ + 174, + 708, + 816, + 720 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We evaluate the optimal $\\sigma$ -VAE which uses an analytic solution for the variance (Section 4). Table 2 shows that it achieves superior results in terms of log-likelihood. We also note that the optimal $\\sigma$ -VAE converges to a good variance estimate instantaneously, which speeds up learning (highlighted in Figure 9 in the Appendix). In addition, we evaluate the per-image optimal $\\sigma$ -VAE, in which a single variance is computed per image. This model achieves significantly higher visual quality. While producing this per-image variance with a neural network would require additional architecture tuning, optimal $\\sigma$ -VAE is extremely simple to implement (it can be implemented simply as changing the axes of summation), not requiring any new tunable parameters. ", + "bbox": [ + 174, + 732, + 825, + 844 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 864, + 318, + 880 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We presented a simple and effective method for learning calibrated decoders, as well as an evaluation of different decoding distributions with several VAE and sequential VAE models. The proposed method outperforms methods that use na¨ıve unit variance Gaussian decoders and tune a heuristic weight $\\beta$ on the KL-divergence loss, as commonly done in prior work. Moreover, it does not use the heuristic weight $\\beta$ , making it easier to train than this prior work. We expect that the simple techniques for learning calibrated decoders can allow practitioners to speed up the development cycle, obtain better results, and reduce the need for manual hyperparameter tuning. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 194, + 285, + 209 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Mart´ın Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, et al. 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" + ], + "image_footnote": [], + "bbox": [ + 204, + 102, + 794, + 328 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Lucas Theis, Aaron van den Oord, and Matthias Bethge. A note on the evaluation of generative ¨ models. ICLR, 2016. ", + "bbox": [ + 173, + 407, + 821, + 436 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Manuel Watter, Jost Springenberg, Joschka Boedecker, and Martin Riedmiller. Embed to control: A locally linear latent dynamics model for control from raw images. In Advances in neural information processing systems, pp. 2746–2754, 2015. ", + "bbox": [ + 173, + 445, + 826, + 489 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Shengjia Zhao, Jiaming Song, and Stefano Ermon. Infovae: Information maximizing variational autoencoders. arXiv preprint arXiv:1706.02262, 2017. ", + "bbox": [ + 173, + 498, + 821, + 527 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Jun-Yan Zhu, Richard Zhang, Deepak Pathak, Trevor Darrell, Alexei A Efros, Oliver Wang, and Eli Shechtman. Toward multimodal image-to-image translation. In Advances in neural information processing systems, pp. 465–476, 2017. ", + "bbox": [ + 174, + 536, + 825, + 579 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A ADDITIONAL EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 607, + 532, + 622 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "In this section, we provide more qualitative results in Figures 7, 6, 8, 5 as well as a graph showing the convergence properties of the variance for different models in Fig. 9. In order to validate our method with a different architecture, we also report performance of different decoders with a small 5-layer convolutional architecture on the CelebA and CIFAR dataset in Table 3. We see that the ordering of the methods is consistent with this smaller architecture. ", + "bbox": [ + 174, + 638, + 825, + 707 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 728, + 416, + 744 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "For the small convolutional network test on SVHN, the encoder has 3 convolutional layers followed by a fully connected layer, while the decoder has a fully connected layer followed by 3 convolutional layers. The $\\beta$ was tuned from 100 to 0.0001 for $\\beta$ -VAE. The number of channels in the convolutional layers starts with 32 and increases 2 times in every layer. The dimension of the latent variable is 20. Adam (Kingma & Ba, 2015) with learning rate of 1e-3 is used for optimization. Batch size of 128 was used and all models were trained for 10 epochs. We additionally evaluate this small convolutional network on CelebA, CIFAR, and Frey Face2 datasets in Table 3. Unit Gaussian prior and Gaussian posteriors with diagonal covariance were used. For the larger hierarchical VAE, we used the official pytorch implementation of (Maaløe et al., 2019). We use the baseline hierarchical VAE with 15 layers of latent variables, without the top-down and bottom-up connections. For the hierarchical VAE and the SVG-LP model, we use the default hyperparameters in the respective implementations. We use the standard train-val-test split for all datasets. All models were trained on a single high-end GPU. We use the official PyTorch implementation of the Inception network to compute FID. All methods are compared on the same hyperparameters. ", + "bbox": [ + 173, + 760, + 825, + 898 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/d38593657ca6b60d33ed0b9cdf75405e10302b180aabe801f09b58dd79319736.jpg", + "image_caption": [ + "Figure 6: Samples from the $\\sigma$ -VAE (left) and the Gaussian VAE (right) on the CelebA dataset, images cropped to the face for clarity. The Gaussian VAE produces blurry results with indistinct face features, while the $\\sigma$ -VAE is able to produce accurate images of faces. " + ], + "image_footnote": [], + "bbox": [ + 205, + 171, + 794, + 398 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/be94a62d54172ebd3bf1676da15571deb510d60d74b84190f350e5d866c362fb.jpg", + "table_caption": [ + "Table 3: Generative modeling performance of the proposed $\\sigma$ -VAE on CelebA, CIFAR, and Frey Face with a smaller model. We see that uncalibrated decoders such as mean-only Gaussian perform poorly. $\\beta$ -VAE allows to calibrate the decoder but needs careful hyperparameter tuning. Calibrated decoders such as categorical or $\\sigma$ -VAE perform best. " + ], + "table_footnote": [], + "table_body": "
CelebA VAECIFAR VAEFrey Face VAE
-logp↓FID↓-logp↓FID↓-logp↓FID↓
Bernoulli VAE Gregor et al. (2015)102.7165.147.7
Categorical VAE;1019550.45;10673124.1<245450.16
bitwise-categorical VAE1101956.361160499.65<317366.77
Logistic mixture VAE;1015461.81;10648100.2<256250.28
Gaussian VAE<2201144.8<1409205.8<726.480.17
β-VAE Higgins et al. (2017)<-194258.73<-1318117.9< -420.037.61
Shared g-VAE (Ours)<-193973.27< （-1830137.8< -49.7842.86
Optimal g-VAE (Ours)<-195161.27< （-183280.9< -162253.36
Opt. per-image g-VAE (Ours)53.1389.8856.07
", + "bbox": [ + 171, + 670, + 869, + 858 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/ca6fc2105698b7956e8b56b977c9c8da04a3379e8a83c6b66b469bc5cb64f51b.jpg", + "image_caption": [ + "Figure 7: Samples from the $\\sigma$ -VAE (top) and the Gaussian VAE (bottom) on the BAIR dataset. Sampled sequences conditioned on two initial frames are shown, and the ground truth sequence is shown at the top. The Gaussian VAE produces blurry robot arm texture and the arm often disappears towards the end of the sequence, while the $\\sigma$ -VAE is able to produce sequences with realistic motion and model the details of the arm texture, such as the gripper. " + ], + "image_footnote": [], + "bbox": [ + 173, + 207, + 825, + 732 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/37438c302fad1c82eb34bd451e5ebc5793a6d0778bfc657860781536da22714b.jpg", + "image_caption": [ + "Figure 8: Samples from the $\\sigma$ -VAE (left) and the Gaussian VAE (right) on the challenging CIFAR dataset. The Gaussian VAE produces blurry results with muted colors, while the $\\sigma$ -VAE models the distribution of shapes in the CIFAR data more faithfully. " + ], + "image_footnote": [], + "bbox": [ + 204, + 131, + 794, + 357 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/8469db8ea7a9237e00486921f5ccc4e7297db5670f610fba5908ade727026f74.jpg", + "image_caption": [ + "Figure 9: Variance convergence speed on SVHN. We see that the shared $\\sigma$ -VAE which optimizes the variance with gradient descent has an initial period of convergence when the variance converges to the region of the optimal value. In contrast, $\\sigma$ -VAE with analytical (optimal) variance quickly learns a good estimate of the variance, which leads to better performance. The unit variance Gaussian $\\beta$ -VAE can be interpreted as having a constant variance determined by $\\beta$ , shown here. Since the variance doesn’t change throughout training, it achieves suboptimal performance. " + ], + "image_footnote": [], + "bbox": [ + 196, + 497, + 797, + 776 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C EMPIRICAL ANALYZIS OF APPROXIMATIONS FOR OPTIMAL $\\sigma$ -VAE ", + "text_level": 1, + "bbox": [ + 171, + 183, + 756, + 199 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The optimal $\\sigma$ -VAE requires computing the following estimate of the variance ", + "bbox": [ + 173, + 215, + 686, + 231 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/0bc0912152c18d52da4d5f676c8faaf55d7382a851ece8a98938080edf41ecbd.jpg", + "text": "$$\n\\sigma ^ { * } = \\underset { \\sigma } { \\arg \\operatorname* { m a x } } \\mathbb { E } _ { x \\sim \\mathrm { D a t a } } \\mathbb { E } _ { q ( z | x ) } \\left[ \\ln p ( x | \\mu _ { \\theta } ( z ) , \\sigma ^ { 2 } I ) \\right] = \\mathbb { E } _ { x \\sim \\mathrm { D a t a } } \\mathbb { E } _ { q ( z | x ) } { \\mathbf { M S E } } ( x , \\mu _ { \\theta } ( z ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 217, + 238, + 779, + 265 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "This requires computing two expectations, with respect to the data in the dataset, and with respect to the encoder distribution. We use MC sampling with one sample per data point to approximate both expectations. Inspired by common practices in VAEs, we use one sample per data point to approximate the inner expectation. On SVHN, the standard error of this approximation is $0 . 2 6 \\%$ of the value of sigma. We further approximate the outer expectation with a single batch instead of the entire dataset. On SVHN, the standard error of this approximation is $2 \\%$ of the value of sigma. We see that both approximations are accurate in practice. The second approximation yields a biased estimate of the evidence lower bound because the same batch is used to approximate the variance and compute the lower bound estimate. However, this bias can be corrected by using a different batch, or with a running average of the variance with an appropriate decay. This running average can also be used to reduce the variance of the estimate and to achieve convergence guarantees, but we did not find it necessary in our experiments. ", + "bbox": [ + 173, + 280, + 825, + 446 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "D ALTERNATIVE DECODER CHOICES ", + "text_level": 1, + "bbox": [ + 173, + 470, + 498, + 486 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We describe the alternative decoders evaluated in Table 2: using the bitwise-categorical, and the logistic mixture distributions. ", + "bbox": [ + 174, + 503, + 418, + 558 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Bitwise-categorical VAE While the 256-way categorical decoder described in Section 3.2 is very powerful due to the ability to specify any possible intensity distribution, it suffers from high computational and memory requirements. Because 256 values need to be kept for each pixel and channel, simply keeping this distribution in memory for one 3-channel $1 0 2 4 \\times 1 0 2 4$ image would require 3 GiB of memory, compared to 0.012 GiB for the Gaussian decoder. Therefore, training deep ", + "bbox": [ + 174, + 578, + 419, + 771 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/6d3699d260ea7e02a4d7ad52c725559580645d723b1769946e7ff698e1a8338e.jpg", + "table_caption": [ + "Table 4: ELBO on discretized data. All distributions except categorical have scalar scale parameters. The $\\sigma$ -VAE performs well on the discretized ELBO metric, performing similarly to a discrete distribution parametrized as a discretized Gaussian or discretized Logistic. Full categorical distribution attains highest likelihood due to having the most statistical power. " + ], + "table_footnote": [], + "table_body": "
CIFAR VAE
- log pdf↓-logp↓FID↓
Categorical VAE<10673137.6
Gaussian VAE<740.5<15131212.7
Gaussianσ-VAE<-896.1<11120136.7
Disc.Gaussian g-VAE<11117136.9
Disc.Logistic g-VAE<11103136.7
", + "bbox": [ + 437, + 628, + 816, + 739 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "neural networks with this full categorical distribution is impractical for high-resolution images or videos. The bitwise-categorical VAE improves the memory complexity by defining the distribution over 256 values in a more compact way. Specifically, it defines a binary distribution over each bit in the pixel intensity value, requiring 8 values in total, one for each bit. This distribution can be thought of as a classifier that predicts the value of each bit in the image separately. In our implementation of the bitwise-categorical likelihood, we convert the image channels to binary format and use the standard binary cross-entropy loss (which reduces to binary log-likelihood since all bits in the image are deterministically either zero or one). While in our experiments the bitwise-categorical distribution did not outperform other choices, it often performs on par with our proposed method. We expect this distribution to be useful due to its generality as it is able to represent values stored in any digital format by converting them into binary. ", + "bbox": [ + 173, + 771, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Logistic mixture VAE For this decoder, we adapt the discretized logistic mixture from Salimans et al. (2017). To define a discrete 256-way distribution, it divides the corresponding continuous distribution into 256 bins, where the probability mass is defined as the integral of the PDF over the corresponding bin. (Kingma et al., 2016) uses the logistic distribution discretized in this manner for the decoder. Salimans et al. (2017) suggests to make all bins except the first and the last be of equal size, whereas the first and the last bin include, respectively, the intervals $( - \\infty , 0 ]$ and $\\lbrack 1 , \\infty )$ . Salimans et al. (2017) further suggests using a mixture of discretized logistics for improved capacity. Our implementation largely follows the one in Salimans et al. (2017), however, we note that the original implementation is not suitable for learning latent variable models, as it generates the channels autoregressively. This will cause the latent variable to lose color information since it can be represented by the autoregressive decoder. We therefore adapt the mixture of discretized logistics to the pure latent variable setup by removing the mean-adjusting coefficients from (Salimans et al., 2017). In our experiments, the logistic mixture outperformed other discrete distributions. ", + "bbox": [ + 173, + 103, + 825, + 284 + ], + "page_idx": 16 + } +] \ No newline at end of file diff --git a/parse/train/nkap3LV7t7O/nkap3LV7t7O_middle.json b/parse/train/nkap3LV7t7O/nkap3LV7t7O_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..5d444fc75ba4c468e6f85d8654f70aa328d6c0f4 --- /dev/null +++ b/parse/train/nkap3LV7t7O/nkap3LV7t7O_middle.json @@ -0,0 +1,44833 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 78, + 407, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 407, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 407, + 97 + ], + "score": 1.0, + "content": "SIMPLE AND EFFECTIVE VAE TRAINING", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 97, + 335, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 335, + 118 + ], + "score": 1.0, + "content": "WITH CALIBRATED DECODERS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 142, + 213, + 469, + 367 + ], + "lines": [ + { + "bbox": [ + 142, + 213, + 470, + 225 + ], + "spans": [ + { + "bbox": [ + 142, + 213, + 470, + 225 + ], + "score": 1.0, + "content": "Variational autoencoders (VAEs) provide an effective and simple method for mod-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 224, + 470, + 237 + ], + "spans": [ + { + "bbox": [ + 141, + 224, + 470, + 237 + ], + "score": 1.0, + "content": "eling complex distributions. 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Our analysis", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 323, + 471, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 323, + 471, + 335 + ], + "score": 1.0, + "content": "covers a range of datasets and several single-image and sequential VAE models.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 470, + 347 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 470, + 347 + ], + "score": 1.0, + "content": "We further propose a simple but novel modification to the commonly used Gaus-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 344, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 469, + 357 + ], + "score": 1.0, + "content": "sian decoder, which computes the prediction variance analytically. We observe", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 356, + 462, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 356, + 462, + 369 + ], + "score": 1.0, + "content": "empirically that using heuristic modifications is not necessary with our method.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 108, + 392, + 205, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 208, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 208, + 408 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "Deep density models based on the variational autoencoder (VAE) (Kingma & Welling, 2014; Rezende", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "et al., 2014) have found ubiquitous use in probabilistic modeling and representation learning as they", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "score": 1.0, + "content": "are both conceptually simple and are able to scale to very complex distributions and large datasets.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 451, + 507, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 507, + 465 + ], + "score": 1.0, + "content": "These VAE techniques are used for tasks such as future frame prediction (Castrejon et al., 2019),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "image segmentation (Kohl et al., 2018), generating speech (Chung et al., 2015) and music (Dhariwal", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "et al., 2020), as well as model-based reinforcement learning (Hafner et al., 2019a). 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This balance", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "trades off fidelity of reconstruction and quality of samples from the model: a model with low rate", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "would not contain enough information to reconstruct the data, while allowing the model to have high", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "rate might lead to unrealistic samples from the prior as the KL-divergence constraint becomes weaker", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "(Alemi et al., 2017; Higgins et al., 2017). 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Moreover, using", + "type": "text" + }, + { + "bbox": [ + 479, + 583, + 505, + 595 + ], + "score": 0.9, + "content": "\\beta \\neq 1", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 595, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 606 + ], + "score": 1.0, + "content": "precludes the appealing interpretation of the VAE objective as a bound on the data likelihood, and is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 604, + 311, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 311, + 618 + ], + "score": 1.0, + "content": "undesirable for applications like density modeling.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "While many architectures for calibrating decoders have been proposed in the literature (Kingma &", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "Welling, 2014; Kingma et al., 2016; Dai & Wipf, 2019), more applied work typically employs VAEs", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 642, + 507, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 507, + 658 + ], + "score": 1.0, + "content": "with uncalibrated decoding distributions, such as Gaussian distributions without a learned variance,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "where the decoder only outputs the mean parameter (Castrejon et al., 2019; Denton & Fergus, 2018;", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "Lee et al., 2019; Babaeizadeh et al., 2018; Lee et al., 2018; Hafner et al., 2019b; Pong et al., 2019;", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Zhu et al., 2017; Pavlakos et al., 2019), or uses other ad-hoc modifications to the objective (Sohn", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "et al., 2015; Henaff et al., 2019). Indeed, it is well known that attempting to learn the variance in a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "Gaussian decoder may lead to numerical instability (Rezende & Viola, 2018; Dai & Wipf, 2019),", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "and na¨ıve approaches often lead to poor results. 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While many methods for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 279, + 470, + 292 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 470, + 292 + ], + "score": 1.0, + "content": "learning calibrated decoders have been proposed, many of the recent papers that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 471, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 471, + 302 + ], + "score": 1.0, + "content": "employ VAEs rely on heuristic hyperparameters and ad-hoc modifications instead.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 301, + 470, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 470, + 313 + ], + "score": 1.0, + "content": "We perform the first comprehensive comparative analysis of calibrated decoder", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 312, + 469, + 324 + ], + "spans": [ + { + "bbox": [ + 142, + 312, + 469, + 324 + ], + "score": 1.0, + "content": "and provide recommendations for simple and effective VAE training. Our analysis", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 323, + 471, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 323, + 471, + 335 + ], + "score": 1.0, + "content": "covers a range of datasets and several single-image and sequential VAE models.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 470, + 347 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 470, + 347 + ], + "score": 1.0, + "content": "We further propose a simple but novel modification to the commonly used Gaus-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 344, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 469, + 357 + ], + "score": 1.0, + "content": "sian decoder, which computes the prediction variance analytically. We observe", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 356, + 462, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 356, + 462, + 369 + ], + "score": 1.0, + "content": "empirically that using heuristic modifications is not necessary with our method.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11.5, + "bbox_fs": [ + 141, + 213, + 471, + 369 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 392, + 205, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 208, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 208, + 408 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "Deep density models based on the variational autoencoder (VAE) (Kingma & Welling, 2014; Rezende", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "et al., 2014) have found ubiquitous use in probabilistic modeling and representation learning as they", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "score": 1.0, + "content": "are both conceptually simple and are able to scale to very complex distributions and large datasets.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 451, + 507, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 507, + 465 + ], + "score": 1.0, + "content": "These VAE techniques are used for tasks such as future frame prediction (Castrejon et al., 2019),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 506, + 475 + ], + "score": 1.0, + "content": "image segmentation (Kohl et al., 2018), generating speech (Chung et al., 2015) and music (Dhariwal", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "et al., 2020), as well as model-based reinforcement learning (Hafner et al., 2019a). However, in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "practice, many of these approaches require careful manual tuning of the balance between two terms", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "that correspond to distortion and rate from information theory (Alemi et al., 2017). This balance", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "trades off fidelity of reconstruction and quality of samples from the model: a model with low rate", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "would not contain enough information to reconstruct the data, while allowing the model to have high", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "rate might lead to unrealistic samples from the prior as the KL-divergence constraint becomes weaker", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "(Alemi et al., 2017; Higgins et al., 2017). While a proper variational lower bound does not expose", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 562 + ], + "score": 1.0, + "content": "any free parameters to control this tradeoff, many prior works heuristically introduce a weight on the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 561, + 268, + 574 + ], + "score": 1.0, + "content": "prior KL-divergence term, often denoted", + "type": "text" + }, + { + "bbox": [ + 268, + 562, + 275, + 573 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 561, + 314, + 574 + ], + "score": 1.0, + "content": ". Usually,", + "type": "text" + }, + { + "bbox": [ + 315, + 562, + 322, + 573 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "needs to be tuned for every dataset and model", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "variant as a hyperparameter, which slows down development and can lead to poor performance as", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 479, + 596 + ], + "score": 1.0, + "content": "finding the optimal value is often prohibitively computationally expensive. Moreover, using", + "type": "text" + }, + { + "bbox": [ + 479, + 583, + 505, + 595 + ], + "score": 0.9, + "content": "\\beta \\neq 1", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 595, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 606 + ], + "score": 1.0, + "content": "precludes the appealing interpretation of the VAE objective as a bound on the data likelihood, and is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 604, + 311, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 311, + 618 + ], + "score": 1.0, + "content": "undesirable for applications like density modeling.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 419, + 507, + 618 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "While many architectures for calibrating decoders have been proposed in the literature (Kingma &", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "Welling, 2014; Kingma et al., 2016; Dai & Wipf, 2019), more applied work typically employs VAEs", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 642, + 507, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 507, + 658 + ], + "score": 1.0, + "content": "with uncalibrated decoding distributions, such as Gaussian distributions without a learned variance,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "where the decoder only outputs the mean parameter (Castrejon et al., 2019; Denton & Fergus, 2018;", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "Lee et al., 2019; Babaeizadeh et al., 2018; Lee et al., 2018; Hafner et al., 2019b; Pong et al., 2019;", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Zhu et al., 2017; Pavlakos et al., 2019), or uses other ad-hoc modifications to the objective (Sohn", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "et al., 2015; Henaff et al., 2019). Indeed, it is well known that attempting to learn the variance in a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "Gaussian decoder may lead to numerical instability (Rezende & Viola, 2018; Dai & Wipf, 2019),", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "and na¨ıve approaches often lead to poor results. As a result, it remains unclear whether practical", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 434, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 434, + 734 + ], + "score": 1.0, + "content": "empirical performance of VAEs actually benefits from calibrated decoders or not.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 621, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "To rectify this, our first contribution is a comparative analysis of various calibrated decoder architec-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "tures and practical recommendations for simple and effective VAE training. We find that, while na¨ıve", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "calibrated decoders often lead to worse results, a careful choice of the decoder distribution can work", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 364, + 129 + ], + "score": 1.0, + "content": "very well, and removes the need to tune the additional parameter", + "type": "text" + }, + { + "bbox": [ + 364, + 116, + 371, + 127 + ], + "score": 0.79, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 115, + 505, + 129 + ], + "score": 1.0, + "content": ". Indeed, we note that the entropy", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 504, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 349, + 138 + ], + "score": 1.0, + "content": "of the decoding distribution controls the mutual information", + "type": "text" + }, + { + "bbox": [ + 350, + 127, + 379, + 138 + ], + "score": 0.92, + "content": "I ( x ; z )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 127, + 504, + 138 + ], + "score": 1.0, + "content": ". Calibrated decoders allow the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 174, + 150 + ], + "score": 1.0, + "content": "model to control", + "type": "text" + }, + { + "bbox": [ + 174, + 137, + 203, + 149 + ], + "score": 0.92, + "content": "I ( x ; z )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "automatically, instead of relying on manual tuning. Our second contribution", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "is a simple but novel technique for optimizing the decoder variance analytically, without requiring the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "decoder network to produce it as an additional output. We call the resulting approach to learning the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 194, + 183 + ], + "score": 1.0, + "content": "Gaussian variance the", + "type": "text" + }, + { + "bbox": [ + 195, + 172, + 202, + 180 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 169, + 321, + 183 + ], + "score": 1.0, + "content": "-VAE. In our experiments, the", + "type": "text" + }, + { + "bbox": [ + 321, + 172, + 328, + 180 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "-VAE outperforms the alternative of learning", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "the variance through gradient descent, while being simpler to implement and extend. We validate our", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 472, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 472, + 205 + ], + "score": 1.0, + "content": "results on several VAE and sequence VAE models and a range of image and video datasets.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 108, + 213, + 211, + 226 + ], + "lines": [ + { + "bbox": [ + 104, + 212, + 213, + 229 + ], + "spans": [ + { + "bbox": [ + 104, + 212, + 213, + 229 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 506, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "Prior work on variational autoencoders has studied a number of different decoder parameterizations.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "Kingma & Welling (2014); Rezende et al. (2014) use the Bernoulli distribution for the binary MNIST", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 258, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 506, + 270 + ], + "score": 1.0, + "content": "data and Kingma & Welling (2014) use Gaussian distributions with learned variance parameter for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 270, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 281 + ], + "score": 1.0, + "content": "grayscale images. However, modeling images with continuous distributions is prone to instability as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 280, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 506, + 292 + ], + "score": 1.0, + "content": "the variance can converge to zero (Rezende & Viola, 2018; Mattei & Frellsen, 2018; Dai & Wipf,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "2019). Some work has attempted to rectify this problem by using dequantization (Gregor et al., 2016),", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "which is theoretically appealing as it is tightly related to the log-likelihood of the original discrete", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 313, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 325 + ], + "score": 1.0, + "content": "data (Theis et al., 2016), optimizing the variance in a two-stage procedure (Arvanitidis et al., 2017),", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "or training a post-hoc prior (Ghosh et al., 2019). Takahashi et al. (2018); Barron (2019) proposed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "score": 1.0, + "content": "more expressive distributions. Additionally, different choices for representing such variance exist,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "including diagonal covariance (Kingma & Welling, 2014; Sønderby et al., 2016; Rolfe, 2016), or a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 357, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 369 + ], + "score": 1.0, + "content": "single shared parameter (Kingma et al., 2016; Dai & Wipf, 2019; Edwards & Storkey, 2016; Rezende", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "& Viola, 2018). We analyze these and notice that learning a single variance parameter shared across", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "images leads to stable training and good performance, without the use of dequantization or even", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 390, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 506, + 402 + ], + "score": 1.0, + "content": "clipping the variance, although these techniques can be used with our decoders; and further improve", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 400, + 333, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 333, + 413 + ], + "score": 1.0, + "content": "the estimation of this variance with an analytic solution.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 507, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 507, + 430 + ], + "score": 1.0, + "content": "Early work on discrete VAE decoders for color images modeled them with the Bernoulli distribution,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "score": 1.0, + "content": "treating the color intensities as probabilities (Gregor et al., 2015). Further work has explored various", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "parameterizations based on discretized continuous distributions, such as discretized logistic (Kingma", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "et al., 2016). More recent work has improved expressivity of the decoder with a mixture of discretized", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "logistics (Chen et al., 2016; Maaløe et al., 2019). However, these models also employ powerful", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "autoregressive decoders (Chen et al., 2016; Gulrajani et al., 2016; Maaløe et al., 2019), and the latent", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 484, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 495 + ], + "score": 1.0, + "content": "variables in these models may not represent all of the significant factors of variation in the data, as", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 494, + 504, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 504, + 506 + ], + "score": 1.0, + "content": "some factors can instead be modeled internally by the autoregressive decoder (Alemi et al., 2017).1", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "While many calibrated decoders have been proposed, outside the core generative modeling community", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 521, + 507, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 507, + 536 + ], + "score": 1.0, + "content": "uncalibrated decoders are ubiquitous. They are used in work on video prediction (Denton & Fergus,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "2018; Castrejon et al., 2019; Lee et al., 2018; Babaeizadeh et al., 2018), image segmentation (Kohl", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 545, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 506, + 557 + ], + "score": 1.0, + "content": "et al., 2018), image-to-image translation (Zhu et al., 2017), 3D human pose (Pavlakos et al., 2019),", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "as well as model-based reinforcement learning (Henaff et al., 2019; Hafner et al., 2019b;a), and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "representation learning (Lee et al., 2019; Watter et al., 2015; Pong et al., 2019). Most of these works", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 249, + 591 + ], + "score": 1.0, + "content": "utilize the heuristic hyperparameter", + "type": "text" + }, + { + "bbox": [ + 249, + 578, + 257, + 588 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "instead, which is undesirable both as the resulting objective is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 286, + 601 + ], + "score": 1.0, + "content": "no longer a bound on the likelihood, and as", + "type": "text" + }, + { + "bbox": [ + 286, + 589, + 294, + 600 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "usually requires extensive tuning. In this work, we", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "analyze the common pitfalls of using calibrated decoders that may have prevented the practitioners", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "from using them, propose a simple and effective analytic way of learning such calibrated distribution,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 621, + 464, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 464, + 634 + ], + "score": 1.0, + "content": "and provide a comprehensive experimental evaluation of different decoding distributions.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 294, + 650 + ], + "score": 1.0, + "content": "Alternative discussions of the hyperparameter", + "type": "text" + }, + { + "bbox": [ + 294, + 639, + 302, + 649 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "are presented by Zhao et al. (2017); Higgins et al.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 649, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 660 + ], + "score": 1.0, + "content": "(2017); Alemi et al. (2017); Achille & Soatto (2018), who show that it controls the amount of", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 239, + 672 + ], + "score": 1.0, + "content": "information in the latent variable,", + "type": "text" + }, + { + "bbox": [ + 240, + 660, + 268, + 672 + ], + "score": 0.93, + "content": "I ( x ; z )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 660, + 505, + 672 + ], + "score": 1.0, + "content": ". Peng et al. (2018); Rezende & Viola (2018) further discuss", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "constrained optimization objectives for VAEs, which also yield a similar hyperparameter. Here, we", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 142, + 694 + ], + "score": 1.0, + "content": "focus on", + "type": "text" + }, + { + "bbox": [ + 143, + 682, + 150, + 693 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "-VAEs with Gaussian decoders with constant variance, as commonly used in recent work,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 693, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 242, + 705 + ], + "score": 1.0, + "content": "and show that the hyperparameter", + "type": "text" + }, + { + "bbox": [ + 242, + 693, + 250, + 704 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 693, + 506, + 705 + ], + "score": 1.0, + "content": "can be incorporated in the decoding likelihood for these models.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 711, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 119, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "1BIVA (Maaløe et al., 2019) uses the Mixture of Logistics decoder proposed in (Salimans et al., 2017) that", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 361, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 361, + 734 + ], + "score": 1.0, + "content": "produces the channels for each pixel autoregressively, see also App D.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "To rectify this, our first contribution is a comparative analysis of various calibrated decoder architec-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "tures and practical recommendations for simple and effective VAE training. We find that, while na¨ıve", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "calibrated decoders often lead to worse results, a careful choice of the decoder distribution can work", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 364, + 129 + ], + "score": 1.0, + "content": "very well, and removes the need to tune the additional parameter", + "type": "text" + }, + { + "bbox": [ + 364, + 116, + 371, + 127 + ], + "score": 0.79, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 115, + 505, + 129 + ], + "score": 1.0, + "content": ". Indeed, we note that the entropy", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 504, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 349, + 138 + ], + "score": 1.0, + "content": "of the decoding distribution controls the mutual information", + "type": "text" + }, + { + "bbox": [ + 350, + 127, + 379, + 138 + ], + "score": 0.92, + "content": "I ( x ; z )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 127, + 504, + 138 + ], + "score": 1.0, + "content": ". Calibrated decoders allow the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 174, + 150 + ], + "score": 1.0, + "content": "model to control", + "type": "text" + }, + { + "bbox": [ + 174, + 137, + 203, + 149 + ], + "score": 0.92, + "content": "I ( x ; z )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "automatically, instead of relying on manual tuning. Our second contribution", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "is a simple but novel technique for optimizing the decoder variance analytically, without requiring the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "decoder network to produce it as an additional output. We call the resulting approach to learning the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 194, + 183 + ], + "score": 1.0, + "content": "Gaussian variance the", + "type": "text" + }, + { + "bbox": [ + 195, + 172, + 202, + 180 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 169, + 321, + 183 + ], + "score": 1.0, + "content": "-VAE. In our experiments, the", + "type": "text" + }, + { + "bbox": [ + 321, + 172, + 328, + 180 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "-VAE outperforms the alternative of learning", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "the variance through gradient descent, while being simpler to implement and extend. We validate our", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 472, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 472, + 205 + ], + "score": 1.0, + "content": "results on several VAE and sequence VAE models and a range of image and video datasets.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 82, + 506, + 205 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 213, + 211, + 226 + ], + "lines": [ + { + "bbox": [ + 104, + 212, + 213, + 229 + ], + "spans": [ + { + "bbox": [ + 104, + 212, + 213, + 229 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 506, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "Prior work on variational autoencoders has studied a number of different decoder parameterizations.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "Kingma & Welling (2014); Rezende et al. (2014) use the Bernoulli distribution for the binary MNIST", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 258, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 506, + 270 + ], + "score": 1.0, + "content": "data and Kingma & Welling (2014) use Gaussian distributions with learned variance parameter for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 270, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 281 + ], + "score": 1.0, + "content": "grayscale images. However, modeling images with continuous distributions is prone to instability as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 280, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 506, + 292 + ], + "score": 1.0, + "content": "the variance can converge to zero (Rezende & Viola, 2018; Mattei & Frellsen, 2018; Dai & Wipf,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "2019). Some work has attempted to rectify this problem by using dequantization (Gregor et al., 2016),", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "which is theoretically appealing as it is tightly related to the log-likelihood of the original discrete", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 313, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 325 + ], + "score": 1.0, + "content": "data (Theis et al., 2016), optimizing the variance in a two-stage procedure (Arvanitidis et al., 2017),", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "or training a post-hoc prior (Ghosh et al., 2019). Takahashi et al. (2018); Barron (2019) proposed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "score": 1.0, + "content": "more expressive distributions. Additionally, different choices for representing such variance exist,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "including diagonal covariance (Kingma & Welling, 2014; Sønderby et al., 2016; Rolfe, 2016), or a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 357, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 369 + ], + "score": 1.0, + "content": "single shared parameter (Kingma et al., 2016; Dai & Wipf, 2019; Edwards & Storkey, 2016; Rezende", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "& Viola, 2018). We analyze these and notice that learning a single variance parameter shared across", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "images leads to stable training and good performance, without the use of dequantization or even", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 390, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 506, + 402 + ], + "score": 1.0, + "content": "clipping the variance, although these techniques can be used with our decoders; and further improve", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 400, + 333, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 333, + 413 + ], + "score": 1.0, + "content": "the estimation of this variance with an analytic solution.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 236, + 506, + 413 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 507, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 507, + 430 + ], + "score": 1.0, + "content": "Early work on discrete VAE decoders for color images modeled them with the Bernoulli distribution,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "score": 1.0, + "content": "treating the color intensities as probabilities (Gregor et al., 2015). Further work has explored various", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "parameterizations based on discretized continuous distributions, such as discretized logistic (Kingma", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "et al., 2016). More recent work has improved expressivity of the decoder with a mixture of discretized", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "logistics (Chen et al., 2016; Maaløe et al., 2019). However, these models also employ powerful", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "autoregressive decoders (Chen et al., 2016; Gulrajani et al., 2016; Maaløe et al., 2019), and the latent", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 484, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 495 + ], + "score": 1.0, + "content": "variables in these models may not represent all of the significant factors of variation in the data, as", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 494, + 504, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 504, + 506 + ], + "score": 1.0, + "content": "some factors can instead be modeled internally by the autoregressive decoder (Alemi et al., 2017).1", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 417, + 507, + 506 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "While many calibrated decoders have been proposed, outside the core generative modeling community", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 521, + 507, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 507, + 536 + ], + "score": 1.0, + "content": "uncalibrated decoders are ubiquitous. They are used in work on video prediction (Denton & Fergus,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "2018; Castrejon et al., 2019; Lee et al., 2018; Babaeizadeh et al., 2018), image segmentation (Kohl", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 545, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 506, + 557 + ], + "score": 1.0, + "content": "et al., 2018), image-to-image translation (Zhu et al., 2017), 3D human pose (Pavlakos et al., 2019),", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "as well as model-based reinforcement learning (Henaff et al., 2019; Hafner et al., 2019b;a), and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "representation learning (Lee et al., 2019; Watter et al., 2015; Pong et al., 2019). Most of these works", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 249, + 591 + ], + "score": 1.0, + "content": "utilize the heuristic hyperparameter", + "type": "text" + }, + { + "bbox": [ + 249, + 578, + 257, + 588 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "instead, which is undesirable both as the resulting objective is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 286, + 601 + ], + "score": 1.0, + "content": "no longer a bound on the likelihood, and as", + "type": "text" + }, + { + "bbox": [ + 286, + 589, + 294, + 600 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "usually requires extensive tuning. In this work, we", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "analyze the common pitfalls of using calibrated decoders that may have prevented the practitioners", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "from using them, propose a simple and effective analytic way of learning such calibrated distribution,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 621, + 464, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 464, + 634 + ], + "score": 1.0, + "content": "and provide a comprehensive experimental evaluation of different decoding distributions.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 511, + 507, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 294, + 650 + ], + "score": 1.0, + "content": "Alternative discussions of the hyperparameter", + "type": "text" + }, + { + "bbox": [ + 294, + 639, + 302, + 649 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "are presented by Zhao et al. (2017); Higgins et al.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 649, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 660 + ], + "score": 1.0, + "content": "(2017); Alemi et al. (2017); Achille & Soatto (2018), who show that it controls the amount of", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 239, + 672 + ], + "score": 1.0, + "content": "information in the latent variable,", + "type": "text" + }, + { + "bbox": [ + 240, + 660, + 268, + 672 + ], + "score": 0.93, + "content": "I ( x ; z )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 660, + 505, + 672 + ], + "score": 1.0, + "content": ". Peng et al. (2018); Rezende & Viola (2018) further discuss", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "constrained optimization objectives for VAEs, which also yield a similar hyperparameter. Here, we", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 142, + 694 + ], + "score": 1.0, + "content": "focus on", + "type": "text" + }, + { + "bbox": [ + 143, + 682, + 150, + 693 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "-VAEs with Gaussian decoders with constant variance, as commonly used in recent work,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 693, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 242, + 705 + ], + "score": 1.0, + "content": "and show that the hyperparameter", + "type": "text" + }, + { + "bbox": [ + 242, + 693, + 250, + 704 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 693, + 506, + 705 + ], + "score": 1.0, + "content": "can be incorporated in the decoding likelihood for these models.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49.5, + "bbox_fs": [ + 105, + 638, + 506, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 332, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 333, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 116, + 93 + ], + "score": 1.0, + "content": "3", + "type": "text" + }, + { + "bbox": [ + 123, + 81, + 333, + 96 + ], + "score": 1.0, + "content": "ANALYSING DECODING DISTRIBUTIONS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 238 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "The generative model of a VAE (Kingma & Welling, 2014; Rezende et al., 2014) with parameters", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 117, + 507, + 130 + ], + "spans": [ + { + "bbox": [ + 107, + 118, + 113, + 127 + ], + "score": 0.76, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 117, + 368, + 130 + ], + "score": 1.0, + "content": "is specified with a prior distribution over the latent variable", + "type": "text" + }, + { + "bbox": [ + 369, + 117, + 392, + 129 + ], + "score": 0.92, + "content": "p _ { \\theta } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 117, + 507, + 130 + ], + "score": 1.0, + "content": ", commonly unit Gaussian,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 220, + 141 + ], + "score": 1.0, + "content": "and a decoding distribution", + "type": "text" + }, + { + "bbox": [ + 221, + 128, + 252, + 140 + ], + "score": 0.91, + "content": "p _ { \\theta } ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 128, + 506, + 141 + ], + "score": 1.0, + "content": ", which for color images is commonly a conditional Gaussian", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 139, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 104, + 139, + 506, + 152 + ], + "score": 1.0, + "content": "parameterized with a neural network. We would like to fit this generative model to a given dataset by", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 496, + 162 + ], + "score": 1.0, + "content": "maximizing the evidence lower bound (ELBO (Neal & Hinton, 1998; Jordan et al., 1999; Kingma", + "type": "text" + }, + { + "bbox": [ + 496, + 150, + 505, + 160 + ], + "score": 0.37, + "content": "\\&", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 451, + 174 + ], + "score": 1.0, + "content": "Welling, 2014; Rezende et al., 2014)), which uses an approximate posterior distribution", + "type": "text" + }, + { + "bbox": [ + 451, + 161, + 483, + 173 + ], + "score": 0.93, + "content": "q _ { \\phi } ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 161, + 506, + 174 + ], + "score": 1.0, + "content": ", also", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "commonly a conditional Gaussian specified with a neural network. In this work, we focus on the form", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 222, + 195 + ], + "score": 1.0, + "content": "of the decoding distribution", + "type": "text" + }, + { + "bbox": [ + 222, + 183, + 254, + 195 + ], + "score": 0.92, + "content": "p _ { \\theta } ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 183, + 505, + 195 + ], + "score": 1.0, + "content": ". To achieve the best results, we want a decoding distribution", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 262, + 206 + ], + "score": 1.0, + "content": "that represents the required probability", + "type": "text" + }, + { + "bbox": [ + 263, + 194, + 290, + 206 + ], + "score": 0.92, + "content": "p ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "accurately In this section, we will review and analyze", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "various choices of decoding distributions that enable better decoder calibration, including expressive", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "decoding distributions that can represent both the prediction of the image and the uncertainty about", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 302, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 302, + 238 + ], + "score": 1.0, + "content": "such prediction, or even multimodal predictions.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 107, + 251, + 227, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 228, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 228, + 265 + ], + "score": 1.0, + "content": "3.1 GAUSSIAN DECODERS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "We first analyse the commonly used Gaussian decoders. We note that the commonly used MSE", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 290, + 295 + ], + "score": 1.0, + "content": "reconstruction loss between the reconstruction", + "type": "text" + }, + { + "bbox": [ + 291, + 284, + 298, + 293 + ], + "score": 0.83, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 284, + 385, + 295 + ], + "score": 1.0, + "content": "and ground truth data", + "type": "text" + }, + { + "bbox": [ + 385, + 285, + 393, + 293 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 284, + 505, + 295 + ], + "score": 1.0, + "content": "is equivalent to the negative", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 294, + 453, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 453, + 307 + ], + "score": 1.0, + "content": "log-likelihood objective with a Gaussian decoding distribution with constant variance:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 310, + 467, + 334 + ], + "lines": [ + { + "bbox": [ + 142, + 310, + 467, + 334 + ], + "spans": [ + { + "bbox": [ + 142, + 310, + 467, + 334 + ], + "score": 0.91, + "content": "- \\ln p ( x | z ) = { \\frac { 1 } { 2 } } | | { \\hat { x } } - x | | ^ { 2 } + D \\ln { \\sqrt { 2 \\pi } } = { \\frac { 1 } { 2 } } | | { \\hat { x } } - x | | ^ { 2 } + c = { \\frac { D } { 2 } } \\mathbf { M } \\mathbf { S } \\mathbf { E } ( { \\hat { x } } , x ) + c ,", + "type": "interline_equation", + "image_path": "c236e8704d065078d3942c790a28d80b4b1fa686a12a6225196ec3965226d373.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 142, + 310, + 467, + 334 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 339, + 504, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 133, + 352 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 339, + 207, + 351 + ], + "score": 0.92, + "content": "p ( x | z ) \\sim \\mathcal { N } ( \\hat { x } , I )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 338, + 270, + 352 + ], + "score": 1.0, + "content": ", the prediction", + "type": "text" + }, + { + "bbox": [ + 270, + 340, + 277, + 349 + ], + "score": 0.83, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 338, + 419, + 352 + ], + "score": 1.0, + "content": "is produced with a neural network", + "type": "text" + }, + { + "bbox": [ + 420, + 339, + 463, + 351 + ], + "score": 0.93, + "content": "\\hat { x } = \\mu _ { \\theta } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 338, + 484, + 352 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 485, + 340, + 494, + 349 + ], + "score": 0.85, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 349, + 205, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 194, + 363 + ], + "score": 1.0, + "content": "the dimensionality of", + "type": "text" + }, + { + "bbox": [ + 194, + 353, + 200, + 360 + ], + "score": 0.8, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 349, + 205, + 363 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 506, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "This demonstrates a drawback of methods that rely simply on the MSE loss (Castrejon et al., 2019;", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "Denton & Fergus, 2018; Lee et al., 2019; Hafner et al., 2019b; Pong et al., 2019; Zhu et al., 2017;", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "score": 1.0, + "content": "Henaff et al., 2019), as it is equivalent to assuming a particular, constant variance of the Gaussian", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "decoding distribution. By learning this variance, we can achieve much better performance due to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "better calibration of the decoder. There are several ways in which we can specify this variance. An", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "expressive way to specify the variance is to specify a diagonal covariance matrix for the image, with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 507, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 507, + 446 + ], + "score": 1.0, + "content": "one value per pixel (Kingma & Welling, 2014; Sønderby et al., 2016; Rolfe, 2016). 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While powerful, we observe in Section 5.3 that this approach attains suboptimal performance,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 503, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 506, + 515 + ], + "score": 1.0, + "content": "and is moreover prone to numerical instability. 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Of particular interest is the interpretation of this parameter. 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We would like to fit this generative model to a given dataset by", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 496, + 162 + ], + "score": 1.0, + "content": "maximizing the evidence lower bound (ELBO (Neal & Hinton, 1998; Jordan et al., 1999; Kingma", + "type": "text" + }, + { + "bbox": [ + 496, + 150, + 505, + 160 + ], + "score": 0.37, + "content": "\\&", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 451, + 174 + ], + "score": 1.0, + "content": "Welling, 2014; Rezende et al., 2014)), which uses an approximate posterior distribution", + "type": "text" + }, + { + "bbox": [ + 451, + 161, + 483, + 173 + ], + "score": 0.93, + "content": "q _ { \\phi } ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 161, + 506, + 174 + ], + "score": 1.0, + "content": ", also", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "commonly a conditional Gaussian specified with a neural network. In this work, we focus on the form", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 222, + 195 + ], + "score": 1.0, + "content": "of the decoding distribution", + "type": "text" + }, + { + "bbox": [ + 222, + 183, + 254, + 195 + ], + "score": 0.92, + "content": "p _ { \\theta } ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 183, + 505, + 195 + ], + "score": 1.0, + "content": ". To achieve the best results, we want a decoding distribution", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 262, + 206 + ], + "score": 1.0, + "content": "that represents the required probability", + "type": "text" + }, + { + "bbox": [ + 263, + 194, + 290, + 206 + ], + "score": 0.92, + "content": "p ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "accurately In this section, we will review and analyze", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "various choices of decoding distributions that enable better decoder calibration, including expressive", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "decoding distributions that can represent both the prediction of the image and the uncertainty about", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 302, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 302, + 238 + ], + "score": 1.0, + "content": "such prediction, or even multimodal predictions.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6.5, + "bbox_fs": [ + 104, + 106, + 507, + 238 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 251, + 227, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 228, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 228, + 265 + ], + "score": 1.0, + "content": "3.1 GAUSSIAN DECODERS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "We first analyse the commonly used Gaussian decoders. We note that the commonly used MSE", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 290, + 295 + ], + "score": 1.0, + "content": "reconstruction loss between the reconstruction", + "type": "text" + }, + { + "bbox": [ + 291, + 284, + 298, + 293 + ], + "score": 0.83, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 284, + 385, + 295 + ], + "score": 1.0, + "content": "and ground truth data", + "type": "text" + }, + { + "bbox": [ + 385, + 285, + 393, + 293 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 284, + 505, + 295 + ], + "score": 1.0, + "content": "is equivalent to the negative", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 294, + 453, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 453, + 307 + ], + "score": 1.0, + "content": "log-likelihood objective with a Gaussian decoding distribution with constant variance:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 272, + 505, + 307 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 310, + 467, + 334 + ], + "lines": [ + { + "bbox": [ + 142, + 310, + 467, + 334 + ], + "spans": [ + { + "bbox": [ + 142, + 310, + 467, + 334 + ], + "score": 0.91, + "content": "- \\ln p ( x | z ) = { \\frac { 1 } { 2 } } | | { \\hat { x } } - x | | ^ { 2 } + D \\ln { \\sqrt { 2 \\pi } } = { \\frac { 1 } { 2 } } | | { \\hat { x } } - x | | ^ { 2 } + c = { \\frac { D } { 2 } } \\mathbf { M } \\mathbf { S } \\mathbf { E } ( { \\hat { x } } , x ) + c ,", + "type": "interline_equation", + "image_path": "c236e8704d065078d3942c790a28d80b4b1fa686a12a6225196ec3965226d373.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 142, + 310, + 467, + 334 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 339, + 504, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 133, + 352 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 339, + 207, + 351 + ], + "score": 0.92, + "content": "p ( x | z ) \\sim \\mathcal { N } ( \\hat { x } , I )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 338, + 270, + 352 + ], + "score": 1.0, + "content": ", the prediction", + "type": "text" + }, + { + "bbox": [ + 270, + 340, + 277, + 349 + ], + "score": 0.83, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 338, + 419, + 352 + ], + "score": 1.0, + "content": "is produced with a neural network", + "type": "text" + }, + { + "bbox": [ + 420, + 339, + 463, + 351 + ], + "score": 0.93, + "content": "\\hat { x } = \\mu _ { \\theta } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 338, + 484, + 352 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 485, + 340, + 494, + 349 + ], + "score": 0.85, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 349, + 205, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 194, + 363 + ], + "score": 1.0, + "content": "the dimensionality of", + "type": "text" + }, + { + "bbox": [ + 194, + 353, + 200, + 360 + ], + "score": 0.8, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 349, + 205, + 363 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 338, + 506, + 363 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 506, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "This demonstrates a drawback of methods that rely simply on the MSE loss (Castrejon et al., 2019;", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "Denton & Fergus, 2018; Lee et al., 2019; Hafner et al., 2019b; Pong et al., 2019; Zhu et al., 2017;", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "score": 1.0, + "content": "Henaff et al., 2019), as it is equivalent to assuming a particular, constant variance of the Gaussian", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "decoding distribution. By learning this variance, we can achieve much better performance due to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "better calibration of the decoder. There are several ways in which we can specify this variance. An", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "expressive way to specify the variance is to specify a diagonal covariance matrix for the image, with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 507, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 507, + 446 + ], + "score": 1.0, + "content": "one value per pixel (Kingma & Welling, 2014; Sønderby et al., 2016; Rolfe, 2016). This can be done,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 265, + 457 + ], + "score": 1.0, + "content": "for example, by letting a neural network", + "type": "text" + }, + { + "bbox": [ + 266, + 446, + 276, + 455 + ], + "score": 0.86, + "content": "\\sigma _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "output the diagonal entries of the covariance matrix given", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 456, + 179, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 168, + 468 + ], + "score": 1.0, + "content": "a latent sample", + "type": "text" + }, + { + "bbox": [ + 169, + 457, + 175, + 465 + ], + "score": 0.73, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 456, + 179, + 468 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 366, + 507, + 468 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 464, + 369, + 479 + ], + "lines": [ + { + "bbox": [ + 242, + 464, + 369, + 479 + ], + "spans": [ + { + "bbox": [ + 242, + 464, + 369, + 479 + ], + "score": 0.95, + "content": "p _ { \\theta } ( x | z ) \\sim \\mathcal { N } \\left( \\mu _ { \\theta } ( z ) , \\sigma _ { \\theta } ( z ) ^ { 2 } \\right) .", + "type": "interline_equation", + "image_path": "4d794c3401c07a472866baac752c1485e32fde12d60d0ce7075bfe703c077245.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 242, + 464, + 369, + 479 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 480, + 506, + 547 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 492 + ], + "score": 1.0, + "content": "This parameterization of the decoding distribution outputs one variance value per each pixel and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 490, + 507, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 507, + 505 + ], + "score": 1.0, + "content": "channel. While powerful, we observe in Section 5.3 that this approach attains suboptimal performance,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 503, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 506, + 515 + ], + "score": 1.0, + "content": "and is moreover prone to numerical instability. Instead, we will find experimentally that a simpler", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 514, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 526 + ], + "score": 1.0, + "content": "parameterization, in which the covariance matrix is specified with a single shared (Kingma et al.,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 524, + 504, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 452, + 537 + ], + "score": 1.0, + "content": "2016; Dai & Wipf, 2019; Edwards & Storkey, 2016; Rezende & Viola, 2018) parameter", + "type": "text" + }, + { + "bbox": [ + 452, + 527, + 459, + 534 + ], + "score": 0.79, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 524, + 471, + 537 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 471, + 525, + 504, + 534 + ], + "score": 0.9, + "content": "\\Sigma = \\sigma I", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 535, + 228, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 228, + 548 + ], + "score": 1.0, + "content": "often works better in practice:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 481, + 507, + 548 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 551, + 367, + 567 + ], + "lines": [ + { + "bbox": [ + 244, + 551, + 367, + 567 + ], + "spans": [ + { + "bbox": [ + 244, + 551, + 367, + 567 + ], + "score": 0.93, + "content": "p _ { \\theta , \\sigma } ( x | z ) \\sim \\mathcal { N } \\left( \\mu _ { \\theta } ( z ) , \\sigma ^ { 2 } I \\right) .", + "type": "interline_equation", + "image_path": "10ece0a186fda45ce7ab58d2efe0f07328f9eccc8c58e98e0fdec58a2f0b2351.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 244, + 551, + 367, + 567 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 506, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 167, + 583 + ], + "score": 1.0, + "content": "The parameter", + "type": "text" + }, + { + "bbox": [ + 168, + 573, + 175, + 581 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 570, + 441, + 583 + ], + "score": 1.0, + "content": "can be optimized together with parameters of the neural network", + "type": "text" + }, + { + "bbox": [ + 441, + 571, + 447, + 581 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "with gradient", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "descent. Of particular interest is the interpretation of this parameter. Writing out the expression for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 593, + 246, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 246, + 604 + ], + "score": 1.0, + "content": "the decoding likelihood, we obtain", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 570, + 506, + 604 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 609, + 510, + 633 + ], + "lines": [ + { + "bbox": [ + 111, + 609, + 510, + 633 + ], + "spans": [ + { + "bbox": [ + 111, + 609, + 510, + 633 + ], + "score": 0.91, + "content": "- \\ln p ( x | z ) = { \\frac { 1 } { 2 \\sigma ^ { 2 } } } | | { \\hat { x } } - x | | ^ { 2 } + D \\ln \\sigma { \\sqrt { 2 \\pi } } = { \\frac { 1 } { 2 \\sigma ^ { 2 } } } | | { \\hat { x } } - x | | ^ { 2 } + D \\ln \\sigma + c = D \\ln \\sigma + { \\frac { D } { 2 \\sigma ^ { 2 } } } \\mathrm { M S E } ( { \\hat { x } } , x ) + c .", + "type": "interline_equation", + "image_path": "e9ce1b76940634dda70fb64b4f302a863d2642241f14f5fc232a274043855271.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 111, + 609, + 510, + 633 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 642, + 324, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 324, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 281, + 655 + ], + "score": 1.0, + "content": "The full objective of the resulting Gaussian", + "type": "text" + }, + { + "bbox": [ + 281, + 645, + 288, + 653 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 642, + 324, + 655 + ], + "score": 1.0, + "content": "-VAE is:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 106, + 642, + 324, + 655 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 659, + 426, + 684 + ], + "lines": [ + { + "bbox": [ + 185, + 659, + 426, + 684 + ], + "spans": [ + { + "bbox": [ + 185, + 659, + 426, + 684 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { \\boldsymbol { \\theta } , \\boldsymbol { \\phi } , \\boldsymbol { \\sigma } } = D \\ln \\sigma + \\frac { D } { 2 \\sigma ^ { 2 } } M S E ( \\hat { x } , { x } ) + D _ { K L } ( q ( \\boldsymbol { z } | { x } ) | | p ( \\boldsymbol { z } ) ) .", + "type": "interline_equation", + "image_path": "1ff481ef555b6fcd22ba8bd4652b3a15d00fec2ca56c1d2e1aa2d3a5b9ba17c2.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 185, + 659, + 426, + 684 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 146, + 700 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 690, + 153, + 698 + ], + "score": 0.73, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "may be viewed as a weighting parameter between the MSE reconstruction term and the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "KL-divergence term in the objective. Moreover, this objective explicitly specifies how to select the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "optimal variance: the variance should be selected to minimize the (weighted) MSE loss while also", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 721, + 275, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 275, + 734 + ], + "score": 1.0, + "content": "minimizing the logarithm of the variance.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 687, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Decoder Calibration It is important that the decoder distribution be calibrated in the statistical", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "sense, that is, the predicted probabilities should correspond to the frequencies of seeing a particular", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 142, + 117 + ], + "score": 1.0, + "content": "value of", + "type": "text" + }, + { + "bbox": [ + 142, + 107, + 149, + 114 + ], + "score": 0.72, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "given that prediction (DeGroot & Fienberg, 1983; Dawid, 1982). The calibration of a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 113, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 113, + 506, + 128 + ], + "score": 1.0, + "content": "neural network can be usually improved by estimating the uncertainty of that prediction (Guo et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 138 + ], + "score": 1.0, + "content": "2017), such as the variance of a Gaussian (Kendall & Gal, 2017). Since the naive MSE loss assumes", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "score": 1.0, + "content": "a constant variance, it does not effectively represent the uncertainty of the prediction, and is often", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 160 + ], + "score": 1.0, + "content": "poorly calibrated. Instead, learning the variance as in Eq. 3 leads to better uncertainty estimation and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "better calibration. In Sec 5.1, we show that learning a good estimate of this uncertainty is crucial for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 171, + 249, + 182 + ], + "spans": [ + { + "bbox": [ + 107, + 171, + 249, + 182 + ], + "score": 1.0, + "content": "the quality of the VAE generations.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 192, + 504, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 168, + 206 + ], + "score": 1.0, + "content": "Connection to", + "type": "text" + }, + { + "bbox": [ + 169, + 194, + 176, + 205 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 192, + 228, + 206 + ], + "score": 1.0, + "content": "-VAE. The", + "type": "text" + }, + { + "bbox": [ + 228, + 194, + 236, + 205 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 192, + 505, + 206 + ], + "score": 1.0, + "content": "-VAE objective (Higgins et al., 2017) for a Gaussian decoder with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 204, + 173, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 173, + 216 + ], + "score": 1.0, + "content": "unit variance is:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 212, + 399, + 236 + ], + "lines": [ + { + "bbox": [ + 211, + 212, + 399, + 236 + ], + "spans": [ + { + "bbox": [ + 211, + 212, + 399, + 236 + ], + "score": 0.94, + "content": "\\mathcal { L } ^ { \\beta } = \\frac { D } { 2 } M S E ( \\hat { x } , x ) + \\beta D _ { K L } ( q ( z | x ) | | p ( z ) ) .", + "type": "interline_equation", + "image_path": "e93051f671722b89c544495fecdd8cb58bb777c81c0a004270b3c3f43e353324.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 211, + 212, + 399, + 236 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "We see that it can be interpreted as a particular case of the objective (3), where the variance is constant", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 160, + 259 + ], + "score": 1.0, + "content": "and the term", + "type": "text" + }, + { + "bbox": [ + 161, + 247, + 188, + 257 + ], + "score": 0.87, + "content": "D \\ln \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 247, + 359, + 259 + ], + "score": 1.0, + "content": "can be ignored during optimization. The", + "type": "text" + }, + { + "bbox": [ + 360, + 248, + 366, + 258 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "-VAE objective is then equivalent", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 124, + 272 + ], + "score": 1.0, + "content": "to a", + "type": "text" + }, + { + "bbox": [ + 125, + 261, + 132, + 269 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 259, + 258, + 272 + ], + "score": 1.0, + "content": "-VAE with a constant variance", + "type": "text" + }, + { + "bbox": [ + 259, + 259, + 306, + 272 + ], + "score": 0.92, + "content": "\\sigma = \\sqrt { \\beta / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "(for a particular learning rate setting). In recent", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 234, + 282 + ], + "score": 1.0, + "content": "work (Zhu et al., 2017; Denton", + "type": "text" + }, + { + "bbox": [ + 234, + 271, + 243, + 280 + ], + "score": 0.33, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 269, + 373, + 282 + ], + "score": 1.0, + "content": "Fergus, 2018; Lee et al., 2019),", + "type": "text" + }, + { + "bbox": [ + 373, + 272, + 380, + 282 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "-VAE models are often used in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 240, + 293 + ], + "score": 1.0, + "content": "this exact regime. By tuning the", + "type": "text" + }, + { + "bbox": [ + 240, + 282, + 248, + 293 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "term, practitioners are able to tune the variance of the decoder,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 419, + 305 + ], + "score": 1.0, + "content": "manually producing a more calibrated decoder. However, by re-interpreting the", + "type": "text" + }, + { + "bbox": [ + 420, + 294, + 426, + 304 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "-VAE objective as a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 312, + 316 + ], + "score": 1.0, + "content": "special case of the VAE and introducing the missing", + "type": "text" + }, + { + "bbox": [ + 312, + 305, + 339, + 313 + ], + "score": 0.63, + "content": "D \\ln \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 302, + 506, + 316 + ], + "score": 1.0, + "content": "term, we can both obtain a valid evidence", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 323, + 327 + ], + "score": 1.0, + "content": "lower bound, and remove the need to manually select", + "type": "text" + }, + { + "bbox": [ + 323, + 316, + 330, + 326 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 313, + 419, + 327 + ], + "score": 1.0, + "content": ". Instead, the variance", + "type": "text" + }, + { + "bbox": [ + 420, + 316, + 427, + 324 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 313, + 505, + 327 + ], + "score": 1.0, + "content": "can instead simply", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 323, + 381, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 381, + 339 + ], + "score": 1.0, + "content": "be learned end-to-end, reducing the need for hyperparameter tuning.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 341, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "An alternative discussion of this connection in the context of linear VAEs is also presented by Lucas", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 201, + 365 + ], + "score": 1.0, + "content": "et al. (2019). While the", + "type": "text" + }, + { + "bbox": [ + 202, + 353, + 209, + 364 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "term is not necessary for good performance if the decoder is calibrated, it", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "can still be employed if desired, such as when the aim is to attain better disentanglement (Higgins", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "et al., 2017) or a particular rate-distortion tradeoff (Alemi et al., 2017). However, we found that with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 385, + 383, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 353, + 398 + ], + "score": 1.0, + "content": "calibrated decoders, the best sample quality is obtained when", + "type": "text" + }, + { + "bbox": [ + 353, + 386, + 378, + 397 + ], + "score": 0.89, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 385, + 383, + 398 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 408, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 422 + ], + "score": 1.0, + "content": "Loss implementation details. For the correct evidence lower bound computation, it is necessary", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 420, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 506, + 432 + ], + "score": 1.0, + "content": "to add the values of the MSE loss and the KL divergence across the dimensions. We observe that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "common implementations of these losses (Denton & Fergus, 2018; Abadi et al., 2016; Paszke et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 442, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 506, + 454 + ], + "score": 1.0, + "content": "2019) use averaging instead, which will lead to poor results if the number of image dimensions is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "significantly different from the number of the latent dimensions. While this can be conveniently", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 463, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 164, + 476 + ], + "score": 1.0, + "content": "ignored in the", + "type": "text" + }, + { + "bbox": [ + 164, + 464, + 171, + 475 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 463, + 457, + 476 + ], + "score": 1.0, + "content": "-VAE regime, where the balance term is tuned manually anyway, for the", + "type": "text" + }, + { + "bbox": [ + 457, + 465, + 464, + 474 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 463, + 506, + 476 + ], + "score": 1.0, + "content": "-VAE it is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 474, + 308, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 308, + 488 + ], + "score": 1.0, + "content": "essential to compute the objective value correctly.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "Variance implementation details. Since the variance is non-negative, we parameterize it logarith-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 506, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 104, + 506, + 151, + 522 + ], + "score": 1.0, + "content": "mically as", + "type": "text" + }, + { + "bbox": [ + 151, + 508, + 191, + 519 + ], + "score": 0.92, + "content": "\\sigma ^ { \\hat { 2 } } = e ^ { 2 \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 506, + 223, + 522 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 223, + 509, + 231, + 519 + ], + "score": 0.76, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 506, + 506, + 522 + ], + "score": 1.0, + "content": "is the logarithm of the standard deviation. For some models, such", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "score": 1.0, + "content": "as per-pixel variance decoders, we observed that it is necessary to restrict the variance range for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 531, + 502, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 502, + 543 + ], + "score": 1.0, + "content": "numerical stability. We do so by using the soft clipping operations proposed by Chua et al. (2018):", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 544, + 457, + 558 + ], + "lines": [ + { + "bbox": [ + 153, + 544, + 457, + 558 + ], + "spans": [ + { + "bbox": [ + 153, + 544, + 457, + 558 + ], + "score": 0.81, + "content": "\\lambda : = \\lambda _ { \\mathrm { m a x } } - \\mathrm { s o f t p l u s } ( \\lambda _ { \\mathrm { m a x } } - \\lambda ) ; \\qquad \\lambda : = \\lambda _ { \\mathrm { m i n } } + \\mathrm { s o f t p l u s } ( \\lambda - \\lambda _ { \\mathrm { m i n } } ) .", + "type": "interline_equation", + "image_path": "069c18d427b4d691f0bd28e1d05bab7551b339d9e5175e7b10afc9c53905f025.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 153, + 544, + 457, + 558 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 560, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 108, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 108, + 559, + 206, + 573 + ], + "score": 1.0, + "content": "We observe that setting", + "type": "text" + }, + { + "bbox": [ + 206, + 560, + 251, + 572 + ], + "score": 0.92, + "content": "\\lambda _ { \\operatorname* { m i n } } = - 6", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "to lower bound the standard deviation to be at least half of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 584 + ], + "score": 1.0, + "content": "distance between allowed color values works well in practice. We also observe that this clipping is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 582, + 287, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 252, + 594 + ], + "score": 1.0, + "content": "unnecessary when learning a shared", + "type": "text" + }, + { + "bbox": [ + 252, + 584, + 259, + 592 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 582, + 287, + 594 + ], + "score": 1.0, + "content": "value.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 108, + 606, + 225, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 226, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 226, + 618 + ], + "score": 1.0, + "content": "3.2 DISCRETE DECODERS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 506, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "It is possible to use discrete decoding distributions to generate images, as color values are commonly", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "restricted to a fixed set of integer pixel intensities (e.g. 0..255). Indeed, for discrete color values,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "discrete distributions are arguably more appropriate. In the most general case, a discrete decoding", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "distribution factorized per each pixel and channel would be specified by a probability mass vector", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 113, + 681 + ], + "score": 0.79, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "with 256 entries, one per each possible intensity value, similarly to a per-pixel classifier of the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 683, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 505, + 694 + ], + "score": 1.0, + "content": "intensity value. We can implement it with a soft-max layer, yielding the following log-likelihood loss", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 693, + 401, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 392, + 705 + ], + "score": 1.0, + "content": "(sometimes called the cross-entropy loss) for a true pixel with intensity", + "type": "text" + }, + { + "bbox": [ + 392, + 694, + 396, + 703 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 693, + 401, + 705 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 707, + 372, + 736 + ], + "lines": [ + { + "bbox": [ + 237, + 707, + 372, + 736 + ], + "spans": [ + { + "bbox": [ + 237, + 707, + 372, + 736 + ], + "score": 0.94, + "content": "- \\ln { p ( x | z ) } = - \\ln { \\frac { \\exp ( \\hat { x } _ { i } ) } { \\sum _ { j } \\exp ( \\hat { x } _ { j } ) } } ,", + "type": "interline_equation", + "image_path": "7a592e2dd0fc7267beca57f9a2ab0de92e4d4abd62ede1bb6299a81ed41d7736.jpg" + } + ] + } + ], + "index": 49.5, + "virtual_lines": [ + { + "bbox": [ + 237, + 707, + 372, + 721.5 + ], + "spans": [], + "index": 49 + }, + { + "bbox": [ + 237, + 721.5, + 372, + 736.0 + ], + "spans": [], + "index": 50 + } + ] + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Decoder Calibration It is important that the decoder distribution be calibrated in the statistical", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "sense, that is, the predicted probabilities should correspond to the frequencies of seeing a particular", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 142, + 117 + ], + "score": 1.0, + "content": "value of", + "type": "text" + }, + { + "bbox": [ + 142, + 107, + 149, + 114 + ], + "score": 0.72, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "given that prediction (DeGroot & Fienberg, 1983; Dawid, 1982). The calibration of a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 113, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 113, + 506, + 128 + ], + "score": 1.0, + "content": "neural network can be usually improved by estimating the uncertainty of that prediction (Guo et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 138 + ], + "score": 1.0, + "content": "2017), such as the variance of a Gaussian (Kendall & Gal, 2017). Since the naive MSE loss assumes", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "score": 1.0, + "content": "a constant variance, it does not effectively represent the uncertainty of the prediction, and is often", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 160 + ], + "score": 1.0, + "content": "poorly calibrated. Instead, learning the variance as in Eq. 3 leads to better uncertainty estimation and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "better calibration. In Sec 5.1, we show that learning a good estimate of this uncertainty is crucial for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 171, + 249, + 182 + ], + "spans": [ + { + "bbox": [ + 107, + 171, + 249, + 182 + ], + "score": 1.0, + "content": "the quality of the VAE generations.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 81, + 506, + 182 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 192, + 504, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 168, + 206 + ], + "score": 1.0, + "content": "Connection to", + "type": "text" + }, + { + "bbox": [ + 169, + 194, + 176, + 205 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 192, + 228, + 206 + ], + "score": 1.0, + "content": "-VAE. The", + "type": "text" + }, + { + "bbox": [ + 228, + 194, + 236, + 205 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 192, + 505, + 206 + ], + "score": 1.0, + "content": "-VAE objective (Higgins et al., 2017) for a Gaussian decoder with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 204, + 173, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 173, + 216 + ], + "score": 1.0, + "content": "unit variance is:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 106, + 192, + 505, + 216 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 212, + 399, + 236 + ], + "lines": [ + { + "bbox": [ + 211, + 212, + 399, + 236 + ], + "spans": [ + { + "bbox": [ + 211, + 212, + 399, + 236 + ], + "score": 0.94, + "content": "\\mathcal { L } ^ { \\beta } = \\frac { D } { 2 } M S E ( \\hat { x } , x ) + \\beta D _ { K L } ( q ( z | x ) | | p ( z ) ) .", + "type": "interline_equation", + "image_path": "e93051f671722b89c544495fecdd8cb58bb777c81c0a004270b3c3f43e353324.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 211, + 212, + 399, + 236 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "We see that it can be interpreted as a particular case of the objective (3), where the variance is constant", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 160, + 259 + ], + "score": 1.0, + "content": "and the term", + "type": "text" + }, + { + "bbox": [ + 161, + 247, + 188, + 257 + ], + "score": 0.87, + "content": "D \\ln \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 247, + 359, + 259 + ], + "score": 1.0, + "content": "can be ignored during optimization. The", + "type": "text" + }, + { + "bbox": [ + 360, + 248, + 366, + 258 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "-VAE objective is then equivalent", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 124, + 272 + ], + "score": 1.0, + "content": "to a", + "type": "text" + }, + { + "bbox": [ + 125, + 261, + 132, + 269 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 259, + 258, + 272 + ], + "score": 1.0, + "content": "-VAE with a constant variance", + "type": "text" + }, + { + "bbox": [ + 259, + 259, + 306, + 272 + ], + "score": 0.92, + "content": "\\sigma = \\sqrt { \\beta / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "(for a particular learning rate setting). In recent", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 234, + 282 + ], + "score": 1.0, + "content": "work (Zhu et al., 2017; Denton", + "type": "text" + }, + { + "bbox": [ + 234, + 271, + 243, + 280 + ], + "score": 0.33, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 269, + 373, + 282 + ], + "score": 1.0, + "content": "Fergus, 2018; Lee et al., 2019),", + "type": "text" + }, + { + "bbox": [ + 373, + 272, + 380, + 282 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "-VAE models are often used in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 240, + 293 + ], + "score": 1.0, + "content": "this exact regime. By tuning the", + "type": "text" + }, + { + "bbox": [ + 240, + 282, + 248, + 293 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "term, practitioners are able to tune the variance of the decoder,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 419, + 305 + ], + "score": 1.0, + "content": "manually producing a more calibrated decoder. However, by re-interpreting the", + "type": "text" + }, + { + "bbox": [ + 420, + 294, + 426, + 304 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "-VAE objective as a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 312, + 316 + ], + "score": 1.0, + "content": "special case of the VAE and introducing the missing", + "type": "text" + }, + { + "bbox": [ + 312, + 305, + 339, + 313 + ], + "score": 0.63, + "content": "D \\ln \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 302, + 506, + 316 + ], + "score": 1.0, + "content": "term, we can both obtain a valid evidence", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 323, + 327 + ], + "score": 1.0, + "content": "lower bound, and remove the need to manually select", + "type": "text" + }, + { + "bbox": [ + 323, + 316, + 330, + 326 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 313, + 419, + 327 + ], + "score": 1.0, + "content": ". Instead, the variance", + "type": "text" + }, + { + "bbox": [ + 420, + 316, + 427, + 324 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 313, + 505, + 327 + ], + "score": 1.0, + "content": "can instead simply", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 323, + 381, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 381, + 339 + ], + "score": 1.0, + "content": "be learned end-to-end, reducing the need for hyperparameter tuning.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 236, + 506, + 339 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 341, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "An alternative discussion of this connection in the context of linear VAEs is also presented by Lucas", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 201, + 365 + ], + "score": 1.0, + "content": "et al. (2019). While the", + "type": "text" + }, + { + "bbox": [ + 202, + 353, + 209, + 364 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "term is not necessary for good performance if the decoder is calibrated, it", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "can still be employed if desired, such as when the aim is to attain better disentanglement (Higgins", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "et al., 2017) or a particular rate-distortion tradeoff (Alemi et al., 2017). However, we found that with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 385, + 383, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 353, + 398 + ], + "score": 1.0, + "content": "calibrated decoders, the best sample quality is obtained when", + "type": "text" + }, + { + "bbox": [ + 353, + 386, + 378, + 397 + ], + "score": 0.89, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 385, + 383, + 398 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 341, + 506, + 398 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 408, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 422 + ], + "score": 1.0, + "content": "Loss implementation details. For the correct evidence lower bound computation, it is necessary", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 420, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 506, + 432 + ], + "score": 1.0, + "content": "to add the values of the MSE loss and the KL divergence across the dimensions. We observe that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "common implementations of these losses (Denton & Fergus, 2018; Abadi et al., 2016; Paszke et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 442, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 506, + 454 + ], + "score": 1.0, + "content": "2019) use averaging instead, which will lead to poor results if the number of image dimensions is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "significantly different from the number of the latent dimensions. While this can be conveniently", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 463, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 164, + 476 + ], + "score": 1.0, + "content": "ignored in the", + "type": "text" + }, + { + "bbox": [ + 164, + 464, + 171, + 475 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 463, + 457, + 476 + ], + "score": 1.0, + "content": "-VAE regime, where the balance term is tuned manually anyway, for the", + "type": "text" + }, + { + "bbox": [ + 457, + 465, + 464, + 474 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 463, + 506, + 476 + ], + "score": 1.0, + "content": "-VAE it is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 474, + 308, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 308, + 488 + ], + "score": 1.0, + "content": "essential to compute the objective value correctly.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 408, + 506, + 488 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "Variance implementation details. Since the variance is non-negative, we parameterize it logarith-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 506, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 104, + 506, + 151, + 522 + ], + "score": 1.0, + "content": "mically as", + "type": "text" + }, + { + "bbox": [ + 151, + 508, + 191, + 519 + ], + "score": 0.92, + "content": "\\sigma ^ { \\hat { 2 } } = e ^ { 2 \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 506, + 223, + 522 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 223, + 509, + 231, + 519 + ], + "score": 0.76, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 506, + 506, + 522 + ], + "score": 1.0, + "content": "is the logarithm of the standard deviation. For some models, such", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "score": 1.0, + "content": "as per-pixel variance decoders, we observed that it is necessary to restrict the variance range for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 531, + 502, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 502, + 543 + ], + "score": 1.0, + "content": "numerical stability. We do so by using the soft clipping operations proposed by Chua et al. (2018):", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 497, + 506, + 543 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 544, + 457, + 558 + ], + "lines": [ + { + "bbox": [ + 153, + 544, + 457, + 558 + ], + "spans": [ + { + "bbox": [ + 153, + 544, + 457, + 558 + ], + "score": 0.81, + "content": "\\lambda : = \\lambda _ { \\mathrm { m a x } } - \\mathrm { s o f t p l u s } ( \\lambda _ { \\mathrm { m a x } } - \\lambda ) ; \\qquad \\lambda : = \\lambda _ { \\mathrm { m i n } } + \\mathrm { s o f t p l u s } ( \\lambda - \\lambda _ { \\mathrm { m i n } } ) .", + "type": "interline_equation", + "image_path": "069c18d427b4d691f0bd28e1d05bab7551b339d9e5175e7b10afc9c53905f025.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 153, + 544, + 457, + 558 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 560, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 108, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 108, + 559, + 206, + 573 + ], + "score": 1.0, + "content": "We observe that setting", + "type": "text" + }, + { + "bbox": [ + 206, + 560, + 251, + 572 + ], + "score": 0.92, + "content": "\\lambda _ { \\operatorname* { m i n } } = - 6", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "to lower bound the standard deviation to be at least half of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 584 + ], + "score": 1.0, + "content": "distance between allowed color values works well in practice. We also observe that this clipping is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 582, + 287, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 252, + 594 + ], + "score": 1.0, + "content": "unnecessary when learning a shared", + "type": "text" + }, + { + "bbox": [ + 252, + 584, + 259, + 592 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 582, + 287, + 594 + ], + "score": 1.0, + "content": "value.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 106, + 559, + 506, + 594 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 606, + 225, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 226, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 226, + 618 + ], + "score": 1.0, + "content": "3.2 DISCRETE DECODERS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 506, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "It is possible to use discrete decoding distributions to generate images, as color values are commonly", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "restricted to a fixed set of integer pixel intensities (e.g. 0..255). Indeed, for discrete color values,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "discrete distributions are arguably more appropriate. In the most general case, a discrete decoding", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "distribution factorized per each pixel and channel would be specified by a probability mass vector", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 113, + 681 + ], + "score": 0.79, + "content": "\\hat { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "with 256 entries, one per each possible intensity value, similarly to a per-pixel classifier of the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 683, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 505, + 694 + ], + "score": 1.0, + "content": "intensity value. We can implement it with a soft-max layer, yielding the following log-likelihood loss", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 693, + 401, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 392, + 705 + ], + "score": 1.0, + "content": "(sometimes called the cross-entropy loss) for a true pixel with intensity", + "type": "text" + }, + { + "bbox": [ + 392, + 694, + 396, + 703 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 693, + 401, + 705 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 627, + 506, + 705 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 707, + 372, + 736 + ], + "lines": [ + { + "bbox": [ + 237, + 707, + 372, + 736 + ], + "spans": [ + { + "bbox": [ + 237, + 707, + 372, + 736 + ], + "score": 0.94, + "content": "- \\ln { p ( x | z ) } = - \\ln { \\frac { \\exp ( \\hat { x } _ { i } ) } { \\sum _ { j } \\exp ( \\hat { x } _ { j } ) } } ,", + "type": "interline_equation", + "image_path": "7a592e2dd0fc7267beca57f9a2ab0de92e4d4abd62ede1bb6299a81ed41d7736.jpg" + } + ] + } + ], + "index": 49.5, + "virtual_lines": [ + { + "bbox": [ + 237, + 707, + 372, + 721.5 + ], + "spans": [], + "index": 49 + }, + { + "bbox": [ + 237, + 721.5, + 372, + 736.0 + ], + "spans": [], + "index": 50 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 65, + 500, + 172 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 65, + 500, + 172 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 65, + 500, + 172 + ], + "spans": [ + { + "bbox": [ + 109, + 65, + 500, + 172 + ], + "score": 0.97, + "type": "image", + "image_path": "a72014d3eeef3fb862e1ae0c45aecac5611320d625332b68e1f68646a77844f0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 65, + 500, + 100.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 100.66666666666666, + 500, + 136.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 136.33333333333331, + 500, + 171.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 176, + 506, + 243 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "score": 1.0, + "content": "Figure 1: Different types of calibrated decoders for Gaussian VAE, model parameters are denoted", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 280, + 200 + ], + "score": 1.0, + "content": "with enclosing squares. Left: both the mean", + "type": "text" + }, + { + "bbox": [ + 280, + 190, + 288, + 199 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 187, + 354, + 200 + ], + "score": 1.0, + "content": "and the variance", + "type": "text" + }, + { + "bbox": [ + 355, + 190, + 362, + 198 + ], + "score": 0.74, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "are output by a neural network with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 104, + 198, + 154, + 211 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 154, + 199, + 160, + 208 + ], + "score": 0.71, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 198, + 199, + 211 + ], + "score": 1.0, + "content": ". Center:", + "type": "text" + }, + { + "bbox": [ + 199, + 200, + 207, + 209 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "-VAE with shared variance, the mean is output by a neural network with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 153, + 222 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 154, + 210, + 160, + 220 + ], + "score": 0.75, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 209, + 456, + 222 + ], + "score": 1.0, + "content": ", but the variance it iself a global parameter. Right: the proposed optimal", + "type": "text" + }, + { + "bbox": [ + 456, + 210, + 463, + 220 + ], + "score": 0.73, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "-VAE, the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 322, + 233 + ], + "score": 1.0, + "content": "mean is output by a neural network with parameters", + "type": "text" + }, + { + "bbox": [ + 323, + 221, + 329, + 230 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 220, + 505, + 233 + ], + "score": 1.0, + "content": ", and the variance is computed analytically", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 231, + 209, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 195, + 244 + ], + "score": 1.0, + "content": "from the training data", + "type": "text" + }, + { + "bbox": [ + 196, + 232, + 205, + 241 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 231, + 209, + 244 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 290 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "We will evaluate these and further choices of discrete decoders, described in Appendix D. We", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 269, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 505, + 280 + ], + "score": 1.0, + "content": "recommend choosing the decoder distribution that best suits the structure of the data, such as discrete", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 279, + 394, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 394, + 291 + ], + "score": 1.0, + "content": "decoders for discrete data and continuous decoders for continuous data.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 105, + 307, + 503, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 322 + ], + "score": 1.0, + "content": "4 OPTIMAL VARIANCE ESTIMATION FOR CALIBRATED GAUSSIAN DECODERS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 332, + 506, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "In this section, we propose a simple but novel analytic way of obtaining a calibrated decoder for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "score": 1.0, + "content": "continuous distributions that further improves performance. The Gaussian decoders with learned", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 355, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 506, + 367 + ], + "score": 1.0, + "content": "variance described in Section 3.1 are calibrated and work better than na¨ıve unit variance decoders.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 162, + 378 + ], + "score": 1.0, + "content": "However, for", + "type": "text" + }, + { + "bbox": [ + 162, + 366, + 169, + 376 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "-VAE optimized with gradient descent or Adam (Kingma & Ba, 2015), we observe", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "that careful learning rate tuning can yield significantly better performance, which is in line with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "prior work that reported poor performance of gradient descent for optimizing Gaussian distributions", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "(Amari, 1998; Peters & Schaal, 2008). A smaller learning rate often produces better performance, but", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 408, + 507, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 408, + 307, + 424 + ], + "score": 1.0, + "content": "slows down the training, as the likelihood values", + "type": "text" + }, + { + "bbox": [ + 307, + 410, + 335, + 421 + ], + "score": 0.92, + "content": "p ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 408, + 507, + 424 + ], + "score": 1.0, + "content": "will be very suboptimal in the beginning.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 419, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 104, + 419, + 352, + 434 + ], + "score": 1.0, + "content": "Instead, here we propose an analytic solution for the value of", + "type": "text" + }, + { + "bbox": [ + 353, + 422, + 360, + 430 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 419, + 506, + 434 + ], + "score": 1.0, + "content": ", which computes it analytically and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 431, + 242, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 242, + 444 + ], + "score": 1.0, + "content": "does not require gradient descent.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 108, + 448, + 503, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 461 + ], + "score": 1.0, + "content": "The maximum likelihood estimate of the variance given a known mean is the average squared distance", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 458, + 170, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 170, + 471 + ], + "score": 1.0, + "content": "from the mean:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 468, + 392, + 489 + ], + "lines": [ + { + "bbox": [ + 218, + 468, + 392, + 489 + ], + "spans": [ + { + "bbox": [ + 218, + 468, + 392, + 489 + ], + "score": 0.93, + "content": "\\boldsymbol { \\sigma } ^ { * } = \\mathop { \\arg \\operatorname* { m a x } } _ { \\boldsymbol { \\sigma } } \\mathcal { N } ( \\boldsymbol { x } | \\mu , \\boldsymbol { \\sigma } ^ { 2 } I ) = \\mathbf { M } \\mathbf { S } \\mathbf { E } ( \\boldsymbol { x } , \\mu ) ,", + "type": "interline_equation", + "image_path": "7a61ad70f46b4e59d8462e381b67b2ffc391be510eb34b1cd6bc7ffa80c065af.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 218, + 468, + 392, + 489 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 506, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 133, + 507 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 492, + 259, + 506 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbf { M S E } ( x , \\mu ) = \\frac { 1 } { D } \\sum _ { i } ( x _ { i } - \\mu _ { i } ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 492, + 506, + 507 + ], + "score": 1.0, + "content": ". Eq. 5 can be easily shown using manual differentiation, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 503, + 471, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 471, + 518 + ], + "score": 1.0, + "content": "is a generalization of the fact that the MLE estimate of the variance is the sample variance.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 521, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 505, + 534 + ], + "score": 1.0, + "content": "The optimal variance for the decoder distribution under the maximum likelihood criterion is then", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "score": 1.0, + "content": "simply the average MSE loss over the data and the encoder distribution. We leverage this to create", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "an optimal analytic solution for the variance. In the batch setting, the optimal variance would be", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "simply the MSE loss, and can be updated after every gradient update for the other parameters of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "score": 1.0, + "content": "decoder. In the mini-batch setting, we use a batchwise estimate of the variance computed for the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 575, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 590 + ], + "score": 1.0, + "content": "current minibatch. We analyze these approximations in Appendix C. At test time, a running average", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 433, + 600 + ], + "score": 1.0, + "content": "of the variance over the training data is used. This method, which we call optimal", + "type": "text" + }, + { + "bbox": [ + 434, + 587, + 462, + 597 + ], + "score": 0.45, + "content": "\\sigma { - } V A E", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 587, + 506, + 600 + ], + "score": 1.0, + "content": ", allows us", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 599, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 610 + ], + "score": 1.0, + "content": "to learn very efficiently as we use the optimal variance estimate at every training step. It is also easier", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "to implement, as no separate optimizer for the variance parameter is needed. If the variance is not", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 619, + 367, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 367, + 633 + ], + "score": 1.0, + "content": "needed at test time, it can also be simply discarded after training.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 644, + 505, + 699 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 188, + 656 + ], + "score": 1.0, + "content": "Per-image optimal", + "type": "text" + }, + { + "bbox": [ + 188, + 645, + 195, + 654 + ], + "score": 0.73, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 644, + 266, + 656 + ], + "score": 1.0, + "content": "-VAE. Optimal", + "type": "text" + }, + { + "bbox": [ + 266, + 645, + 273, + 654 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "-VAE uses a single variance value shared across all data", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 223, + 669 + ], + "score": 1.0, + "content": "points. 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This approach", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 688, + 497, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 497, + 700 + ], + "score": 1.0, + "content": "can be interpreted as variational variance prediction in the framework of Stirn & Knowles (2020).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 716, + 257, + 728 + ], + "lines": [ + { + "bbox": [ + 105, + 716, + 258, + 730 + ], + "spans": [ + { + "bbox": [ + 105, + 716, + 258, + 730 + ], + "score": 1.0, + "content": "5 EXPERIMENTAL RESULTS", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 65, + 500, + 172 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 65, + 500, + 172 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 65, + 500, + 172 + ], + "spans": [ + { + "bbox": [ + 109, + 65, + 500, + 172 + ], + "score": 0.97, + "type": "image", + "image_path": "a72014d3eeef3fb862e1ae0c45aecac5611320d625332b68e1f68646a77844f0.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 65, + 500, + 100.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 100.66666666666666, + 500, + 136.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 136.33333333333331, + 500, + 171.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 176, + 506, + 243 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "score": 1.0, + "content": "Figure 1: Different types of calibrated decoders for Gaussian VAE, model parameters are denoted", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 280, + 200 + ], + "score": 1.0, + "content": "with enclosing squares. Left: both the mean", + "type": "text" + }, + { + "bbox": [ + 280, + 190, + 288, + 199 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 187, + 354, + 200 + ], + "score": 1.0, + "content": "and the variance", + "type": "text" + }, + { + "bbox": [ + 355, + 190, + 362, + 198 + ], + "score": 0.74, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "are output by a neural network with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 104, + 198, + 154, + 211 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 154, + 199, + 160, + 208 + ], + "score": 0.71, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 198, + 199, + 211 + ], + "score": 1.0, + "content": ". Center:", + "type": "text" + }, + { + "bbox": [ + 199, + 200, + 207, + 209 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "-VAE with shared variance, the mean is output by a neural network with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 153, + 222 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 154, + 210, + 160, + 220 + ], + "score": 0.75, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 209, + 456, + 222 + ], + "score": 1.0, + "content": ", but the variance it iself a global parameter. Right: the proposed optimal", + "type": "text" + }, + { + "bbox": [ + 456, + 210, + 463, + 220 + ], + "score": 0.73, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "-VAE, the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 322, + 233 + ], + "score": 1.0, + "content": "mean is output by a neural network with parameters", + "type": "text" + }, + { + "bbox": [ + 323, + 221, + 329, + 230 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 220, + 505, + 233 + ], + "score": 1.0, + "content": ", and the variance is computed analytically", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 231, + 209, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 195, + 244 + ], + "score": 1.0, + "content": "from the training data", + "type": "text" + }, + { + "bbox": [ + 196, + 232, + 205, + 241 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 231, + 209, + 244 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 290 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "We will evaluate these and further choices of discrete decoders, described in Appendix D. We", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 269, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 505, + 280 + ], + "score": 1.0, + "content": "recommend choosing the decoder distribution that best suits the structure of the data, such as discrete", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 279, + 394, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 394, + 291 + ], + "score": 1.0, + "content": "decoders for discrete data and continuous decoders for continuous data.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 257, + 505, + 291 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 307, + 503, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 322 + ], + "score": 1.0, + "content": "4 OPTIMAL VARIANCE ESTIMATION FOR CALIBRATED GAUSSIAN DECODERS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 332, + 506, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "In this section, we propose a simple but novel analytic way of obtaining a calibrated decoder for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "score": 1.0, + "content": "continuous distributions that further improves performance. The Gaussian decoders with learned", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 355, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 506, + 367 + ], + "score": 1.0, + "content": "variance described in Section 3.1 are calibrated and work better than na¨ıve unit variance decoders.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 162, + 378 + ], + "score": 1.0, + "content": "However, for", + "type": "text" + }, + { + "bbox": [ + 162, + 366, + 169, + 376 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "-VAE optimized with gradient descent or Adam (Kingma & Ba, 2015), we observe", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "that careful learning rate tuning can yield significantly better performance, which is in line with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "prior work that reported poor performance of gradient descent for optimizing Gaussian distributions", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "(Amari, 1998; Peters & Schaal, 2008). A smaller learning rate often produces better performance, but", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 408, + 507, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 408, + 307, + 424 + ], + "score": 1.0, + "content": "slows down the training, as the likelihood values", + "type": "text" + }, + { + "bbox": [ + 307, + 410, + 335, + 421 + ], + "score": 0.92, + "content": "p ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 408, + 507, + 424 + ], + "score": 1.0, + "content": "will be very suboptimal in the beginning.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 419, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 104, + 419, + 352, + 434 + ], + "score": 1.0, + "content": "Instead, here we propose an analytic solution for the value of", + "type": "text" + }, + { + "bbox": [ + 353, + 422, + 360, + 430 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 419, + 506, + 434 + ], + "score": 1.0, + "content": ", which computes it analytically and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 431, + 242, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 242, + 444 + ], + "score": 1.0, + "content": "does not require gradient descent.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 333, + 507, + 444 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 448, + 503, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 461 + ], + "score": 1.0, + "content": "The maximum likelihood estimate of the variance given a known mean is the average squared distance", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 458, + 170, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 170, + 471 + ], + "score": 1.0, + "content": "from the mean:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 106, + 447, + 505, + 471 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 468, + 392, + 489 + ], + "lines": [ + { + "bbox": [ + 218, + 468, + 392, + 489 + ], + "spans": [ + { + "bbox": [ + 218, + 468, + 392, + 489 + ], + "score": 0.93, + "content": "\\boldsymbol { \\sigma } ^ { * } = \\mathop { \\arg \\operatorname* { m a x } } _ { \\boldsymbol { \\sigma } } \\mathcal { N } ( \\boldsymbol { x } | \\mu , \\boldsymbol { \\sigma } ^ { 2 } I ) = \\mathbf { M } \\mathbf { S } \\mathbf { E } ( \\boldsymbol { x } , \\mu ) ,", + "type": "interline_equation", + "image_path": "7a61ad70f46b4e59d8462e381b67b2ffc391be510eb34b1cd6bc7ffa80c065af.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 218, + 468, + 392, + 489 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 506, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 133, + 507 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 492, + 259, + 506 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbf { M S E } ( x , \\mu ) = \\frac { 1 } { D } \\sum _ { i } ( x _ { i } - \\mu _ { i } ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 492, + 506, + 507 + ], + "score": 1.0, + "content": ". Eq. 5 can be easily shown using manual differentiation, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 503, + 471, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 471, + 518 + ], + "score": 1.0, + "content": "is a generalization of the fact that the MLE estimate of the variance is the sample variance.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 492, + 506, + 518 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 520, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 521, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 505, + 534 + ], + "score": 1.0, + "content": "The optimal variance for the decoder distribution under the maximum likelihood criterion is then", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 545 + ], + "score": 1.0, + "content": "simply the average MSE loss over the data and the encoder distribution. We leverage this to create", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "an optimal analytic solution for the variance. In the batch setting, the optimal variance would be", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "simply the MSE loss, and can be updated after every gradient update for the other parameters of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 579 + ], + "score": 1.0, + "content": "decoder. In the mini-batch setting, we use a batchwise estimate of the variance computed for the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 575, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 590 + ], + "score": 1.0, + "content": "current minibatch. We analyze these approximations in Appendix C. At test time, a running average", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 433, + 600 + ], + "score": 1.0, + "content": "of the variance over the training data is used. This method, which we call optimal", + "type": "text" + }, + { + "bbox": [ + 434, + 587, + 462, + 597 + ], + "score": 0.45, + "content": "\\sigma { - } V A E", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 587, + 506, + 600 + ], + "score": 1.0, + "content": ", allows us", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 599, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 610 + ], + "score": 1.0, + "content": "to learn very efficiently as we use the optimal variance estimate at every training step. It is also easier", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "to implement, as no separate optimizer for the variance parameter is needed. 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Figure 3 shows the qualitative results from this experiment.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 74, + "bbox_fs": [ + 200, + 718, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "the variance than is possible with manual search, improving the likelihood (as measured by the lower", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 466, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 466, + 106 + ], + "score": 1.0, + "content": "bound) and the visual quality. Figure 3 shows the qualitative results from this experiment.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 506, + 123 + ], + "score": 1.0, + "content": "We further validate our results on both single-image and sequential VAE models on a range of datasets", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "in Table 2 and Figure 2. Single-sample ELBO values are reported, and ELBO values on discretized", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "data are reported for discrete distributions. We see that learning a shared variance in a Gaussian", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 177, + 156 + ], + "score": 1.0, + "content": "decoders (shared", + "type": "text" + }, + { + "bbox": [ + 178, + 144, + 185, + 154 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "-VAE) outperforms the na¨ıve unit variance decoder (Gaussian VAE) as well as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 149, + 167 + ], + "score": 1.0, + "content": "tuning the", + "type": "text" + }, + { + "bbox": [ + 149, + 155, + 156, + 165 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "constant for the Gaussian VAE manually. We also see that calibrated discrete decoders,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "score": 1.0, + "content": "such as full categorical distribution or mixture of discretized logistics, perform better than the na¨ıve", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "Gaussian VAE. Using Bernoulli distribution by treating the color intensities as probabilities (Gregor", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "et al., 2015; Watter et al., 2015) performs poorly. Our results further improve upon the sequence VAE", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 196, + 507, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 434, + 212 + ], + "score": 1.0, + "content": "method of Denton & Fergus (2018), which uses a unit variance Gaussian with the", + "type": "text" + }, + { + "bbox": [ + 434, + 199, + 441, + 209 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 196, + 507, + 212 + ], + "score": 1.0, + "content": "-VAE objective.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 107, + 226, + 465, + 247 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 468, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 468, + 237 + ], + "score": 1.0, + "content": "5.2 HOW DOES LEARNING CALIBRATED DECODERS IMPACT THE LATENT VARIABLE", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 129, + 237, + 241, + 248 + ], + "spans": [ + { + "bbox": [ + 129, + 237, + 241, + 248 + ], + "score": 1.0, + "content": "INFORMATION CONTENT?", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "We saw above that calibrated decoders result in higher log-likelihood bounds. Are calibrated", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 472, + 282 + ], + "score": 1.0, + "content": "decoders also beneficial for representation learning? 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Our results further improve upon the sequence VAE", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 196, + 507, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 434, + 212 + ], + "score": 1.0, + "content": "method of Denton & Fergus (2018), which uses a unit variance Gaussian with the", + "type": "text" + }, + { + "bbox": [ + 434, + 199, + 441, + 209 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 196, + 507, + 212 + ], + "score": 1.0, + "content": "-VAE objective.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 110, + 507, + 212 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 226, + 465, + 247 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 468, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 468, + 237 + ], + "score": 1.0, + "content": "5.2 HOW DOES LEARNING CALIBRATED DECODERS IMPACT THE LATENT VARIABLE", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 129, + 237, + 241, + 248 + ], + "spans": [ + { + "bbox": [ + 129, + 237, + 241, + 248 + ], + "score": 1.0, + "content": "INFORMATION CONTENT?", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 270 + ], + "score": 1.0, + "content": "We saw above that calibrated decoders result in higher log-likelihood bounds. 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(2017), who show that", + "type": "text" + }, + { + "bbox": [ + 462, + 710, + 469, + 721 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "controls", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 105, + 719, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 470, + 733 + ], + "score": 1.0, + "content": "the rate-distortion trade-off. Here, we show that the crucial trade-off also controlled by", + "type": "text" + }, + { + "bbox": [ + 471, + 721, + 478, + 732 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 719, + 505, + 733 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 73 + }, + { + "bbox": [ + 106, + 297, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 310 + ], + "score": 1.0, + "content": "trade-off between two components of the rate itself, which control expressivity of representations and", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 308, + 403, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 403, + 321 + ], + "score": 1.0, + "content": "the match between the variational and the prior distributions, respectively.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 72.5, + "bbox_fs": [ + 105, + 709, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 109, + 145, + 502, + 284 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 136 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 507, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 343, + 93 + ], + "score": 1.0, + "content": "Table 2: Generative modeling performance of the proposed", + "type": "text" + }, + { + "bbox": [ + 343, + 82, + 350, + 91 + ], + "score": 0.79, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 79, + 507, + 93 + ], + "score": 1.0, + "content": "-VAE on different models and datasets.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 442, + 104 + ], + "score": 1.0, + "content": "For SVG, we compare with the original method (Denton & Fergus, 2018), which uses", + "type": "text" + }, + { + "bbox": [ + 442, + 91, + 450, + 102 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 90, + 506, + 104 + ], + "score": 1.0, + "content": "-VAE. 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Calibrated decoders such as categorical or", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 124, + 503, + 136 + ], + "spans": [ + { + "bbox": [ + 107, + 125, + 114, + 134 + ], + "score": 0.73, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 124, + 503, + 136 + ], + "score": 1.0, + "content": "-VAE perform best. [1] Gregor et al. (2015), [2] Takahashi et al. (2018), [3] Higgins et al. (2017).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "table_body", + "bbox": [ + 109, + 145, + 502, + 284 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 145, + 502, + 284 + ], + "spans": [ + { + "bbox": [ + 109, + 145, + 502, + 284 + ], + "score": 0.986, + "html": "
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First, we find that this method often", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "diverges very quickly due to numerical instability, as the network is able to predict certain pixels", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 408, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 422 + ], + "score": 1.0, + "content": "with very high certainty, leading to degenerate variances. In contrast, learning a shared variance is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 418, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 434 + ], + "score": 1.0, + "content": "always numerically stable in our experiments. We can rectify this numerical instability by bounding", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 429, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 445 + ], + "score": 1.0, + "content": "the output variance (Section 3.1). However, even with bounded variance, we observe that learning", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "per-pixel variances leads to poor results in Table 2. While the per-pixel variance achieves a good", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 452, + 457, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 457, + 466 + ], + "score": 1.0, + "content": "ELBO value, it produces very poor samples, as measured by FID and visual inspection.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "We see that the specific form of learned variance: a shared variance, a per-image variance, or a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "per-pixel variance, can lead to very different performance in practice. We hypothesize the per-pixel", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "score": 1.0, + "content": "decoder performs poorly as it incentivizes the model to focus on particular pixels that can be predicted", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "well, instead of focusing equally on all parts of the image. This is consistent with prior work on", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "denoising diffusion models which noted that likelihood-based models place too much focus on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "imperceptible details, which leads to deteriorated results (Ho et al., 2020). The shared and per-image", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 534, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 548 + ], + "score": 1.0, + "content": "variance models mitigate this issue at the cost of introducing more bias, and work better in practice.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 107, + 561, + 500, + 571 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 502, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 502, + 572 + ], + "score": 1.0, + "content": "5.4 CAN AN ANALYTIC SOLUTION FOR OPTIMAL VARIANCE FURTHER IMPROVE LEARNING?", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 204, + 593 + ], + "score": 1.0, + "content": "We evaluate the optimal", + "type": "text" + }, + { + "bbox": [ + 204, + 582, + 211, + 591 + ], + "score": 0.7, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 581, + 506, + 593 + ], + "score": 1.0, + "content": "-VAE which uses an analytic solution for the variance (Section 4). Table 2", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 592, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 604 + ], + "score": 1.0, + "content": "shows that it achieves superior results in terms of log-likelihood. We also note that the optimal", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 107, + 604, + 114, + 612 + ], + "score": 0.74, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "-VAE converges to a good variance estimate instantaneously, which speeds up learning (highlighted", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 614, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 425, + 626 + ], + "score": 1.0, + "content": "in Figure 9 in the Appendix). In addition, we evaluate the per-image optimal", + "type": "text" + }, + { + "bbox": [ + 425, + 615, + 432, + 624 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 614, + 506, + 626 + ], + "score": 1.0, + "content": "-VAE, in which a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "single variance is computed per image. This model achieves significantly higher visual quality. While", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 634, + 507, + 650 + ], + "spans": [ + { + "bbox": [ + 104, + 634, + 507, + 650 + ], + "score": 1.0, + "content": "producing this per-image variance with a neural network would require additional architecture tuning,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 138, + 659 + ], + "score": 1.0, + "content": "optimal", + "type": "text" + }, + { + "bbox": [ + 139, + 648, + 146, + 657 + ], + "score": 0.73, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 646, + 506, + 659 + ], + "score": 1.0, + "content": "-VAE is extremely simple to implement (it can be implemented simply as changing the axes", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 657, + 340, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 340, + 670 + ], + "score": 1.0, + "content": "of summation), not requiring any new tunable parameters.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 108, + 685, + 195, + 697 + ], + "lines": [ + { + "bbox": [ + 104, + 682, + 197, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 682, + 197, + 700 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We presented a simple and effective method for learning calibrated decoders, as well as an evaluation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 720, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 504, + 732 + ], + "score": 1.0, + "content": "of different decoding distributions with several VAE and sequential VAE models. 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Calibrated decoders such as categorical or", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 124, + 503, + 136 + ], + "spans": [ + { + "bbox": [ + 107, + 125, + 114, + 134 + ], + "score": 0.73, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 124, + 503, + 136 + ], + "score": 1.0, + "content": "-VAE perform best. [1] Gregor et al. (2015), [2] Takahashi et al. (2018), [3] Higgins et al. (2017).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "table_body", + "bbox": [ + 109, + 145, + 502, + 284 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 145, + 502, + 284 + ], + "spans": [ + { + "bbox": [ + 109, + 145, + 502, + 284 + ], + "score": 0.986, + "html": "
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First, we find that this method often", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "diverges very quickly due to numerical instability, as the network is able to predict certain pixels", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 408, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 422 + ], + "score": 1.0, + "content": "with very high certainty, leading to degenerate variances. In contrast, learning a shared variance is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 418, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 434 + ], + "score": 1.0, + "content": "always numerically stable in our experiments. We can rectify this numerical instability by bounding", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 429, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 445 + ], + "score": 1.0, + "content": "the output variance (Section 3.1). However, even with bounded variance, we observe that learning", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "per-pixel variances leads to poor results in Table 2. While the per-pixel variance achieves a good", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 452, + 457, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 457, + 466 + ], + "score": 1.0, + "content": "ELBO value, it produces very poor samples, as measured by FID and visual inspection.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 365, + 506, + 466 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "We see that the specific form of learned variance: a shared variance, a per-image variance, or a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "per-pixel variance, can lead to very different performance in practice. We hypothesize the per-pixel", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "score": 1.0, + "content": "decoder performs poorly as it incentivizes the model to focus on particular pixels that can be predicted", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "well, instead of focusing equally on all parts of the image. This is consistent with prior work on", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "denoising diffusion models which noted that likelihood-based models place too much focus on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "imperceptible details, which leads to deteriorated results (Ho et al., 2020). The shared and per-image", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 534, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 548 + ], + "score": 1.0, + "content": "variance models mitigate this issue at the cost of introducing more bias, and work better in practice.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 468, + 506, + 548 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 561, + 500, + 571 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 502, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 502, + 572 + ], + "score": 1.0, + "content": "5.4 CAN AN ANALYTIC SOLUTION FOR OPTIMAL VARIANCE FURTHER IMPROVE LEARNING?", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 204, + 593 + ], + "score": 1.0, + "content": "We evaluate the optimal", + "type": "text" + }, + { + "bbox": [ + 204, + 582, + 211, + 591 + ], + "score": 0.7, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 581, + 506, + 593 + ], + "score": 1.0, + "content": "-VAE which uses an analytic solution for the variance (Section 4). Table 2", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 592, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 604 + ], + "score": 1.0, + "content": "shows that it achieves superior results in terms of log-likelihood. We also note that the optimal", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 107, + 604, + 114, + 612 + ], + "score": 0.74, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "-VAE converges to a good variance estimate instantaneously, which speeds up learning (highlighted", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 614, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 425, + 626 + ], + "score": 1.0, + "content": "in Figure 9 in the Appendix). In addition, we evaluate the per-image optimal", + "type": "text" + }, + { + "bbox": [ + 425, + 615, + 432, + 624 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 614, + 506, + 626 + ], + "score": 1.0, + "content": "-VAE, in which a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "single variance is computed per image. This model achieves significantly higher visual quality. 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All methods", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 284, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 284, + 128 + ], + "score": 1.0, + "content": "are compared on the same hyperparameters.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 105, + 145, + 463, + 158 + ], + "lines": [ + { + "bbox": [ + 105, + 144, + 465, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 425, + 159 + ], + "score": 1.0, + "content": "C EMPIRICAL ANALYZIS OF APPROXIMATIONS FOR OPTIMAL", + "type": "text" + }, + { + "bbox": [ + 426, + 147, + 434, + 156 + ], + "score": 0.69, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 144, + 465, + 159 + ], + "score": 1.0, + "content": "-VAE", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 420, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 171, + 420, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 157, + 185 + ], + "score": 1.0, + "content": "The optimal", + "type": "text" + }, + { + "bbox": [ + 157, + 173, + 164, + 181 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 171, + 420, + 185 + ], + "score": 1.0, + "content": "-VAE requires computing the following estimate of the variance", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 189, + 477, + 210 + ], + "lines": [ + { + "bbox": [ + 133, + 189, + 477, + 210 + ], + "spans": [ + { + "bbox": [ + 133, + 189, + 477, + 210 + ], + "score": 0.86, + "content": "\\sigma ^ { * } = \\underset { \\sigma } { \\arg \\operatorname* { m a x } } \\mathbb { E } _ { x \\sim \\mathrm { D a t a } } \\mathbb { E } _ { q ( z | x ) } \\left[ \\ln p ( x | \\mu _ { \\theta } ( z ) , \\sigma ^ { 2 } I ) \\right] = \\mathbb { E } _ { x \\sim \\mathrm { D a t a } } \\mathbb { E } _ { q ( z | x ) } { \\mathbf { M S E } } ( x , \\mu _ { \\theta } ( z ) ) .", + "type": "interline_equation", + "image_path": "0bc0912152c18d52da4d5f676c8faaf55d7382a851ece8a98938080edf41ecbd.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 133, + 189, + 477, + 210 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 222, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 104, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 104, + 222, + 506, + 236 + ], + "score": 1.0, + "content": "This requires computing two expectations, with respect to the data in the dataset, and with respect", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "to the encoder distribution. We use MC sampling with one sample per data point to approximate", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 505, + 257 + ], + "score": 1.0, + "content": "both expectations. Inspired by common practices in VAEs, we use one sample per data point to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 477, + 268 + ], + "score": 1.0, + "content": "approximate the inner expectation. On SVHN, the standard error of this approximation is", + "type": "text" + }, + { + "bbox": [ + 477, + 255, + 505, + 266 + ], + "score": 0.88, + "content": "0 . 2 6 \\%", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 267, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 506, + 279 + ], + "score": 1.0, + "content": "of the value of sigma. We further approximate the outer expectation with a single batch instead of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 400, + 290 + ], + "score": 1.0, + "content": "the entire dataset. On SVHN, the standard error of this approximation is", + "type": "text" + }, + { + "bbox": [ + 401, + 277, + 416, + 288 + ], + "score": 0.88, + "content": "2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 277, + 506, + 290 + ], + "score": 1.0, + "content": "of the value of sigma.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 301 + ], + "score": 1.0, + "content": "We see that both approximations are accurate in practice. The second approximation yields a biased", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "estimate of the evidence lower bound because the same batch is used to approximate the variance and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 505, + 324 + ], + "score": 1.0, + "content": "compute the lower bound estimate. However, this bias can be corrected by using a different batch, or", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "score": 1.0, + "content": "with a running average of the variance with an appropriate decay. This running average can also be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 332, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 506, + 344 + ], + "score": 1.0, + "content": "used to reduce the variance of the estimate and to achieve convergence guarantees, but we did not", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 343, + 253, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 253, + 356 + ], + "score": 1.0, + "content": "find it necessary in our experiments.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 106, + 373, + 305, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 306, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 306, + 387 + ], + "score": 1.0, + "content": "D ALTERNATIVE DECODER CHOICES", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 399, + 256, + 442 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 257, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 257, + 410 + ], + "score": 1.0, + "content": "We describe the alternative decoders", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 408, + 257, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 257, + 422 + ], + "score": 1.0, + "content": "evaluated in Table 2: using the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 421, + 257, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 257, + 433 + ], + "score": 1.0, + "content": "bitwise-categorical, and the logistic", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 432, + 195, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 195, + 443 + ], + "score": 1.0, + "content": "mixture distributions.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 257, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 258, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 258, + 469 + ], + "score": 1.0, + "content": "Bitwise-categorical VAE While", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 468, + 258, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 258, + 481 + ], + "score": 1.0, + "content": "the 256-way categorical decoder", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 477, + 258, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 258, + 493 + ], + "score": 1.0, + "content": "described in Section 3.2 is very", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 258, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 258, + 503 + ], + "score": 1.0, + "content": "powerful due to the ability to specify", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 502, + 258, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 258, + 514 + ], + "score": 1.0, + "content": "any possible intensity distribution,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 258, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 258, + 525 + ], + "score": 1.0, + "content": "it suffers from high computational", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 524, + 258, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 258, + 535 + ], + "score": 1.0, + "content": "and memory requirements. 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This running average can also be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 332, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 506, + 344 + ], + "score": 1.0, + "content": "used to reduce the variance of the estimate and to achieve convergence guarantees, but we did not", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 343, + 253, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 253, + 356 + ], + "score": 1.0, + "content": "find it necessary in our experiments.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 222, + 506, + 356 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 373, + 305, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 306, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 306, + 387 + ], + "score": 1.0, + "content": "D ALTERNATIVE DECODER CHOICES", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 399, + 256, + 442 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 257, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 257, + 410 + ], + "score": 1.0, + "content": "We describe the alternative decoders", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 408, + 257, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 257, + 422 + ], + "score": 1.0, + "content": "evaluated in Table 2: using the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 421, + 257, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 257, + 433 + ], + "score": 1.0, + "content": "bitwise-categorical, and the logistic", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 432, + 195, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 195, + 443 + ], + "score": 1.0, + "content": "mixture distributions.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 398, + 257, + 443 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 257, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 258, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 258, + 469 + ], + "score": 1.0, + "content": "Bitwise-categorical VAE While", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 468, + 258, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 258, + 481 + ], + "score": 1.0, + "content": "the 256-way categorical decoder", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 477, + 258, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 258, + 493 + ], + "score": 1.0, + "content": "described in Section 3.2 is very", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 258, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 258, + 503 + ], + "score": 1.0, + "content": "powerful due to the ability to specify", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 502, + 258, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 258, + 514 + ], + "score": 1.0, + "content": "any possible intensity distribution,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 258, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 258, + 525 + ], + "score": 1.0, + "content": "it suffers from high computational", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 524, + 258, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 258, + 535 + ], + "score": 1.0, + "content": "and memory requirements. 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CelebA HVAESVHN VAECIFAR HVAEBAIR SVG
-logp↓FID↓-logp↓FID↓-logp↓FID↓-logp↓FID↓
Bernoulli VAE[1]177.643.26284.5122.6
Categorical VAE<635971.5<917946.13<7179101.7N/AN/A
Bitwise-categorical VAE<906766.61<1080033.84<939091.2<4874446.13
Logistic mixture VAE<793265.3<908543.19<8443143.1<4061642.94
Gaussian VAE<7173186.5<2184112.5<7186293.7<-1037935.64
Per-pixel g-VAE<-7814159.3<2184114.7<-7222131<-1405141.98
Student-t VAE [2]<-840171.06<-365970.4<-7419123.6■
β-VAE [3]<-271361.6<-318627.93<-331103<-1347234.64
Shared g-VAE<-637460.7<-334922.25<-5435116.1<-1397434.24
Optimal g-VAE<-844660.3< （-333327.25<-5677101.4<-1417334.13
Opt. per-image g-VAE66.0126.28104.033.21
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CelebA VAECIFAR VAEFrey Face VAE
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Bernoulli VAE Gregor et al. (2015)102.7165.147.7
Categorical VAE;1019550.45;10673124.1<245450.16
bitwise-categorical VAE1101956.361160499.65<317366.77
Logistic mixture VAE;1015461.81;10648100.2<256250.28
Gaussian VAE<2201144.8<1409205.8<726.480.17
β-VAE Higgins et al. (2017)<-194258.73<-1318117.9< -420.037.61
Shared g-VAE (Ours)<-193973.27< （-1830137.8< -49.7842.86
Optimal g-VAE (Ours)<-195161.27< （-183280.9< -162253.36
Opt. per-image g-VAE (Ours)53.1389.8856.07
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Categorical VAE<10673137.6
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STOCHASTIC OPTIMIZATION + +Rahul Kidambi∗1, Praneeth Netrapalli2, Prateek Jain2 and Sham M. Kakade1 + +1 University of Washington Seattle 2 Microsoft Research India rkidambi@uw.edu, {praneeth, prajain}@microsoft.com, sham@cs.washington.edu + +# ABSTRACT + +Momentum based stochastic gradient methods such as heavy ball (HB) and Nesterov’s accelerated gradient descent (NAG) method are widely used in practice for training deep networks and other supervised learning models, as they often provide significant improvements over stochastic gradient descent (SGD). Rigorously speaking, “fast gradient” methods have provable improvements over gradient descent only for the deterministic case, where the gradients are exact. In the stochastic case, the popular explanations for their wide applicability is that when these fast gradient methods are applied in the stochastic case, they partially mimic their exact gradient counterparts, resulting in some practical gain. This work provides a counterpoint to this belief by proving that there exist simple problem instances where these methods cannot outperform SGD despite the best setting of its parameters. These negative problem instances are, in an informal sense, generic; they do not look like carefully constructed pathological instances. These results suggest (along with empirical evidence) that HB or NAG’s practical performance gains are a by-product of mini-batching. + +Furthermore, this work provides a viable (and provable) alternative, which, on the same set of problem instances, significantly improves over HB, NAG, and SGD’s performance. This algorithm, referred to as Accelerated Stochastic Gradient Descent (ASGD), is a simple to implement stochastic algorithm, based on a relatively less popular variant of Nesterov’s Acceleration. Extensive empirical results in this paper show that ASGD has performance gains over HB, NAG, and SGD. The code implementing the ASGD Algorithm can be found here1. + +# 1 INTRODUCTION + +First order optimization methods, which access a function (to be optimized) through its gradient or an unbiased approximation of its gradient, are the workhorses for modern large scale optimization problems, which include training the current state-of-the-art deep neural networks. Gradient descent (Cauchy, 1847) is the simplest first order method that is used heavily in practice. However, it is known that for the class of smooth convex functions as well as some simple non-smooth problems (Nesterov, 2012a)), gradient descent is suboptimal (Nesterov, 2004) and there exists a class of algorithms called fast gradient/momentum based methods which achieve optimal convergence guarantees. The heavy ball method (Polyak, 1964) and Nesterov’s accelerated gradient descent (Nesterov, 1983) are two of the most popular methods in this category. + +On the other hand, training deep neural networks on large scale datasets have been possible through the use of Stochastic Gradient Descent (SGD) (Robbins & Monro, 1951), which samples a random subset of training data to compute gradient estimates that are then used to optimize the objective function. The advantages of SGD for large scale optimization and the related issues of tradeoffs between computational and statistical efficiency was highlighted in Bottou & Bousquet (2007). + +The above mentioned theoretical advantages of fast gradient methods (Polyak, 1964; Nesterov, 1983) (albeit for smooth convex problems) coupled with cheap to compute stochastic gradient estimates led to the influential work of Sutskever et al. (2013), which demonstrated the empirical advantages possessed by SGD when augmented with the momentum machinery. This work has led to widespread adoption of momentum methods for training deep neural nets; so much so that, in the context of neural network training, gradient descent often refers to momentum methods. + +But, there is a subtle difference between classical momentum methods and their implementation in practice – classical momentum methods work in the exact first order oracle model (Nesterov, 2004), i.e., they employ exact gradients (computed on the full training dataset), while in practice (Sutskever et al., 2013), they are implemented with stochastic gradients (estimated from a randomly sampled mini-batch of training data). This leads to a natural question: + +“Are momentum methods optimal even in the stochastic first order oracle (SFO) model, where we access stochastic gradients computed on a small constant sized minibatches (or a batchsize of 1?)” + +Even disregarding the question of optimality of momentum methods in the SFO model, it is not even known if momentum methods (say, Polyak (1964); Nesterov (1983)) provide any provable improvement over SGD in this model. While these are open questions, a recent effort of Jain et al. (2017) showed that improving upon SGD (in the stochastic first order oracle) is rather subtle as there exists problem instances in SFO model where it is not possible to improve upon SGD, even information theoretically. Jain et al. (2017) studied a variant of Nesterov’s accelerated gradient updates (Nesterov, 2012b) for stochastic linear regression and show that their method improves upon SGD wherever it is information theoretically admissible. Through out this paper, we refer to the algorithm of Jain et al. (2017) as Accelerated Stochastic Gradient Method (ASGD) while we refer to a stochastic version of the most widespread form of Nesterov’s method (Nesterov, 1983) as NAG; HB denotes a stochastic version of the heavy ball method (Polyak, 1964). Critically, while Jain et al. (2017) shows that ASGD improves on SGD in any information-theoretically admissible regime, it is still not known whether HB and NAG can achieve a similar performance gain. + +A key contribution of this work is to show that HB does not provide similar performance gains over SGD even when it is informationally-theoretically admissible. That is, we provide a problem instance where it is indeed possible to improve upon SGD (and ASGD achieves this improvement), but HB cannot achieve any improvement over SGD. We validate this claim empirically as well. In fact, we provide empirical evidence to the claim that NAG also do not achieve any improvement over SGD for several problems where ASGD can still achieve better rates of convergence. + +This raises a question about why HB and NAG provide better performance than SGD in practice (Sutskever et al., 2013), especially for training deep networks. Our conclusion (that is well supported by our theoretical result) is that HB and NAG’s improved performance is attributed to mini-batching and hence, these methods will often struggle to improve over SGD with small constant batch sizes. This is in stark contrast to methods like ASGD, which is designed to improve over SGD across both small or large mini-batch sizes. In fact, based on our experiments, we observe that on the task of training deep residual networks (He et al., 2016a) on the cifar-10 dataset, we note that ASGD offers noticeable improvements by achieving $5 - 7 \%$ better test error over HB and NAG even with commonly used batch sizes like 128 during the initial stages of the optimization. + +# 1.1 CONTRIBUTIONS + +The contributions of this paper are as follows. + +1. In Section 3, we prove that HB is not optimal in the SFO model. In particular, there exist linear regression problems for which the performance of HB (with any step size and momentum) is either the same or worse than that of SGD while ASGD improves upon both of them. +2. Experiments on several linear regression problems suggest that the suboptimality of HB in the SFO model is not restricted to special cases – it is rather widespread. Empirically, the same holds true for NAG as well (Section 5). +3. The above observations suggest that the only reason for the superiority of momentum methods in practice is mini-batching, which reduces the variance in stochastic gradients and moves the SFO closer to the exact first order oracle. This conclusion is supported by em + +# Algorithm 1 HB: Heavy ball with a SFO + +# Algorithm 2 NAG: Nesterov’s AGD with a SFO + +Require: Initial $w _ { 0 }$ , stepsize $\delta$ , momentum $\alpha$ + +Require: Initial $w _ { 0 }$ , stepsize $\delta$ , momentum $\alpha$ + +1: $v _ { 0 } w _ { 0 }$ ; $t \gets 0$ /\*Set $v _ { 0 }$ to $\boldsymbol { w _ { 0 } } ^ { * } /$ +2: while $w _ { t }$ not converged do +3: $v _ { t + 1 } \gets w _ { t } - \delta \cdot \tilde { \widehat { \nabla } } f _ { t } ( w _ { t } ) / { * } \mathrm { S G D ~ s t e p } ^ { { * } / }$ +4: $w _ { t + 1 } = ( 1 + \alpha ) v _ { t + 1 } - \alpha v _ { t } / \ast \mathrm { S u m }$ of SGD +step and previous iterate\*/ +5: $t \gets t + 1$ + +/\*Return the last iterate\*/ /\*Return the last iterate\*/ + +pirical evidence through training deep residual networks on cifar-10, with a batch size of 8 (see Section 5.3). + +4. We present an intuitive and easier to tune version of ASGD (see Section 4) and show that ASGD can provide significantly faster convergence to a reasonable accuracy than SGD, HB, NAG, while still providing favorable or comparable asymptotic accuracy as these methods, particularly on several deep learning problems. + +Hence, the take-home message of this paper is: HB and NAG are not optimal in the SFO model. The only reason for the superiority of momentum methods in practice is mini-batching. ASGD provides a distinct advantage in training deep networks over SGD, HB and NAG. + +# 2 NOTATION + +We denote matrices by bold-face capital letters and vectors by lower-case letters. $f ( w ) =$ $1 / n \sum _ { i } f _ { i } ( w )$ denotes the function to optimize w.r.t. model parameters $w$ . $\nabla f ( w )$ denotes exact gradient of $f$ at $w$ while $\widehat { \nabla } f _ { t } ( \boldsymbol { w } )$ denotes a stochastic gradient of $f$ . That is, $\widehat { \nabla } f _ { t } ( w _ { t } ) = \nabla f _ { i _ { t } } ( w )$ where $i _ { t }$ is sampled uniformly at random from $[ 1 , \ldots , n ]$ . For linear regression, $f _ { i } ( w ) = 0 . 5 \cdot ( b _ { i } -$ $\langle w , a _ { i } \rangle ) ^ { 2 }$ where $b _ { i } \in \Re$ is the target and $a _ { i } \in \Re ^ { d }$ is the covariate, and $\widehat { \nabla } f _ { t } ( w _ { t } ) = - \big ( b _ { t } - \langle w _ { t } , a _ { t } \rangle \big ) a _ { t }$ . In this case, $\mathbf { H } = \mathbb { E } \left[ a a ^ { \top } \right]$ denotes the Hessian of $f$ and $\begin{array} { r } { \kappa = \frac { \lambda _ { 1 } ( \mathbf { H } ) } { \lambda _ { d } ( \mathbf { H } ) } } \end{array}$ denotes it’s condition number. + +Algorithm 1 provides a pseudo-code of HB method (Polyak, 1964). $w _ { t } - w _ { t - 1 }$ is the momentum term and $\alpha$ denotes the momentum parameter. Next iterate $w _ { t + 1 }$ is obtained by a linear combination of the SGD update and the momentum term. Algorithm 2 provides pseudo-code of a stochastic version of the most commonly used form of Nesterov’s accelerated gradient descent (Nesterov, 1983). + +# 3 SUBOPTIMALITY OF HEAVY BALL METHOD + +In this section, we show that there exists linear regression problems where the performance of HB (Algorithm 1) is no better than that of SGD, while ASGD significantly improves upon SGD’s performance. Let us now describe the problem instance. + +Fix $w ^ { \ast } \in \mathbb { R } ^ { 2 }$ and let $( a , b ) \sim \mathcal { D }$ be a sample from the distribution such that: + +$$ +\begin{array} { r } { a = \left\{ \begin{array} { l l } { \sigma _ { 1 } \cdot z \cdot e _ { 1 } \mathrm { w . p . ~ } 0 . 5 } \\ { \sigma _ { 2 } \cdot z \cdot e _ { 2 } \mathrm { w . p . ~ } 0 . 5 , } \end{array} \right. \qquad \mathrm { a n d } \qquad b = \left. w ^ { * } , a \right. , } \end{array} +$$ + +where $e _ { 1 } , e _ { 2 } \in \mathbb { R } ^ { 2 }$ are canonical basis vectors, $\sigma _ { 1 } > \sigma _ { 2 } > 0$ . Let $z$ be a random variable such that E $\left[ z ^ { 2 } \right] = 2$ and $\mathbb { E } \left[ z ^ { 4 } \right] = 2 c \geq 4$ . Hence, we have: $\Xi \left[ ( a ^ { ( i ) } ) ^ { 2 } \right] = \sigma _ { i } ^ { 2 } , \mathbb { E } \left[ ( a ^ { ( i ) } ) ^ { 4 } \right] = c \sigma _ { i } ^ { 4 }$ , for $i = 1 , \dot { 2 }$ . Now, our goal is to minimize: + +$$ +f ( \boldsymbol { w } ) \stackrel { \mathrm { d e f } } { = } 0 . 5 \cdot \mathbb { E } \left[ \left( \left. \boldsymbol { w } ^ { * } , \boldsymbol { a } \right. - \boldsymbol { b } \right) ^ { 2 } \right] \mathrm { , ~ H e s s i a n \ } \mathbf { H } \stackrel { \mathrm { d e f } } { = } \mathbb { E } \left[ \boldsymbol { a } \boldsymbol { a } ^ { \top } \right] = \left[ \sigma _ { 1 } ^ { 2 } \quad 0 _ { 2 } ^ { 2 } \right] . +$$ + +Let $\kappa$ and $\tilde { \kappa }$ denote the computational and statistical condition numbers – see Jain et al. (2017) for definitions. For the problem above, we have $\begin{array} { r } { \kappa = \frac { c \sigma _ { 1 } ^ { 2 } } { \sigma _ { 2 } ^ { 2 } } } \end{array}$ and $\tilde { \kappa } = c$ . Then we obtain following convergence rates for SGD and ASGD when applied to the above given problem instance: + +Input: Initial $w _ { 0 }$ , short step $\delta$ , long step parameter $\kappa \geq 1$ , statistical advantage parameter $\xi \le \sqrt { \kappa }$ +1: $\bar { w } _ { 0 } w _ { 0 }$ ; $t \gets 0$ /\*Set running average to $\boldsymbol { w _ { 0 } } ^ { * } /$ +2: α ← 1 − 0.72·ξ /\*Set momentum value\*/ +3: while $w _ { t }$ not converged do +4: $\begin{array} { r } { \bar { w } _ { t + 1 } \alpha \cdot \bar { w } _ { t } + \overline { { ( 1 - \alpha ) \cdot \Big ( w _ { t } - \frac { \kappa \cdot \delta } { 0 . 7 } \cdot \widehat \nabla f _ { t } ( w _ { t } ) \Big ) } } } \end{array}$ /\*Update the running average as a weighted average of previous running average and a long step gradient $^ { * } /$ +5: $\begin{array} { r } { \overline { { w _ { t + 1 } } } \frac { 0 . 7 } { 0 . 7 + ( 1 - \alpha ) } \cdot ( w _ { t } - \delta \cdot \widehat { \nabla } f _ { t } ( w _ { t } ) ) + \frac { 1 - \alpha } { 0 . 7 + ( 1 - \alpha ) } \cdot \bar { w } _ { t + 1 } } \end{array}$ /\*Update the iterate as weighted average of current running average and short step gradient\*/ +6: $t \gets t + 1$ + +Output: $w _ { t }$ /\*Return the last iterate\*/ + +Corollary 1 (of Theorem 1 of Jain et al. (2016)). Let $w _ { t } ^ { S G D }$ be the $t ^ { t h }$ iterate of SGD on the above problem with starting point $w _ { 0 }$ and stepsize cσ 2 t . The error of $w _ { t } ^ { S G D }$ can be bounded as, + +$$ +\mathbb { E } \left[ f \left( w _ { t } ^ { S G D } \right) \right] - f \left( w _ { * } \right) \leq \exp \left( \frac { - t } { \kappa } \right) \left( f \left( w _ { 0 } \right) - f \left( w _ { * } \right) \right) . +$$ + +On the other hand, ASGD achieves the following superior rate. + +Corollary 2 (of Theorem 1 of Jain et al. (2017)). Let $w _ { t } ^ { A S G D }$ be the $t ^ { t h }$ iterate of ASGD on the above problem with starting point $w _ { 0 }$ and appropriate parameters. The error of $\dot { w } _ { t } ^ { A S G D }$ can be bounded as, + +$$ +\mathbb { E } \left[ f \left( w _ { t } ^ { A S G D } \right) \right] - f \left( w _ { * } \right) \le \mathrm { p o l y } ( \kappa ) \exp \left( \frac { - t } { \sqrt { \kappa \tilde { \kappa } } } \right) \left( f \left( w _ { 0 } \right) - f \left( w _ { * } \right) \right) . +$$ + +Note that for a given problem/input distribution $\tilde { \kappa } = c$ is a constant while $\begin{array} { r } { \kappa = \frac { c \sigma _ { 1 } ^ { 2 } } { \sigma _ { 2 } ^ { 2 } } } \end{array}$ can be arbitrarily large. Note that $\kappa > \tilde { \kappa } = c$ . Hence, ASGD improves upon rate of SGD by a factor of $\sqrt { \kappa }$ . The following proposition, which is the main result of this section, establishes that HB (Algorithm 1) cannot provide a similar improvement over SGD as what ASGD offers. In fact, we show no matter the choice of parameters of HB, its performance does not improve over SGD by more than a constant. + +Proposition 3. Let $w _ { t } ^ { H B }$ be the $t ^ { t h }$ iterate of HB (Algorithm $I$ ) on the above problem with starting point $w _ { 0 }$ . For any choice of stepsize $\delta$ and momentum $\alpha \in [ 0 , 1 ]$ , $\exists T$ large enough such that $\forall t \geq T$ , we have, + +$$ +\mathbb { E } \left[ f \left( w _ { t } ^ { H B } \right) \right] - f \left( w _ { * } \right) \ge C ( \kappa , \delta , \alpha ) \cdot \exp \left( \frac { - 5 0 0 t } { \kappa } \right) \left( f \left( w _ { 0 } \right) - f \left( w _ { * } \right) \right) , +$$ + +where $C ( \kappa , \delta , \alpha )$ depends on $\kappa , \delta$ and $\alpha$ (but not on $t$ ). + +Thus, to obtain $\widehat { w }$ s.t. $\| \widehat { \boldsymbol { w } } - \boldsymbol { w } ^ { * } \| \le \epsilon$ , HB requires $\Omega ( \kappa \log { \frac { 1 } { \epsilon } } )$ samples and iterations. On the other hand, ASGD can obtain $\epsilon$ -approximation to $w ^ { * }$ in $\mathcal { O } ( \sqrt { \kappa } \log \kappa \log \frac { 1 } { \epsilon } )$ iterations. We note that the gains offered by ASGD are meaningful when $\kappa > \mathcal { O } ( c )$ (Jain et al., 2017); otherwise, all the algorithms including SGD achieve nearly the same rates (upto constant factors). While we do not prove it theoretically, we observe empirically that for the same problem instance, NAG also obtains nearly same rate as HB and SGD. We conjecture that a lower bound for NAG can be established using a similar proof technique as that of HB (i.e. Proposition 3). We also believe that the constant in the lower bound described in proposition 3 can be improved to some small number $( \leq 5 )$ . + +# 4 ALGORITHM + +We will now present and explain an intuitive version of ASGD (pseudo code in Algorithm 3). The algorithm takes three inputs: short step $\delta$ , long step parameter $\kappa$ and statistical advantage parameter $\xi$ . The short step $\delta$ is precisely the same as the step size in SGD, HB or NAG. For convex problems, this scales inversely with the smoothness of the function. The long step parameter $\kappa$ is intended to give an estimate of the ratio of the largest and smallest curvatures of the function; for convex functions, this is just the condition number. The statistical advantage parameter $\xi$ captures trade√ off between statistical and computational condition numbers – in the deterministic case, $\xi = \sqrt { \kappa }$ and ASGD is equivalent to NAG, while in the high stochasticity regime, $\xi$ is much smaller. The algorithm maintains two iterates: descent iterate $w _ { t }$ and a running average $\bar { w } _ { t }$ . The running average is a weighted average of the previous average and a long gradient step from the descent iterate, while the descent iterate is updated as a convex combination of short gradient step from the descent iterate and the running average. The idea is that since the algorithm takes a long step as well as short step and an appropriate average of both of them, it can make progress on different directions at a similar pace. Appendix B shows the equivalence between Algorithm 3 and ASGD as proposed in Jain et al. (2017). Note that the constant 0.7 appearing in Algorithm 3 has no special significance. Jain et al. (2017) require it to be smaller than $\sqrt { 1 / 6 }$ but any constant smaller than 1 seems to work in practice. + +# 5 EXPERIMENTS + +We now present our experimental results exploring performance of SGD, HB, NAG and ASGD. Our experiments are geared towards answering the following questions: + +• Even for linear regression, is the suboptimality of HB restricted to specific distributions in Section 3 or does it hold for more general distributions as well? Is the same true of NAG? +What is the reason for the superiority of HB and NAG in practice? Is it because momentum methods have better performance that SGD for stochastic gradients or due to minibatching? Does this superiority hold even for small minibatches? +• How does the performance of ASGD compare to that of SGD, HB and NAG, when training deep networks? + +Section 5.1 and parts of Section 5.2 address the first two questions. Section 5.2 and 5.3 address Question 2 partially and the last question. We use Matlab to conduct experiments presented in Section 5.1 and use PyTorch (pytorch, 2017) for our deep networks related experiments. Pytorch code implementing the ASGD algorithm can be found at https://github.com/rahulkidambi/AccSGD. + +# 5.1 LINEAR REGRESSION + +In this section, we will present results on performance of the four optimization methods (SGD, HB, NAG, and ASGD) for linear regression problems. We consider two different class of linear regression problems, both of them in two dimensions. Given $\kappa$ which stands for condition number, we consider the following two distributions: + +$a = e _ { 1 }$ w.p. 0.5 and $\textstyle a = { \frac { 2 } { \kappa } } \cdot e _ { 2 }$ with 0.5; $e _ { i }$ is the $i ^ { t h }$ + +Gaussian : $a \in \mathbb { R } ^ { 2 }$ is distributed as a Gaussian random vector with covariance matrix $\left[ { \begin{array} { c c } { 1 } & { 0 } \\ { 0 } & { { \frac { 1 } { \kappa } } } \end{array} } \right] .$ + +We fix a randomly generated $w ^ { \ast } \in \mathbb { R } ^ { 2 }$ and for both the distributions above, we let $b = \langle w ^ { * } , a \rangle$ . We vary $\kappa$ from $\{ \mathbf { \bar { 2 } ^ { 4 } } , 2 ^ { 5 } , . . . , 2 ^ { 1 2 } \}$ and for each $\kappa$ in this set, we run 100 independent runs of all four methods, each for a total of $t = 5 \kappa$ iterations. We define that the algorithm converges if there is no error in the second half (i.e. after $2 . 5 \kappa$ updates) that exceeds the starting error - this is reasonable since we expect geometric convergence of the initial error. + +Unlike ASGD and SGD, we do not know optimal learning rate and momentum parameters for NAG and HB in the stochastic gradient model. So, we perform a grid search over the values of the learning rate and momentum parameters. In particular, we lay a $1 0 \times 1 0$ grid in $[ 0 , 1 ] \times [ 0 , 1 ]$ for learning rate and momentum and run NAG and HB. Then, for each grid point, we consider the subset of 100 trials that converged and computed the final error using these. Finally, the parameters that yield the minimal error are chosen for NAG and HB, and these numbers are reported. We measure convergence performance of a method using: + +$$ +{ \mathrm { r a t e } } = { \frac { \log ( f ( w _ { 0 } ) ) - \log ( f ( w _ { t } ) ) } { t } } , +$$ + +![](images/3d3b10853538102302f9fd8281848da346ad9275d7a5233ce0344ccdced8ae59.jpg) +Figure 1: Plot of 1/rate (refer equation (1)) vs condition number $( \kappa )$ for various methods for the linear regression problem. Discrete distribution in the left, Gaussian to the right. + +Table 1: Slopes (i.e. $\gamma$ ) obtained by fitting a line to the curves in Figure 1. A value of $\gamma$ indicates that the error decays at a rate of exp $\left( { \frac { - t } { \kappa ^ { \gamma } } } \right)$ . A smaller value of $\gamma$ indicates a faster rate of error decay. + +
AlgorithmSlope-discreteSlope 1 Gaussian
SGD0.93020.8745
HB NAG0.85220.8769
0.980.9494
0.54800.5127
+ +We compute the rate (1) for all the algorithms with varying condition number $\kappa$ . Given a rate vs $\kappa$ plot for a method, we compute it’s slope (denoted as $\gamma$ ) using linear regression. Table 1 presents the estimated slopes (i.e. $\gamma$ ) for various methods for both the discrete and the Gaussian case. The slope values clearly show that the rate of SGD, HB and NAG have a nearly linear dependence on √ $\kappa$ while that of ASGD seems to scale linearly with $\sqrt { \kappa }$ . + +# 5.2 DEEP AUTOENCODERS FOR MNIST + +In this section, we present experimental results on training deep autoencoders for the mnist dataset, and we closely follow the setup of Hinton & Salakhutdinov (2006). This problem is a standard benchmark for evaluating the performance of different optimization algorithms e.g., Martens (2010); Sutskever et al. (2013); Martens $\&$ Grosse (2015); Reddi et al. (2017). The network architecture follows previous work (Hinton & Salakhutdinov, 2006) and is represented as $7 8 4 - 1 0 0 0 - 5 0 0 -$ $2 5 0 - 3 0 - 2 5 0 - 5 0 0 - 1 0 0 0 - 7 8 4$ with the first and last 784 nodes representing the input and output respectively. All hidden/output nodes employ sigmoid activations except for the layer with 30 nodes which employs linear activations and we use MSE loss. Initialization follows the scheme of Martens (2010), also employed in Sutskever et al. (2013); Martens & Grosse (2015). We perform training with two minibatch sizes $- 1$ and 8. The runs with minibatch size of 1 were run for 30 epochs while the runs with minibatch size of 8 were run for 50 epochs. For each of SGD, HB, NAG and ASGD, a grid search over learning rate, momentum and long step parameter (whichever is applicable) was done and best parameters were chosen based on achieving the smallest training error in the same protocol followed by Sutskever et al. (2013). The grid was extended whenever the best parameter fell at the edge of a grid. For the parameters chosen by grid search, we perform 10 runs with different seeds and averaged the results. The results are presented in Figures 2 and 3. Note that the final loss values reported are suboptimal compared to those in published literature e.g., Sutskever et al. (2013); while Sutskever et al. (2013) report results after 750000 updates with a large batch size of 200 (which implies a total of $7 5 0 0 0 0 \times 2 0 0 = 1 5 0 \mathbf { M }$ gradient evaluations), whereas, our results are after 1.8M updates of SGD with a batch size 1 (which is just 1.8M gradient evaluations). + +Effect of minibatch sizes: While HB and NAG decay the loss faster compared to SGD for a minibatch size of 8 (Figure 2), this superior decay rate does not hold for a minibatch size of 1 (Figure 3). This supports our intuitions from the stochastic linear regression setting, where we demonstrate that HB and NAG are suboptimal in the stochastic first order oracle model. + +![](images/e7e9ca2132cb1ddd893c549fd607c122fbf8832557608c7292e2d72b3058de61.jpg) +Figure 2: Training loss (left) and test loss (right) while training deep autoencoder for mnist with minibatch size 8. Clearly, ASGD matches performance of NAG and outperforms SGD on the test data. HB also outperforms SGD. + +![](images/1de51d3d68f2b8fcd7d7f8f74749949493d74864a33ffe1b0da66f947386a0f1.jpg) +Figure 3: Training loss (left) and test loss (right) while training deep autoencoder for mnist with minibatch size 1. Interestingly, SGD, HB and NAG, all decrease the loss at a similar rate, while ASGD decays at a faster rate. + +Comparison of ASGD with momentum methods: While ASGD performs slightly better than NAG for batch size 8 in the training error (Figure 2), ASGD decays the error at a faster rate compared to all the three other methods for a batch size of 1 (Figure 3). + +# 5.3 DEEP RESIDUAL NETWORKS FOR CIFAR-10 + +We will now present experimental results on training deep residual networks (He et al., 2016b) with pre-activation blocks He et al. (2016a) for classifying images in cifar-10 (Krizhevsky & Hinton, 2009); the network we use has 44 layers (dubbed preresnet-44). The code for this section was downloaded from preresnet (2017). One of the most distinct characteristics of this experiment compared to our previous experiments is learning rate decay. We use a validation set based decay scheme, wherein, after every 3 epochs, we decay the learning rate by a certain factor (which we grid search on) if the validation zero one error does not decrease by at least a certain amount (precise numbers are provided in the appendix since they vary across batch sizes). Due to space constraints, we present only a subset of training error plots. Please see Appendix C.3 for some more plots on training errors. + +Effect of minibatch sizes: Our first experiment tries to understand how the performance of HB and NAG compare with that of SGD and how it varies with minibatch sizes. Figure 4 presents the test zero one error for minibatch sizes of 8 and 128. While training with batch size 8 was done for 40 epochs, with batch size 128, it was done for 120 epochs. We perform a grid search over all parameters for each of these algorithms. See Appendix C.3 for details on the grid search parameters. We observe that final error achieved by SGD, HB and NAG are all very close for both batch sizes. While NAG exhibits a superior rate of convergence compared to SGD and HB for batch size 128, this superior rate of convergence disappears for a batch size of 8. + +Comparison of ASGD with momentum methods: The next experiment tries to understand how ASGD compares with HB and NAG. The errors achieved by various methods when we do + +![](images/db5f311c1e2dbe4b962051b725dbf673899c05a23640369f13b18c9e0f70e473.jpg) +Figure 4: Test zero one loss for batch size 128 (left), batch size 8 (center) and training function value for batch size 8 (right) for SGD, HB and NAG. + +![](images/9082f0ff9ebf68aa4e0752f19fd7b95ba1e73a84b488c3c29ff243cc33fb06fa.jpg) +Figure 5: Test zero one loss for batch size 128 (left), batch size 8 (center) and training function value for batch size 8 (right) for ASGD compared to HB. In the above plots, both ASGD and ASGD-HbParams refer to ASGD run with the learning rate and decay schedule of HB. ASGD-Fully-Optimized refers to ASGD where learning rate and decay schedule were also selected by grid search. + +grid search over all parameters are presented in Table 2. Note that the final test errors for batch size 128 are better than those for batch size 8 since the former was run for 120 epochs while the latter was run only for 40 epochs (due to time constraints). + +
AlgorithmFinal test error-batch size 128Final test error-batch size 8
SGD8.32± 0.219.57±0.18
HB7.98 ± 0.199.28± 0.25
NAG7.63 ± 0.189.07 ±0.18
ASGD7.23 ± 0.228.52 ± 0.16
+ +Table 2: Final test errors achieved by various methods for batch sizes of 128 and 8. The hyperparameters have been chosen by grid search. + +While the final error achieved by ASGD is similar/favorable compared to all other methods, we are also interested in understanding whether ASGD has a superior convergence speed. For this experiment, we need to address the issue of differing learning rates used by various algorithms and different iterations where they decay learning rates. So, for each of HB and NAG, we choose the learning rate and decay factors by grid search, use these values for ASGD and do grid search only over long step parameter $\kappa$ and momentum $\alpha$ for ASGD. The results are presented in Figures 5 and 6. For batch size 128, ASGD decays error at a faster rate compared to both HB and NAG. For batch size 8, while we see a superior convergence of ASGD compared to NAG, we do not see this superiority over HB. The reason for this turns out to be that the learning rate for HB, which we also use for ASGD, turns out to be quite suboptimal for ASGD. So, for batch size 8, we also compare fully optimized (i.e., grid search over learning rate as well) ASGD with HB. The superiority of ASGD over HB is clear from this comparison. These results suggest that ASGD decays error at a faster rate compared to HB and NAG across different batch sizes. + +# 6 RELATED WORK + +First order oracle methods: The primary method in this family is Gradient Descent (GD) (Cauchy, 1847). As mentioned previously, GD is suboptimal for smooth convex optimization (Nesterov, + +![](images/1520b13e594816b4bd1f8bfe08d8102111f7964499e85c5692f7d19723eb2535.jpg) +Figure 6: Test zero one loss for batch size 128 (left), batch size 8 (center) and training function value for batch size 8 (right) for ASGD compared to NAG. In the above plots, ASGD was run with the learning rate and decay schedule of NAG. Other parameters were selected by grid search. + +2004), and this is addressed using momentum methods such as the Heavy Ball method (Polyak, +1964) (for quadratics), and Nesterov’s Accelerated gradient descent (Nesterov, 1983). + +Stochastic first order methods and noise stability: The simplest method employing the SFO is SGD (Robbins & Monro, 1951); the effectiveness of SGD has been immense, and its applicability goes well beyond optimizing convex objectives. Accelerating SGD is a tricky proposition given the instability of fast gradient methods in dealing with noise, as evidenced by several negative results which consider statistical (Proakis, 1974; Polyak, 1987; Roy & Shynk, 1990), numerical (Paige, 1971; Greenbaum, 1989) and adversarial errors (d’Aspremont, 2008; Devolder et al., 2014). A result of Jain et al. (2017) developed the first provably accelerated SGD method for linear regression which achieved minimax rates, inspired by a method of Nesterov (2012b). Schemes of Ghadimi & Lan (2012; 2013); Dieuleveut et al. (2016), which indicate acceleration is possible with noisy gradients do not hold in the SFO model satisfied by algorithms that are run in practice (see Jain et al. (2017) for more details). + +While HB (Polyak, 1964) and NAG (Nesterov, 1983) are known to be effective in case of exact first order oracle, for the SFO, the theoretical performance of HB and NAG is not well understood. + +Understanding Stochastic Heavy Ball: Understanding HB’s performance with inexact gradients has been considered in efforts spanning several decades, in many communities like controls, optimization and signal processing. Polyak (1987) considered HB with noisy gradients and concluded that the improvements offered by HB with inexact gradients vanish unless strong assumptions on the inexactness was considered; an instance of this is when the variance of inexactness decreased as the iterates approach the minimizer. Proakis (1974); Roy & Shynk (1990); Sharma et al. (1998) suggest that the improved non-asymptotic rates offered by stochastic HB arose at the cost of worse asymptotic behavior. We resolve these unquantified improvements on rates as being just constant factors over SGD, in stark contrast to the gains offered by ASGD. Loizou & Richtarik ´ (2017) state their method as Stochastic HB but require stochastic gradients that nearly behave as exact gradients; indeed, their rates match that of the standard HB method (Polyak, 1964). Such rates are not information theoretically possible (see Jain et al. (2017)), especially with a batch size of 1 or even with constant sized minibatches. + +Accelerated and Fast Methods for finite-sums: There have been developments pertaining to faster methods for finite-sums (also known as offline stochastic optimization): amongst these are methods such as SDCA (Shalev-Shwartz & Zhang, 2012), SAG (Roux et al., 2012), SVRG (Johnson & Zhang, 2013), SAGA (Defazio et al., 2014), which offer linear convergence rates for strongly convex finite-sums, improving over SGD’s sub-linear rates (Rakhlin et al., 2012). These methods have been improved using accelerated variants (Shalev-Shwartz & Zhang, 2014; Frostig et al., 2015a; Lin et al., 2015; Defazio, 2016; Allen-Zhu, 2016). Note that these methods require storing the entire training set in memory and taking multiple passes over the same for guaranteed progress. Furthermore, these methods require computing a batch gradient or require memory requirements (typically $\Omega ( \lfloor$ training data points|)). For deep learning problems, data augmentation is often deemed necessary for achieving good performance; this implies computing quantities such as batch gradient (or storage necessities) over this augmented dataset is often infeasible. Such requirements are mitigated by the use of simple streaming methods such as SGD, ASGD, HB, NAG. For other technical distinctions between the offline and online stochastic methods refer to Frostig et al. (2015b). + +Practical methods for training deep networks: Momentum based methods employed with stochastic gradients (Sutskever et al., 2013) have become standard and very popular in practice. These schemes tend to outperform standard SGD on several important practical problems. As previously mentioned, we attribute this improvement to effect of mini-batching rather than improvement offered by HB or NAG in the SFO model. Schemes such as Adagrad (Duchi et al., 2011), RMSProp (Tieleman & Hinton, 2012), Adam (Kingma & Ba, 2014) represent an important and useful class of algorithms. The advantages offered by these methods are orthogonal to the advantages offered by fast gradient methods; it is an important direction to explore augmenting these methods with ASGD as opposed to standard HB or NAG based acceleration schemes. + +Chaudhari et al. (2017) proposed Entropy-SGD, which is an altered objective that adds a local strong convexity term to the actual empirical risk objective, with an aim to improve generalization. However, we do not understand convergence rates for convex problems or the generalization ability of this technique in a rigorous manner. Chaudhari et al. (2017) propose to use SGD in their procedure but mention that they employ the HB/NAG method in their implementation for achieving better performance. Naturally, we can use ASGD in this context. Path normalized SGD (Neyshabur et al., 2015) is a variant of SGD that alters the metric on which the weights are optimized. As noted in their paper, path normalized SGD could be improved using HB/NAG (or even the ASGD method). + +# 7 CONCLUSIONS AND FUTURE DIRECTIONS + +In this paper, we show that the performance gain of HB over SGD in stochastic setting is attributed to mini-batching rather than the algorithm’s ability to accelerate with stochastic gradients. Concretely, we provide a formal proof that for several easy problem instances, HB does not outperform SGD despite large condition number of the problem; we observe this trend for NAG in our experiments. In contrast, ASGD (Jain et al., 2017) provides significant improvement over SGD for these problem instances. We observe similar trends when training a resnet on cifar-10 and an autoencoder on mnist. This work motivates several directions such as understanding the behavior of ASGD on domains such as NLP, and developing automatic momentum tuning schemes (Zhang et al., 2017). + +# ACKNOWLEDGMENTS + +Sham Kakade acknowledges funding from NSF Awards CCF-1703574 and CCF-1740551. + +# REFERENCES + +Zeyuan Allen-Zhu. Katyusha: The first direct acceleration of stochastic gradient methods. CoRR, abs/1603.05953, 2016. + +Leon Bottou and Olivier Bousquet. The tradeoffs of large scale learning. In ´ NIPS 20, 2007. + +Louis Augustin Cauchy. Methode g ´ en´ erale pour la r ´ esolution des syst ´ emes d’ ´ equations simultanees. ´ C. R. Acad. Sci. Paris, 1847. + +Pratik Chaudhari, Anna Choromanska, Stefano Soatto, Yann LeCun, Carlo Baldassi, Christian Borgs, Jennifer Chayes, Levent Sagun, and Riccardo Zecchina. Entropy-sgd: Biasing gradient descent into wide valleys. CoRR, abs/1611.01838, 2017. + +Alexandre d’Aspremont. Smooth optimization with approximate gradient. 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CoRR, abs/1706.03471, 2017. + +# A SUBOPTIMALITY OF HB: PROOF OF PROPOSITION 3 + +Before proceeding to the proof, we introduce some additional notation. Let $\pmb { \theta } _ { t + 1 } ^ { ( j ) }$ denote the concatenated and centered estimates in the $j ^ { \mathrm { t h } }$ direction for $j = 1 , 2$ . + +$$ +\begin{array} { r } { \pmb { \theta } _ { t + 1 } ^ { ( j ) } \stackrel { \mathrm { d e f } } { = } \left[ \mathbf { w } _ { t + 1 } ^ { ( j ) } - ( \mathbf { w } ^ { * } ) ^ { ( j ) } \right] , \quad j = 1 , 2 . } \end{array} +$$ + +Since the distribution over $x$ is such that the coordinates are decoupled, we see that $\pmb { \theta } _ { t + 1 } ^ { ( j ) }$ can be written in terms of $\pmb { \theta } _ { t } ^ { ( j ) }$ as: + +$$ +\pmb { \theta } _ { t + 1 } ^ { ( j ) } = \widehat { \mathbf { A } } _ { t + 1 } ^ { ( j ) } \pmb { \theta } _ { t } ^ { ( j ) } , \mathrm { w i t h } \widehat { \mathbf { A } } _ { t + 1 } ^ { ( j ) } = \left[ \frac { 1 + \alpha - \delta ( a _ { t + 1 } ^ { ( j ) } ) ^ { 2 } } { 1 } - \alpha \right] . +$$ + +Let $\Phi _ { t + 1 } ^ { ( j ) } \ { \stackrel { \mathrm { d e f } } { = } } \ \mathbb { E } \left[ \pmb { \theta } _ { t + 1 } ^ { ( j ) } \otimes \pmb { \theta } _ { t + 1 } ^ { ( j ) } \right]$ denote the covariance matrix of $\pmb { \theta } _ { t + 1 } ^ { ( j ) }$ . We have $\Phi _ { t + 1 } ^ { ( j ) } = B ^ { ( j ) } \Phi _ { t } ^ { ( j ) }$ with, $B ^ { ( j ) }$ defined as + +$$ +\begin{array} { r l } { \mathfrak { L } ( \mathfrak { j } ) \overset { \mathrm { d e f } } { = } \left[ \begin{array} { c c c c } { \mathbb { E } \left[ ( 1 + \alpha - \delta ( a ^ { ( j ) } ) ^ { 2 } ) ^ { 2 } \right] } & { \mathbb { E } \left[ - \alpha ( 1 + \alpha - \delta ( a ^ { ( j ) } ) ^ { 2 } ) \right] } & { \mathbb { E } \left[ - \alpha ( 1 + \alpha - \delta ( a ^ { ( j ) } ) ^ { 2 } \right] } & { \alpha ^ { 2 } } \\ { \mathbb { E } \left[ ( 1 + \alpha - \delta ( a ^ { ( j ) } ) ^ { 2 } ) \right] } & { 0 } & { - \alpha } & { 0 } \\ { \mathbb { E } \left[ ( 1 + \alpha - \delta ( a ^ { ( j ) } ) ^ { 2 } ) \right] } & { - \alpha } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 0 } & { 0 } \end{array} \right] , } & { } \\ { = \left[ \begin{array} { c c c c } { ( 1 + \alpha - \delta \sigma _ { j } ^ { 2 } ) ^ { 2 } + ( c - 1 ) ( \delta \sigma _ { j } ^ { 2 } ) ^ { 2 } } & { - \alpha ( 1 + \alpha - \delta \sigma _ { j } ^ { 2 } ) } & { - \alpha ( 1 + \alpha - \delta \sigma _ { j } ^ { 2 } ) } & { \alpha ^ { 2 } } \\ { ( 1 + \alpha - \delta \sigma _ { j } ^ { 2 } ) } & { 0 } & { - \alpha } & { 0 } \\ { ( 1 + \alpha - \delta \sigma _ { j } ^ { 2 } ) } & { - \alpha } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 0 } & { 0 } \end{array} \right] . } \end{array} +$$ + +We prove Proposition 3 by showing that for any choice of stepsize and momentum, either of the two holds: + +• $B ^ { ( 1 ) }$ has an eigenvalue larger than 1, or, • the largest eigenvalue of $B ^ { ( 2 ) }$ is greater than $1 - { \frac { 5 0 0 } { \kappa } }$ . + +This is formalized in the following two lemmas. + +Lemma 4. If the stepsize $\delta$ is such that $\delta \sigma _ { 1 } ^ { 2 } \geq \frac { 2 \left( 1 - \alpha ^ { 2 } \right) } { c + ( c - 2 ) \alpha }$ , then $\boldsymbol { B } ^ { ( 1 ) }$ has an eigenvalue $\geq 1$ + +Lemma 5. If the stepsize $\delta$ is such that $\begin{array} { r } { \delta \sigma _ { 1 } ^ { 2 } < \frac { 2 \left( 1 - \alpha ^ { 2 } \right) } { c + ( c - 2 ) \alpha } } \end{array}$ , then $B ^ { ( 2 ) }$ has an eigenvalue of magnitude $\textstyle { \ge } 1 - { \frac { 5 0 0 } { \kappa } }$ + +Given this notation, we can now consider the $j ^ { t h }$ dimension without the superscripts; when needed, they will be made clear in the exposition. Denoting $x \stackrel { \mathrm { d e f } } { = } \delta \sigma ^ { 2 }$ and $t \stackrel { \mathrm { d e f } } { = } 1 + \alpha - x$ , we have: + +$$ +\begin{array} { r } { B = \left[ \begin{array} { c c c c } { t ^ { 2 } + ( c - 1 ) x ^ { 2 } } & { - \alpha t } & { - \alpha t } & { \alpha ^ { 2 } } \\ { t } & { 0 } & { - \alpha } & { 0 } \\ { t } & { - \alpha } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 0 } & { 0 } \end{array} \right] } \end{array} +$$ + +# A.1 PROOF + +The analysis goes via computation of the characteristic polynomial of $\boldsymbol { B }$ and evaluating it at different values to obtain bounds on its roots. + +Lemma 6. The characteristic polynomial of $\boldsymbol { B }$ is: + +$$ +D ( z ) = z ^ { 4 } - ( t ^ { 2 } + ( c - 1 ) x ^ { 2 } ) z ^ { 3 } + ( 2 \alpha t ^ { 2 } - 2 \alpha ^ { 2 } ) z ^ { 2 } + ( - t ^ { 2 } + ( c - 1 ) x ^ { 2 } ) \alpha ^ { 2 } z + \alpha ^ { 4 } . +$$ + +Proof. We first begin by writing out the expression for the determinant: + +$$ +D e t ( B - z { \mathcal { Z } } ) = \left| \begin{array} { c c c c } { t ^ { 2 } + ( c - 1 ) x ^ { 2 } - z } & { - \alpha t } & { - \alpha t } & { \alpha ^ { 2 } } \\ { t } & { - z } & { - \alpha } & { 0 } \\ { t } & { - \alpha } & { - z } & { 0 } \\ { 1 } & { 0 } & { 0 } & { - z } \end{array} \right| . +$$ + +expanding along the first column, we have: + +$$ +\begin{array} { r l } & { \gamma _ { e t } ( B - z \mathcal { Z } ) = ( t ^ { 2 } + ( c - 1 ) x ^ { 2 } - z ) ( \alpha ^ { 2 } z - z ^ { 3 } ) - t ( - \alpha t z ^ { 2 } + \alpha ^ { 2 } t z ) + t ( - \alpha t ( \alpha z ) + z \cdot \alpha t z ) - ( z \cdot \alpha ^ { 2 } z - \alpha t \mathcal { Z } ) } \\ & { \qquad = ( t ^ { 2 } + ( c - 1 ) x ^ { 2 } - z ) ( \alpha ^ { 2 } z - z ^ { 3 } ) - 2 t ( \alpha ^ { 2 } t z - \alpha t z ^ { 2 } ) - ( \alpha ^ { 2 } z ^ { 2 } - \alpha ^ { 4 } ) . } \end{array} +$$ + +Expanding the terms yields the expression in the lemma. + +The next corollary follows by some simple arithmetic manipulations. + +Corollary 7. Substituting $z = 1 - \tau$ in the characteristic equation of Lemma $6$ , we have: + +$$ +\begin{array} { r l } & { D ( 1 - \tau ) = \tau ^ { 4 } + \tau ^ { 3 } ( - 4 + t ^ { 2 } + ( c - 1 ) x ^ { 2 } ) + \tau ^ { 2 } ( 6 - 3 t ^ { 2 } - 3 ( c - 1 ) x ^ { 2 } - 2 \alpha ^ { 2 } + 2 \alpha t ^ { 2 } ) } \\ & { \qquad + \tau ( - 4 + 3 t ^ { 2 } + 3 ( c - 1 ) x ^ { 2 } + 4 \alpha ^ { 2 } - 4 \alpha t ^ { 2 } - ( c - 1 ) x ^ { 2 } \alpha ^ { 2 } + t ^ { 2 } \alpha ^ { 2 } ) } \\ & { \qquad + ( 1 - t ^ { 2 } - ( c - 1 ) x ^ { 2 } - 2 \alpha ^ { 2 } + 2 \alpha t ^ { 2 } + ( c - 1 ) x ^ { 2 } \alpha ^ { 2 } - t ^ { 2 } \alpha ^ { 2 } + \alpha ^ { 4 } ) } \\ & { \qquad = \tau ^ { 4 } + \tau ^ { 3 } [ - ( 3 + \alpha ) ( 1 - \alpha ) - 2 x ( 1 + \alpha ) + c x ^ { 2 } ] } \\ & { \qquad + \tau ^ { 2 } [ ( 3 - 4 \alpha - \alpha ^ { 2 } + 2 \alpha ^ { 3 } ) - 2 x ( 1 + \alpha ) ( 2 \alpha - 3 ) + x ^ { 2 } ( 2 \alpha - 3 c ) ] } \\ & { \qquad + \tau [ - ( 1 - \alpha ) ^ { 2 } ( 1 - \alpha ^ { 2 } ) - 2 x ( 3 - \alpha ) ( 1 - \alpha ^ { 2 } ) + x ^ { 2 } ( 3 c - 4 \alpha + ( 2 - c ) \alpha ^ { 2 } ) ] } \\ & { \qquad + x ( 1 - \alpha ) [ 2 ( 1 - \alpha ^ { 2 } ) - x ( c + ( c - 2 ) \alpha ) ] . } \end{array} +$$ + +Proof of Lemma 4. The first observation necessary to prove the lemma is that the characteristic polynomial $D ( z )$ approaches $\infty$ as $z \infty$ , i.e., $\begin{array} { r } { \operatorname* { l i m } _ { z \infty } D ( z ) = + \infty } \end{array}$ . + +Next, we evaluate the characteristic polynomial at 1, i.e. compute $D ( 1 )$ . This follows in a straightforward manner from corollary (7) by substituting $\tau = 0$ in equation (2), and this yields, + +$$ +D ( 1 ) = ( 1 - \alpha ) x \cdot \bigg ( 2 ( 1 - \alpha ^ { 2 } ) - x ( 1 - \alpha ) - ( c - 1 ) x ( 1 + \alpha ) \bigg ) . +$$ + +As $\alpha < 1$ , $x = \delta \sigma ^ { 2 } > 0$ , we have the following by setting $D ( 1 ) \leq 0$ and solving for $x$ : + +$$ +x \geq \frac { 2 ( 1 - \alpha ^ { 2 } ) } { c + ( c - 2 ) \alpha } . +$$ + +Since $D ( 1 ) \leq 0$ and $D ( z ) \geq 0$ as $z \infty$ , there exists a root of $D ( \cdot )$ which is $\geq 1$ . + +Remark 8. The above characterization is striking in the sense that for any $c > 1$ , increasing the momentum parameter $\alpha$ naturally requires the reduction in the step size $\delta$ to permit the convergence of the algorithm, which is not observed when fast gradient methods are employed in deterministic optimization. For instance, in the case of deterministic optimization, setting $c = 1$ yields $\delta \sigma _ { 1 } ^ { 2 } <$ $2 ( 1 + \alpha )$ . On the other hand, when employing the stochastic heavy ball method with $x ^ { ( j ) } = 2 \sigma _ { j } ^ { 2 }$ , we have the condition that $c = 2$ , and this implies, $\begin{array} { r } { \delta \sigma _ { 1 } ^ { 2 } < \frac { 2 ( 1 - \alpha ^ { 2 } ) } { 2 } = 1 - \alpha ^ { 2 } } \end{array}$ . + +We now prove Lemma 5. We first consider the large momentum setting. + +Lemma 9. When the momentum parameter $\alpha$ is set such that $1 - 4 5 0 / \kappa \leq \alpha \leq 1 , ~ $ $\boldsymbol { B }$ has an eigenvalue of magnitude ≥ 1 − 450κ . + +Proof. This follows easily from the fact that $\begin{array} { r } { \operatorname* { d e t } ( \boldsymbol { B } ) = \alpha ^ { 4 } = \prod _ { j = 1 } ^ { 4 } \lambda _ { j } ( \boldsymbol { B } ) \le ( \lambda _ { \operatorname* { m a x } } ( \boldsymbol { B } ) ) ^ { 4 } } \end{array}$ , thus implying $1 - 4 5 0 / \kappa \leq \alpha \leq | \lambda _ { \mathrm { m a x } } ( \beta ) |$ . + +Remark 10. Note that the above lemma holds for any value of the learning rate $\delta$ , and holds for every eigen direction of $\mathbf { H }$ . Thus, for “large” values of momentum, the behavior of stochastic heavy ball does degenerate to the behavior of stochastic gradient descent. + +We now consider the setting where momentum is bounded away from 1. + +Corollary 11. Consider $B ^ { ( 2 ) }$ , by substituting $= l / \kappa , x = \delta \lambda _ { \mathrm { m i n } } = c ( \delta \sigma _ { 1 } ^ { 2 } ) / \kappa$ in equation (2) and accumulating terms in varying powers of $1 / \kappa$ , we obtain: + +$$ +\begin{array} { l } { { G ( l ) \stackrel { d e f } { = } \frac { c ^ { 3 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \kappa ^ { 5 } } + l ^ { 4 } - 2 c ( \delta \sigma _ { 1 } ^ { 2 } ) l ^ { 3 } ( 1 + \alpha ) + ( 2 \alpha - 3 c ) c ^ { 2 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 2 } } } \\ { { \ + \ \frac { - ( 3 + \alpha ) ( 1 - \alpha ) l ^ { 3 } - 2 ( 1 + \alpha ) ( 2 \alpha - 3 ) c ( \delta \sigma _ { 1 } ^ { 2 } ) l ^ { 2 } + ( 3 c - 4 \alpha + ( 2 - c ) \alpha ^ { 2 } ) c ^ { 2 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \kappa ^ { 3 } } } } \\ { { \ + \ \frac { ( 3 - 4 \alpha - \alpha ^ { 2 } + 2 \alpha ^ { 3 } ) l ^ { 2 } - 2 c ( \delta \sigma _ { 1 } ^ { 2 } ) l ( 3 - \alpha ) ( 1 - \alpha ^ { 2 } ) - c ^ { 2 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } ( 1 - \alpha ) ( c + ( c - 2 ) \alpha ) } { \kappa ^ { 2 } } } } \\ { { \ + \ \frac { - ( 1 - \alpha ) ^ { 2 } ( 1 - \alpha ^ { 2 } ) l + 2 c ( \delta \sigma _ { 1 } ^ { 2 } ) ( 1 - \alpha ) ( 1 - \alpha ^ { 2 } ) } { \kappa } } } \end{array} +$$ + +Lemma 12. Let $2 < c < 3 0 0 0$ , $\textstyle 0 \leq \alpha \leq 1 - { \frac { 4 5 0 } { \kappa } }$ , $\begin{array} { r } { l = 1 + \frac { 2 c ( \delta \sigma _ { 1 } ^ { 2 } ) } { 1 - \alpha } } \end{array}$ 2c(δσ21) . Then, G(l) ≤ 0. + +Proof. Since $\begin{array} { r } { ( \delta \sigma _ { 1 } ^ { 2 } ) \le \frac { 2 ( 1 - \alpha ^ { 2 } ) } { c + ( c - 2 ) \alpha } } \end{array}$ , this implies $\begin{array} { r } { \frac { ( \delta \sigma _ { 1 } ^ { 2 } ) } { 1 - \alpha } \leq \frac { 2 ( 1 + \alpha ) } { c + ( c - 2 ) \alpha } \leq \frac { 4 } { c } } \end{array}$ , thus implying, $1 \leq l \leq 9$ + +Substituting the value of $l$ in equation (3), the coefficient of $\mathcal { O } ( 1 / \kappa )$ is $- ( 1 - \alpha ) ^ { 3 } ( 1 + \alpha )$ . + +We will bound this term along with $( 3 - 4 \alpha - \alpha ^ { 2 } + 2 \alpha ^ { 3 } ) l ^ { 2 } / \kappa ^ { 2 } = ( 1 - \alpha ) ^ { 2 } ( 3 + 2 \alpha ) l ^ { 2 } / \kappa ^ { 2 }$ to obtain: + +$$ +\begin{array} { r l } & { \frac { - ( 1 - \alpha ) ^ { 3 } ( 1 + \alpha ) } { \kappa } + \frac { ( 1 - \alpha ) ^ { 2 } ( 3 + 2 \alpha ) l ^ { 2 } } { \kappa ^ { 2 } } \leq \frac { - ( 1 - \alpha ) ^ { 3 } ( 1 + \alpha ) } { \kappa } + \frac { 4 0 5 ( 1 - \alpha ) ^ { 2 } } { \kappa ^ { 2 } } } \\ & { \qquad \leq \frac { ( 1 - \alpha ) ^ { 2 } } { \kappa } \bigg ( \frac { 4 0 5 } { \kappa } - ( 1 - \alpha ^ { 2 } ) \bigg ) } \\ & { \qquad \leq \frac { ( 1 - \alpha ) ^ { 2 } } { \kappa } \bigg ( \frac { 4 0 5 } { \kappa } - ( 1 - \alpha ) \bigg ) \leq - \frac { 4 5 \cdot 4 5 0 ^ { 2 } } { \kappa ^ { 4 } } , } \end{array} +$$ + +where, we use the fact that $\alpha < 1 , l \le 9$ . The natural implication of this bound is that the terms that are lower order, such as $\mathcal { O } ( 1 / \kappa ^ { 4 } )$ and $\mathcal { O } ( 1 / \kappa ^ { 5 } )$ will be negative owing to the large constant above. Let us verify that this is indeed the case by considering the terms having powers of $\mathcal { O } ( 1 / \kappa ^ { 4 } )$ and $\mathcal { O } ( 1 / \kappa ^ { 5 } )$ from equation (3): + +$$ +\begin{array} { r l } & { \frac { c ^ { 3 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \kappa ^ { 5 } } + \frac { l ^ { 4 } - 2 c ( \delta \sigma _ { 1 } ^ { 2 } ) l ^ { 3 } ( 1 + \alpha ) + ( 2 \alpha - 3 c ) c ^ { 2 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 2 } } { \kappa ^ { 4 } } - \frac { 4 5 \cdot 4 5 0 ^ { 2 } } { \kappa ^ { 4 } } } \\ & { \leq \frac { c ^ { 3 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \kappa ^ { 5 } } + \frac { l ^ { 4 } } { \kappa ^ { 4 } } - \frac { 4 5 \cdot 4 5 0 ^ { 2 } } { \kappa ^ { 4 } } } \\ & { \leq \frac { c l ^ { 3 } } { \kappa ^ { 5 } } + \frac { ( 9 ^ { 4 } - ( 4 5 \cdot 4 5 0 ^ { 2 } ) ) } { \kappa ^ { 4 } } \leq \frac { 9 ^ { 3 } c + 9 ^ { 4 } - ( 4 5 \cdot 4 5 0 ^ { 2 } ) } { \kappa ^ { 4 } } } \end{array} +$$ + +The expression above evaluates to $\leq 0$ given an upperbound on the value of $c$ . The expression above follows from the fact that $l \leq 9 , \kappa \geq 1$ . + +Next, consider the terms involving $\mathcal { O } ( 1 / \kappa ^ { 3 } )$ and $\mathcal { O } ( 1 / \kappa ^ { 2 } )$ , in particular, + +$$ +\begin{array} { r l } & { \frac { ( 3 c - 4 \alpha + ( 2 - c ) \alpha ^ { 2 } ) c ^ { 2 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } l } { \kappa ^ { 3 } } - \frac { c ^ { 2 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } ( 1 - \alpha ) ( c + ( c - 2 ) \alpha ) } { \kappa ^ { 2 } } } \\ & { \leq \frac { c ^ { 2 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \kappa ^ { 2 } } ( \frac { l ( 3 c + 2 ) } { \kappa } - ( 1 - \alpha ) ( c + ( c - 2 ) \alpha ) ) } \\ & { \leq \frac { c ^ { 2 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \kappa ^ { 2 } } ( \frac { 5 \epsilon } { \kappa } - ( 1 - \alpha ) ( c + ( c - 2 ) \alpha ) ) } \\ & { \leq \frac { c ^ { 2 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \kappa ^ { 2 } } ( \frac { 5 \epsilon l } { \kappa } - ( 1 - \alpha ) c ) } \\ & { \leq \frac { c ^ { 3 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \kappa ^ { 2 } } ( \frac { 5 l } { \kappa } - \frac { 4 5 0 } { \kappa } ) } \\ & { \leq \frac { c ^ { 3 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \kappa ^ { 2 } } ( \frac { 1 - \delta ^ { 2 } } { \kappa } ) } \\ & { \leq \frac { c ^ { 3 } ( \delta \sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \kappa ^ { 2 } } \cdot \frac { - 4 4 5 0 } { \kappa } \leq 0 . } \end{array} +$$ + +Next, + +$$ +\begin{array} { r l } & { \frac { - 2 ( 1 + \alpha ) ( 2 \alpha - 3 ) e ( \delta \sigma _ { 1 } ^ { 2 } ) l ^ { 2 } } { \kappa ^ { 3 } } - \frac { 2 c ( \delta \sigma _ { 1 } ^ { 2 } ) l ( 3 - \alpha ) ( 1 - \alpha ^ { 2 } ) } { \kappa ^ { 2 } } } \\ & { \leq \frac { 2 ( 1 + \alpha ) c ( \delta \sigma _ { 1 } ^ { 2 } ) l } { \kappa ^ { 2 } } \Big ( \frac { - ( 2 \alpha - 3 ) l } { \kappa } - ( 3 - \alpha ) ( 1 - \alpha ) \Big ) } \\ & { \leq \frac { 2 ( 1 + \alpha ) c ( \delta \sigma _ { 1 } ^ { 2 } ) l } { \kappa ^ { 2 } } \Big ( \frac { 3 l } { \kappa } - 2 ( 1 - \alpha ) \Big ) } \\ & { \leq \frac { 2 ( 1 + \alpha ) c ( \delta \sigma _ { 1 } ^ { 2 } ) l } { \kappa ^ { 2 } } \Big ( \frac { 3 l } { \kappa } - \frac { 2 \cdot 4 5 0 } { \kappa } \Big ) } \\ & { \leq \frac { 2 ( 1 + \alpha ) c ( \delta \sigma _ { 1 } ^ { 2 } ) l } { \kappa ^ { 2 } } \Big ( \frac { 3 \cdot 2 7 } { \kappa } - \frac { 2 \cdot 4 5 0 } { \kappa } \Big ) \leq 0 . } \end{array} +$$ + +In both these cases, we used the fact that remaining terms are negative. $\textstyle \alpha \leq 1 - { \frac { 4 5 0 } { \kappa } }$ implying $\begin{array} { r } { - ( 1 - \alpha ) \le \frac { - 4 5 0 } { \kappa } } \end{array}$ . Finally, other + +Before rounding up the proof of the proposition, we need the following lemma to ensure that our lower bounds on the largest eigenvalue of $\boldsymbol { B }$ indeed affect the algorithm’s rates and are true irrespective of where the algorithm is begun. Note that this allows our result to be much stronger than typical optimization lowerbounds that rely on specific initializations to ensure a component along the largest eigendirection of the update operator, for which bounds are proven. + +Lemma 13. For any starting iterate $\mathbf { w } _ { 0 } \neq \mathbf { w } ^ { * }$ , the HB method produces a non-zero component along the largest eigen direction of $\boldsymbol { B }$ . + +Proof. We note that in a similar manner as other proofs, it suffices to argue for each dimension of the problem separately. But before we start looking at each dimension separately, let us consider the $\bar { j } ^ { \mathrm { t h } }$ dimension, and detail the approach we use to prove the claim: the idea is to examine the subspace spanned by covariance $\mathbb { E } \left[ \pmb { \theta } _ { . } ^ { ( j ) } \otimes \pmb { \theta } _ { . } ^ { ( j ) } \right]$ of the iterates $\pmb { \theta } _ { 0 } ^ { ( j ) } , \pmb { \theta } _ { 1 } ^ { ( j ) } , \pmb { \theta } _ { 2 } ^ { ( j ) } , . . . ,$ for every starting iterate $\pmb { \theta } _ { 0 } ^ { ( j ) } \neq \left[ 0 , 0 \right] ^ { \top }$ and prove that the largest eigenvector of the expected operator $B ^ { ( j ) }$ is not orthogonal to this subspace. This implies that there exists a non-zero component of $\mathbb { E } \left[ \pmb { \theta } _ { \cdot } ^ { ( j ) } \otimes \pmb { \theta } _ { \cdot } ^ { ( j ) } \right]$ in the largest eigen direction of $B ^ { ( j ) }$ , and this decays at a rate that is at best $\lambda _ { \operatorname* { m a x } } ( B ^ { ( j ) } )$ . + +Since $\boldsymbol { B } ^ { ( j ) } ~ \in ~ \mathbb { R } ^ { 4 \times 4 }$ , we begin by examining the expected covariance spanned by the iterates $\pmb { \theta } _ { 0 } ^ { ( j ) } , \pmb { \theta } _ { 1 } ^ { ( j ) } , \pmb { \theta } _ { 2 } ^ { ( j ) } , \pmb { \theta } _ { 3 } ^ { ( j ) }$ . Let $\mathbf { w } _ { 0 } ^ { ( j ) } - ( \mathbf { w } ^ { * } ) ^ { ( j ) } = \mathbf { \bar { w } } _ { - 1 } ^ { ( j ) } - ( \mathbf { w } ^ { * } ) ^ { ( j ) } = k ^ { ( j ) }$ . Now, this implies $\theta _ { 0 } ^ { ( j ) } =$ $\boldsymbol { k } ^ { ( j ) } \cdot \left[ 1 , 1 \right] ^ { \top }$ . Then, + +$$ +\pmb { \theta } _ { 1 } ^ { ( j ) } = k ^ { ( j ) } \widehat { \mathbf { A } } _ { 1 } ^ { ( j ) } \left[ 1 \right] , \mathrm { w i t h } \widehat { \mathbf { A } } _ { 1 } ^ { ( j ) } = \left[ 1 + \alpha - \delta \widehat { \mathbf { H } } _ { 1 } ^ { ( j ) } \quad - \alpha \right] , \mathrm { w h e r e } \widehat { \mathbf { H } } _ { 1 } ^ { ( j ) } = \big ( a _ { 1 } ^ { ( j ) } \big ) ^ { 2 } . +$$ + +This implies that $k$ just appears as a scale factor. This in turn implies that in order to analyze the subspace spanned by the covariance of iterates $\theta _ { 0 } ^ { ( j ) } , \theta _ { 1 } ^ { ( j ) } , . . . ,$ , we can assume $k ^ { ( j ) } = 1$ without any loss in generality. This implies, $\pmb { \theta } _ { 0 } ^ { ( j ) } = \left[ 1 , 1 \right] ^ { \top }$ . Note that with this in place, we see that we can now drop the superscript $j$ that represents the dimension, since the analysis decouples across the dimensions $j \in \{ 1 , 2 \}$ . Furthermore, let the entries of the vector $\pmb { \theta } _ { k }$ be represented as $\pmb { \theta } _ { k }$ def = $\left[ \theta _ { k 1 } \quad \theta _ { k 2 } \right] ^ { \top }$ Next, denote $1 + \alpha - \delta \widehat { \mathbf { H } } _ { k } = \widehat { t } _ { k }$ . This implies, + +$$ +\begin{array} { r } \widehat { \mathbf { A } } _ { k } = \left[ \begin{array} { c c } { \widehat { t } _ { k } } & { - \alpha \right] . } \end{array} \end{array} +$$ + +Furthermore, + +$$ +\begin{array} { r } { \pmb \theta _ { 1 } = \widehat { \mathbf A } _ { 1 } \pmb \theta _ { 0 } = \left[ \hat { t } _ { 1 } - \alpha \right] , \pmb \theta _ { 2 } = \widehat { \mathbf A } _ { 2 } \pmb \theta _ { 1 } = \left[ \hat { t } _ { 2 } ( \hat { t } _ { 1 } - \alpha ) - \alpha \right] , } \\ { \pmb \theta _ { 3 } = \widehat { \mathbf A } _ { 3 } \pmb \theta _ { 2 } = \left[ \hat { t } _ { 3 } ( \hat { t } _ { 2 } ( \hat { t } _ { 1 } - \alpha ) - \alpha ) - \alpha ( \hat { t } _ { 1 } - \alpha ) \right] . } \end{array} +$$ + +Let us consider the vectorized form of $\Phi _ { j } = \mathbb { E } \left[ \pmb { \theta } _ { j } \otimes \pmb { \theta } _ { j } \right]$ , and we denote this as vec $( \Phi _ { j } )$ . Note that $\mathrm { v e c } ( \Phi _ { j } )$ makes $\Phi _ { j }$ become a column vector of size $4 \times 1$ . Now, consider vec $( \Phi _ { j } )$ for $\bar { \boldsymbol { j } } = 0 , 1 , 2 , 3$ and concatenate these to form a matrix that we denote as $\mathcal { D }$ , i.e. + +$$ +\mathcal { D } = \left[ \mathrm { v e c } ( \Phi _ { 0 } ) \mathrm { v e c } ( \Phi _ { 1 } ) \mathrm { v e c } ( \Phi _ { 2 } ) \mathrm { v e c } ( \Phi _ { 3 } ) \right] . +$$ + +Now, since we note that $\Phi _ { j }$ is a symmetric $2 \times 2$ matrix, $\mathcal { D }$ should contain two identical rows implying that it has an eigenvalue that is zero and a corresponding eigenvector that is $\begin{array} { r } { \left[ { 0 \mathrm { ~ \ t ~ { ~ - } 1 / { \sqrt { 2 } } ~ } } { \hat { 1 } } / { \sqrt { ( 2 \tau ) } } \mathrm { ~ \ t ~ { ~ } } \right] ^ { \top } } \end{array}$ . It turns out that this is also an eigenvector of $\boldsymbol { B }$ with an eigenvalue $\alpha$ . Note that $\operatorname* { d e t } ( B ) = \alpha ^ { 4 }$ . This implies there are two cases that we need to consider: (i) when all eigenvalues of $\boldsymbol { B }$ have the same magnitude $( = \alpha )$ . In this case, we are already done, because there exists at least one non zero eigenvalue of $\mathcal { D }$ and this should have some component along one of the eigenvectors of $\boldsymbol { B }$ and we know that all eigenvectors have eigenvalues with a magnitude equal to $\lambda _ { \mathrm { m a x } } ( B )$ . Thus, there exists an iterate which has a non-zero component along the largest eigendirection of $\boldsymbol { B }$ . (ii) the second case is the situation when we have eigenvalues with different magnitudes. In this case, note that $\operatorname* { d e t } ( \mathcal B ) = \alpha ^ { 4 } < ( \lambda _ { \operatorname* { m a x } } ( \mathcal B ) ) ^ { 4 }$ implying $\bar { \lambda } _ { \mathrm { m a x } } ( B ) > \alpha$ . In this case, we need to prove that $\mathcal { D }$ spans a three-dimensional subspace; if it does, it contains a component along the largest eigendirection of $\boldsymbol { B }$ which will round up the proof. Since we need to understand whether $\mathcal { D }$ spans a three dimensional subspace, we can consider a different (yet related) matrix, which we call $\mathcal { R }$ and this is defined as: + +$$ +\mathcal { R } \stackrel { \mathrm { d e f } } { = } \mathbb { E } \left( \begin{array} { c c c } { \theta _ { 0 1 } ^ { 2 } } & { \theta _ { 1 1 } ^ { 2 } } & { \theta _ { 2 1 } ^ { 2 } } \\ { \theta _ { 0 1 } \theta _ { 0 2 } } & { \theta _ { 1 1 } \theta _ { 1 2 } } & { \theta _ { 2 1 } \theta _ { 2 2 } } \\ { \theta _ { 0 2 } ^ { 2 } } & { \theta _ { 1 2 } ^ { 2 } } & { \theta _ { 2 2 } ^ { 2 } } \end{array} \right) +$$ + +Given the expressions for $\{ \pmb { \theta } _ { j } \} _ { j = 0 } ^ { 3 }$ (by definition of $\pmb { \theta } _ { 0 }$ and using equation 4), we can substitute to see that $\mathcal { R }$ has the following expression: + +$$ +\mathcal { R } = \left[ { 1 \atop 1 } \begin{array} { c c } { { \mathbb { E } \left[ ( \hat { t } _ { 1 } - \alpha ) ^ { 2 } \right] } } & { { \mathbb { E } \left[ ( \hat { t } _ { 2 } ( \hat { t } _ { 1 } - \alpha ) - \alpha ) ^ { 2 } \right] } } \\ { { \mathbb { E } \left[ \hat { t } _ { 1 } - \alpha \right] } } & { { \mathbb { E } \left[ ( ( \hat { t } _ { 2 } ( \hat { t } _ { 1 } - \alpha ) - \alpha ) ) ( \hat { t } _ { 1 } - \alpha ) \right] } } \\ { { 1 } } & { { \mathbb { E } \left[ ( \hat { t } _ { 1 } - \alpha ) ^ { 2 } \right] } } \end{array} \right] . +$$ + +If we compute and prove that $\operatorname* { d e t } ( \mathcal { R } ) \neq 0$ , we are done since that implies that $\mathcal { R }$ has three non-zero eigenvalues. + +This implies, we first define the following: let $q _ { \gamma } = ( t - \gamma ) ^ { 2 } + ( c - 1 ) x ^ { 2 }$ . Then, $\mathcal { R }$ can be expressed as: + +$$ +\begin{array} { r l } & { \mathrm { d e t } ( \mathcal { R } ) = \mathrm { d e t } { \left( \left[ \begin{array} { l l l } { 1 } & { q _ { \alpha } } & { 2 q _ { \alpha } - 2 \alpha t ( t - \alpha ) + \alpha ^ { 2 } } \\ { 1 } & { t - \alpha } & { t q _ { \alpha } - \alpha ( t - \alpha ) } \end{array} \right] \right) } } \\ & { \qquad = \mathrm { d e t } { \left( \left[ \begin{array} { l l l } { 1 } & { q _ { \alpha } } & { 2 \alpha } \\ { 1 } & { 1 } & { \phi _ { \alpha } } \\ { 1 } & { 1 } & { \phi _ { \alpha } } \\ { 1 } & { t - \alpha } & { q _ { \alpha } - \alpha ( t - \alpha ) - 2 \alpha t ( t - \alpha ) + \alpha ^ { 2 } } \\ { 1 } & { 1 } & { \theta _ { \alpha } } \end{array} \right] \right) } } \\ & { \qquad = \mathrm { d e t } { \left( \left[ \begin{array} { l l l } { 1 } & { q _ { \alpha } - 1 } & { q _ { \alpha } ( q _ { \alpha } - q _ { \alpha } ) - 2 \alpha t ( t - \alpha ) + \alpha ^ { 2 } } \\ { 1 } & { t - \alpha - 1 } & { 1 } & { 0 } \\ { 1 } & { 0 } & { t q _ { \alpha } - \alpha ( t - \alpha ) - 0 } & { ( t - \alpha ) q _ { \alpha } } \end{array} \right] \right) } } \\ & { \qquad = \mathrm { d e t } { \left( \left[ \begin{array} { l l l } { 0 } & { q _ { \alpha } - 1 } & { q _ { \alpha } ( q _ { \alpha } - q _ { \alpha } ) - ( t - \alpha ) + \alpha ^ { 2 } } \\ { 0 } & { 1 } & { \theta _ { \alpha } } \end{array} \right] \right) } } \\ & { \qquad = \mathrm { d e t } { \left( \left[ \begin{array} { l l l } { 0 } & { q _ { \alpha } - 1 } & { q _ { \alpha } ( q _ { \alpha } - q _ { \alpha } ) - 2 \alpha t ( t - \alpha ) + \alpha ^ { 2 } } \\ { 0 } & { t - \alpha - 1 } & { 1 } \end{array} \right] \right) } } \end{array} +$$ + +Note: (i) $q _ { \alpha } - 1 = ( t - \alpha ) ^ { 2 } - 1 + ( c - 1 ) x ^ { 2 } = ( 1 - x ) ^ { 2 } - 1 + ( c - 1 ) x ^ { 2 } = - 2 x + x ^ { 2 } + ( c - 1 ) x ^ { 2 } = 0$ $- 2 x + c x ^ { 2 }$ . + +(ii) $t - \alpha - 1 = - x$ +(iv) (iii) $\begin{array} { r l } & { \alpha ( q _ { \alpha } - ( t - \alpha ) ) = \alpha ( ( t - \alpha ) ^ { 2 } - ( t - \alpha ) + ( c - 1 ) x ^ { 2 } ) = \alpha ( ( 1 - x ) ( - x ) + ( c - 1 ) x ^ { 2 } ) = \alpha x ( - 1 + c x ) } \\ & { q _ { 0 } - q _ { \alpha } = t ^ { 2 } - ( t - \alpha ) ^ { 2 } = \alpha ( 2 t - \alpha ) = 2 t \alpha - \alpha ^ { 2 } . } \end{array}$ +Then, + +$$ +\begin{array} { r } { ( 2 \alpha t - \alpha ^ { 2 } ) q _ { \alpha } - 2 \alpha t ( t - \alpha ) + \alpha ^ { 2 } = 2 t \alpha ( q _ { \alpha } - ( t - \alpha ) ) + \alpha ^ { 2 } ( 1 - q _ { \alpha } ) } \\ { = 2 t \alpha ( - x + c x ^ { 2 } ) - \alpha ^ { 2 } ( - 2 x + c x ^ { 2 } ) } \end{array} +$$ + +$$ +\begin{array} { r l } & { = - 2 t \alpha x + 2 x \alpha ^ { 2 } + 2 t \alpha c x ^ { 2 } - c \alpha ^ { 2 } x ^ { 2 } } \\ & { = 2 \alpha x ( - t + \alpha ) + c \alpha x ^ { 2 } ( 2 t - \alpha ) } \\ & { = - 2 \alpha x ( 1 - x ) + 2 c \alpha x ^ { 2 } ( 1 - x ) + c \alpha ^ { 2 } x ^ { 2 } } \\ & { = 2 \alpha x ( 1 - x ) ( - 1 + c x ) + c \alpha ^ { 2 } x ^ { 2 } . } \end{array} +$$ + +Then, + +$$ +{ \begin{array} { r l } & { \operatorname* { d e t } ( \mathcal { R } ) = \operatorname* { d e t } { \left( \begin{array} { l l l } { 0 } & { x ( c x - 2 ) } & { 2 \alpha x ( 1 - x ) ( - 1 + c x ) + c \alpha ^ { 2 } x ^ { 2 } } \\ { 0 } & { - x } & { \alpha x ( c x - 1 ) } \\ { 1 } & { 0 } & { 0 } \end{array} \right) } } \\ & { \qquad = x ^ { 2 } \alpha \operatorname* { d e t } \left( { \left[ \begin{array} { l l l } { 0 } & { ( c x - 2 ) } & { c \alpha x + 2 ( 1 - x ) ( c x - 1 ) } \\ { 0 } & { - 1 } & { c x - 1 } \\ { 1 } & { 0 } & { 0 } \end{array} \right] } \right) } \\ & { \qquad = x ^ { 3 } \alpha \operatorname* { d e t } \left( { \left[ \begin{array} { l l l } { 0 } & { c } & { c \alpha - 2 ( c x - 1 ) } \\ { 0 } & { - 1 } & { c x - 1 } \\ { 1 } & { 0 } & { 0 } \end{array} \right] } \right) } \end{array} } +$$ + +Then, + +$$ +\begin{array} { c } { { \operatorname * { d e t } ( \mathcal { R } ) = x ^ { 3 } \alpha \bigg ( c ( - 1 + c x ) - 2 ( - 1 + c x ) + c \alpha \bigg ) } } \\ { { = \alpha x ^ { 3 } \bigg ( ( c - 2 ) ( - 1 + c x ) + c \alpha \bigg ) } } \end{array} +$$ + +Note that this determinant can be zero when + +$$ +\alpha = \frac { ( c - 2 ) ( 1 - c x ) } { c } . +$$ + +We show this is not possible by splitting our argument into two parts, one about the convergent regime of the algorithm (where, $\begin{array} { r } { \delta \sigma _ { 1 } ^ { 2 } < \frac { 2 ( 1 - \alpha ^ { 2 } ) } { c + ( c - 2 ) \alpha } ) } \end{array}$ and the other about the divergent regime. + +Let us first provide a proof for the convergent regime of the algorithm. For this regime, let the chosen $\delta$ be represented as $\delta ^ { + }$ . Now, for the smaller eigen direction, $x = \delta ^ { + } \lambda _ { \mathrm { m i n } } = c \bar { \delta ^ { + } } \sigma _ { 1 } ^ { 2 } / \kappa$ . Suppose $\alpha$ was chosen as per equation 5, + +$$ +\begin{array} { c } { { \displaystyle \frac { c \alpha } { c - 2 } = 1 - \frac { c ^ { 2 } \delta ^ { + } \sigma _ { 1 } ^ { 2 } } { \kappa } } } \\ { { \implies \delta ^ { + } \sigma _ { 1 } ^ { 2 } = \displaystyle \frac { \kappa } { c ^ { 2 } } - \frac { \kappa \alpha } { c ( c - 2 ) } . } } \end{array} +$$ + +We will now prove that δ+σ21 = κc ( 1c is much larger than one allowed by the convergence of the HB updates, i.e., $\begin{array} { r } { \delta \sigma _ { 1 } ^ { 2 } < \frac { 2 ( 1 - \alpha ^ { 2 } ) } { c + ( c - 2 ) \alpha } \le \frac { 2 ( 1 - \alpha ^ { 2 } ) } { c } } \end{array}$ . In particular, if we prove that $\textstyle { \frac { \kappa } { c } } { \bigl ( } { \frac { 1 } { c } } - { \frac { \alpha } { c - 2 } } { \bigr ) } >$ 2(1−α2) for any admissible value of α, we are done. + +$$ +\begin{array} { c } { { \displaystyle \frac \kappa c ( \frac 1 c - \frac \alpha { c - 2 } ) > \frac { 2 ( 1 - \alpha ^ { 2 } ) } { c } } } \\ { { \Leftrightarrow \displaystyle \frac \kappa c - \frac { \kappa \alpha } { c - 2 } > 2 - 2 \alpha ^ { 2 } } } \\ { { \Leftrightarrow \displaystyle \frac \kappa c - \frac { \kappa \alpha } { c - 2 } > \frac \kappa c - \frac { \kappa \alpha } { c } > 2 - 2 \alpha ^ { 2 } } } \\ { { \Leftrightarrow \kappa - \kappa \alpha > 2 c - 2 c \alpha ^ { 2 } } } \\ { { \Leftrightarrow 2 c \alpha ^ { 2 } - \kappa \alpha + ( \kappa - 2 c ) > 0 . } } \end{array} +$$ + +The two roots of this quadratic equation are $\alpha ^ { + } = \textstyle { \frac { \kappa } { 2 c } } - 1$ and $\alpha ^ { - } = 1$ . Note that $\kappa \geq \widetilde { \kappa } = c$ ; note that there is not much any method gains over SGD if $\kappa = \mathcal { O } ( c )$ . And, for any $\kappa \geq 4 c$ , note, $\alpha ^ { + } > \alpha ^ { - }$ , indicating that the above equation holds true if $\begin{array} { r } { \alpha > \alpha ^ { + } = \frac { \kappa } { 2 c } - 1 } \end{array}$ or if $\alpha < \alpha ^ { - } = 1$ . The latter condition is true and hence the proposition that $\delta ^ { + } \sigma _ { 1 } ^ { 2 } > \frac { 2 ( 1 - \alpha ^ { 2 } ) } { c + ( c - 2 ) \alpha }$ is true. + +We need to prove that the determinant does not vanish in the divergent regime for rounding up the proof to the lemma. + +Now, let us consider the divergent regime of the algorithm, i.e., when, $\begin{array} { r } { \delta \sigma _ { 1 } ^ { 2 } > \frac { 2 ( 1 - \alpha ^ { 2 } ) } { c + ( c - 2 ) \alpha } } \end{array}$ . Furthermore, for the larger eigendirection, the determinant is zero when $\begin{array} { r } { \delta \sigma _ { 1 } ^ { 2 } = \frac { 1 - \frac { c \alpha } { c - 2 } } { c } = \frac { 1 } { c } - \frac { \alpha } { c - 2 } } \end{array}$ (obtained by substituting $x = \delta \sigma _ { 1 } ^ { 2 }$ in equation 5). If we show that $\textstyle { \frac { 2 ( 1 - \alpha ^ { 2 } ) } { c + ( c - 2 ) \alpha } } > { \frac { 1 } { c } } - { \frac { \alpha } { c - 2 } }$ for all admissible values of $c$ , we are done. We will explore this in greater detail: + +$$ +\begin{array} { c } { { \frac { 2 ( 1 - \alpha ^ { 2 } ) } { c + ( c - 2 ) \alpha } > \displaystyle \frac { 1 } { c } - \frac { \alpha } { c - 2 } } } \\ { { \Leftrightarrow 2 ( 1 - \alpha ^ { 2 } ) \geq 1 + \displaystyle \frac { c - 2 } { c } \alpha - \displaystyle \frac { c } { c - 2 } \alpha - \alpha ^ { 2 } } } \\ { { \Leftrightarrow 1 - \alpha ^ { 2 } \geq \displaystyle \frac { - 4 ( c - 1 ) } { c ( c - 2 ) } \alpha } } \\ { { \Leftrightarrow c ^ { 2 } - 2 c - \alpha ^ { 2 } c ^ { 2 } + 2 c \alpha ^ { 2 } \geq - 4 c \alpha + 4 \alpha } } \\ { { \Leftrightarrow c ^ { 2 } ( 1 - \alpha ^ { 2 } ) - 2 c ( 1 - \alpha ^ { 2 } - 2 \alpha ) - 4 \alpha \geq 0 . } } \end{array} +$$ + +considering the quadratic in the left hand size and solving it for $c$ , we have: + +$$ +\begin{array} { l } { { c ^ { \pm } = \frac { 2 ( 1 - \alpha ^ { 2 } - 2 \alpha ) \pm \sqrt { 4 ( 1 - \alpha ^ { 2 } - 2 \alpha ) ^ { 2 } + 1 6 \alpha ( 1 - \alpha ^ { 2 } ) } } { 2 ( 1 - \alpha ^ { 2 } ) } } } \\ { { { } ~ = \frac { ( 1 - \alpha ^ { 2 } - 2 \alpha ) \pm \sqrt { ( 1 - \alpha ^ { 2 } - 2 \alpha ) ^ { 2 } + 4 \alpha ( 1 - \alpha ^ { 2 } ) } } { ( 1 - \alpha ^ { 2 } ) } } } \\ { { { } ~ = \frac { ( 1 - \alpha ^ { 2 } - 2 \alpha ) \pm \sqrt { 1 + \alpha ^ { 4 } + 4 \alpha ^ { 2 } - 2 \alpha ^ { 2 } - 4 \alpha + 4 \alpha ^ { 3 } + 4 \alpha ( 1 - \alpha ^ { 2 } ) } } { ( 1 - \alpha ^ { 2 } ) } } } \\ { { { } ~ = \frac { ( 1 - \alpha ^ { 2 } - 2 \alpha ) \pm ( 1 + \alpha ^ { 2 } ) } { ( 1 - \alpha ^ { 2 } ) } } } \end{array} +$$ + +This holds true iff + +$$ +c \leq c ^ { - } = \frac { - 2 \alpha ( 1 + \alpha ) } { 1 - \alpha ^ { 2 } } = \frac { - 2 \alpha } { 1 - \alpha } , +$$ + +or iff, + +$$ +c \geq c ^ { + } = { \frac { 2 ( 1 - \alpha ) } { 1 - \alpha ^ { 2 } } } = { \frac { 2 } { 1 + \alpha } } . +$$ + +Which is true automatically since $c > 2$ . This completes the proof of the lemma. + +We are now ready to prove Lemma 5. + +Proof of Lemma 5. Combining Lemmas 9 and 12, we see that no matter what stepsize and momentum we choose, $\boldsymbol { B } ^ { ( j ) }$ has an eigenvalue of magnitude at least $1 - { \frac { 5 0 0 } { \kappa } }$ for some $j \in \{ 1 , 2 \}$ . This proves the lemma. □ + +# B EQUIVALENCE OF ALGORITHM 3 AND ASGD + +We begin by writing out the updates of ASGD as written out in Jain et al. (2017), which starts with two iterates $\widehat { a } _ { 0 }$ and $\widehat { d } _ { 0 }$ , and from time $t = 0 , 1 , . . . T - 1$ implements the following updates: + +$$ +\begin{array} { r l r } & { } & { \widehat { b } _ { t } = \alpha _ { 1 } \widehat { a } _ { t } + ( 1 - \alpha _ { 1 } ) \widehat { d } _ { t } } \\ & { } & { \widehat { a } _ { t + 1 } = \widehat { b } _ { t } - \delta _ { 1 } \widehat { \nabla } f _ { t + 1 } ( \widehat { b } _ { t } ) } \\ & { } & { \widehat { c } _ { t } = \beta _ { 1 } \widehat { b } _ { t } + ( 1 - \beta _ { 1 } ) \widehat { d } _ { t } } \\ & { } & { \widehat { d } _ { t + 1 } = \widehat { c } _ { t } - \gamma _ { 1 } \widehat { \nabla } f _ { t + 1 } ( \widehat { b } _ { t } ) . } \end{array} +$$ + +Next, we specify the step sizes $\beta _ { 1 } = c _ { 3 } ^ { 2 } / \sqrt { \kappa \widetilde { \kappa } }$ , $\alpha _ { 1 } = c _ { 3 } / ( c _ { 3 } + \beta )$ , $\gamma _ { 1 } = \beta / ( c _ { 3 } \lambda _ { \operatorname* { m i n } } )$ and $\delta _ { 1 } = 1 / R ^ { 2 }$ , where $\kappa = R ^ { 2 } / \lambda _ { \operatorname* { m i n } }$ e. Note that the step sizes in the paper of Jain et al. (2017) with $c _ { 1 }$ in their paper set to 1 yields the step sizes above. Now, substituting equation 8 in equation 9 and substituting the value of $\gamma _ { 1 }$ , we have: + +$$ +\begin{array} { r l } & { \widehat { d } _ { t + 1 } = \beta _ { 1 } \left( \widehat { b } _ { t } - \frac { 1 } { c _ { 3 } \lambda _ { \operatorname* { m i n } } } \hat { \nabla } f _ { t + 1 } ( \widehat { b } _ { t } ) \right) + ( 1 - \beta _ { 1 } ) \widehat { d } _ { t } } \\ & { \qquad = \beta _ { 1 } \left( \widehat { b } _ { t } - \frac { \delta \kappa } { c _ { 3 } } \hat { \nabla } f _ { t + 1 } ( \widehat { b } _ { t } ) \right) + ( 1 - \beta _ { 1 } ) \widehat { d } _ { t } . } \end{array} +$$ + +We see that $\widehat { d } _ { t + 1 }$ is precisely the update of the running average $\bar { w } _ { t + 1 }$ in the ASGD method employed in this paper. + +We now update $\widehat { b } _ { t }$ to become $\widehat { b } _ { t + 1 }$ and this can be done by writing out equation 6 at $t + 1$ , i.e: + +$$ +\begin{array} { r l } & { \widehat { b } _ { t + 1 } = \alpha _ { 1 } \widehat { a } _ { t + 1 } + ( 1 - \alpha _ { 1 } ) \widehat { d } _ { t + 1 } } \\ & { \qquad = \alpha _ { 1 } \left( \widehat { b } _ { t } - \delta _ { 1 } \widehat { \nabla } f _ { t + 1 } ( \widehat { b } _ { t } ) \right) + ( 1 - \alpha _ { 1 } ) \widehat { d } _ { t + 1 } . } \end{array} +$$ + +By substituting the value of $\alpha _ { 1 }$ we note that this is indeed the update of the iterate as a convex combination of the current running average and a short gradient step as written in this paper. In this paper, we set $c _ { 3 }$ to be equal to 0.7, and any constant less than 1 works. In terms of variables, we note that $\alpha$ in this paper’s algorithm description maps to $1 - \beta _ { 1 }$ . + +# C MORE DETAILS ON EXPERIMENTS + +In this section, we will present more details on our experimental setup. + +# C.1 LINEAR REGRESSION + +In this section, we will present some more results on our experiments on the linear regression problem. Just as in Appendix A, it is indeed possible to compute the expected error of all the algorithms among SGD, HB, NAG and ASGD, by tracking certain covariance matrices which evolve as linear systems. For SGD, for instance, denoting $\Phi _ { t } ^ { S G D } \stackrel { \mathrm { d e f } } { = } \mathbb { E } \left[ \left( \mathbf { w } _ { t } ^ { S G D } - w ^ { * } \right) \otimes \left( \mathbf { w } _ { t } ^ { S G D } - w ^ { * } \right) \right]$ , we see that ΦSGDt+1 $\Phi _ { t + 1 } ^ { S G D } \ : = \ : B \circ \Phi _ { t } ^ { S G D }$ , where $\boldsymbol { B }$ is a linear operator acting on $d \times d$ matrices such that ${ \mathcal { B } } \circ M { \stackrel { \mathrm { d e f } } { = } } M - \delta H M - \delta M H + \delta ^ { 2 } \mathbb { E } \left[ \left. x , M x \right. x x ^ { \top } \right]$ . Similarly, HB, NAG and ASGD also have corresponding operators (see Appendix A for more details on the operator corresponding to HB). The largest magnitude of the eigenvalues of these matrices indicate the rate of decay achieved by the particular algorithm – smaller it is compared to 1, faster the decay. + +We now detail the range of parameters explored for these results: the condition number $\kappa$ was varied from $\{ 2 ^ { 4 } , 2 ^ { 5 } , . . , \bar { 2 } ^ { 2 8 } \}$ for all the optimization methods and for both the discrete and gaussian problem. For each of these experiments, we draw 1000 samples and compute the empirical estimate of the fourth moment tensor. For NAG and HB, we did a very fine grid search by sampling 50 values in the interval $( 0 , 1 ]$ for both the learning rate and the momentum parameter and chose the parameter setting that yielded the smallest $\lambda _ { \mathrm { m a x } } ( B )$ that is less than 1 (so that it falls in the range of convergence of the algorithm). As for SGD and ASGD, we employed a learning rate of $1 / 3$ for the Gaussian case and a step size of 0.9 for the discrete case. The statistical advantage parameter of ASGD was chosen to be $\sqrt { 3 \kappa / 2 }$ for the Gaussian case and $\sqrt { 2 \kappa / 3 }$ for the Discrete case, and the a long step parameters of $3 \kappa$ and $2 \kappa$ were chosen for the Gaussian and Discrete case respectively. The reason it appears as if we choose a parameter above the theoretically maximal allowed value of the advantage parameter is because the definition of $\kappa$ is different in this case. The $\kappa$ we speak about for this experiment is $\lambda _ { \operatorname* { m a x } } / \lambda _ { \operatorname* { m i n } }$ unlike the condition number for the stochastic optimization problem. In a manner similar to actually running the algorithms (the results of whose are presented in the main paper), we also note that we can compute the rate as in equation 1 and join all these rates using a curve and estimate its slope (in the log scale). This result is indicated in table 3. + +Figure 7 presents these results, where for each method, we did grid search over all parameters and chose parameters that give smallest $\lambda _ { \operatorname* { m a x } }$ . We see the same pattern as in Figure 1 from actual runs – SGD,HB and NAG all have linear dependence on condition number $\kappa$ , while ASGD has a dependence of $\sqrt { \kappa }$ . + +![](images/2ce4c7a58a43a49aac0ae0911e2b09d770a20e8605b3023da8d2c7e23eef036f.jpg) +Figure 7: Expected rate of error decay (equation 1) vs condition number for various methods for the linear regression problem. Left is for discrete distribution and right is for Gaussian distribution. +Table 3: Slopes (i.e. $\gamma$ ) obtained by fitting a line to the curves in Figure 7. A value of $\gamma$ indicates that the error decays at a rate of exp $\left( { \frac { - t } { \kappa ^ { \gamma } } } \right)$ . A smaller value of $\gamma$ indicates a faster rate of error decay. + +
AlgorithmSlope - discreteSlope- Gaussian
SGD0.99900.9995
HB1.03400.9989
NAG1.06271.0416
ASGD0.49230.4906
+ +# C.2 AUTOENCODERS FOR MNIST + +We begin by noting that the learning rates tend to vary as we vary batch sizes, which is something that is known in theory (Jain et al., 2016). Furthermore, we extend the grid especially whenever our best parameters of a baseline method tends to land at the edge of a grid. The parameter ranges explored by our grid search are: + +Batch Size 1: (parameters chosen by running for 20 epochs) + +• SGD: learning rate: $\{ 0 . 0 1 , 0 . 0 1 { \sqrt { 1 0 } } , 0 . 1 , 0 . 1 { \sqrt { 1 0 } } , 1 , { \sqrt { 1 0 } } , 5 , 1 0 , 2 0 , 1 0 { \sqrt { 1 0 } } , 4 0 , 6 0 , 8 0 , 1 0 0 .$ +• NAG/HB: learning rate: $\{ 0 . 0 1 \sqrt { 1 0 } , 0 . 1 , 0 . 1 \sqrt { 1 0 } , 1 , \sqrt { 1 0 } , 1 0 \}$ , momentum $\left\{ 0 , 0 . 5 , 0 . 7 5 , 0 . 9 , 0 . 9 5 , 0 . 9 7 \right\}$ . +• ASGD: learning rate: $\{ 2 . 5 , 5 \}$ , long step $\{ 1 0 0 . 0 , 1 0 0 0 . 0 \}$ , advantage parameter $\{ 2 . 5 , 5 . 0 , 1 0 . 0 , 2 0 . 0 \}$ . + +Batch Size 8: (parameters chosen by running for 50 epochs) + +• SGD: learning rate: $\{ 0 . 0 0 1 , 0 . 0 0 1 \sqrt { 1 0 . 0 } , 0 . 0 1 , 0 . 0 1 \sqrt { 1 0 } , 0 . 1 , 0 . 1 \sqrt { 1 0 } , 1 , \sqrt { 1 0 } , 5 , 1 0 \}$ , $1 0 \sqrt { 1 0 } , 4 0 , 6 0 , 8 0 , 1 0 0 , 1 2 0 , 1 4 0 \}$ . +• NAG/HB: learning rate: $\{ 5 . 0 , 1 0 . 0 , 2 0 . 0 , 1 0 \sqrt { 1 0 } , 4 0 , 6 0 \} .$ , momentum $\{ 0 , 0 . 2 5 , 0 . 5 , 0 . 7 5 , 0 . 9 , 0 . 9 5 \}$ . +• ASGD: learning rate $\{ 4 0 , 6 0 \}$ . For a long step of 100, advantage parameters of $\{ 1 . 5 , 2 , 2 . 5 , 5 , 1 \bar { 0 } , 2 0 \}$ . For a long step of 1000, we swept over advantage parameters of $\{ 2 . 5 , 5 , 1 0 \}$ . + +# C.3 DEEP RESIDUAL NETWORKS FOR CIFAR-10 + +In this section, we will provide more details on our experiments on cifar-10, as well as present some additional results. We used a weight decay of 0.0005 in all our experiments. The grid search parameters we used for various algorithms are as follows. Note that the ranges in which parameters such as learning rate need to be searched differ based on batch size (Jain et al., 2016). Furthermore, we tend to extrapolate the grid search whenever a parameter (except for the learning rate decay factor) at the edge of the grid has been chosen; this is done so that we always tend to lie in the interior of the grid that we have searched on. Note that for the purposes of the grid search, we choose a hold out set from the training data and add it in to the training data after the parameters are chosen, for the final run. + +Batch Size 8: Note: (i) parameters chosen by running for 40 epochs and picking the grid search parameter that yields the smallest validation $0 / 1$ error. (ii) The validation set decay scheme that we use is that if the validation error does not decay by at least $1 \%$ every three passes over the data, we cut the learning rate by a constant factor (which is grid searched as described below). The minimal learning rate to use is fixed to be $6 . 2 5 \times 1 0 ^ { - 5 }$ , so that we do not decay far too many times and curtail progress prematurely. + +• SGD: learning rate: $\left\{ 0 . 0 0 3 3 , 0 . 0 1 , 0 . 0 3 3 , 0 . 1 , 0 . 3 3 \right\}$ , learning rate decay factor $\{ 5 , 1 0 \}$ . +• NAG/HB: learning rate: $\{ 0 . 0 0 1 , 0 . 0 0 3 3 , 0 . 0 1 , 0 . 0 3 3 \}$ , momentum $\{ 0 . 8 , 0 . 9 , 0 . 9 5 , 0 . 9 7 \}$ , learning rate decay factor $\{ 5 , 1 0 \}$ . +• ASGD: learning rate $\{ 0 . 0 1 , 0 . 0 3 3 0 , 0 . 1 \}$ , long step $\{ 1 0 0 0 , 1 0 0 0 0 , 5 0 0 0 0 \}$ , advantage parameter $\{ 5 , 1 0 \}$ , learning rate decay factor $\{ 5 , 1 0 \}$ . + +Batch Size 128: Note: (i) parameters chosen by running for 120 epochs and picking the grid search parameter that yields the smallest validation $0 \dot { / } 1$ error. (ii) The validation set decay scheme that we use is that if the validation error does not decay by at least $0 . 2 \%$ every four passes over the data, we cut the learning rate by a constant factor (which is grid searched as described below). The minimal learning rate to use is fixed to be $1 \times 1 0 ^ { - 3 }$ , so that we do not decay far too many times and curtail progress prematurely. + +• SGD: learning rate: $\{ 0 . 0 1 , 0 . 0 3 , 0 . 0 9 , 0 . 2 7 , 0 . 8 1 \}$ , learning rate decay factor $\{ 2 , { \sqrt { 1 0 } } , 5 \}$ . +• NAG/HB: learning rate: $\{ 0 . 0 1 , 0 . 0 3 , 0 . 0 9 , 0 . 2 7 \}$ , momentum $\left. 0 . 5 , 0 . 8 , 0 . 9 , 0 . 9 5 , 0 . 9 7 \right.$ , learning rate decay factor $\{ 2 , { \sqrt { 1 0 } } , 5 \}$ . +• ASGD: learning rate $\{ 0 . 0 1 , 0 . 0 3 , 0 . 0 9 , 0 . 2 7 \}$ , long step √ $\{ 1 0 0 , 1 0 0 0 , 1 0 0 0 0 \}$ , advantage parameter $\{ 5 , 1 0 , 2 0 \}$ , learning rate decay factor $\{ 2 , { \sqrt { 1 0 } } , 5 \}$ . + +As a final remark, for any comparison across algorithms, such as, (i) ASGD vs. NAG, (ii) ASGD vs HB, we fix the starting learning rate, learning rate decay factor and decay schedule chosen by the best grid search run of NAG/HB respectively and perform a grid search over the long step and advantage parameter of ASGD. In a similar manner, when we compare (iii) SGD vs NAG or, (iv) SGD vs. HB, we choose the learning rate, learning rate decay factor and decay schedule of SGD and simply sweep over the momentum parameter of NAG or HB and choose the momentum that offers the best validation error. + +We now present plots of training function value for different algorithms and batch sizes. + +Effect of minibatch sizes: Figure 8 plots training function value for batch sizes of 128 and 8 for SGD, HB and NAG. We notice that in the initial stages of training, NAG obtains substantial improvements compared to SGD and HB for batch size 128 but not for batch size 8. Towards the end of training however, NAG starts decreasing the training function value rapidly for both the batch sizes. The reason for this phenomenon is not clear. Note however, that at this point, the test error has already stabilized and the algorithms are just overfitting to the data. + +Comparison of ASGD with momentum methods: We now present the training error plots for ASGD compared to HB and NAG in Figures 9 and 10 respectively. As mentioned earlier, in order to see a clear trend, we constrain the learning rate and decay schedule of ASGD to be the same as that of HB and NAG respectively, which themselves were learned using grid search. We see similar trends as in the validation error plots from Figures 5 and 6. Please see the figures and their captions for more details. + +![](images/d4f9589382a35d59086f9e033e6141ac77583966f826f72c83135bd3dea92e69.jpg) +Figure 8: Training loss for batch sizes 128 and 8 respectively for SGD, HB and NAG. + +![](images/2974afd1ee80f1ce8e17340db6d644072d3bfc4ff9a48fb040725052e5588668.jpg) +Figure 9: Training function value for ASGD compared to HB for batch sizes 128 and 8 respectively. + +![](images/c730a68098f40ebfb306a2227df07eefd31a0cc84e8d3ba245c31b1e5fcb3db8.jpg) +Figure 10: Training function value for ASGD compared to NAG for batch size 128 and 8 respectively. \ No newline at end of file diff --git a/parse/train/rJTutzbA-/rJTutzbA-_content_list.json b/parse/train/rJTutzbA-/rJTutzbA-_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..4c1da0e850df197a6d5dedb4bca0dba92b49e120 --- /dev/null +++ b/parse/train/rJTutzbA-/rJTutzbA-_content_list.json @@ -0,0 +1,3288 @@ +[ + { + "type": "text", + "text": "ON THE INSUFFICIENCY OF EXISTING MOMENTUM SCHEMES FOR STOCHASTIC OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 176, + 101, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Rahul Kidambi∗1, Praneeth Netrapalli2, Prateek Jain2 and Sham M. Kakade1 ", + "bbox": [ + 184, + 169, + 746, + 185 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 University of Washington Seattle 2 Microsoft Research India rkidambi@uw.edu, {praneeth, prajain}@microsoft.com, sham@cs.washington.edu ", + "bbox": [ + 184, + 190, + 673, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 276, + 544, + 291 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Momentum based stochastic gradient methods such as heavy ball (HB) and Nesterov’s accelerated gradient descent (NAG) method are widely used in practice for training deep networks and other supervised learning models, as they often provide significant improvements over stochastic gradient descent (SGD). Rigorously speaking, “fast gradient” methods have provable improvements over gradient descent only for the deterministic case, where the gradients are exact. In the stochastic case, the popular explanations for their wide applicability is that when these fast gradient methods are applied in the stochastic case, they partially mimic their exact gradient counterparts, resulting in some practical gain. This work provides a counterpoint to this belief by proving that there exist simple problem instances where these methods cannot outperform SGD despite the best setting of its parameters. These negative problem instances are, in an informal sense, generic; they do not look like carefully constructed pathological instances. These results suggest (along with empirical evidence) that HB or NAG’s practical performance gains are a by-product of mini-batching. ", + "bbox": [ + 233, + 309, + 764, + 517 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Furthermore, this work provides a viable (and provable) alternative, which, on the same set of problem instances, significantly improves over HB, NAG, and SGD’s performance. This algorithm, referred to as Accelerated Stochastic Gradient Descent (ASGD), is a simple to implement stochastic algorithm, based on a relatively less popular variant of Nesterov’s Acceleration. Extensive empirical results in this paper show that ASGD has performance gains over HB, NAG, and SGD. The code implementing the ASGD Algorithm can be found here1. ", + "bbox": [ + 233, + 521, + 764, + 617 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 648, + 336, + 664 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "First order optimization methods, which access a function (to be optimized) through its gradient or an unbiased approximation of its gradient, are the workhorses for modern large scale optimization problems, which include training the current state-of-the-art deep neural networks. Gradient descent (Cauchy, 1847) is the simplest first order method that is used heavily in practice. However, it is known that for the class of smooth convex functions as well as some simple non-smooth problems (Nesterov, 2012a)), gradient descent is suboptimal (Nesterov, 2004) and there exists a class of algorithms called fast gradient/momentum based methods which achieve optimal convergence guarantees. The heavy ball method (Polyak, 1964) and Nesterov’s accelerated gradient descent (Nesterov, 1983) are two of the most popular methods in this category. ", + "bbox": [ + 174, + 681, + 825, + 806 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "On the other hand, training deep neural networks on large scale datasets have been possible through the use of Stochastic Gradient Descent (SGD) (Robbins & Monro, 1951), which samples a random subset of training data to compute gradient estimates that are then used to optimize the objective function. The advantages of SGD for large scale optimization and the related issues of tradeoffs between computational and statistical efficiency was highlighted in Bottou & Bousquet (2007). ", + "bbox": [ + 174, + 814, + 823, + 883 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The above mentioned theoretical advantages of fast gradient methods (Polyak, 1964; Nesterov, 1983) (albeit for smooth convex problems) coupled with cheap to compute stochastic gradient estimates led to the influential work of Sutskever et al. (2013), which demonstrated the empirical advantages possessed by SGD when augmented with the momentum machinery. This work has led to widespread adoption of momentum methods for training deep neural nets; so much so that, in the context of neural network training, gradient descent often refers to momentum methods. ", + "bbox": [ + 174, + 103, + 823, + 188 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "But, there is a subtle difference between classical momentum methods and their implementation in practice – classical momentum methods work in the exact first order oracle model (Nesterov, 2004), i.e., they employ exact gradients (computed on the full training dataset), while in practice (Sutskever et al., 2013), they are implemented with stochastic gradients (estimated from a randomly sampled mini-batch of training data). This leads to a natural question: ", + "bbox": [ + 174, + 194, + 823, + 263 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "“Are momentum methods optimal even in the stochastic first order oracle (SFO) model, where we access stochastic gradients computed on a small constant sized minibatches (or a batchsize of 1?)” ", + "bbox": [ + 176, + 271, + 820, + 299 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Even disregarding the question of optimality of momentum methods in the SFO model, it is not even known if momentum methods (say, Polyak (1964); Nesterov (1983)) provide any provable improvement over SGD in this model. While these are open questions, a recent effort of Jain et al. (2017) showed that improving upon SGD (in the stochastic first order oracle) is rather subtle as there exists problem instances in SFO model where it is not possible to improve upon SGD, even information theoretically. Jain et al. (2017) studied a variant of Nesterov’s accelerated gradient updates (Nesterov, 2012b) for stochastic linear regression and show that their method improves upon SGD wherever it is information theoretically admissible. Through out this paper, we refer to the algorithm of Jain et al. (2017) as Accelerated Stochastic Gradient Method (ASGD) while we refer to a stochastic version of the most widespread form of Nesterov’s method (Nesterov, 1983) as NAG; HB denotes a stochastic version of the heavy ball method (Polyak, 1964). Critically, while Jain et al. (2017) shows that ASGD improves on SGD in any information-theoretically admissible regime, it is still not known whether HB and NAG can achieve a similar performance gain. ", + "bbox": [ + 174, + 306, + 825, + 486 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A key contribution of this work is to show that HB does not provide similar performance gains over SGD even when it is informationally-theoretically admissible. That is, we provide a problem instance where it is indeed possible to improve upon SGD (and ASGD achieves this improvement), but HB cannot achieve any improvement over SGD. We validate this claim empirically as well. In fact, we provide empirical evidence to the claim that NAG also do not achieve any improvement over SGD for several problems where ASGD can still achieve better rates of convergence. ", + "bbox": [ + 174, + 493, + 825, + 577 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This raises a question about why HB and NAG provide better performance than SGD in practice (Sutskever et al., 2013), especially for training deep networks. Our conclusion (that is well supported by our theoretical result) is that HB and NAG’s improved performance is attributed to mini-batching and hence, these methods will often struggle to improve over SGD with small constant batch sizes. This is in stark contrast to methods like ASGD, which is designed to improve over SGD across both small or large mini-batch sizes. In fact, based on our experiments, we observe that on the task of training deep residual networks (He et al., 2016a) on the cifar-10 dataset, we note that ASGD offers noticeable improvements by achieving $5 - 7 \\%$ better test error over HB and NAG even with commonly used batch sizes like 128 during the initial stages of the optimization. ", + "bbox": [ + 174, + 584, + 825, + 709 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.1 CONTRIBUTIONS ", + "text_level": 1, + "bbox": [ + 176, + 719, + 333, + 734 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The contributions of this paper are as follows. ", + "bbox": [ + 176, + 746, + 473, + 761 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. In Section 3, we prove that HB is not optimal in the SFO model. In particular, there exist linear regression problems for which the performance of HB (with any step size and momentum) is either the same or worse than that of SGD while ASGD improves upon both of them. \n2. Experiments on several linear regression problems suggest that the suboptimality of HB in the SFO model is not restricted to special cases – it is rather widespread. Empirically, the same holds true for NAG as well (Section 5). \n3. The above observations suggest that the only reason for the superiority of momentum methods in practice is mini-batching, which reduces the variance in stochastic gradients and moves the SFO closer to the exact first order oracle. This conclusion is supported by em", + "bbox": [ + 210, + 772, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Algorithm 1 HB: Heavy ball with a SFO ", + "text_level": 1, + "bbox": [ + 176, + 118, + 444, + 132 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Algorithm 2 NAG: Nesterov’s AGD with a SFO ", + "text_level": 1, + "bbox": [ + 511, + 118, + 830, + 132 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Require: Initial $w _ { 0 }$ , stepsize $\\delta$ , momentum $\\alpha$ ", + "bbox": [ + 176, + 136, + 472, + 150 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Require: Initial $w _ { 0 }$ , stepsize $\\delta$ , momentum $\\alpha$ ", + "bbox": [ + 511, + 136, + 812, + 150 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1: $v _ { 0 } w _ { 0 }$ ; $t \\gets 0$ /\\*Set $v _ { 0 }$ to $\\boldsymbol { w _ { 0 } } ^ { * } /$ \n2: while $w _ { t }$ not converged do \n3: $v _ { t + 1 } \\gets w _ { t } - \\delta \\cdot \\tilde { \\widehat { \\nabla } } f _ { t } ( w _ { t } ) / { * } \\mathrm { S G D ~ s t e p } ^ { { * } / }$ \n4: $w _ { t + 1 } = ( 1 + \\alpha ) v _ { t + 1 } - \\alpha v _ { t } / \\ast \\mathrm { S u m }$ of SGD \nstep and previous iterate\\*/ \n5: $t \\gets t + 1$ ", + "bbox": [ + 514, + 150, + 834, + 234 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "/\\*Return the last iterate\\*/ /\\*Return the last iterate\\*/ ", + "bbox": [ + 333, + 236, + 501, + 248 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 666, + 236, + 834, + 250 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "pirical evidence through training deep residual networks on cifar-10, with a batch size of 8 (see Section 5.3). ", + "bbox": [ + 228, + 279, + 823, + 308 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4. We present an intuitive and easier to tune version of ASGD (see Section 4) and show that ASGD can provide significantly faster convergence to a reasonable accuracy than SGD, HB, NAG, while still providing favorable or comparable asymptotic accuracy as these methods, particularly on several deep learning problems. ", + "bbox": [ + 212, + 313, + 825, + 368 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Hence, the take-home message of this paper is: HB and NAG are not optimal in the SFO model. The only reason for the superiority of momentum methods in practice is mini-batching. ASGD provides a distinct advantage in training deep networks over SGD, HB and NAG. ", + "bbox": [ + 176, + 380, + 825, + 422 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 NOTATION ", + "text_level": 1, + "bbox": [ + 174, + 434, + 294, + 450 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We denote matrices by bold-face capital letters and vectors by lower-case letters. $f ( w ) =$ $1 / n \\sum _ { i } f _ { i } ( w )$ denotes the function to optimize w.r.t. model parameters $w$ . $\\nabla f ( w )$ denotes exact gradient of $f$ at $w$ while $\\widehat { \\nabla } f _ { t } ( \\boldsymbol { w } )$ denotes a stochastic gradient of $f$ . That is, $\\widehat { \\nabla } f _ { t } ( w _ { t } ) = \\nabla f _ { i _ { t } } ( w )$ where $i _ { t }$ is sampled uniformly at random from $[ 1 , \\ldots , n ]$ . For linear regression, $f _ { i } ( w ) = 0 . 5 \\cdot ( b _ { i } -$ $\\langle w , a _ { i } \\rangle ) ^ { 2 }$ where $b _ { i } \\in \\Re$ is the target and $a _ { i } \\in \\Re ^ { d }$ is the covariate, and $\\widehat { \\nabla } f _ { t } ( w _ { t } ) = - \\big ( b _ { t } - \\langle w _ { t } , a _ { t } \\rangle \\big ) a _ { t }$ . In this case, $\\mathbf { H } = \\mathbb { E } \\left[ a a ^ { \\top } \\right]$ denotes the Hessian of $f$ and $\\begin{array} { r } { \\kappa = \\frac { \\lambda _ { 1 } ( \\mathbf { H } ) } { \\lambda _ { d } ( \\mathbf { H } ) } } \\end{array}$ denotes it’s condition number. ", + "bbox": [ + 173, + 467, + 825, + 564 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Algorithm 1 provides a pseudo-code of HB method (Polyak, 1964). $w _ { t } - w _ { t - 1 }$ is the momentum term and $\\alpha$ denotes the momentum parameter. Next iterate $w _ { t + 1 }$ is obtained by a linear combination of the SGD update and the momentum term. Algorithm 2 provides pseudo-code of a stochastic version of the most commonly used form of Nesterov’s accelerated gradient descent (Nesterov, 1983). ", + "bbox": [ + 173, + 569, + 825, + 626 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 SUBOPTIMALITY OF HEAVY BALL METHOD ", + "text_level": 1, + "bbox": [ + 173, + 637, + 573, + 655 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we show that there exists linear regression problems where the performance of HB (Algorithm 1) is no better than that of SGD, while ASGD significantly improves upon SGD’s performance. Let us now describe the problem instance. ", + "bbox": [ + 173, + 670, + 825, + 712 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Fix $w ^ { \\ast } \\in \\mathbb { R } ^ { 2 }$ and let $( a , b ) \\sim \\mathcal { D }$ be a sample from the distribution such that: ", + "bbox": [ + 174, + 718, + 671, + 734 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/af1fe3c9ae11009fbc2189b008ba43663a44233ef1ca444cd9aabf2ef145bf53.jpg", + "text": "$$\n\\begin{array} { r } { a = \\left\\{ \\begin{array} { l l } { \\sigma _ { 1 } \\cdot z \\cdot e _ { 1 } \\mathrm { w . p . ~ } 0 . 5 } \\\\ { \\sigma _ { 2 } \\cdot z \\cdot e _ { 2 } \\mathrm { w . p . ~ } 0 . 5 , } \\end{array} \\right. \\qquad \\mathrm { a n d } \\qquad b = \\left. w ^ { * } , a \\right. , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 312, + 742, + 684, + 777 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $e _ { 1 } , e _ { 2 } \\in \\mathbb { R } ^ { 2 }$ are canonical basis vectors, $\\sigma _ { 1 } > \\sigma _ { 2 } > 0$ . Let $z$ be a random variable such that E $\\left[ z ^ { 2 } \\right] = 2$ and $\\mathbb { E } \\left[ z ^ { 4 } \\right] = 2 c \\geq 4$ . Hence, we have: $\\Xi \\left[ ( a ^ { ( i ) } ) ^ { 2 } \\right] = \\sigma _ { i } ^ { 2 } , \\mathbb { E } \\left[ ( a ^ { ( i ) } ) ^ { 4 } \\right] = c \\sigma _ { i } ^ { 4 }$ , for $i = 1 , \\dot { 2 }$ . Now, our goal is to minimize: ", + "bbox": [ + 173, + 785, + 825, + 828 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/26873c0ead4debcb0af7abe9d44636183f9100018d946ddceb34642f7916b3d0.jpg", + "text": "$$\nf ( \\boldsymbol { w } ) \\stackrel { \\mathrm { d e f } } { = } 0 . 5 \\cdot \\mathbb { E } \\left[ \\left( \\left. \\boldsymbol { w } ^ { * } , \\boldsymbol { a } \\right. - \\boldsymbol { b } \\right) ^ { 2 } \\right] \\mathrm { , ~ H e s s i a n \\ } \\mathbf { H } \\stackrel { \\mathrm { d e f } } { = } \\mathbb { E } \\left[ \\boldsymbol { a } \\boldsymbol { a } ^ { \\top } \\right] = \\left[ \\sigma _ { 1 } ^ { 2 } \\quad 0 _ { 2 } ^ { 2 } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 258, + 833, + 738, + 868 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let $\\kappa$ and $\\tilde { \\kappa }$ denote the computational and statistical condition numbers – see Jain et al. (2017) for definitions. For the problem above, we have $\\begin{array} { r } { \\kappa = \\frac { c \\sigma _ { 1 } ^ { 2 } } { \\sigma _ { 2 } ^ { 2 } } } \\end{array}$ and $\\tilde { \\kappa } = c$ . Then we obtain following convergence rates for SGD and ASGD when applied to the above given problem instance: ", + "bbox": [ + 173, + 873, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Input: Initial $w _ { 0 }$ , short step $\\delta$ , long step parameter $\\kappa \\geq 1$ , statistical advantage parameter $\\xi \\le \\sqrt { \\kappa }$ \n1: $\\bar { w } _ { 0 } w _ { 0 }$ ; $t \\gets 0$ /\\*Set running average to $\\boldsymbol { w _ { 0 } } ^ { * } /$ \n2: α ← 1 − 0.72·ξ /\\*Set momentum value\\*/ \n3: while $w _ { t }$ not converged do \n4: $\\begin{array} { r } { \\bar { w } _ { t + 1 } \\alpha \\cdot \\bar { w } _ { t } + \\overline { { ( 1 - \\alpha ) \\cdot \\Big ( w _ { t } - \\frac { \\kappa \\cdot \\delta } { 0 . 7 } \\cdot \\widehat \\nabla f _ { t } ( w _ { t } ) \\Big ) } } } \\end{array}$ /\\*Update the running average as a weighted average of previous running average and a long step gradient $^ { * } /$ \n5: $\\begin{array} { r } { \\overline { { w _ { t + 1 } } } \\frac { 0 . 7 } { 0 . 7 + ( 1 - \\alpha ) } \\cdot ( w _ { t } - \\delta \\cdot \\widehat { \\nabla } f _ { t } ( w _ { t } ) ) + \\frac { 1 - \\alpha } { 0 . 7 + ( 1 - \\alpha ) } \\cdot \\bar { w } _ { t + 1 } } \\end{array}$ /\\*Update the iterate as weighted average of current running average and short step gradient\\*/ \n6: $t \\gets t + 1$ ", + "bbox": [ + 176, + 121, + 825, + 267 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Output: $w _ { t }$ /\\*Return the last iterate\\*/ ", + "bbox": [ + 174, + 268, + 264, + 282 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 655, + 267, + 823, + 281 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Corollary 1 (of Theorem 1 of Jain et al. (2016)). Let $w _ { t } ^ { S G D }$ be the $t ^ { t h }$ iterate of SGD on the above problem with starting point $w _ { 0 }$ and stepsize cσ 2 t . The error of $w _ { t } ^ { S G D }$ can be bounded as, ", + "bbox": [ + 173, + 310, + 823, + 344 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b30294f1e21f3409bb3e144fb0eb94c356390ce17e81d7c46e0c66dbcecabc43.jpg", + "text": "$$\n\\mathbb { E } \\left[ f \\left( w _ { t } ^ { S G D } \\right) \\right] - f \\left( w _ { * } \\right) \\leq \\exp \\left( \\frac { - t } { \\kappa } \\right) \\left( f \\left( w _ { 0 } \\right) - f \\left( w _ { * } \\right) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 300, + 351, + 696, + 386 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "On the other hand, ASGD achieves the following superior rate. ", + "bbox": [ + 173, + 398, + 586, + 414 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Corollary 2 (of Theorem 1 of Jain et al. (2017)). Let $w _ { t } ^ { A S G D }$ be the $t ^ { t h }$ iterate of ASGD on the above problem with starting point $w _ { 0 }$ and appropriate parameters. The error of $\\dot { w } _ { t } ^ { A S G D }$ can be bounded as, ", + "bbox": [ + 173, + 415, + 825, + 459 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6dd8c8d023ad022cab755d475f6c6e8f6efdd3fcd5383300d1b87cfe80f81a20.jpg", + "text": "$$\n\\mathbb { E } \\left[ f \\left( w _ { t } ^ { A S G D } \\right) \\right] - f \\left( w _ { * } \\right) \\le \\mathrm { p o l y } ( \\kappa ) \\exp \\left( \\frac { - t } { \\sqrt { \\kappa \\tilde { \\kappa } } } \\right) \\left( f \\left( w _ { 0 } \\right) - f \\left( w _ { * } \\right) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 259, + 463, + 735, + 498 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Note that for a given problem/input distribution $\\tilde { \\kappa } = c$ is a constant while $\\begin{array} { r } { \\kappa = \\frac { c \\sigma _ { 1 } ^ { 2 } } { \\sigma _ { 2 } ^ { 2 } } } \\end{array}$ can be arbitrarily large. Note that $\\kappa > \\tilde { \\kappa } = c$ . Hence, ASGD improves upon rate of SGD by a factor of $\\sqrt { \\kappa }$ . The following proposition, which is the main result of this section, establishes that HB (Algorithm 1) cannot provide a similar improvement over SGD as what ASGD offers. In fact, we show no matter the choice of parameters of HB, its performance does not improve over SGD by more than a constant. ", + "bbox": [ + 174, + 513, + 825, + 592 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Proposition 3. Let $w _ { t } ^ { H B }$ be the $t ^ { t h }$ iterate of HB (Algorithm $I$ ) on the above problem with starting point $w _ { 0 }$ . For any choice of stepsize $\\delta$ and momentum $\\alpha \\in [ 0 , 1 ]$ , $\\exists T$ large enough such that $\\forall t \\geq T$ , we have, ", + "bbox": [ + 173, + 594, + 825, + 637 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2bd934bd656cc5053bdfd788fb196e86a945c179b3f1da9d64e16654a2903dd3.jpg", + "text": "$$\n\\mathbb { E } \\left[ f \\left( w _ { t } ^ { H B } \\right) \\right] - f \\left( w _ { * } \\right) \\ge C ( \\kappa , \\delta , \\alpha ) \\cdot \\exp \\left( \\frac { - 5 0 0 t } { \\kappa } \\right) \\left( f \\left( w _ { 0 } \\right) - f \\left( w _ { * } \\right) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 253, + 636, + 741, + 671 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $C ( \\kappa , \\delta , \\alpha )$ depends on $\\kappa , \\delta$ and $\\alpha$ (but not on $t$ ). ", + "bbox": [ + 174, + 676, + 529, + 691 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Thus, to obtain $\\widehat { w }$ s.t. $\\| \\widehat { \\boldsymbol { w } } - \\boldsymbol { w } ^ { * } \\| \\le \\epsilon$ , HB requires $\\Omega ( \\kappa \\log { \\frac { 1 } { \\epsilon } } )$ samples and iterations. On the other hand, ASGD can obtain $\\epsilon$ -approximation to $w ^ { * }$ in $\\mathcal { O } ( \\sqrt { \\kappa } \\log \\kappa \\log \\frac { 1 } { \\epsilon } )$ iterations. We note that the gains offered by ASGD are meaningful when $\\kappa > \\mathcal { O } ( c )$ (Jain et al., 2017); otherwise, all the algorithms including SGD achieve nearly the same rates (upto constant factors). While we do not prove it theoretically, we observe empirically that for the same problem instance, NAG also obtains nearly same rate as HB and SGD. We conjecture that a lower bound for NAG can be established using a similar proof technique as that of HB (i.e. Proposition 3). We also believe that the constant in the lower bound described in proposition 3 can be improved to some small number $( \\leq 5 )$ . ", + "bbox": [ + 173, + 700, + 825, + 818 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 ALGORITHM ", + "text_level": 1, + "bbox": [ + 174, + 837, + 312, + 852 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We will now present and explain an intuitive version of ASGD (pseudo code in Algorithm 3). The algorithm takes three inputs: short step $\\delta$ , long step parameter $\\kappa$ and statistical advantage parameter $\\xi$ . The short step $\\delta$ is precisely the same as the step size in SGD, HB or NAG. For convex problems, this scales inversely with the smoothness of the function. The long step parameter $\\kappa$ is intended to give an estimate of the ratio of the largest and smallest curvatures of the function; for convex functions, this is just the condition number. The statistical advantage parameter $\\xi$ captures trade√ off between statistical and computational condition numbers – in the deterministic case, $\\xi = \\sqrt { \\kappa }$ and ASGD is equivalent to NAG, while in the high stochasticity regime, $\\xi$ is much smaller. The algorithm maintains two iterates: descent iterate $w _ { t }$ and a running average $\\bar { w } _ { t }$ . The running average is a weighted average of the previous average and a long gradient step from the descent iterate, while the descent iterate is updated as a convex combination of short gradient step from the descent iterate and the running average. The idea is that since the algorithm takes a long step as well as short step and an appropriate average of both of them, it can make progress on different directions at a similar pace. Appendix B shows the equivalence between Algorithm 3 and ASGD as proposed in Jain et al. (2017). Note that the constant 0.7 appearing in Algorithm 3 has no special significance. Jain et al. (2017) require it to be smaller than $\\sqrt { 1 / 6 }$ but any constant smaller than 1 seems to work in practice. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 273 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 286, + 326, + 303 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now present our experimental results exploring performance of SGD, HB, NAG and ASGD. Our experiments are geared towards answering the following questions: ", + "bbox": [ + 173, + 318, + 823, + 347 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• Even for linear regression, is the suboptimality of HB restricted to specific distributions in Section 3 or does it hold for more general distributions as well? Is the same true of NAG? \nWhat is the reason for the superiority of HB and NAG in practice? Is it because momentum methods have better performance that SGD for stochastic gradients or due to minibatching? Does this superiority hold even for small minibatches? \n• How does the performance of ASGD compare to that of SGD, HB and NAG, when training deep networks? ", + "bbox": [ + 215, + 358, + 825, + 465 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Section 5.1 and parts of Section 5.2 address the first two questions. Section 5.2 and 5.3 address Question 2 partially and the last question. We use Matlab to conduct experiments presented in Section 5.1 and use PyTorch (pytorch, 2017) for our deep networks related experiments. Pytorch code implementing the ASGD algorithm can be found at https://github.com/rahulkidambi/AccSGD. ", + "bbox": [ + 174, + 477, + 825, + 534 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5.1 LINEAR REGRESSION ", + "text_level": 1, + "bbox": [ + 174, + 550, + 364, + 564 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, we will present results on performance of the four optimization methods (SGD, HB, NAG, and ASGD) for linear regression problems. We consider two different class of linear regression problems, both of them in two dimensions. Given $\\kappa$ which stands for condition number, we consider the following two distributions: ", + "bbox": [ + 173, + 575, + 825, + 632 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$a = e _ { 1 }$ w.p. 0.5 and $\\textstyle a = { \\frac { 2 } { \\kappa } } \\cdot e _ { 2 }$ with 0.5; $e _ { i }$ is the $i ^ { t h }$ ", + "bbox": [ + 174, + 637, + 728, + 655 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Gaussian : $a \\in \\mathbb { R } ^ { 2 }$ is distributed as a Gaussian random vector with covariance matrix $\\left[ { \\begin{array} { c c } { 1 } & { 0 } \\\\ { 0 } & { { \\frac { 1 } { \\kappa } } } \\end{array} } \\right] .$ ", + "bbox": [ + 174, + 662, + 797, + 695 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We fix a randomly generated $w ^ { \\ast } \\in \\mathbb { R } ^ { 2 }$ and for both the distributions above, we let $b = \\langle w ^ { * } , a \\rangle$ . We vary $\\kappa$ from $\\{ \\mathbf { \\bar { 2 } ^ { 4 } } , 2 ^ { 5 } , . . . , 2 ^ { 1 2 } \\}$ and for each $\\kappa$ in this set, we run 100 independent runs of all four methods, each for a total of $t = 5 \\kappa$ iterations. We define that the algorithm converges if there is no error in the second half (i.e. after $2 . 5 \\kappa$ updates) that exceeds the starting error - this is reasonable since we expect geometric convergence of the initial error. ", + "bbox": [ + 174, + 700, + 823, + 771 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Unlike ASGD and SGD, we do not know optimal learning rate and momentum parameters for NAG and HB in the stochastic gradient model. So, we perform a grid search over the values of the learning rate and momentum parameters. In particular, we lay a $1 0 \\times 1 0$ grid in $[ 0 , 1 ] \\times [ 0 , 1 ]$ for learning rate and momentum and run NAG and HB. Then, for each grid point, we consider the subset of 100 trials that converged and computed the final error using these. Finally, the parameters that yield the minimal error are chosen for NAG and HB, and these numbers are reported. We measure convergence performance of a method using: ", + "bbox": [ + 174, + 777, + 825, + 876 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/b9dc3438e9ba56ee3d2a40a0d96780a48908dc1caffc92e0d02e001399705b0d.jpg", + "text": "$$\n{ \\mathrm { r a t e } } = { \\frac { \\log ( f ( w _ { 0 } ) ) - \\log ( f ( w _ { t } ) ) } { t } } ,\n$$", + "text_format": "latex", + "bbox": [ + 383, + 882, + 611, + 912 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/3d3b10853538102302f9fd8281848da346ad9275d7a5233ce0344ccdced8ae59.jpg", + "image_caption": [ + "Figure 1: Plot of 1/rate (refer equation (1)) vs condition number $( \\kappa )$ for various methods for the linear regression problem. Discrete distribution in the left, Gaussian to the right. " + ], + "image_footnote": [], + "bbox": [ + 210, + 114, + 774, + 277 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/20c9005f45559a82fc61edff90798770e1b625d070525ebf93e41eab421e7f44.jpg", + "table_caption": [ + "Table 1: Slopes (i.e. $\\gamma$ ) obtained by fitting a line to the curves in Figure 1. A value of $\\gamma$ indicates that the error decays at a rate of exp $\\left( { \\frac { - t } { \\kappa ^ { \\gamma } } } \\right)$ . A smaller value of $\\gamma$ indicates a faster rate of error decay. " + ], + "table_footnote": [], + "table_body": "
AlgorithmSlope-discreteSlope 1 Gaussian
SGD0.93020.8745
HB NAG0.85220.8769
0.980.9494
0.54800.5127
", + "bbox": [ + 325, + 327, + 671, + 409 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We compute the rate (1) for all the algorithms with varying condition number $\\kappa$ . Given a rate vs $\\kappa$ plot for a method, we compute it’s slope (denoted as $\\gamma$ ) using linear regression. Table 1 presents the estimated slopes (i.e. $\\gamma$ ) for various methods for both the discrete and the Gaussian case. The slope values clearly show that the rate of SGD, HB and NAG have a nearly linear dependence on √ $\\kappa$ while that of ASGD seems to scale linearly with $\\sqrt { \\kappa }$ . ", + "bbox": [ + 173, + 477, + 825, + 546 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.2 DEEP AUTOENCODERS FOR MNIST ", + "text_level": 1, + "bbox": [ + 176, + 556, + 462, + 570 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we present experimental results on training deep autoencoders for the mnist dataset, and we closely follow the setup of Hinton & Salakhutdinov (2006). This problem is a standard benchmark for evaluating the performance of different optimization algorithms e.g., Martens (2010); Sutskever et al. (2013); Martens $\\&$ Grosse (2015); Reddi et al. (2017). The network architecture follows previous work (Hinton & Salakhutdinov, 2006) and is represented as $7 8 4 - 1 0 0 0 - 5 0 0 -$ $2 5 0 - 3 0 - 2 5 0 - 5 0 0 - 1 0 0 0 - 7 8 4$ with the first and last 784 nodes representing the input and output respectively. All hidden/output nodes employ sigmoid activations except for the layer with 30 nodes which employs linear activations and we use MSE loss. Initialization follows the scheme of Martens (2010), also employed in Sutskever et al. (2013); Martens & Grosse (2015). We perform training with two minibatch sizes $- 1$ and 8. The runs with minibatch size of 1 were run for 30 epochs while the runs with minibatch size of 8 were run for 50 epochs. For each of SGD, HB, NAG and ASGD, a grid search over learning rate, momentum and long step parameter (whichever is applicable) was done and best parameters were chosen based on achieving the smallest training error in the same protocol followed by Sutskever et al. (2013). The grid was extended whenever the best parameter fell at the edge of a grid. For the parameters chosen by grid search, we perform 10 runs with different seeds and averaged the results. The results are presented in Figures 2 and 3. Note that the final loss values reported are suboptimal compared to those in published literature e.g., Sutskever et al. (2013); while Sutskever et al. (2013) report results after 750000 updates with a large batch size of 200 (which implies a total of $7 5 0 0 0 0 \\times 2 0 0 = 1 5 0 \\mathbf { M }$ gradient evaluations), whereas, our results are after 1.8M updates of SGD with a batch size 1 (which is just 1.8M gradient evaluations). ", + "bbox": [ + 173, + 583, + 825, + 861 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Effect of minibatch sizes: While HB and NAG decay the loss faster compared to SGD for a minibatch size of 8 (Figure 2), this superior decay rate does not hold for a minibatch size of 1 (Figure 3). This supports our intuitions from the stochastic linear regression setting, where we demonstrate that HB and NAG are suboptimal in the stochastic first order oracle model. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/e7e9ca2132cb1ddd893c549fd607c122fbf8832557608c7292e2d72b3058de61.jpg", + "image_caption": [ + "Figure 2: Training loss (left) and test loss (right) while training deep autoencoder for mnist with minibatch size 8. Clearly, ASGD matches performance of NAG and outperforms SGD on the test data. HB also outperforms SGD. " + ], + "image_footnote": [], + "bbox": [ + 215, + 116, + 772, + 277 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/1de51d3d68f2b8fcd7d7f8f74749949493d74864a33ffe1b0da66f947386a0f1.jpg", + "image_caption": [ + "Figure 3: Training loss (left) and test loss (right) while training deep autoencoder for mnist with minibatch size 1. Interestingly, SGD, HB and NAG, all decrease the loss at a similar rate, while ASGD decays at a faster rate. " + ], + "image_footnote": [], + "bbox": [ + 215, + 342, + 772, + 503 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparison of ASGD with momentum methods: While ASGD performs slightly better than NAG for batch size 8 in the training error (Figure 2), ASGD decays the error at a faster rate compared to all the three other methods for a batch size of 1 (Figure 3). ", + "bbox": [ + 176, + 565, + 821, + 607 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.3 DEEP RESIDUAL NETWORKS FOR CIFAR-10 ", + "text_level": 1, + "bbox": [ + 174, + 617, + 527, + 632 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We will now present experimental results on training deep residual networks (He et al., 2016b) with pre-activation blocks He et al. (2016a) for classifying images in cifar-10 (Krizhevsky & Hinton, 2009); the network we use has 44 layers (dubbed preresnet-44). The code for this section was downloaded from preresnet (2017). One of the most distinct characteristics of this experiment compared to our previous experiments is learning rate decay. We use a validation set based decay scheme, wherein, after every 3 epochs, we decay the learning rate by a certain factor (which we grid search on) if the validation zero one error does not decrease by at least a certain amount (precise numbers are provided in the appendix since they vary across batch sizes). Due to space constraints, we present only a subset of training error plots. Please see Appendix C.3 for some more plots on training errors. ", + "bbox": [ + 173, + 645, + 825, + 770 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Effect of minibatch sizes: Our first experiment tries to understand how the performance of HB and NAG compare with that of SGD and how it varies with minibatch sizes. Figure 4 presents the test zero one error for minibatch sizes of 8 and 128. While training with batch size 8 was done for 40 epochs, with batch size 128, it was done for 120 epochs. We perform a grid search over all parameters for each of these algorithms. See Appendix C.3 for details on the grid search parameters. We observe that final error achieved by SGD, HB and NAG are all very close for both batch sizes. While NAG exhibits a superior rate of convergence compared to SGD and HB for batch size 128, this superior rate of convergence disappears for a batch size of 8. ", + "bbox": [ + 173, + 777, + 825, + 888 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparison of ASGD with momentum methods: The next experiment tries to understand how ASGD compares with HB and NAG. The errors achieved by various methods when we do ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/db5f311c1e2dbe4b962051b725dbf673899c05a23640369f13b18c9e0f70e473.jpg", + "image_caption": [ + "Figure 4: Test zero one loss for batch size 128 (left), batch size 8 (center) and training function value for batch size 8 (right) for SGD, HB and NAG. " + ], + "image_footnote": [], + "bbox": [ + 189, + 113, + 800, + 229 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/9082f0ff9ebf68aa4e0752f19fd7b95ba1e73a84b488c3c29ff243cc33fb06fa.jpg", + "image_caption": [ + "Figure 5: Test zero one loss for batch size 128 (left), batch size 8 (center) and training function value for batch size 8 (right) for ASGD compared to HB. In the above plots, both ASGD and ASGD-HbParams refer to ASGD run with the learning rate and decay schedule of HB. ASGD-Fully-Optimized refers to ASGD where learning rate and decay schedule were also selected by grid search. " + ], + "image_footnote": [], + "bbox": [ + 191, + 279, + 799, + 396 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/8c3f21b358d38819e5f3442d751b0ffec52249458241bcf1e2529c0d62583b13.jpg", + "table_caption": [ + "grid search over all parameters are presented in Table 2. Note that the final test errors for batch size 128 are better than those for batch size 8 since the former was run for 120 epochs while the latter was run only for 40 epochs (due to time constraints). " + ], + "table_footnote": [], + "table_body": "
AlgorithmFinal test error-batch size 128Final test error-batch size 8
SGD8.32± 0.219.57±0.18
HB7.98 ± 0.199.28± 0.25
NAG7.63 ± 0.189.07 ±0.18
ASGD7.23 ± 0.228.52 ± 0.16
", + "bbox": [ + 236, + 526, + 761, + 608 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 2: Final test errors achieved by various methods for batch sizes of 128 and 8. The hyperparameters have been chosen by grid search. ", + "bbox": [ + 174, + 618, + 821, + 646 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "While the final error achieved by ASGD is similar/favorable compared to all other methods, we are also interested in understanding whether ASGD has a superior convergence speed. For this experiment, we need to address the issue of differing learning rates used by various algorithms and different iterations where they decay learning rates. So, for each of HB and NAG, we choose the learning rate and decay factors by grid search, use these values for ASGD and do grid search only over long step parameter $\\kappa$ and momentum $\\alpha$ for ASGD. The results are presented in Figures 5 and 6. For batch size 128, ASGD decays error at a faster rate compared to both HB and NAG. For batch size 8, while we see a superior convergence of ASGD compared to NAG, we do not see this superiority over HB. The reason for this turns out to be that the learning rate for HB, which we also use for ASGD, turns out to be quite suboptimal for ASGD. So, for batch size 8, we also compare fully optimized (i.e., grid search over learning rate as well) ASGD with HB. The superiority of ASGD over HB is clear from this comparison. These results suggest that ASGD decays error at a faster rate compared to HB and NAG across different batch sizes. ", + "bbox": [ + 173, + 664, + 825, + 844 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 861, + 344, + 877 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "First order oracle methods: The primary method in this family is Gradient Descent (GD) (Cauchy, 1847). As mentioned previously, GD is suboptimal for smooth convex optimization (Nesterov, ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/1520b13e594816b4bd1f8bfe08d8102111f7964499e85c5692f7d19723eb2535.jpg", + "image_caption": [ + "Figure 6: Test zero one loss for batch size 128 (left), batch size 8 (center) and training function value for batch size 8 (right) for ASGD compared to NAG. In the above plots, ASGD was run with the learning rate and decay schedule of NAG. Other parameters were selected by grid search. " + ], + "image_footnote": [], + "bbox": [ + 189, + 113, + 799, + 229 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "2004), and this is addressed using momentum methods such as the Heavy Ball method (Polyak, \n1964) (for quadratics), and Nesterov’s Accelerated gradient descent (Nesterov, 1983). ", + "bbox": [ + 176, + 311, + 821, + 339 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Stochastic first order methods and noise stability: The simplest method employing the SFO is SGD (Robbins & Monro, 1951); the effectiveness of SGD has been immense, and its applicability goes well beyond optimizing convex objectives. Accelerating SGD is a tricky proposition given the instability of fast gradient methods in dealing with noise, as evidenced by several negative results which consider statistical (Proakis, 1974; Polyak, 1987; Roy & Shynk, 1990), numerical (Paige, 1971; Greenbaum, 1989) and adversarial errors (d’Aspremont, 2008; Devolder et al., 2014). A result of Jain et al. (2017) developed the first provably accelerated SGD method for linear regression which achieved minimax rates, inspired by a method of Nesterov (2012b). Schemes of Ghadimi & Lan (2012; 2013); Dieuleveut et al. (2016), which indicate acceleration is possible with noisy gradients do not hold in the SFO model satisfied by algorithms that are run in practice (see Jain et al. (2017) for more details). ", + "bbox": [ + 174, + 347, + 825, + 500 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "While HB (Polyak, 1964) and NAG (Nesterov, 1983) are known to be effective in case of exact first order oracle, for the SFO, the theoretical performance of HB and NAG is not well understood. ", + "bbox": [ + 176, + 507, + 823, + 535 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Understanding Stochastic Heavy Ball: Understanding HB’s performance with inexact gradients has been considered in efforts spanning several decades, in many communities like controls, optimization and signal processing. Polyak (1987) considered HB with noisy gradients and concluded that the improvements offered by HB with inexact gradients vanish unless strong assumptions on the inexactness was considered; an instance of this is when the variance of inexactness decreased as the iterates approach the minimizer. Proakis (1974); Roy & Shynk (1990); Sharma et al. (1998) suggest that the improved non-asymptotic rates offered by stochastic HB arose at the cost of worse asymptotic behavior. We resolve these unquantified improvements on rates as being just constant factors over SGD, in stark contrast to the gains offered by ASGD. Loizou & Richtarik ´ (2017) state their method as Stochastic HB but require stochastic gradients that nearly behave as exact gradients; indeed, their rates match that of the standard HB method (Polyak, 1964). Such rates are not information theoretically possible (see Jain et al. (2017)), especially with a batch size of 1 or even with constant sized minibatches. ", + "bbox": [ + 174, + 541, + 825, + 722 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Accelerated and Fast Methods for finite-sums: There have been developments pertaining to faster methods for finite-sums (also known as offline stochastic optimization): amongst these are methods such as SDCA (Shalev-Shwartz & Zhang, 2012), SAG (Roux et al., 2012), SVRG (Johnson & Zhang, 2013), SAGA (Defazio et al., 2014), which offer linear convergence rates for strongly convex finite-sums, improving over SGD’s sub-linear rates (Rakhlin et al., 2012). These methods have been improved using accelerated variants (Shalev-Shwartz & Zhang, 2014; Frostig et al., 2015a; Lin et al., 2015; Defazio, 2016; Allen-Zhu, 2016). Note that these methods require storing the entire training set in memory and taking multiple passes over the same for guaranteed progress. Furthermore, these methods require computing a batch gradient or require memory requirements (typically $\\Omega ( \\lfloor$ training data points|)). For deep learning problems, data augmentation is often deemed necessary for achieving good performance; this implies computing quantities such as batch gradient (or storage necessities) over this augmented dataset is often infeasible. Such requirements are mitigated by the use of simple streaming methods such as SGD, ASGD, HB, NAG. For other technical distinctions between the offline and online stochastic methods refer to Frostig et al. (2015b). ", + "bbox": [ + 173, + 729, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Practical methods for training deep networks: Momentum based methods employed with stochastic gradients (Sutskever et al., 2013) have become standard and very popular in practice. These schemes tend to outperform standard SGD on several important practical problems. As previously mentioned, we attribute this improvement to effect of mini-batching rather than improvement offered by HB or NAG in the SFO model. Schemes such as Adagrad (Duchi et al., 2011), RMSProp (Tieleman & Hinton, 2012), Adam (Kingma & Ba, 2014) represent an important and useful class of algorithms. The advantages offered by these methods are orthogonal to the advantages offered by fast gradient methods; it is an important direction to explore augmenting these methods with ASGD as opposed to standard HB or NAG based acceleration schemes. ", + "bbox": [ + 174, + 103, + 825, + 228 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Chaudhari et al. (2017) proposed Entropy-SGD, which is an altered objective that adds a local strong convexity term to the actual empirical risk objective, with an aim to improve generalization. However, we do not understand convergence rates for convex problems or the generalization ability of this technique in a rigorous manner. Chaudhari et al. (2017) propose to use SGD in their procedure but mention that they employ the HB/NAG method in their implementation for achieving better performance. Naturally, we can use ASGD in this context. Path normalized SGD (Neyshabur et al., 2015) is a variant of SGD that alters the metric on which the weights are optimized. As noted in their paper, path normalized SGD could be improved using HB/NAG (or even the ASGD method). ", + "bbox": [ + 174, + 236, + 825, + 347 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "7 CONCLUSIONS AND FUTURE DIRECTIONS ", + "text_level": 1, + "bbox": [ + 176, + 368, + 553, + 385 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we show that the performance gain of HB over SGD in stochastic setting is attributed to mini-batching rather than the algorithm’s ability to accelerate with stochastic gradients. Concretely, we provide a formal proof that for several easy problem instances, HB does not outperform SGD despite large condition number of the problem; we observe this trend for NAG in our experiments. In contrast, ASGD (Jain et al., 2017) provides significant improvement over SGD for these problem instances. We observe similar trends when training a resnet on cifar-10 and an autoencoder on mnist. This work motivates several directions such as understanding the behavior of ASGD on domains such as NLP, and developing automatic momentum tuning schemes (Zhang et al., 2017). ", + "bbox": [ + 174, + 400, + 825, + 512 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 530, + 326, + 542 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Sham Kakade acknowledges funding from NSF Awards CCF-1703574 and CCF-1740551. 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", + "bbox": [ + 174, + 886, + 821, + 915 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A SUBOPTIMALITY OF HB: PROOF OF PROPOSITION 3 ", + "text_level": 1, + "bbox": [ + 174, + 101, + 642, + 119 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Before proceeding to the proof, we introduce some additional notation. Let $\\pmb { \\theta } _ { t + 1 } ^ { ( j ) }$ denote the concatenated and centered estimates in the $j ^ { \\mathrm { t h } }$ direction for $j = 1 , 2$ . ", + "bbox": [ + 174, + 133, + 821, + 166 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/2d7f7d93663979170a21ec6947f999faf51d397b4be388611d75c371d13ec849.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { \\theta } _ { t + 1 } ^ { ( j ) } \\stackrel { \\mathrm { d e f } } { = } \\left[ \\mathbf { w } _ { t + 1 } ^ { ( j ) } - ( \\mathbf { w } ^ { * } ) ^ { ( j ) } \\right] , \\quad j = 1 , 2 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 364, + 174, + 632, + 217 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Since the distribution over $x$ is such that the coordinates are decoupled, we see that $\\pmb { \\theta } _ { t + 1 } ^ { ( j ) }$ can be written in terms of $\\pmb { \\theta } _ { t } ^ { ( j ) }$ as: ", + "bbox": [ + 174, + 227, + 825, + 263 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/1e9087439006256f226453054d397c1ad759b357bf7d62525d93d55832d52800.jpg", + "text": "$$\n\\pmb { \\theta } _ { t + 1 } ^ { ( j ) } = \\widehat { \\mathbf { A } } _ { t + 1 } ^ { ( j ) } \\pmb { \\theta } _ { t } ^ { ( j ) } , \\mathrm { w i t h } \\widehat { \\mathbf { A } } _ { t + 1 } ^ { ( j ) } = \\left[ \\frac { 1 + \\alpha - \\delta ( a _ { t + 1 } ^ { ( j ) } ) ^ { 2 } } { 1 } - \\alpha \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 295, + 272, + 699, + 308 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Let $\\Phi _ { t + 1 } ^ { ( j ) } \\ { \\stackrel { \\mathrm { d e f } } { = } } \\ \\mathbb { E } \\left[ \\pmb { \\theta } _ { t + 1 } ^ { ( j ) } \\otimes \\pmb { \\theta } _ { t + 1 } ^ { ( j ) } \\right]$ denote the covariance matrix of $\\pmb { \\theta } _ { t + 1 } ^ { ( j ) }$ . We have $\\Phi _ { t + 1 } ^ { ( j ) } = B ^ { ( j ) } \\Phi _ { t } ^ { ( j ) }$ with, $B ^ { ( j ) }$ defined as ", + "bbox": [ + 173, + 315, + 821, + 357 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/df99d5a25cb95a22a2ffc994ee26982d39bf54529fa4f5f383d6af4fcc810dcc.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathfrak { L } ( \\mathfrak { j } ) \\overset { \\mathrm { d e f } } { = } \\left[ \\begin{array} { c c c c } { \\mathbb { E } \\left[ ( 1 + \\alpha - \\delta ( a ^ { ( j ) } ) ^ { 2 } ) ^ { 2 } \\right] } & { \\mathbb { E } \\left[ - \\alpha ( 1 + \\alpha - \\delta ( a ^ { ( j ) } ) ^ { 2 } ) \\right] } & { \\mathbb { E } \\left[ - \\alpha ( 1 + \\alpha - \\delta ( a ^ { ( j ) } ) ^ { 2 } \\right] } & { \\alpha ^ { 2 } } \\\\ { \\mathbb { E } \\left[ ( 1 + \\alpha - \\delta ( a ^ { ( j ) } ) ^ { 2 } ) \\right] } & { 0 } & { - \\alpha } & { 0 } \\\\ { \\mathbb { E } \\left[ ( 1 + \\alpha - \\delta ( a ^ { ( j ) } ) ^ { 2 } ) \\right] } & { - \\alpha } & { 0 } & { 0 } \\\\ { 1 } & { 0 } & { 0 } & { 0 } \\end{array} \\right] , } & { } \\\\ { = \\left[ \\begin{array} { c c c c } { ( 1 + \\alpha - \\delta \\sigma _ { j } ^ { 2 } ) ^ { 2 } + ( c - 1 ) ( \\delta \\sigma _ { j } ^ { 2 } ) ^ { 2 } } & { - \\alpha ( 1 + \\alpha - \\delta \\sigma _ { j } ^ { 2 } ) } & { - \\alpha ( 1 + \\alpha - \\delta \\sigma _ { j } ^ { 2 } ) } & { \\alpha ^ { 2 } } \\\\ { ( 1 + \\alpha - \\delta \\sigma _ { j } ^ { 2 } ) } & { 0 } & { - \\alpha } & { 0 } \\\\ { ( 1 + \\alpha - \\delta \\sigma _ { j } ^ { 2 } ) } & { - \\alpha } & { 0 } & { 0 } \\\\ { 1 } & { 0 } & { 0 } & { 0 } \\end{array} \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 362, + 825, + 494 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We prove Proposition 3 by showing that for any choice of stepsize and momentum, either of the two holds: ", + "bbox": [ + 169, + 502, + 825, + 532 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• $B ^ { ( 1 ) }$ has an eigenvalue larger than 1, or, • the largest eigenvalue of $B ^ { ( 2 ) }$ is greater than $1 - { \\frac { 5 0 0 } { \\kappa } }$ . ", + "bbox": [ + 214, + 542, + 583, + 583 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "This is formalized in the following two lemmas. ", + "bbox": [ + 173, + 594, + 490, + 609 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Lemma 4. If the stepsize $\\delta$ is such that $\\delta \\sigma _ { 1 } ^ { 2 } \\geq \\frac { 2 \\left( 1 - \\alpha ^ { 2 } \\right) } { c + ( c - 2 ) \\alpha }$ , then $\\boldsymbol { B } ^ { ( 1 ) }$ has an eigenvalue $\\geq 1$ ", + "bbox": [ + 166, + 614, + 766, + 640 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Lemma 5. If the stepsize $\\delta$ is such that $\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } < \\frac { 2 \\left( 1 - \\alpha ^ { 2 } \\right) } { c + ( c - 2 ) \\alpha } } \\end{array}$ , then $B ^ { ( 2 ) }$ has an eigenvalue of magnitude $\\textstyle { \\ge } 1 - { \\frac { 5 0 0 } { \\kappa } }$ ", + "bbox": [ + 173, + 645, + 825, + 686 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Given this notation, we can now consider the $j ^ { t h }$ dimension without the superscripts; when needed, they will be made clear in the exposition. Denoting $x \\stackrel { \\mathrm { d e f } } { = } \\delta \\sigma ^ { 2 }$ and $t \\stackrel { \\mathrm { d e f } } { = } 1 + \\alpha - x$ , we have: ", + "bbox": [ + 173, + 699, + 823, + 734 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/7a3d15dc1dc2a465faf69d0d032b93a68286dfd83e1add214990fb4de90fc945.jpg", + "text": "$$\n\\begin{array} { r } { B = \\left[ \\begin{array} { c c c c } { t ^ { 2 } + ( c - 1 ) x ^ { 2 } } & { - \\alpha t } & { - \\alpha t } & { \\alpha ^ { 2 } } \\\\ { t } & { 0 } & { - \\alpha } & { 0 } \\\\ { t } & { - \\alpha } & { 0 } & { 0 } \\\\ { 1 } & { 0 } & { 0 } & { 0 } \\end{array} \\right] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 357, + 741, + 640, + 801 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 PROOF ", + "text_level": 1, + "bbox": [ + 173, + 819, + 266, + 833 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The analysis goes via computation of the characteristic polynomial of $\\boldsymbol { B }$ and evaluating it at different values to obtain bounds on its roots. ", + "bbox": [ + 174, + 845, + 825, + 875 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Lemma 6. The characteristic polynomial of $\\boldsymbol { B }$ is: ", + "bbox": [ + 173, + 880, + 501, + 895 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/f4144805b38ad8db88f573b95f9fd054cd912f90e52958624cb428369ce3c519.jpg", + "text": "$$\nD ( z ) = z ^ { 4 } - ( t ^ { 2 } + ( c - 1 ) x ^ { 2 } ) z ^ { 3 } + ( 2 \\alpha t ^ { 2 } - 2 \\alpha ^ { 2 } ) z ^ { 2 } + ( - t ^ { 2 } + ( c - 1 ) x ^ { 2 } ) \\alpha ^ { 2 } z + \\alpha ^ { 4 } .\n$$", + "text_format": "latex", + "bbox": [ + 220, + 901, + 779, + 921 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. We first begin by writing out the expression for the determinant: ", + "bbox": [ + 173, + 103, + 643, + 119 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/ea76d69feaee8e0f22640391a543c329bef208829a17e3ee0b71a625e9d2db1f.jpg", + "text": "$$\nD e t ( B - z { \\mathcal { Z } } ) = \\left| \\begin{array} { c c c c } { t ^ { 2 } + ( c - 1 ) x ^ { 2 } - z } & { - \\alpha t } & { - \\alpha t } & { \\alpha ^ { 2 } } \\\\ { t } & { - z } & { - \\alpha } & { 0 } \\\\ { t } & { - \\alpha } & { - z } & { 0 } \\\\ { 1 } & { 0 } & { 0 } & { - z } \\end{array} \\right| .\n$$", + "text_format": "latex", + "bbox": [ + 303, + 123, + 692, + 184 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "expanding along the first column, we have: ", + "bbox": [ + 174, + 190, + 457, + 205 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/13db8664f5f384bd3705cf4d39cce332c87323864434569fa790e37f71bc0b1b.jpg", + "text": "$$\n\\begin{array} { r l } & { \\gamma _ { e t } ( B - z \\mathcal { Z } ) = ( t ^ { 2 } + ( c - 1 ) x ^ { 2 } - z ) ( \\alpha ^ { 2 } z - z ^ { 3 } ) - t ( - \\alpha t z ^ { 2 } + \\alpha ^ { 2 } t z ) + t ( - \\alpha t ( \\alpha z ) + z \\cdot \\alpha t z ) - ( z \\cdot \\alpha ^ { 2 } z - \\alpha t \\mathcal { Z } ) } \\\\ & { \\qquad = ( t ^ { 2 } + ( c - 1 ) x ^ { 2 } - z ) ( \\alpha ^ { 2 } z - z ^ { 3 } ) - 2 t ( \\alpha ^ { 2 } t z - \\alpha t z ^ { 2 } ) - ( \\alpha ^ { 2 } z ^ { 2 } - \\alpha ^ { 4 } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 210, + 839, + 251 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Expanding the terms yields the expression in the lemma. ", + "bbox": [ + 176, + 255, + 549, + 270 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The next corollary follows by some simple arithmetic manipulations. ", + "bbox": [ + 171, + 284, + 632, + 299 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Corollary 7. Substituting $z = 1 - \\tau$ in the characteristic equation of Lemma $6$ , we have: ", + "bbox": [ + 171, + 303, + 753, + 318 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/1dcd7a6dadb1f017f50da83ae925f64de04e3cc31c884cb27b30f5e8c299e4a5.jpg", + "text": "$$\n\\begin{array} { r l } & { D ( 1 - \\tau ) = \\tau ^ { 4 } + \\tau ^ { 3 } ( - 4 + t ^ { 2 } + ( c - 1 ) x ^ { 2 } ) + \\tau ^ { 2 } ( 6 - 3 t ^ { 2 } - 3 ( c - 1 ) x ^ { 2 } - 2 \\alpha ^ { 2 } + 2 \\alpha t ^ { 2 } ) } \\\\ & { \\qquad + \\tau ( - 4 + 3 t ^ { 2 } + 3 ( c - 1 ) x ^ { 2 } + 4 \\alpha ^ { 2 } - 4 \\alpha t ^ { 2 } - ( c - 1 ) x ^ { 2 } \\alpha ^ { 2 } + t ^ { 2 } \\alpha ^ { 2 } ) } \\\\ & { \\qquad + ( 1 - t ^ { 2 } - ( c - 1 ) x ^ { 2 } - 2 \\alpha ^ { 2 } + 2 \\alpha t ^ { 2 } + ( c - 1 ) x ^ { 2 } \\alpha ^ { 2 } - t ^ { 2 } \\alpha ^ { 2 } + \\alpha ^ { 4 } ) } \\\\ & { \\qquad = \\tau ^ { 4 } + \\tau ^ { 3 } [ - ( 3 + \\alpha ) ( 1 - \\alpha ) - 2 x ( 1 + \\alpha ) + c x ^ { 2 } ] } \\\\ & { \\qquad + \\tau ^ { 2 } [ ( 3 - 4 \\alpha - \\alpha ^ { 2 } + 2 \\alpha ^ { 3 } ) - 2 x ( 1 + \\alpha ) ( 2 \\alpha - 3 ) + x ^ { 2 } ( 2 \\alpha - 3 c ) ] } \\\\ & { \\qquad + \\tau [ - ( 1 - \\alpha ) ^ { 2 } ( 1 - \\alpha ^ { 2 } ) - 2 x ( 3 - \\alpha ) ( 1 - \\alpha ^ { 2 } ) + x ^ { 2 } ( 3 c - 4 \\alpha + ( 2 - c ) \\alpha ^ { 2 } ) ] } \\\\ & { \\qquad + x ( 1 - \\alpha ) [ 2 ( 1 - \\alpha ^ { 2 } ) - x ( c + ( c - 2 ) \\alpha ) ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 207, + 321, + 785, + 465 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof of Lemma 4. The first observation necessary to prove the lemma is that the characteristic polynomial $D ( z )$ approaches $\\infty$ as $z \\infty$ , i.e., $\\begin{array} { r } { \\operatorname* { l i m } _ { z \\infty } D ( z ) = + \\infty } \\end{array}$ . ", + "bbox": [ + 174, + 476, + 823, + 506 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Next, we evaluate the characteristic polynomial at 1, i.e. compute $D ( 1 )$ . This follows in a straightforward manner from corollary (7) by substituting $\\tau = 0$ in equation (2), and this yields, ", + "bbox": [ + 174, + 511, + 825, + 540 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/e0bd88d6953eb39b22592e4d54f9d03958f770d27b69406b5c6350d5b00c653d.jpg", + "text": "$$\nD ( 1 ) = ( 1 - \\alpha ) x \\cdot \\bigg ( 2 ( 1 - \\alpha ^ { 2 } ) - x ( 1 - \\alpha ) - ( c - 1 ) x ( 1 + \\alpha ) \\bigg ) .\n$$", + "text_format": "latex", + "bbox": [ + 281, + 545, + 717, + 580 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "As $\\alpha < 1$ , $x = \\delta \\sigma ^ { 2 } > 0$ , we have the following by setting $D ( 1 ) \\leq 0$ and solving for $x$ : ", + "bbox": [ + 171, + 587, + 743, + 603 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/068e1807ff1d226a23c3f534bdac1e43bcbb26cf8aa58bf3a0da2d40f7fb882c.jpg", + "text": "$$\nx \\geq \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } .\n$$", + "text_format": "latex", + "bbox": [ + 434, + 609, + 562, + 643 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Since $D ( 1 ) \\leq 0$ and $D ( z ) \\geq 0$ as $z \\infty$ , there exists a root of $D ( \\cdot )$ which is $\\geq 1$ . ", + "bbox": [ + 173, + 648, + 717, + 666 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Remark 8. The above characterization is striking in the sense that for any $c > 1$ , increasing the momentum parameter $\\alpha$ naturally requires the reduction in the step size $\\delta$ to permit the convergence of the algorithm, which is not observed when fast gradient methods are employed in deterministic optimization. For instance, in the case of deterministic optimization, setting $c = 1$ yields $\\delta \\sigma _ { 1 } ^ { 2 } <$ $2 ( 1 + \\alpha )$ . On the other hand, when employing the stochastic heavy ball method with $x ^ { ( j ) } = 2 \\sigma _ { j } ^ { 2 }$ , we have the condition that $c = 2$ , and this implies, $\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } < \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { 2 } = 1 - \\alpha ^ { 2 } } \\end{array}$ . ", + "bbox": [ + 173, + 671, + 826, + 767 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We now prove Lemma 5. We first consider the large momentum setting. ", + "bbox": [ + 173, + 775, + 647, + 791 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Lemma 9. When the momentum parameter $\\alpha$ is set such that $1 - 4 5 0 / \\kappa \\leq \\alpha \\leq 1 , ~ $ $\\boldsymbol { B }$ has an eigenvalue of magnitude ≥ 1 − 450κ . ", + "bbox": [ + 173, + 795, + 825, + 825 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. This follows easily from the fact that $\\begin{array} { r } { \\operatorname* { d e t } ( \\boldsymbol { B } ) = \\alpha ^ { 4 } = \\prod _ { j = 1 } ^ { 4 } \\lambda _ { j } ( \\boldsymbol { B } ) \\le ( \\lambda _ { \\operatorname* { m a x } } ( \\boldsymbol { B } ) ) ^ { 4 } } \\end{array}$ , thus implying $1 - 4 5 0 / \\kappa \\leq \\alpha \\leq | \\lambda _ { \\mathrm { m a x } } ( \\beta ) |$ . ", + "bbox": [ + 173, + 840, + 825, + 876 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Remark 10. Note that the above lemma holds for any value of the learning rate $\\delta$ , and holds for every eigen direction of $\\mathbf { H }$ . Thus, for “large” values of momentum, the behavior of stochastic heavy ball does degenerate to the behavior of stochastic gradient descent. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We now consider the setting where momentum is bounded away from 1. ", + "bbox": [ + 173, + 103, + 647, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Corollary 11. Consider $B ^ { ( 2 ) }$ , by substituting $= l / \\kappa , x = \\delta \\lambda _ { \\mathrm { m i n } } = c ( \\delta \\sigma _ { 1 } ^ { 2 } ) / \\kappa$ in equation (2) and accumulating terms in varying powers of $1 / \\kappa$ , we obtain: ", + "bbox": [ + 174, + 122, + 825, + 154 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/2e49ae20187a91b888fee3c37c47a6bfa117379d22ff04dc6574ea64371c31ed.jpg", + "text": "$$\n\\begin{array} { l } { { G ( l ) \\stackrel { d e f } { = } \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \\kappa ^ { 5 } } + l ^ { 4 } - 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ^ { 3 } ( 1 + \\alpha ) + ( 2 \\alpha - 3 c ) c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 2 } } } \\\\ { { \\ + \\ \\frac { - ( 3 + \\alpha ) ( 1 - \\alpha ) l ^ { 3 } - 2 ( 1 + \\alpha ) ( 2 \\alpha - 3 ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ^ { 2 } + ( 3 c - 4 \\alpha + ( 2 - c ) \\alpha ^ { 2 } ) c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \\kappa ^ { 3 } } } } \\\\ { { \\ + \\ \\frac { ( 3 - 4 \\alpha - \\alpha ^ { 2 } + 2 \\alpha ^ { 3 } ) l ^ { 2 } - 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ( 3 - \\alpha ) ( 1 - \\alpha ^ { 2 } ) - c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } ( 1 - \\alpha ) ( c + ( c - 2 ) \\alpha ) } { \\kappa ^ { 2 } } } } \\\\ { { \\ + \\ \\frac { - ( 1 - \\alpha ) ^ { 2 } ( 1 - \\alpha ^ { 2 } ) l + 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) ( 1 - \\alpha ) ( 1 - \\alpha ^ { 2 } ) } { \\kappa } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 183, + 156, + 810, + 287 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Lemma 12. Let $2 < c < 3 0 0 0$ , $\\textstyle 0 \\leq \\alpha \\leq 1 - { \\frac { 4 5 0 } { \\kappa } }$ , $\\begin{array} { r } { l = 1 + \\frac { 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) } { 1 - \\alpha } } \\end{array}$ 2c(δσ21) . Then, G(l) ≤ 0. ", + "bbox": [ + 173, + 295, + 723, + 318 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. Since $\\begin{array} { r } { ( \\delta \\sigma _ { 1 } ^ { 2 } ) \\le \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } } \\end{array}$ , this implies $\\begin{array} { r } { \\frac { ( \\delta \\sigma _ { 1 } ^ { 2 } ) } { 1 - \\alpha } \\leq \\frac { 2 ( 1 + \\alpha ) } { c + ( c - 2 ) \\alpha } \\leq \\frac { 4 } { c } } \\end{array}$ , thus implying, $1 \\leq l \\leq 9$ ", + "bbox": [ + 171, + 330, + 797, + 356 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Substituting the value of $l$ in equation (3), the coefficient of $\\mathcal { O } ( 1 / \\kappa )$ is $- ( 1 - \\alpha ) ^ { 3 } ( 1 + \\alpha )$ . ", + "bbox": [ + 176, + 361, + 769, + 377 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We will bound this term along with $( 3 - 4 \\alpha - \\alpha ^ { 2 } + 2 \\alpha ^ { 3 } ) l ^ { 2 } / \\kappa ^ { 2 } = ( 1 - \\alpha ) ^ { 2 } ( 3 + 2 \\alpha ) l ^ { 2 } / \\kappa ^ { 2 }$ to obtain: ", + "bbox": [ + 178, + 381, + 821, + 398 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/c227990b778998dcd360275c899bb2dfc87c9724d75f1bfe40777d978027adb7.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { - ( 1 - \\alpha ) ^ { 3 } ( 1 + \\alpha ) } { \\kappa } + \\frac { ( 1 - \\alpha ) ^ { 2 } ( 3 + 2 \\alpha ) l ^ { 2 } } { \\kappa ^ { 2 } } \\leq \\frac { - ( 1 - \\alpha ) ^ { 3 } ( 1 + \\alpha ) } { \\kappa } + \\frac { 4 0 5 ( 1 - \\alpha ) ^ { 2 } } { \\kappa ^ { 2 } } } \\\\ & { \\qquad \\leq \\frac { ( 1 - \\alpha ) ^ { 2 } } { \\kappa } \\bigg ( \\frac { 4 0 5 } { \\kappa } - ( 1 - \\alpha ^ { 2 } ) \\bigg ) } \\\\ & { \\qquad \\leq \\frac { ( 1 - \\alpha ) ^ { 2 } } { \\kappa } \\bigg ( \\frac { 4 0 5 } { \\kappa } - ( 1 - \\alpha ) \\bigg ) \\leq - \\frac { 4 5 \\cdot 4 5 0 ^ { 2 } } { \\kappa ^ { 4 } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 405, + 797, + 510 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where, we use the fact that $\\alpha < 1 , l \\le 9$ . The natural implication of this bound is that the terms that are lower order, such as $\\mathcal { O } ( 1 / \\kappa ^ { 4 } )$ and $\\mathcal { O } ( 1 / \\kappa ^ { 5 } )$ will be negative owing to the large constant above. Let us verify that this is indeed the case by considering the terms having powers of $\\mathcal { O } ( 1 / \\kappa ^ { 4 } )$ and $\\mathcal { O } ( 1 / \\kappa ^ { 5 } )$ from equation (3): ", + "bbox": [ + 176, + 512, + 825, + 569 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/d9c4930d752d40ea7b9abf18ee1e20a7394ed6064f0d4d22b6da3dd963b47530.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \\kappa ^ { 5 } } + \\frac { l ^ { 4 } - 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ^ { 3 } ( 1 + \\alpha ) + ( 2 \\alpha - 3 c ) c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 2 } } { \\kappa ^ { 4 } } - \\frac { 4 5 \\cdot 4 5 0 ^ { 2 } } { \\kappa ^ { 4 } } } \\\\ & { \\leq \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \\kappa ^ { 5 } } + \\frac { l ^ { 4 } } { \\kappa ^ { 4 } } - \\frac { 4 5 \\cdot 4 5 0 ^ { 2 } } { \\kappa ^ { 4 } } } \\\\ & { \\leq \\frac { c l ^ { 3 } } { \\kappa ^ { 5 } } + \\frac { ( 9 ^ { 4 } - ( 4 5 \\cdot 4 5 0 ^ { 2 } ) ) } { \\kappa ^ { 4 } } \\leq \\frac { 9 ^ { 3 } c + 9 ^ { 4 } - ( 4 5 \\cdot 4 5 0 ^ { 2 } ) } { \\kappa ^ { 4 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 251, + 574, + 746, + 675 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The expression above evaluates to $\\leq 0$ given an upperbound on the value of $c$ . The expression above follows from the fact that $l \\leq 9 , \\kappa \\geq 1$ . ", + "bbox": [ + 173, + 678, + 825, + 708 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Next, consider the terms involving $\\mathcal { O } ( 1 / \\kappa ^ { 3 } )$ and $\\mathcal { O } ( 1 / \\kappa ^ { 2 } )$ , in particular, ", + "bbox": [ + 174, + 712, + 648, + 728 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/4ca8e3746ecda1b419bb3c5576e6b565c92c10c25b1ae667713f3542cc658b64.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { ( 3 c - 4 \\alpha + ( 2 - c ) \\alpha ^ { 2 } ) c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l } { \\kappa ^ { 3 } } - \\frac { c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } ( 1 - \\alpha ) ( c + ( c - 2 ) \\alpha ) } { \\kappa ^ { 2 } } } \\\\ & { \\leq \\frac { c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { l ( 3 c + 2 ) } { \\kappa } - ( 1 - \\alpha ) ( c + ( c - 2 ) \\alpha ) ) } \\\\ & { \\leq \\frac { c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { 5 \\epsilon } { \\kappa } - ( 1 - \\alpha ) ( c + ( c - 2 ) \\alpha ) ) } \\\\ & { \\leq \\frac { c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { 5 \\epsilon l } { \\kappa } - ( 1 - \\alpha ) c ) } \\\\ & { \\leq \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { 5 l } { \\kappa } - \\frac { 4 5 0 } { \\kappa } ) } \\\\ & { \\leq \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { 1 - \\delta ^ { 2 } } { \\kappa } ) } \\\\ & { \\leq \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } \\cdot \\frac { - 4 4 5 0 } { \\kappa } \\leq 0 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 733, + 727, + 933 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Next, ", + "bbox": [ + 173, + 103, + 212, + 117 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/c6d3a6cbc597db87d0d5141e5bf9a7de29a1c946fa9e13405ca548749641087c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { - 2 ( 1 + \\alpha ) ( 2 \\alpha - 3 ) e ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ^ { 2 } } { \\kappa ^ { 3 } } - \\frac { 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ( 3 - \\alpha ) ( 1 - \\alpha ^ { 2 } ) } { \\kappa ^ { 2 } } } \\\\ & { \\leq \\frac { 2 ( 1 + \\alpha ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l } { \\kappa ^ { 2 } } \\Big ( \\frac { - ( 2 \\alpha - 3 ) l } { \\kappa } - ( 3 - \\alpha ) ( 1 - \\alpha ) \\Big ) } \\\\ & { \\leq \\frac { 2 ( 1 + \\alpha ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l } { \\kappa ^ { 2 } } \\Big ( \\frac { 3 l } { \\kappa } - 2 ( 1 - \\alpha ) \\Big ) } \\\\ & { \\leq \\frac { 2 ( 1 + \\alpha ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l } { \\kappa ^ { 2 } } \\Big ( \\frac { 3 l } { \\kappa } - \\frac { 2 \\cdot 4 5 0 } { \\kappa } \\Big ) } \\\\ & { \\leq \\frac { 2 ( 1 + \\alpha ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l } { \\kappa ^ { 2 } } \\Big ( \\frac { 3 \\cdot 2 7 } { \\kappa } - \\frac { 2 \\cdot 4 5 0 } { \\kappa } \\Big ) \\leq 0 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 303, + 122, + 692, + 299 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In both these cases, we used the fact that remaining terms are negative. $\\textstyle \\alpha \\leq 1 - { \\frac { 4 5 0 } { \\kappa } }$ implying $\\begin{array} { r } { - ( 1 - \\alpha ) \\le \\frac { - 4 5 0 } { \\kappa } } \\end{array}$ . Finally, other ", + "bbox": [ + 171, + 303, + 825, + 333 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Before rounding up the proof of the proposition, we need the following lemma to ensure that our lower bounds on the largest eigenvalue of $\\boldsymbol { B }$ indeed affect the algorithm’s rates and are true irrespective of where the algorithm is begun. Note that this allows our result to be much stronger than typical optimization lowerbounds that rely on specific initializations to ensure a component along the largest eigendirection of the update operator, for which bounds are proven. ", + "bbox": [ + 173, + 347, + 825, + 417 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Lemma 13. For any starting iterate $\\mathbf { w } _ { 0 } \\neq \\mathbf { w } ^ { * }$ , the HB method produces a non-zero component along the largest eigen direction of $\\boldsymbol { B }$ . ", + "bbox": [ + 173, + 421, + 823, + 450 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof. We note that in a similar manner as other proofs, it suffices to argue for each dimension of the problem separately. But before we start looking at each dimension separately, let us consider the $\\bar { j } ^ { \\mathrm { t h } }$ dimension, and detail the approach we use to prove the claim: the idea is to examine the subspace spanned by covariance $\\mathbb { E } \\left[ \\pmb { \\theta } _ { . } ^ { ( j ) } \\otimes \\pmb { \\theta } _ { . } ^ { ( j ) } \\right]$ of the iterates $\\pmb { \\theta } _ { 0 } ^ { ( j ) } , \\pmb { \\theta } _ { 1 } ^ { ( j ) } , \\pmb { \\theta } _ { 2 } ^ { ( j ) } , . . . ,$ for every starting iterate $\\pmb { \\theta } _ { 0 } ^ { ( j ) } \\neq \\left[ 0 , 0 \\right] ^ { \\top }$ and prove that the largest eigenvector of the expected operator $B ^ { ( j ) }$ is not orthogonal to this subspace. This implies that there exists a non-zero component of $\\mathbb { E } \\left[ \\pmb { \\theta } _ { \\cdot } ^ { ( j ) } \\otimes \\pmb { \\theta } _ { \\cdot } ^ { ( j ) } \\right]$ in the largest eigen direction of $B ^ { ( j ) }$ , and this decays at a rate that is at best $\\lambda _ { \\operatorname* { m a x } } ( B ^ { ( j ) } )$ . ", + "bbox": [ + 173, + 464, + 826, + 590 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Since $\\boldsymbol { B } ^ { ( j ) } ~ \\in ~ \\mathbb { R } ^ { 4 \\times 4 }$ , we begin by examining the expected covariance spanned by the iterates $\\pmb { \\theta } _ { 0 } ^ { ( j ) } , \\pmb { \\theta } _ { 1 } ^ { ( j ) } , \\pmb { \\theta } _ { 2 } ^ { ( j ) } , \\pmb { \\theta } _ { 3 } ^ { ( j ) }$ . Let $\\mathbf { w } _ { 0 } ^ { ( j ) } - ( \\mathbf { w } ^ { * } ) ^ { ( j ) } = \\mathbf { \\bar { w } } _ { - 1 } ^ { ( j ) } - ( \\mathbf { w } ^ { * } ) ^ { ( j ) } = k ^ { ( j ) }$ . Now, this implies $\\theta _ { 0 } ^ { ( j ) } =$ $\\boldsymbol { k } ^ { ( j ) } \\cdot \\left[ 1 , 1 \\right] ^ { \\top }$ . Then, ", + "bbox": [ + 173, + 595, + 825, + 648 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/075e58cbc0117e80b6adfaa51e1cf1a4b122387aa3cfe9a35517173163888eb2.jpg", + "text": "$$\n\\pmb { \\theta } _ { 1 } ^ { ( j ) } = k ^ { ( j ) } \\widehat { \\mathbf { A } } _ { 1 } ^ { ( j ) } \\left[ 1 \\right] , \\mathrm { w i t h } \\widehat { \\mathbf { A } } _ { 1 } ^ { ( j ) } = \\left[ 1 + \\alpha - \\delta \\widehat { \\mathbf { H } } _ { 1 } ^ { ( j ) } \\quad - \\alpha \\right] , \\mathrm { w h e r e } \\widehat { \\mathbf { H } } _ { 1 } ^ { ( j ) } = \\big ( a _ { 1 } ^ { ( j ) } \\big ) ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 223, + 654, + 772, + 689 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "This implies that $k$ just appears as a scale factor. This in turn implies that in order to analyze the subspace spanned by the covariance of iterates $\\theta _ { 0 } ^ { ( j ) } , \\theta _ { 1 } ^ { ( j ) } , . . . ,$ , we can assume $k ^ { ( j ) } = 1$ without any loss in generality. This implies, $\\pmb { \\theta } _ { 0 } ^ { ( j ) } = \\left[ 1 , 1 \\right] ^ { \\top }$ . Note that with this in place, we see that we can now drop the superscript $j$ that represents the dimension, since the analysis decouples across the dimensions $j \\in \\{ 1 , 2 \\}$ . Furthermore, let the entries of the vector $\\pmb { \\theta } _ { k }$ be represented as $\\pmb { \\theta } _ { k }$ def = $\\left[ \\theta _ { k 1 } \\quad \\theta _ { k 2 } \\right] ^ { \\top }$ Next, denote $1 + \\alpha - \\delta \\widehat { \\mathbf { H } } _ { k } = \\widehat { t } _ { k }$ . This implies, ", + "bbox": [ + 173, + 694, + 826, + 795 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/563194c085e318134ba1dcd4c84395874345a02b1d6ed4fcb6126b19077e91b2.jpg", + "text": "$$\n\\begin{array} { r } \\widehat { \\mathbf { A } } _ { k } = \\left[ \\begin{array} { c c } { \\widehat { t } _ { k } } & { - \\alpha \\right] . } \\end{array} \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 436, + 799, + 562, + 835 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Furthermore, ", + "bbox": [ + 173, + 839, + 261, + 853 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/0614d54ea272ff955e32aeeab22d751a11862b3257157be460f37fd70aae708d.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb \\theta _ { 1 } = \\widehat { \\mathbf A } _ { 1 } \\pmb \\theta _ { 0 } = \\left[ \\hat { t } _ { 1 } - \\alpha \\right] , \\pmb \\theta _ { 2 } = \\widehat { \\mathbf A } _ { 2 } \\pmb \\theta _ { 1 } = \\left[ \\hat { t } _ { 2 } ( \\hat { t } _ { 1 } - \\alpha ) - \\alpha \\right] , } \\\\ { \\pmb \\theta _ { 3 } = \\widehat { \\mathbf A } _ { 3 } \\pmb \\theta _ { 2 } = \\left[ \\hat { t } _ { 3 } ( \\hat { t } _ { 2 } ( \\hat { t } _ { 1 } - \\alpha ) - \\alpha ) - \\alpha ( \\hat { t } _ { 1 } - \\alpha ) \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 858, + 699, + 931 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Let us consider the vectorized form of $\\Phi _ { j } = \\mathbb { E } \\left[ \\pmb { \\theta } _ { j } \\otimes \\pmb { \\theta } _ { j } \\right]$ , and we denote this as vec $( \\Phi _ { j } )$ . Note that $\\mathrm { v e c } ( \\Phi _ { j } )$ makes $\\Phi _ { j }$ become a column vector of size $4 \\times 1$ . Now, consider vec $( \\Phi _ { j } )$ for $\\bar { \\boldsymbol { j } } = 0 , 1 , 2 , 3$ and concatenate these to form a matrix that we denote as $\\mathcal { D }$ , i.e. ", + "bbox": [ + 174, + 102, + 825, + 146 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/d69f4ec225f119d9ce46cad40495b635c267c73ffe69a3ea0e4729eb7c48c06f.jpg", + "text": "$$\n\\mathcal { D } = \\left[ \\mathrm { v e c } ( \\Phi _ { 0 } ) \\mathrm { v e c } ( \\Phi _ { 1 } ) \\mathrm { v e c } ( \\Phi _ { 2 } ) \\mathrm { v e c } ( \\Phi _ { 3 } ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 151, + 663, + 169 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Now, since we note that $\\Phi _ { j }$ is a symmetric $2 \\times 2$ matrix, $\\mathcal { D }$ should contain two identical rows implying that it has an eigenvalue that is zero and a corresponding eigenvector that is $\\begin{array} { r } { \\left[ { 0 \\mathrm { ~ \\ t ~ { ~ - } 1 / { \\sqrt { 2 } } ~ } } { \\hat { 1 } } / { \\sqrt { ( 2 \\tau ) } } \\mathrm { ~ \\ t ~ { ~ } } \\right] ^ { \\top } } \\end{array}$ . It turns out that this is also an eigenvector of $\\boldsymbol { B }$ with an eigenvalue $\\alpha$ . Note that $\\operatorname* { d e t } ( B ) = \\alpha ^ { 4 }$ . This implies there are two cases that we need to consider: (i) when all eigenvalues of $\\boldsymbol { B }$ have the same magnitude $( = \\alpha )$ . In this case, we are already done, because there exists at least one non zero eigenvalue of $\\mathcal { D }$ and this should have some component along one of the eigenvectors of $\\boldsymbol { B }$ and we know that all eigenvectors have eigenvalues with a magnitude equal to $\\lambda _ { \\mathrm { m a x } } ( B )$ . Thus, there exists an iterate which has a non-zero component along the largest eigendirection of $\\boldsymbol { B }$ . (ii) the second case is the situation when we have eigenvalues with different magnitudes. In this case, note that $\\operatorname* { d e t } ( \\mathcal B ) = \\alpha ^ { 4 } < ( \\lambda _ { \\operatorname* { m a x } } ( \\mathcal B ) ) ^ { 4 }$ implying $\\bar { \\lambda } _ { \\mathrm { m a x } } ( B ) > \\alpha$ . In this case, we need to prove that $\\mathcal { D }$ spans a three-dimensional subspace; if it does, it contains a component along the largest eigendirection of $\\boldsymbol { B }$ which will round up the proof. Since we need to understand whether $\\mathcal { D }$ spans a three dimensional subspace, we can consider a different (yet related) matrix, which we call $\\mathcal { R }$ and this is defined as: ", + "bbox": [ + 173, + 174, + 826, + 376 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/3c20cce8c2c0d6134445db18bfd15fcba74adcbd9ac0bdca4df8f04fef5db123.jpg", + "text": "$$\n\\mathcal { R } \\stackrel { \\mathrm { d e f } } { = } \\mathbb { E } \\left( \\begin{array} { c c c } { \\theta _ { 0 1 } ^ { 2 } } & { \\theta _ { 1 1 } ^ { 2 } } & { \\theta _ { 2 1 } ^ { 2 } } \\\\ { \\theta _ { 0 1 } \\theta _ { 0 2 } } & { \\theta _ { 1 1 } \\theta _ { 1 2 } } & { \\theta _ { 2 1 } \\theta _ { 2 2 } } \\\\ { \\theta _ { 0 2 } ^ { 2 } } & { \\theta _ { 1 2 } ^ { 2 } } & { \\theta _ { 2 2 } ^ { 2 } } \\end{array} \\right)\n$$", + "text_format": "latex", + "bbox": [ + 369, + 377, + 629, + 429 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Given the expressions for $\\{ \\pmb { \\theta } _ { j } \\} _ { j = 0 } ^ { 3 }$ (by definition of $\\pmb { \\theta } _ { 0 }$ and using equation 4), we can substitute to see that $\\mathcal { R }$ has the following expression: ", + "bbox": [ + 171, + 434, + 823, + 463 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/fff99e0533348a2104f79eb84a2054e261225711c13efbd3cf425173bcd8318a.jpg", + "text": "$$\n\\mathcal { R } = \\left[ { 1 \\atop 1 } \\begin{array} { c c } { { \\mathbb { E } \\left[ ( \\hat { t } _ { 1 } - \\alpha ) ^ { 2 } \\right] } } & { { \\mathbb { E } \\left[ ( \\hat { t } _ { 2 } ( \\hat { t } _ { 1 } - \\alpha ) - \\alpha ) ^ { 2 } \\right] } } \\\\ { { \\mathbb { E } \\left[ \\hat { t } _ { 1 } - \\alpha \\right] } } & { { \\mathbb { E } \\left[ ( ( \\hat { t } _ { 2 } ( \\hat { t } _ { 1 } - \\alpha ) - \\alpha ) ) ( \\hat { t } _ { 1 } - \\alpha ) \\right] } } \\\\ { { 1 } } & { { \\mathbb { E } \\left[ ( \\hat { t } _ { 1 } - \\alpha ) ^ { 2 } \\right] } } \\end{array} \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 294, + 468, + 702, + 520 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "If we compute and prove that $\\operatorname* { d e t } ( \\mathcal { R } ) \\neq 0$ , we are done since that implies that $\\mathcal { R }$ has three non-zero eigenvalues. ", + "bbox": [ + 173, + 525, + 823, + 554 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "This implies, we first define the following: let $q _ { \\gamma } = ( t - \\gamma ) ^ { 2 } + ( c - 1 ) x ^ { 2 }$ . Then, $\\mathcal { R }$ can be expressed as: ", + "bbox": [ + 173, + 559, + 823, + 588 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/5e6b21a6080326b5942c4da37a2a03ce47ec3a1c714e266fd9d1fcec5fed47b3.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { d e t } ( \\mathcal { R } ) = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 1 } & { q _ { \\alpha } } & { 2 q _ { \\alpha } - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 1 } & { t - \\alpha } & { t q _ { \\alpha } - \\alpha ( t - \\alpha ) } \\end{array} \\right] \\right) } } \\\\ & { \\qquad = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 1 } & { q _ { \\alpha } } & { 2 \\alpha } \\\\ { 1 } & { 1 } & { \\phi _ { \\alpha } } \\\\ { 1 } & { 1 } & { \\phi _ { \\alpha } } \\\\ { 1 } & { t - \\alpha } & { q _ { \\alpha } - \\alpha ( t - \\alpha ) - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 1 } & { 1 } & { \\theta _ { \\alpha } } \\end{array} \\right] \\right) } } \\\\ & { \\qquad = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 1 } & { q _ { \\alpha } - 1 } & { q _ { \\alpha } ( q _ { \\alpha } - q _ { \\alpha } ) - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 1 } & { t - \\alpha - 1 } & { 1 } & { 0 } \\\\ { 1 } & { 0 } & { t q _ { \\alpha } - \\alpha ( t - \\alpha ) - 0 } & { ( t - \\alpha ) q _ { \\alpha } } \\end{array} \\right] \\right) } } \\\\ & { \\qquad = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 0 } & { q _ { \\alpha } - 1 } & { q _ { \\alpha } ( q _ { \\alpha } - q _ { \\alpha } ) - ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 0 } & { 1 } & { \\theta _ { \\alpha } } \\end{array} \\right] \\right) } } \\\\ & { \\qquad = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 0 } & { q _ { \\alpha } - 1 } & { q _ { \\alpha } ( q _ { \\alpha } - q _ { \\alpha } ) - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 0 } & { t - \\alpha - 1 } & { 1 } \\end{array} \\right] \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 267, + 590, + 728, + 791 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Note: (i) $q _ { \\alpha } - 1 = ( t - \\alpha ) ^ { 2 } - 1 + ( c - 1 ) x ^ { 2 } = ( 1 - x ) ^ { 2 } - 1 + ( c - 1 ) x ^ { 2 } = - 2 x + x ^ { 2 } + ( c - 1 ) x ^ { 2 } = 0$ $- 2 x + c x ^ { 2 }$ . ", + "bbox": [ + 174, + 796, + 823, + 827 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "(ii) $t - \\alpha - 1 = - x$ \n(iv) (iii) $\\begin{array} { r l } & { \\alpha ( q _ { \\alpha } - ( t - \\alpha ) ) = \\alpha ( ( t - \\alpha ) ^ { 2 } - ( t - \\alpha ) + ( c - 1 ) x ^ { 2 } ) = \\alpha ( ( 1 - x ) ( - x ) + ( c - 1 ) x ^ { 2 } ) = \\alpha x ( - 1 + c x ) } \\\\ & { q _ { 0 } - q _ { \\alpha } = t ^ { 2 } - ( t - \\alpha ) ^ { 2 } = \\alpha ( 2 t - \\alpha ) = 2 t \\alpha - \\alpha ^ { 2 } . } \\end{array}$ \nThen, ", + "bbox": [ + 173, + 828, + 823, + 883 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/20fbd1362c636fb537b4b2079712bb5c9e76d381bbdf6e4400c3cc2cb4011a96.jpg", + "text": "$$\n\\begin{array} { r } { ( 2 \\alpha t - \\alpha ^ { 2 } ) q _ { \\alpha } - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } = 2 t \\alpha ( q _ { \\alpha } - ( t - \\alpha ) ) + \\alpha ^ { 2 } ( 1 - q _ { \\alpha } ) } \\\\ { = 2 t \\alpha ( - x + c x ^ { 2 } ) - \\alpha ^ { 2 } ( - 2 x + c x ^ { 2 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 886, + 717, + 928 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/c2d87a6adadba871e5f064c44116e20243172461ca53c598ba5613e87d00c556.jpg", + "text": "$$\n\\begin{array} { r l } & { = - 2 t \\alpha x + 2 x \\alpha ^ { 2 } + 2 t \\alpha c x ^ { 2 } - c \\alpha ^ { 2 } x ^ { 2 } } \\\\ & { = 2 \\alpha x ( - t + \\alpha ) + c \\alpha x ^ { 2 } ( 2 t - \\alpha ) } \\\\ & { = - 2 \\alpha x ( 1 - x ) + 2 c \\alpha x ^ { 2 } ( 1 - x ) + c \\alpha ^ { 2 } x ^ { 2 } } \\\\ & { = 2 \\alpha x ( 1 - x ) ( - 1 + c x ) + c \\alpha ^ { 2 } x ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 467, + 102, + 753, + 181 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Then, ", + "bbox": [ + 173, + 185, + 215, + 200 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/a16691d3c7ee37009e76b6c2fee6d01aeac1460ff850eb443b092b9570c3a0ef.jpg", + "text": "$$\n{ \\begin{array} { r l } & { \\operatorname* { d e t } ( \\mathcal { R } ) = \\operatorname* { d e t } { \\left( \\begin{array} { l l l } { 0 } & { x ( c x - 2 ) } & { 2 \\alpha x ( 1 - x ) ( - 1 + c x ) + c \\alpha ^ { 2 } x ^ { 2 } } \\\\ { 0 } & { - x } & { \\alpha x ( c x - 1 ) } \\\\ { 1 } & { 0 } & { 0 } \\end{array} \\right) } } \\\\ & { \\qquad = x ^ { 2 } \\alpha \\operatorname* { d e t } \\left( { \\left[ \\begin{array} { l l l } { 0 } & { ( c x - 2 ) } & { c \\alpha x + 2 ( 1 - x ) ( c x - 1 ) } \\\\ { 0 } & { - 1 } & { c x - 1 } \\\\ { 1 } & { 0 } & { 0 } \\end{array} \\right] } \\right) } \\\\ & { \\qquad = x ^ { 3 } \\alpha \\operatorname* { d e t } \\left( { \\left[ \\begin{array} { l l l } { 0 } & { c } & { c \\alpha - 2 ( c x - 1 ) } \\\\ { 0 } & { - 1 } & { c x - 1 } \\\\ { 1 } & { 0 } & { 0 } \\end{array} \\right] } \\right) } \\end{array} }\n$$", + "text_format": "latex", + "bbox": [ + 267, + 205, + 728, + 348 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Then, ", + "bbox": [ + 173, + 353, + 215, + 367 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/0eadf542bbb122afce66a44b53d459f6100df1faae43af7754b8b176cf312392.jpg", + "text": "$$\n\\begin{array} { c } { { \\operatorname * { d e t } ( \\mathcal { R } ) = x ^ { 3 } \\alpha \\bigg ( c ( - 1 + c x ) - 2 ( - 1 + c x ) + c \\alpha \\bigg ) } } \\\\ { { = \\alpha x ^ { 3 } \\bigg ( ( c - 2 ) ( - 1 + c x ) + c \\alpha \\bigg ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 330, + 372, + 666, + 444 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Note that this determinant can be zero when ", + "bbox": [ + 174, + 446, + 465, + 462 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/3eb8355657aeb44dfcbd395c2082fbf6fd540dce6e01864969c1289e7447458a.jpg", + "text": "$$\n\\alpha = \\frac { ( c - 2 ) ( 1 - c x ) } { c } .\n$$", + "text_format": "latex", + "bbox": [ + 424, + 464, + 573, + 496 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We show this is not possible by splitting our argument into two parts, one about the convergent regime of the algorithm (where, $\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } < \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } ) } \\end{array}$ and the other about the divergent regime. ", + "bbox": [ + 174, + 500, + 825, + 536 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Let us first provide a proof for the convergent regime of the algorithm. For this regime, let the chosen $\\delta$ be represented as $\\delta ^ { + }$ . Now, for the smaller eigen direction, $x = \\delta ^ { + } \\lambda _ { \\mathrm { m i n } } = c \\bar { \\delta ^ { + } } \\sigma _ { 1 } ^ { 2 } / \\kappa$ . Suppose $\\alpha$ was chosen as per equation 5, ", + "bbox": [ + 174, + 541, + 825, + 585 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/de1df55d734357c1c9e66322d4865ff165cc66c9c2614aecd7423dde5b5f49cf.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\frac { c \\alpha } { c - 2 } = 1 - \\frac { c ^ { 2 } \\delta ^ { + } \\sigma _ { 1 } ^ { 2 } } { \\kappa } } } \\\\ { { \\implies \\delta ^ { + } \\sigma _ { 1 } ^ { 2 } = \\displaystyle \\frac { \\kappa } { c ^ { 2 } } - \\frac { \\kappa \\alpha } { c ( c - 2 ) } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 400, + 590, + 599, + 655 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We will now prove that δ+σ21 = κc ( 1c is much larger than one allowed by the convergence of the HB updates, i.e., $\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } < \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } \\le \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c } } \\end{array}$ . In particular, if we prove that $\\textstyle { \\frac { \\kappa } { c } } { \\bigl ( } { \\frac { 1 } { c } } - { \\frac { \\alpha } { c - 2 } } { \\bigr ) } >$ 2(1−α2) for any admissible value of α, we are done. ", + "bbox": [ + 173, + 659, + 825, + 719 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/4209fa860481162363cfcba17cb63e9b25abe1e5f76a87c55d10cbe7223d24b4.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\frac \\kappa c ( \\frac 1 c - \\frac \\alpha { c - 2 } ) > \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c } } } \\\\ { { \\Leftrightarrow \\displaystyle \\frac \\kappa c - \\frac { \\kappa \\alpha } { c - 2 } > 2 - 2 \\alpha ^ { 2 } } } \\\\ { { \\Leftrightarrow \\displaystyle \\frac \\kappa c - \\frac { \\kappa \\alpha } { c - 2 } > \\frac \\kappa c - \\frac { \\kappa \\alpha } { c } > 2 - 2 \\alpha ^ { 2 } } } \\\\ { { \\Leftrightarrow \\kappa - \\kappa \\alpha > 2 c - 2 c \\alpha ^ { 2 } } } \\\\ { { \\Leftrightarrow 2 c \\alpha ^ { 2 } - \\kappa \\alpha + ( \\kappa - 2 c ) > 0 . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 344, + 724, + 647, + 856 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The two roots of this quadratic equation are $\\alpha ^ { + } = \\textstyle { \\frac { \\kappa } { 2 c } } - 1$ and $\\alpha ^ { - } = 1$ . Note that $\\kappa \\geq \\widetilde { \\kappa } = c$ ; note that there is not much any method gains over SGD if $\\kappa = \\mathcal { O } ( c )$ . And, for any $\\kappa \\geq 4 c$ , note, $\\alpha ^ { + } > \\alpha ^ { - }$ , indicating that the above equation holds true if $\\begin{array} { r } { \\alpha > \\alpha ^ { + } = \\frac { \\kappa } { 2 c } - 1 } \\end{array}$ or if $\\alpha < \\alpha ^ { - } = 1$ . The latter condition is true and hence the proposition that $\\delta ^ { + } \\sigma _ { 1 } ^ { 2 } > \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha }$ is true. ", + "bbox": [ + 173, + 861, + 826, + 928 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We need to prove that the determinant does not vanish in the divergent regime for rounding up the proof to the lemma. ", + "bbox": [ + 173, + 103, + 825, + 132 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Now, let us consider the divergent regime of the algorithm, i.e., when, $\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } > \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } } \\end{array}$ . Furthermore, for the larger eigendirection, the determinant is zero when $\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } = \\frac { 1 - \\frac { c \\alpha } { c - 2 } } { c } = \\frac { 1 } { c } - \\frac { \\alpha } { c - 2 } } \\end{array}$ (obtained by substituting $x = \\delta \\sigma _ { 1 } ^ { 2 }$ in equation 5). If we show that $\\textstyle { \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } } > { \\frac { 1 } { c } } - { \\frac { \\alpha } { c - 2 } }$ for all admissible values of $c$ , we are done. We will explore this in greater detail: ", + "bbox": [ + 173, + 138, + 826, + 217 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/401b2fa244d3b195ef2a5a6b2f13fcfb4f6525d18fb05b869c3d40152ea3178a.jpg", + "text": "$$\n\\begin{array} { c } { { \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } > \\displaystyle \\frac { 1 } { c } - \\frac { \\alpha } { c - 2 } } } \\\\ { { \\Leftrightarrow 2 ( 1 - \\alpha ^ { 2 } ) \\geq 1 + \\displaystyle \\frac { c - 2 } { c } \\alpha - \\displaystyle \\frac { c } { c - 2 } \\alpha - \\alpha ^ { 2 } } } \\\\ { { \\Leftrightarrow 1 - \\alpha ^ { 2 } \\geq \\displaystyle \\frac { - 4 ( c - 1 ) } { c ( c - 2 ) } \\alpha } } \\\\ { { \\Leftrightarrow c ^ { 2 } - 2 c - \\alpha ^ { 2 } c ^ { 2 } + 2 c \\alpha ^ { 2 } \\geq - 4 c \\alpha + 4 \\alpha } } \\\\ { { \\Leftrightarrow c ^ { 2 } ( 1 - \\alpha ^ { 2 } ) - 2 c ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) - 4 \\alpha \\geq 0 . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 258, + 220, + 736, + 363 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "considering the quadratic in the left hand size and solving it for $c$ , we have: ", + "bbox": [ + 173, + 367, + 666, + 382 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/ae8558b43deba1c0f65d2bf390e9ee9196ec81de566ff02ddc856a88fe2909fd.jpg", + "text": "$$\n\\begin{array} { l } { { c ^ { \\pm } = \\frac { 2 ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) \\pm \\sqrt { 4 ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) ^ { 2 } + 1 6 \\alpha ( 1 - \\alpha ^ { 2 } ) } } { 2 ( 1 - \\alpha ^ { 2 } ) } } } \\\\ { { { } ~ = \\frac { ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) \\pm \\sqrt { ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) ^ { 2 } + 4 \\alpha ( 1 - \\alpha ^ { 2 } ) } } { ( 1 - \\alpha ^ { 2 } ) } } } \\\\ { { { } ~ = \\frac { ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) \\pm \\sqrt { 1 + \\alpha ^ { 4 } + 4 \\alpha ^ { 2 } - 2 \\alpha ^ { 2 } - 4 \\alpha + 4 \\alpha ^ { 3 } + 4 \\alpha ( 1 - \\alpha ^ { 2 } ) } } { ( 1 - \\alpha ^ { 2 } ) } } } \\\\ { { { } ~ = \\frac { ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) \\pm ( 1 + \\alpha ^ { 2 } ) } { ( 1 - \\alpha ^ { 2 } ) } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 248, + 387, + 748, + 535 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "This holds true iff ", + "bbox": [ + 174, + 539, + 295, + 554 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/562c8cb38fb814a43502622bc28811015cf0482d08d354218f24c09b304f3432.jpg", + "text": "$$\nc \\leq c ^ { - } = \\frac { - 2 \\alpha ( 1 + \\alpha ) } { 1 - \\alpha ^ { 2 } } = \\frac { - 2 \\alpha } { 1 - \\alpha } ,\n$$", + "text_format": "latex", + "bbox": [ + 385, + 558, + 611, + 588 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "or iff, ", + "bbox": [ + 173, + 593, + 214, + 608 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/5183796f3ff4c4c7c38f9ea246d689041c470c4cd14b7849b8936f32cfdc455e.jpg", + "text": "$$\nc \\geq c ^ { + } = { \\frac { 2 ( 1 - \\alpha ) } { 1 - \\alpha ^ { 2 } } } = { \\frac { 2 } { 1 + \\alpha } } .\n$$", + "text_format": "latex", + "bbox": [ + 397, + 612, + 599, + 645 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Which is true automatically since $c > 2$ . This completes the proof of the lemma. ", + "bbox": [ + 174, + 648, + 700, + 665 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We are now ready to prove Lemma 5. ", + "bbox": [ + 176, + 679, + 419, + 694 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proof of Lemma 5. Combining Lemmas 9 and 12, we see that no matter what stepsize and momentum we choose, $\\boldsymbol { B } ^ { ( j ) }$ has an eigenvalue of magnitude at least $1 - { \\frac { 5 0 0 } { \\kappa } }$ for some $j \\in \\{ 1 , 2 \\}$ . This proves the lemma. □ ", + "bbox": [ + 174, + 708, + 825, + 753 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B EQUIVALENCE OF ALGORITHM 3 AND ASGD ", + "text_level": 1, + "bbox": [ + 174, + 772, + 588, + 790 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We begin by writing out the updates of ASGD as written out in Jain et al. (2017), which starts with two iterates $\\widehat { a } _ { 0 }$ and $\\widehat { d } _ { 0 }$ , and from time $t = 0 , 1 , . . . T - 1$ implements the following updates: ", + "bbox": [ + 173, + 803, + 821, + 835 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/ddeca56d9e7b854d0dc0e34d661bc923e33374f1f00c93564287c1a65170c120.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { \\widehat { b } _ { t } = \\alpha _ { 1 } \\widehat { a } _ { t } + ( 1 - \\alpha _ { 1 } ) \\widehat { d } _ { t } } \\\\ & { } & { \\widehat { a } _ { t + 1 } = \\widehat { b } _ { t } - \\delta _ { 1 } \\widehat { \\nabla } f _ { t + 1 } ( \\widehat { b } _ { t } ) } \\\\ & { } & { \\widehat { c } _ { t } = \\beta _ { 1 } \\widehat { b } _ { t } + ( 1 - \\beta _ { 1 } ) \\widehat { d } _ { t } } \\\\ & { } & { \\widehat { d } _ { t + 1 } = \\widehat { c } _ { t } - \\gamma _ { 1 } \\widehat { \\nabla } f _ { t + 1 } ( \\widehat { b } _ { t } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 406, + 840, + 589, + 928 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Next, we specify the step sizes $\\beta _ { 1 } = c _ { 3 } ^ { 2 } / \\sqrt { \\kappa \\widetilde { \\kappa } }$ , $\\alpha _ { 1 } = c _ { 3 } / ( c _ { 3 } + \\beta )$ , $\\gamma _ { 1 } = \\beta / ( c _ { 3 } \\lambda _ { \\operatorname* { m i n } } )$ and $\\delta _ { 1 } = 1 / R ^ { 2 }$ , where $\\kappa = R ^ { 2 } / \\lambda _ { \\operatorname* { m i n } }$ e. Note that the step sizes in the paper of Jain et al. (2017) with $c _ { 1 }$ in their paper set to 1 yields the step sizes above. Now, substituting equation 8 in equation 9 and substituting the value of $\\gamma _ { 1 }$ , we have: ", + "bbox": [ + 174, + 102, + 825, + 160 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/436d2b99f7356ae7befe7bc769245732cd69a9850d47a86cfd7e5b20851b9a81.jpg", + "text": "$$\n\\begin{array} { r l } & { \\widehat { d } _ { t + 1 } = \\beta _ { 1 } \\left( \\widehat { b } _ { t } - \\frac { 1 } { c _ { 3 } \\lambda _ { \\operatorname* { m i n } } } \\hat { \\nabla } f _ { t + 1 } ( \\widehat { b } _ { t } ) \\right) + ( 1 - \\beta _ { 1 } ) \\widehat { d } _ { t } } \\\\ & { \\qquad = \\beta _ { 1 } \\left( \\widehat { b } _ { t } - \\frac { \\delta \\kappa } { c _ { 3 } } \\hat { \\nabla } f _ { t + 1 } ( \\widehat { b } _ { t } ) \\right) + ( 1 - \\beta _ { 1 } ) \\widehat { d } _ { t } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 325, + 164, + 669, + 237 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We see that $\\widehat { d } _ { t + 1 }$ is precisely the update of the running average $\\bar { w } _ { t + 1 }$ in the ASGD method employed in this paper. ", + "bbox": [ + 171, + 242, + 825, + 272 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We now update $\\widehat { b } _ { t }$ to become $\\widehat { b } _ { t + 1 }$ and this can be done by writing out equation 6 at $t + 1$ , i.e: ", + "bbox": [ + 171, + 279, + 787, + 296 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/265d66dacb4c6ae7532fa2778a10f860acbc96d81447052b4f3e15acc9ee3207.jpg", + "text": "$$\n\\begin{array} { r l } & { \\widehat { b } _ { t + 1 } = \\alpha _ { 1 } \\widehat { a } _ { t + 1 } + ( 1 - \\alpha _ { 1 } ) \\widehat { d } _ { t + 1 } } \\\\ & { \\qquad = \\alpha _ { 1 } \\left( \\widehat { b } _ { t } - \\delta _ { 1 } \\widehat { \\nabla } f _ { t + 1 } ( \\widehat { b } _ { t } ) \\right) + ( 1 - \\alpha _ { 1 } ) \\widehat { d } _ { t + 1 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 334, + 300, + 661, + 351 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "By substituting the value of $\\alpha _ { 1 }$ we note that this is indeed the update of the iterate as a convex combination of the current running average and a short gradient step as written in this paper. In this paper, we set $c _ { 3 }$ to be equal to 0.7, and any constant less than 1 works. In terms of variables, we note that $\\alpha$ in this paper’s algorithm description maps to $1 - \\beta _ { 1 }$ . ", + "bbox": [ + 173, + 354, + 825, + 410 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C MORE DETAILS ON EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 430, + 490, + 446 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In this section, we will present more details on our experimental setup. ", + "bbox": [ + 174, + 460, + 637, + 477 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.1 LINEAR REGRESSION ", + "text_level": 1, + "bbox": [ + 174, + 492, + 367, + 507 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In this section, we will present some more results on our experiments on the linear regression problem. Just as in Appendix A, it is indeed possible to compute the expected error of all the algorithms among SGD, HB, NAG and ASGD, by tracking certain covariance matrices which evolve as linear systems. For SGD, for instance, denoting $\\Phi _ { t } ^ { S G D } \\stackrel { \\mathrm { d e f } } { = } \\mathbb { E } \\left[ \\left( \\mathbf { w } _ { t } ^ { S G D } - w ^ { * } \\right) \\otimes \\left( \\mathbf { w } _ { t } ^ { S G D } - w ^ { * } \\right) \\right]$ , we see that ΦSGDt+1 $\\Phi _ { t + 1 } ^ { S G D } \\ : = \\ : B \\circ \\Phi _ { t } ^ { S G D }$ , where $\\boldsymbol { B }$ is a linear operator acting on $d \\times d$ matrices such that ${ \\mathcal { B } } \\circ M { \\stackrel { \\mathrm { d e f } } { = } } M - \\delta H M - \\delta M H + \\delta ^ { 2 } \\mathbb { E } \\left[ \\left. x , M x \\right. x x ^ { \\top } \\right]$ . Similarly, HB, NAG and ASGD also have corresponding operators (see Appendix A for more details on the operator corresponding to HB). The largest magnitude of the eigenvalues of these matrices indicate the rate of decay achieved by the particular algorithm – smaller it is compared to 1, faster the decay. ", + "bbox": [ + 173, + 518, + 825, + 659 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We now detail the range of parameters explored for these results: the condition number $\\kappa$ was varied from $\\{ 2 ^ { 4 } , 2 ^ { 5 } , . . , \\bar { 2 } ^ { 2 8 } \\}$ for all the optimization methods and for both the discrete and gaussian problem. For each of these experiments, we draw 1000 samples and compute the empirical estimate of the fourth moment tensor. For NAG and HB, we did a very fine grid search by sampling 50 values in the interval $( 0 , 1 ]$ for both the learning rate and the momentum parameter and chose the parameter setting that yielded the smallest $\\lambda _ { \\mathrm { m a x } } ( B )$ that is less than 1 (so that it falls in the range of convergence of the algorithm). As for SGD and ASGD, we employed a learning rate of $1 / 3$ for the Gaussian case and a step size of 0.9 for the discrete case. The statistical advantage parameter of ASGD was chosen to be $\\sqrt { 3 \\kappa / 2 }$ for the Gaussian case and $\\sqrt { 2 \\kappa / 3 }$ for the Discrete case, and the a long step parameters of $3 \\kappa$ and $2 \\kappa$ were chosen for the Gaussian and Discrete case respectively. The reason it appears as if we choose a parameter above the theoretically maximal allowed value of the advantage parameter is because the definition of $\\kappa$ is different in this case. The $\\kappa$ we speak about for this experiment is $\\lambda _ { \\operatorname* { m a x } } / \\lambda _ { \\operatorname* { m i n } }$ unlike the condition number for the stochastic optimization problem. In a manner similar to actually running the algorithms (the results of whose are presented in the main paper), we also note that we can compute the rate as in equation 1 and join all these rates using a curve and estimate its slope (in the log scale). This result is indicated in table 3. ", + "bbox": [ + 173, + 665, + 825, + 888 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Figure 7 presents these results, where for each method, we did grid search over all parameters and chose parameters that give smallest $\\lambda _ { \\operatorname* { m a x } }$ . We see the same pattern as in Figure 1 from actual runs – SGD,HB and NAG all have linear dependence on condition number $\\kappa$ , while ASGD has a dependence of $\\sqrt { \\kappa }$ . ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 133 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/2ce4c7a58a43a49aac0ae0911e2b09d770a20e8605b3023da8d2c7e23eef036f.jpg", + "image_caption": [ + "Figure 7: Expected rate of error decay (equation 1) vs condition number for various methods for the linear regression problem. Left is for discrete distribution and right is for Gaussian distribution. ", + "Table 3: Slopes (i.e. $\\gamma$ ) obtained by fitting a line to the curves in Figure 7. A value of $\\gamma$ indicates that the error decays at a rate of exp $\\left( { \\frac { - t } { \\kappa ^ { \\gamma } } } \\right)$ . A smaller value of $\\gamma$ indicates a faster rate of error decay. " + ], + "image_footnote": [], + "bbox": [ + 187, + 160, + 790, + 337 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/17bf7e686bbe60ac6c31ae4df2d7082fa189ac57751ae25411dfa620c73a0619.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
AlgorithmSlope - discreteSlope- Gaussian
SGD0.99900.9995
HB1.03400.9989
NAG1.06271.0416
ASGD0.49230.4906
", + "bbox": [ + 321, + 405, + 676, + 489 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "C.2 AUTOENCODERS FOR MNIST ", + "text_level": 1, + "bbox": [ + 176, + 558, + 423, + 573 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We begin by noting that the learning rates tend to vary as we vary batch sizes, which is something that is known in theory (Jain et al., 2016). Furthermore, we extend the grid especially whenever our best parameters of a baseline method tends to land at the edge of a grid. The parameter ranges explored by our grid search are: ", + "bbox": [ + 174, + 585, + 825, + 641 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Batch Size 1: (parameters chosen by running for 20 epochs) ", + "bbox": [ + 173, + 647, + 571, + 662 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• SGD: learning rate: $\\{ 0 . 0 1 , 0 . 0 1 { \\sqrt { 1 0 } } , 0 . 1 , 0 . 1 { \\sqrt { 1 0 } } , 1 , { \\sqrt { 1 0 } } , 5 , 1 0 , 2 0 , 1 0 { \\sqrt { 1 0 } } , 4 0 , 6 0 , 8 0 , 1 0 0 .$ \n• NAG/HB: learning rate: $\\{ 0 . 0 1 \\sqrt { 1 0 } , 0 . 1 , 0 . 1 \\sqrt { 1 0 } , 1 , \\sqrt { 1 0 } , 1 0 \\}$ , momentum $\\left\\{ 0 , 0 . 5 , 0 . 7 5 , 0 . 9 , 0 . 9 5 , 0 . 9 7 \\right\\}$ . \n• ASGD: learning rate: $\\{ 2 . 5 , 5 \\}$ , long step $\\{ 1 0 0 . 0 , 1 0 0 0 . 0 \\}$ , advantage parameter $\\{ 2 . 5 , 5 . 0 , 1 0 . 0 , 2 0 . 0 \\}$ . ", + "bbox": [ + 215, + 674, + 828, + 765 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Batch Size 8: (parameters chosen by running for 50 epochs) ", + "bbox": [ + 173, + 775, + 571, + 791 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• SGD: learning rate: $\\{ 0 . 0 0 1 , 0 . 0 0 1 \\sqrt { 1 0 . 0 } , 0 . 0 1 , 0 . 0 1 \\sqrt { 1 0 } , 0 . 1 , 0 . 1 \\sqrt { 1 0 } , 1 , \\sqrt { 1 0 } , 5 , 1 0 \\}$ , $1 0 \\sqrt { 1 0 } , 4 0 , 6 0 , 8 0 , 1 0 0 , 1 2 0 , 1 4 0 \\}$ . \n• NAG/HB: learning rate: $\\{ 5 . 0 , 1 0 . 0 , 2 0 . 0 , 1 0 \\sqrt { 1 0 } , 4 0 , 6 0 \\} .$ , momentum $\\{ 0 , 0 . 2 5 , 0 . 5 , 0 . 7 5 , 0 . 9 , 0 . 9 5 \\}$ . \n• ASGD: learning rate $\\{ 4 0 , 6 0 \\}$ . For a long step of 100, advantage parameters of $\\{ 1 . 5 , 2 , 2 . 5 , 5 , 1 \\bar { 0 } , 2 0 \\}$ . For a long step of 1000, we swept over advantage parameters of $\\{ 2 . 5 , 5 , 1 0 \\}$ . ", + "bbox": [ + 215, + 801, + 826, + 921 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "C.3 DEEP RESIDUAL NETWORKS FOR CIFAR-10 ", + "text_level": 1, + "bbox": [ + 178, + 103, + 526, + 117 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "In this section, we will provide more details on our experiments on cifar-10, as well as present some additional results. We used a weight decay of 0.0005 in all our experiments. The grid search parameters we used for various algorithms are as follows. Note that the ranges in which parameters such as learning rate need to be searched differ based on batch size (Jain et al., 2016). Furthermore, we tend to extrapolate the grid search whenever a parameter (except for the learning rate decay factor) at the edge of the grid has been chosen; this is done so that we always tend to lie in the interior of the grid that we have searched on. Note that for the purposes of the grid search, we choose a hold out set from the training data and add it in to the training data after the parameters are chosen, for the final run. ", + "bbox": [ + 173, + 130, + 825, + 256 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Batch Size 8: Note: (i) parameters chosen by running for 40 epochs and picking the grid search parameter that yields the smallest validation $0 / 1$ error. (ii) The validation set decay scheme that we use is that if the validation error does not decay by at least $1 \\%$ every three passes over the data, we cut the learning rate by a constant factor (which is grid searched as described below). The minimal learning rate to use is fixed to be $6 . 2 5 \\times 1 0 ^ { - 5 }$ , so that we do not decay far too many times and curtail progress prematurely. ", + "bbox": [ + 174, + 262, + 825, + 345 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "• SGD: learning rate: $\\left\\{ 0 . 0 0 3 3 , 0 . 0 1 , 0 . 0 3 3 , 0 . 1 , 0 . 3 3 \\right\\}$ , learning rate decay factor $\\{ 5 , 1 0 \\}$ . \n• NAG/HB: learning rate: $\\{ 0 . 0 0 1 , 0 . 0 0 3 3 , 0 . 0 1 , 0 . 0 3 3 \\}$ , momentum $\\{ 0 . 8 , 0 . 9 , 0 . 9 5 , 0 . 9 7 \\}$ , learning rate decay factor $\\{ 5 , 1 0 \\}$ . \n• ASGD: learning rate $\\{ 0 . 0 1 , 0 . 0 3 3 0 , 0 . 1 \\}$ , long step $\\{ 1 0 0 0 , 1 0 0 0 0 , 5 0 0 0 0 \\}$ , advantage parameter $\\{ 5 , 1 0 \\}$ , learning rate decay factor $\\{ 5 , 1 0 \\}$ . ", + "bbox": [ + 215, + 358, + 825, + 443 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Batch Size 128: Note: (i) parameters chosen by running for 120 epochs and picking the grid search parameter that yields the smallest validation $0 \\dot { / } 1$ error. (ii) The validation set decay scheme that we use is that if the validation error does not decay by at least $0 . 2 \\%$ every four passes over the data, we cut the learning rate by a constant factor (which is grid searched as described below). The minimal learning rate to use is fixed to be $1 \\times 1 0 ^ { - 3 }$ , so that we do not decay far too many times and curtail progress prematurely. ", + "bbox": [ + 174, + 454, + 825, + 537 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "• SGD: learning rate: $\\{ 0 . 0 1 , 0 . 0 3 , 0 . 0 9 , 0 . 2 7 , 0 . 8 1 \\}$ , learning rate decay factor $\\{ 2 , { \\sqrt { 1 0 } } , 5 \\}$ . \n• NAG/HB: learning rate: $\\{ 0 . 0 1 , 0 . 0 3 , 0 . 0 9 , 0 . 2 7 \\}$ , momentum $\\left. 0 . 5 , 0 . 8 , 0 . 9 , 0 . 9 5 , 0 . 9 7 \\right.$ , learning rate decay factor $\\{ 2 , { \\sqrt { 1 0 } } , 5 \\}$ . \n• ASGD: learning rate $\\{ 0 . 0 1 , 0 . 0 3 , 0 . 0 9 , 0 . 2 7 \\}$ , long step √ $\\{ 1 0 0 , 1 0 0 0 , 1 0 0 0 0 \\}$ , advantage parameter $\\{ 5 , 1 0 , 2 0 \\}$ , learning rate decay factor $\\{ 2 , { \\sqrt { 1 0 } } , 5 \\}$ . ", + "bbox": [ + 215, + 551, + 823, + 640 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "As a final remark, for any comparison across algorithms, such as, (i) ASGD vs. NAG, (ii) ASGD vs HB, we fix the starting learning rate, learning rate decay factor and decay schedule chosen by the best grid search run of NAG/HB respectively and perform a grid search over the long step and advantage parameter of ASGD. In a similar manner, when we compare (iii) SGD vs NAG or, (iv) SGD vs. HB, we choose the learning rate, learning rate decay factor and decay schedule of SGD and simply sweep over the momentum parameter of NAG or HB and choose the momentum that offers the best validation error. ", + "bbox": [ + 174, + 651, + 825, + 748 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We now present plots of training function value for different algorithms and batch sizes. ", + "bbox": [ + 174, + 756, + 750, + 770 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Effect of minibatch sizes: Figure 8 plots training function value for batch sizes of 128 and 8 for SGD, HB and NAG. We notice that in the initial stages of training, NAG obtains substantial improvements compared to SGD and HB for batch size 128 but not for batch size 8. Towards the end of training however, NAG starts decreasing the training function value rapidly for both the batch sizes. The reason for this phenomenon is not clear. Note however, that at this point, the test error has already stabilized and the algorithms are just overfitting to the data. ", + "bbox": [ + 174, + 776, + 825, + 861 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Comparison of ASGD with momentum methods: We now present the training error plots for ASGD compared to HB and NAG in Figures 9 and 10 respectively. As mentioned earlier, in order to see a clear trend, we constrain the learning rate and decay schedule of ASGD to be the same as that of HB and NAG respectively, which themselves were learned using grid search. We see similar trends as in the validation error plots from Figures 5 and 6. Please see the figures and their captions for more details. ", + "bbox": [ + 174, + 868, + 823, + 924 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/d4f9589382a35d59086f9e033e6141ac77583966f826f72c83135bd3dea92e69.jpg", + "image_caption": [ + "Figure 8: Training loss for batch sizes 128 and 8 respectively for SGD, HB and NAG. " + ], + "image_footnote": [], + "bbox": [ + 184, + 116, + 790, + 290 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 345, + 825, + 375 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/2974afd1ee80f1ce8e17340db6d644072d3bfc4ff9a48fb040725052e5588668.jpg", + "image_caption": [ + "Figure 9: Training function value for ASGD compared to HB for batch sizes 128 and 8 respectively. " + ], + "image_footnote": [], + "bbox": [ + 184, + 402, + 790, + 577 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/c730a68098f40ebfb306a2227df07eefd31a0cc84e8d3ba245c31b1e5fcb3db8.jpg", + "image_caption": [ + "Figure 10: Training function value for ASGD compared to NAG for batch size 128 and 8 respectively. " + ], + "image_footnote": [], + "bbox": [ + 184, + 652, + 790, + 825 + ], + "page_idx": 22 + } +] \ No newline at end of file diff --git a/parse/train/rJTutzbA-/rJTutzbA-_middle.json b/parse/train/rJTutzbA-/rJTutzbA-_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..da38317224c373eca177c8211c49353aec40eb4b --- /dev/null +++ b/parse/train/rJTutzbA-/rJTutzbA-_middle.json @@ -0,0 +1,65842 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 80, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 79, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 505, + 97 + ], + "score": 1.0, + "content": "ON THE INSUFFICIENCY OF EXISTING MOMENTUM", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 100, + 426, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 426, + 117 + ], + "score": 1.0, + "content": "SCHEMES FOR STOCHASTIC OPTIMIZATION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 113, + 134, + 457, + 147 + ], + "lines": [ + { + "bbox": [ + 111, + 133, + 460, + 148 + ], + "spans": [ + { + "bbox": [ + 111, + 133, + 460, + 148 + ], + "score": 1.0, + "content": "Rahul Kidambi∗1, Praneeth Netrapalli2, Prateek Jain2 and Sham M. 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Rigorously", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 289, + 470, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 470, + 302 + ], + "score": 1.0, + "content": "speaking, “fast gradient” methods have provable improvements over gradient de-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 301, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 301, + 469, + 312 + ], + "score": 1.0, + "content": "scent only for the deterministic case, where the gradients are exact. 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This work provides", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 344, + 470, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 470, + 357 + ], + "score": 1.0, + "content": "a counterpoint to this belief by proving that there exist simple problem instances", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 354, + 469, + 368 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 469, + 368 + ], + "score": 1.0, + "content": "where these methods cannot outperform SGD despite the best setting of its pa-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 366, + 469, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 469, + 378 + ], + "score": 1.0, + "content": "rameters. 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The advantages of SGD for large scale optimization and the related issues of tradeoffs", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 689, + 486, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 486, + 700 + ], + "score": 1.0, + "content": "between computational and statistical efficiency was highlighted in Bottou & Bousquet (2007).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 644, + 506, + 700 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 94 + ], + "score": 1.0, + "content": "The above mentioned theoretical advantages of fast gradient methods (Polyak, 1964; Nesterov,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "1983) (albeit for smooth convex problems) coupled with cheap to compute stochastic gradient es-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "timates led to the influential work of Sutskever et al. (2013), which demonstrated the empirical", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "advantages possessed by SGD when augmented with the momentum machinery. This work has led", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "score": 1.0, + "content": "to widespread adoption of momentum methods for training deep neural nets; so much so that, in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 460, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 460, + 149 + ], + "score": 1.0, + "content": "context of neural network training, gradient descent often refers to momentum methods.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 504, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "But, there is a subtle difference between classical momentum methods and their implementation in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "practice – classical momentum methods work in the exact first order oracle model (Nesterov, 2004),", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "i.e., they employ exact gradients (computed on the full training dataset), while in practice (Sutskever", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "et al., 2013), they are implemented with stochastic gradients (estimated from a randomly sampled", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 351, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 351, + 210 + ], + "score": 1.0, + "content": "mini-batch of training data). This leads to a natural question:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 108, + 215, + 502, + 237 + ], + "lines": [ + { + "bbox": [ + 108, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 108, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "“Are momentum methods optimal even in the stochastic first order oracle (SFO) model, where we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 504, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 504, + 239 + ], + "score": 1.0, + "content": "access stochastic gradients computed on a small constant sized minibatches (or a batchsize of 1?)”", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "Even disregarding the question of optimality of momentum methods in the SFO model, it is not", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "even known if momentum methods (say, Polyak (1964); Nesterov (1983)) provide any provable", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "improvement over SGD in this model. While these are open questions, a recent effort of Jain et al.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "(2017) showed that improving upon SGD (in the stochastic first order oracle) is rather subtle as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 287, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 298 + ], + "score": 1.0, + "content": "there exists problem instances in SFO model where it is not possible to improve upon SGD, even", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 298, + 504, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 504, + 309 + ], + "score": 1.0, + "content": "information theoretically. Jain et al. (2017) studied a variant of Nesterov’s accelerated gradient", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 308, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 322 + ], + "score": 1.0, + "content": "updates (Nesterov, 2012b) for stochastic linear regression and show that their method improves", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "score": 1.0, + "content": "upon SGD wherever it is information theoretically admissible. Through out this paper, we refer to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "the algorithm of Jain et al. (2017) as Accelerated Stochastic Gradient Method (ASGD) while we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "refer to a stochastic version of the most widespread form of Nesterov’s method (Nesterov, 1983) as", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "NAG; HB denotes a stochastic version of the heavy ball method (Polyak, 1964). Critically, while", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "Jain et al. (2017) shows that ASGD improves on SGD in any information-theoretically admissible", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 470, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 470, + 387 + ], + "score": 1.0, + "content": "regime, it is still not known whether HB and NAG can achieve a similar performance gain.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "A key contribution of this work is to show that HB does not provide similar performance gains", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "over SGD even when it is informationally-theoretically admissible. That is, we provide a problem", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "instance where it is indeed possible to improve upon SGD (and ASGD achieves this improvement),", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 425, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 436 + ], + "score": 1.0, + "content": "but HB cannot achieve any improvement over SGD. We validate this claim empirically as well. In", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "fact, we provide empirical evidence to the claim that NAG also do not achieve any improvement", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 468, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 468, + 459 + ], + "score": 1.0, + "content": "over SGD for several problems where ASGD can still achieve better rates of convergence.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 562 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "This raises a question about why HB and NAG provide better performance than SGD in practice", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "(Sutskever et al., 2013), especially for training deep networks. Our conclusion (that is well supported", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 483, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 499 + ], + "score": 1.0, + "content": "by our theoretical result) is that HB and NAG’s improved performance is attributed to mini-batching", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "and hence, these methods will often struggle to improve over SGD with small constant batch sizes.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "score": 1.0, + "content": "This is in stark contrast to methods like ASGD, which is designed to improve over SGD across", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "both small or large mini-batch sizes. In fact, based on our experiments, we observe that on the task", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "of training deep residual networks (He et al., 2016a) on the cifar-10 dataset, we note that ASGD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 289, + 552 + ], + "score": 1.0, + "content": "offers noticeable improvements by achieving", + "type": "text" + }, + { + "bbox": [ + 289, + 540, + 321, + 551 + ], + "score": 0.9, + "content": "5 - 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "better test error over HB and NAG even with", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 551, + 429, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 429, + 564 + ], + "score": 1.0, + "content": "commonly used batch sizes like 128 during the initial stages of the optimization.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 108, + 570, + 204, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 205, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 205, + 583 + ], + "score": 1.0, + "content": "1.1 CONTRIBUTIONS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 591, + 290, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 590, + 291, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 291, + 605 + ], + "score": 1.0, + "content": "The contributions of this paper are as follows.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 129, + 612, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 130, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 130, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "1. In Section 3, we prove that HB is not optimal in the SFO model. In particular, there", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 142, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 142, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "exist linear regression problems for which the performance of HB (with any step size and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 140, + 633, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 140, + 633, + 505, + 649 + ], + "score": 1.0, + "content": "momentum) is either the same or worse than that of SGD while ASGD improves upon", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 141, + 645, + 198, + 658 + ], + "spans": [ + { + "bbox": [ + 141, + 645, + 198, + 658 + ], + "score": 1.0, + "content": "both of them.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 128, + 660, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 128, + 660, + 505, + 674 + ], + "score": 1.0, + "content": "2. Experiments on several linear regression problems suggest that the suboptimality of HB in", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 141, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 141, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "the SFO model is not restricted to special cases – it is rather widespread. Empirically, the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 141, + 682, + 325, + 696 + ], + "spans": [ + { + "bbox": [ + 141, + 682, + 325, + 696 + ], + "score": 1.0, + "content": "same holds true for NAG as well (Section 5).", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 128, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 128, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "3. The above observations suggest that the only reason for the superiority of momentum meth-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 142, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 142, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "ods in practice is mini-batching, which reduces the variance in stochastic gradients and", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 141, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 141, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "moves the SFO closer to the exact first order oracle. This conclusion is supported by em-", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 47.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 94 + ], + "score": 1.0, + "content": "The above mentioned theoretical advantages of fast gradient methods (Polyak, 1964; Nesterov,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "1983) (albeit for smooth convex problems) coupled with cheap to compute stochastic gradient es-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "timates led to the influential work of Sutskever et al. (2013), which demonstrated the empirical", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "advantages possessed by SGD when augmented with the momentum machinery. This work has led", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "score": 1.0, + "content": "to widespread adoption of momentum methods for training deep neural nets; so much so that, in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 460, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 460, + 149 + ], + "score": 1.0, + "content": "context of neural network training, gradient descent often refers to momentum methods.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 81, + 506, + 149 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 504, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "But, there is a subtle difference between classical momentum methods and their implementation in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "practice – classical momentum methods work in the exact first order oracle model (Nesterov, 2004),", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "i.e., they employ exact gradients (computed on the full training dataset), while in practice (Sutskever", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "et al., 2013), they are implemented with stochastic gradients (estimated from a randomly sampled", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 351, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 351, + 210 + ], + "score": 1.0, + "content": "mini-batch of training data). This leads to a natural question:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 154, + 505, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 215, + 502, + 237 + ], + "lines": [ + { + "bbox": [ + 108, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 108, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "“Are momentum methods optimal even in the stochastic first order oracle (SFO) model, where we", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 504, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 504, + 239 + ], + "score": 1.0, + "content": "access stochastic gradients computed on a small constant sized minibatches (or a batchsize of 1?)”", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 214, + 505, + 239 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "Even disregarding the question of optimality of momentum methods in the SFO model, it is not", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "even known if momentum methods (say, Polyak (1964); Nesterov (1983)) provide any provable", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "improvement over SGD in this model. While these are open questions, a recent effort of Jain et al.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "(2017) showed that improving upon SGD (in the stochastic first order oracle) is rather subtle as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 287, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 298 + ], + "score": 1.0, + "content": "there exists problem instances in SFO model where it is not possible to improve upon SGD, even", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 298, + 504, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 504, + 309 + ], + "score": 1.0, + "content": "information theoretically. Jain et al. (2017) studied a variant of Nesterov’s accelerated gradient", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 308, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 322 + ], + "score": 1.0, + "content": "updates (Nesterov, 2012b) for stochastic linear regression and show that their method improves", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "score": 1.0, + "content": "upon SGD wherever it is information theoretically admissible. Through out this paper, we refer to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "the algorithm of Jain et al. (2017) as Accelerated Stochastic Gradient Method (ASGD) while we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "refer to a stochastic version of the most widespread form of Nesterov’s method (Nesterov, 1983) as", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "NAG; HB denotes a stochastic version of the heavy ball method (Polyak, 1964). Critically, while", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 376 + ], + "score": 1.0, + "content": "Jain et al. (2017) shows that ASGD improves on SGD in any information-theoretically admissible", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 470, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 470, + 387 + ], + "score": 1.0, + "content": "regime, it is still not known whether HB and NAG can achieve a similar performance gain.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 243, + 506, + 387 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "A key contribution of this work is to show that HB does not provide similar performance gains", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "over SGD even when it is informationally-theoretically admissible. That is, we provide a problem", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "instance where it is indeed possible to improve upon SGD (and ASGD achieves this improvement),", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 425, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 436 + ], + "score": 1.0, + "content": "but HB cannot achieve any improvement over SGD. We validate this claim empirically as well. In", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "fact, we provide empirical evidence to the claim that NAG also do not achieve any improvement", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 468, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 468, + 459 + ], + "score": 1.0, + "content": "over SGD for several problems where ASGD can still achieve better rates of convergence.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 390, + 506, + 459 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 562 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "This raises a question about why HB and NAG provide better performance than SGD in practice", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "(Sutskever et al., 2013), especially for training deep networks. Our conclusion (that is well supported", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 483, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 499 + ], + "score": 1.0, + "content": "by our theoretical result) is that HB and NAG’s improved performance is attributed to mini-batching", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "and hence, these methods will often struggle to improve over SGD with small constant batch sizes.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "score": 1.0, + "content": "This is in stark contrast to methods like ASGD, which is designed to improve over SGD across", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "both small or large mini-batch sizes. In fact, based on our experiments, we observe that on the task", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "of training deep residual networks (He et al., 2016a) on the cifar-10 dataset, we note that ASGD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 289, + 552 + ], + "score": 1.0, + "content": "offers noticeable improvements by achieving", + "type": "text" + }, + { + "bbox": [ + 289, + 540, + 321, + 551 + ], + "score": 0.9, + "content": "5 - 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "better test error over HB and NAG even with", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 551, + 429, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 429, + 564 + ], + "score": 1.0, + "content": "commonly used batch sizes like 128 during the initial stages of the optimization.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 463, + 506, + 564 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 570, + 204, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 205, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 205, + 583 + ], + "score": 1.0, + "content": "1.1 CONTRIBUTIONS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 591, + 290, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 590, + 291, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 291, + 605 + ], + "score": 1.0, + "content": "The contributions of this paper are as follows.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42, + "bbox_fs": [ + 106, + 590, + 291, + 605 + ] + }, + { + "type": "list", + "bbox": [ + 129, + 612, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 130, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 130, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "1. In Section 3, we prove that HB is not optimal in the SFO model. In particular, there", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 142, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "exist linear regression problems for which the performance of HB (with any step size and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 140, + 633, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 140, + 633, + 505, + 649 + ], + "score": 1.0, + "content": "momentum) is either the same or worse than that of SGD while ASGD improves upon", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 141, + 645, + 198, + 658 + ], + "spans": [ + { + "bbox": [ + 141, + 645, + 198, + 658 + ], + "score": 1.0, + "content": "both of them.", + "type": "text" + } + ], + "index": 46, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 660, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 128, + 660, + 505, + 674 + ], + "score": 1.0, + "content": "2. Experiments on several linear regression problems suggest that the suboptimality of HB in", + "type": "text" + } + ], + "index": 47, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 141, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "the SFO model is not restricted to special cases – it is rather widespread. Empirically, the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 141, + 682, + 325, + 696 + ], + "spans": [ + { + "bbox": [ + 141, + 682, + 325, + 696 + ], + "score": 1.0, + "content": "same holds true for NAG as well (Section 5).", + "type": "text" + } + ], + "index": 49, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 128, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "3. The above observations suggest that the only reason for the superiority of momentum meth-", + "type": "text" + } + ], + "index": 50, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 142, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "ods in practice is mini-batching, which reduces the variance in stochastic gradients and", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 141, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 141, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "moves the SFO closer to the exact first order oracle. 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Next iterate", + "type": "text" + }, + { + "bbox": [ + 326, + 464, + 347, + 474 + ], + "score": 0.89, + "content": "w _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "is obtained by a linear combination of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "the SGD update and the momentum term. Algorithm 2 provides pseudo-code of a stochastic version", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 484, + 483, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 483, + 496 + ], + "score": 1.0, + "content": "of the most commonly used form of Nesterov’s accelerated gradient descent (Nesterov, 1983).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 106, + 505, + 351, + 519 + ], + "lines": [ + { + "bbox": [ + 104, + 504, + 351, + 521 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 351, + 521 + ], + "score": 1.0, + "content": "3 SUBOPTIMALITY OF HEAVY BALL METHOD", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "In this section, we show that there exists linear regression problems where the performance", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "of HB (Algorithm 1) is no better than that of SGD, while ASGD significantly improves upon SGD’s", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 552, + 332, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 332, + 566 + ], + "score": 1.0, + "content": "performance. Let us now describe the problem instance.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 411, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 411, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 122, + 583 + ], + "score": 1.0, + "content": "Fix", + "type": "text" + }, + { + "bbox": [ + 122, + 569, + 159, + 580 + ], + "score": 0.91, + "content": "w ^ { \\ast } \\in \\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 569, + 190, + 583 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 191, + 570, + 234, + 582 + ], + "score": 0.93, + "content": "( a , b ) \\sim \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 569, + 411, + 583 + ], + "score": 1.0, + "content": "be a sample from the distribution such that:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 588, + 419, + 616 + ], + "lines": [ + { + "bbox": [ + 191, + 588, + 419, + 616 + ], + "spans": [ + { + "bbox": [ + 191, + 588, + 419, + 616 + ], + "score": 0.9, + "content": "\\begin{array} { r } { a = \\left\\{ \\begin{array} { l l } { \\sigma _ { 1 } \\cdot z \\cdot e _ { 1 } \\mathrm { w . p . ~ } 0 . 5 } \\\\ { \\sigma _ { 2 } \\cdot z \\cdot e _ { 2 } \\mathrm { w . p . ~ } 0 . 5 , } \\end{array} \\right. \\qquad \\mathrm { a n d } \\qquad b = \\left. w ^ { * } , a \\right. , } \\end{array}", + "type": "interline_equation", + "image_path": "af1fe3c9ae11009fbc2189b008ba43663a44233ef1ca444cd9aabf2ef145bf53.jpg" + } + ] + } + ], + "index": 38.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 588, + 419, + 602.0 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 191, + 602.0, + 419, + 616.0 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 505, + 656 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 134, + 634 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 621, + 185, + 633 + ], + "score": 0.9, + "content": "e _ { 1 } , e _ { 2 } \\in \\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 621, + 303, + 634 + ], + "score": 1.0, + "content": "are canonical basis vectors,", + "type": "text" + }, + { + "bbox": [ + 303, + 623, + 362, + 633 + ], + "score": 0.9, + "content": "\\sigma _ { 1 } > \\sigma _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 621, + 386, + 634 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 386, + 624, + 393, + 632 + ], + "score": 0.78, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "be a random variable such", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 631, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 104, + 631, + 133, + 647 + ], + "score": 1.0, + "content": "that E", + "type": "text" + }, + { + "bbox": [ + 133, + 633, + 172, + 646 + ], + "score": 0.86, + "content": "\\left[ z ^ { 2 } \\right] = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 631, + 191, + 647 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 632, + 262, + 646 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left[ z ^ { 4 } \\right] = 2 c \\geq 4", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 631, + 342, + 647 + ], + "score": 1.0, + "content": ". Hence, we have:", + "type": "text" + }, + { + "bbox": [ + 343, + 632, + 487, + 646 + ], + "score": 0.89, + "content": "\\Xi \\left[ ( a ^ { ( i ) } ) ^ { 2 } \\right] = \\sigma _ { i } ^ { 2 } , \\mathbb { E } \\left[ ( a ^ { ( i ) } ) ^ { 4 } \\right] = c \\sigma _ { i } ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 631, + 506, + 647 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 643, + 264, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 139, + 655 + ], + "score": 0.84, + "content": "i = 1 , \\dot { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 643, + 264, + 656 + ], + "score": 1.0, + "content": ". Now, our goal is to minimize:", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 660, + 452, + 688 + ], + "lines": [ + { + "bbox": [ + 158, + 660, + 452, + 688 + ], + "spans": [ + { + "bbox": [ + 158, + 660, + 452, + 688 + ], + "score": 0.89, + "content": "f ( \\boldsymbol { w } ) \\stackrel { \\mathrm { d e f } } { = } 0 . 5 \\cdot \\mathbb { E } \\left[ \\left( \\left. \\boldsymbol { w } ^ { * } , \\boldsymbol { a } \\right. - \\boldsymbol { b } \\right) ^ { 2 } \\right] \\mathrm { , ~ H e s s i a n \\ } \\mathbf { H } \\stackrel { \\mathrm { d e f } } { = } \\mathbb { E } \\left[ \\boldsymbol { a } \\boldsymbol { a } ^ { \\top } \\right] = \\left[ \\sigma _ { 1 } ^ { 2 } \\quad 0 _ { 2 } ^ { 2 } \\right] .", + "type": "interline_equation", + "image_path": "26873c0ead4debcb0af7abe9d44636183f9100018d946ddceb34642f7916b3d0.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 158, + 660, + 452, + 669.3333333333334 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 158, + 669.3333333333334, + 452, + 678.6666666666667 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 158, + 678.6666666666667, + 452, + 688.0000000000001 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 692, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 692, + 504, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 123, + 704 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 695, + 131, + 703 + ], + "score": 0.77, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 692, + 150, + 704 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 151, + 693, + 158, + 703 + ], + "score": 0.78, + "content": "\\tilde { \\kappa }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 692, + 504, + 704 + ], + "score": 1.0, + "content": "denote the computational and statistical condition numbers – see Jain et al. (2017)", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 701, + 504, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 306, + 723 + ], + "score": 1.0, + "content": "for definitions. For the problem above, we have", + "type": "text" + }, + { + "bbox": [ + 306, + 703, + 343, + 722 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\kappa = \\frac { c \\sigma _ { 1 } ^ { 2 } } { \\sigma _ { 2 } ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 701, + 363, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 363, + 707, + 391, + 717 + ], + "score": 0.86, + "content": "\\tilde { \\kappa } = c", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 701, + 504, + 723 + ], + "score": 1.0, + "content": ". Then we obtain following", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 468, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 468, + 733 + ], + "score": 1.0, + "content": "convergence rates for SGD and ASGD when applied to the above given problem instance:", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] 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We present an intuitive and easier to tune version of ASGD (see Section 4) and show", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 141, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "that ASGD can provide significantly faster convergence to a reasonable accuracy than", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 269, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 141, + 269, + 506, + 283 + ], + "score": 1.0, + "content": "SGD, HB, NAG, while still providing favorable or comparable asymptotic accuracy as", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 281, + 393, + 293 + ], + "spans": [ + { + "bbox": [ + 142, + 281, + 393, + 293 + ], + "score": 1.0, + "content": "these methods, particularly on several deep learning problems.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 128, + 248, + 506, + 293 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 301, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 301, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 315 + ], + "score": 1.0, + "content": "Hence, the take-home message of this paper is: HB and NAG are not optimal in the SFO model. The", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "only reason for the superiority of momentum methods in practice is mini-batching. ASGD provides", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 323, + 396, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 396, + 335 + ], + "score": 1.0, + "content": "a distinct advantage in training deep networks over SGD, HB and NAG.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 301, + 505, + 335 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 344, + 180, + 357 + ], + "lines": [ + { + "bbox": [ + 104, + 343, + 182, + 361 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 182, + 361 + ], + "score": 1.0, + "content": "2 NOTATION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 504, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 466, + 383 + ], + "score": 1.0, + "content": "We denote matrices by bold-face capital letters and vectors by lower-case letters.", + "type": "text" + }, + { + "bbox": [ + 466, + 370, + 504, + 382 + ], + "score": 0.89, + "content": "f ( w ) =", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 380, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 107, + 381, + 164, + 394 + ], + "score": 0.92, + "content": "1 / n \\sum _ { i } f _ { i } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 380, + 400, + 394 + ], + "score": 1.0, + "content": "denotes the function to optimize w.r.t. model parameters", + "type": "text" + }, + { + "bbox": [ + 401, + 383, + 409, + 391 + ], + "score": 0.62, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 380, + 415, + 394 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 415, + 381, + 446, + 393 + ], + "score": 0.9, + "content": "\\nabla f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 380, + 506, + 394 + ], + "score": 1.0, + "content": "denotes exact", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 504, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 153, + 408 + ], + "score": 1.0, + "content": "gradient of", + "type": "text" + }, + { + "bbox": [ + 153, + 395, + 161, + 406 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 393, + 172, + 408 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 172, + 396, + 181, + 405 + ], + "score": 0.73, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 393, + 207, + 408 + ], + "score": 1.0, + "content": "while", + "type": "text" + }, + { + "bbox": [ + 207, + 392, + 240, + 407 + ], + "score": 0.93, + "content": "\\widehat { \\nabla } f _ { t } ( \\boldsymbol { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 393, + 371, + 408 + ], + "score": 1.0, + "content": "denotes a stochastic gradient of", + "type": "text" + }, + { + "bbox": [ + 372, + 395, + 379, + 406 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 393, + 418, + 408 + ], + "score": 1.0, + "content": ". That is,", + "type": "text" + }, + { + "bbox": [ + 418, + 393, + 504, + 406 + ], + "score": 0.9, + "content": "\\widehat { \\nabla } f _ { t } ( w _ { t } ) = \\nabla f _ { i _ { t } } ( w )", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 404, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 133, + 419 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 406, + 141, + 417 + ], + "score": 0.86, + "content": "i _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 404, + 294, + 419 + ], + "score": 1.0, + "content": "is sampled uniformly at random from", + "type": "text" + }, + { + "bbox": [ + 295, + 406, + 334, + 418 + ], + "score": 0.94, + "content": "[ 1 , \\ldots , n ]", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 404, + 426, + 419 + ], + "score": 1.0, + "content": ". For linear regression,", + "type": "text" + }, + { + "bbox": [ + 426, + 406, + 505, + 418 + ], + "score": 0.88, + "content": "f _ { i } ( w ) = 0 . 5 \\cdot ( b _ { i } -", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 417, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 107, + 418, + 144, + 431 + ], + "score": 0.92, + "content": "\\langle w , a _ { i } \\rangle ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 417, + 171, + 433 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 172, + 419, + 200, + 430 + ], + "score": 0.91, + "content": "b _ { i } \\in \\Re", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 417, + 264, + 433 + ], + "score": 1.0, + "content": "is the target and", + "type": "text" + }, + { + "bbox": [ + 264, + 418, + 298, + 430 + ], + "score": 0.92, + "content": "a _ { i } \\in \\Re ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 417, + 379, + 433 + ], + "score": 1.0, + "content": "is the covariate, and", + "type": "text" + }, + { + "bbox": [ + 379, + 417, + 501, + 431 + ], + "score": 0.9, + "content": "\\widehat { \\nabla } f _ { t } ( w _ { t } ) = - \\big ( b _ { t } - \\langle w _ { t } , a _ { t } \\rangle \\big ) a _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 417, + 505, + 433 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 430, + 502, + 448 + ], + "spans": [ + { + "bbox": [ + 104, + 430, + 156, + 448 + ], + "score": 1.0, + "content": "In this case,", + "type": "text" + }, + { + "bbox": [ + 156, + 431, + 213, + 446 + ], + "score": 0.93, + "content": "\\mathbf { H } = \\mathbb { E } \\left[ a a ^ { \\top } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 430, + 307, + 448 + ], + "score": 1.0, + "content": "denotes the Hessian of", + "type": "text" + }, + { + "bbox": [ + 308, + 433, + 315, + 444 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 430, + 333, + 448 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 333, + 430, + 377, + 447 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\kappa = \\frac { \\lambda _ { 1 } ( \\mathbf { H } ) } { \\lambda _ { d } ( \\mathbf { H } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 430, + 502, + 448 + ], + "score": 1.0, + "content": "denotes it’s condition number.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 369, + 506, + 448 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 373, + 464 + ], + "score": 1.0, + "content": "Algorithm 1 provides a pseudo-code of HB method (Polyak, 1964).", + "type": "text" + }, + { + "bbox": [ + 373, + 453, + 414, + 463 + ], + "score": 0.89, + "content": "w _ { t } - w _ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "is the momentum term", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 123, + 475 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 464, + 132, + 472 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 462, + 325, + 475 + ], + "score": 1.0, + "content": "denotes the momentum parameter. Next iterate", + "type": "text" + }, + { + "bbox": [ + 326, + 464, + 347, + 474 + ], + "score": 0.89, + "content": "w _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "is obtained by a linear combination of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "the SGD update and the momentum term. Algorithm 2 provides pseudo-code of a stochastic version", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 484, + 483, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 483, + 496 + ], + "score": 1.0, + "content": "of the most commonly used form of Nesterov’s accelerated gradient descent (Nesterov, 1983).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 451, + 506, + 496 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 505, + 351, + 519 + ], + "lines": [ + { + "bbox": [ + 104, + 504, + 351, + 521 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 351, + 521 + ], + "score": 1.0, + "content": "3 SUBOPTIMALITY OF HEAVY BALL METHOD", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "In this section, we show that there exists linear regression problems where the performance", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "of HB (Algorithm 1) is no better than that of SGD, while ASGD significantly improves upon SGD’s", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 552, + 332, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 332, + 566 + ], + "score": 1.0, + "content": "performance. Let us now describe the problem instance.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 531, + 505, + 566 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 411, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 411, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 122, + 583 + ], + "score": 1.0, + "content": "Fix", + "type": "text" + }, + { + "bbox": [ + 122, + 569, + 159, + 580 + ], + "score": 0.91, + "content": "w ^ { \\ast } \\in \\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 569, + 190, + 583 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 191, + 570, + 234, + 582 + ], + "score": 0.93, + "content": "( a , b ) \\sim \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 569, + 411, + 583 + ], + "score": 1.0, + "content": "be a sample from the distribution such that:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 569, + 411, + 583 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 588, + 419, + 616 + ], + "lines": [ + { + "bbox": [ + 191, + 588, + 419, + 616 + ], + "spans": [ + { + "bbox": [ + 191, + 588, + 419, + 616 + ], + "score": 0.9, + "content": "\\begin{array} { r } { a = \\left\\{ \\begin{array} { l l } { \\sigma _ { 1 } \\cdot z \\cdot e _ { 1 } \\mathrm { w . p . ~ } 0 . 5 } \\\\ { \\sigma _ { 2 } \\cdot z \\cdot e _ { 2 } \\mathrm { w . p . ~ } 0 . 5 , } \\end{array} \\right. \\qquad \\mathrm { a n d } \\qquad b = \\left. w ^ { * } , a \\right. , } \\end{array}", + "type": "interline_equation", + "image_path": "af1fe3c9ae11009fbc2189b008ba43663a44233ef1ca444cd9aabf2ef145bf53.jpg" + } + ] + } + ], + "index": 38.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 588, + 419, + 602.0 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 191, + 602.0, + 419, + 616.0 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 505, + 656 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 134, + 634 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 621, + 185, + 633 + ], + "score": 0.9, + "content": "e _ { 1 } , e _ { 2 } \\in \\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 621, + 303, + 634 + ], + "score": 1.0, + "content": "are canonical basis vectors,", + "type": "text" + }, + { + "bbox": [ + 303, + 623, + 362, + 633 + ], + "score": 0.9, + "content": "\\sigma _ { 1 } > \\sigma _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 621, + 386, + 634 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 386, + 624, + 393, + 632 + ], + "score": 0.78, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "be a random variable such", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 631, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 104, + 631, + 133, + 647 + ], + "score": 1.0, + "content": "that E", + "type": "text" + }, + { + "bbox": [ + 133, + 633, + 172, + 646 + ], + "score": 0.86, + "content": "\\left[ z ^ { 2 } \\right] = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 631, + 191, + 647 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 632, + 262, + 646 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left[ z ^ { 4 } \\right] = 2 c \\geq 4", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 631, + 342, + 647 + ], + "score": 1.0, + "content": ". Hence, we have:", + "type": "text" + }, + { + "bbox": [ + 343, + 632, + 487, + 646 + ], + "score": 0.89, + "content": "\\Xi \\left[ ( a ^ { ( i ) } ) ^ { 2 } \\right] = \\sigma _ { i } ^ { 2 } , \\mathbb { E } \\left[ ( a ^ { ( i ) } ) ^ { 4 } \\right] = c \\sigma _ { i } ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 631, + 506, + 647 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 643, + 264, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 139, + 655 + ], + "score": 0.84, + "content": "i = 1 , \\dot { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 643, + 264, + 656 + ], + "score": 1.0, + "content": ". Now, our goal is to minimize:", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 621, + 506, + 656 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 660, + 452, + 688 + ], + "lines": [ + { + "bbox": [ + 158, + 660, + 452, + 688 + ], + "spans": [ + { + "bbox": [ + 158, + 660, + 452, + 688 + ], + "score": 0.89, + "content": "f ( \\boldsymbol { w } ) \\stackrel { \\mathrm { d e f } } { = } 0 . 5 \\cdot \\mathbb { E } \\left[ \\left( \\left. \\boldsymbol { w } ^ { * } , \\boldsymbol { a } \\right. - \\boldsymbol { b } \\right) ^ { 2 } \\right] \\mathrm { , ~ H e s s i a n \\ } \\mathbf { H } \\stackrel { \\mathrm { d e f } } { = } \\mathbb { E } \\left[ \\boldsymbol { a } \\boldsymbol { a } ^ { \\top } \\right] = \\left[ \\sigma _ { 1 } ^ { 2 } \\quad 0 _ { 2 } ^ { 2 } \\right] .", + "type": "interline_equation", + "image_path": "26873c0ead4debcb0af7abe9d44636183f9100018d946ddceb34642f7916b3d0.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 158, + 660, + 452, + 669.3333333333334 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 158, + 669.3333333333334, + 452, + 678.6666666666667 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 158, + 678.6666666666667, + 452, + 688.0000000000001 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 692, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 692, + 504, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 123, + 704 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 695, + 131, + 703 + ], + "score": 0.77, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 692, + 150, + 704 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 151, + 693, + 158, + 703 + ], + "score": 0.78, + "content": "\\tilde { \\kappa }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 692, + 504, + 704 + ], + "score": 1.0, + "content": "denote the computational and statistical condition numbers – see Jain et al. (2017)", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 701, + 504, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 306, + 723 + ], + "score": 1.0, + "content": "for definitions. For the problem above, we have", + "type": "text" + }, + { + "bbox": [ + 306, + 703, + 343, + 722 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\kappa = \\frac { c \\sigma _ { 1 } ^ { 2 } } { \\sigma _ { 2 } ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 701, + 363, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 363, + 707, + 391, + 717 + ], + "score": 0.86, + "content": "\\tilde { \\kappa } = c", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 701, + 504, + 723 + ], + "score": 1.0, + "content": ". Then we obtain following", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 468, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 468, + 733 + ], + "score": 1.0, + "content": "convergence rates for SGD and ASGD when applied to the above given problem instance:", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 692, + 504, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 96, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 106, + 96, + 502, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 167, + 109 + ], + "score": 1.0, + "content": "Input: Initial", + "type": "text" + }, + { + "bbox": [ + 168, + 99, + 180, + 108 + ], + "score": 0.79, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 96, + 225, + 109 + ], + "score": 1.0, + "content": ", short step", + "type": "text" + }, + { + "bbox": [ + 225, + 97, + 231, + 107 + ], + "score": 0.69, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 96, + 315, + 109 + ], + "score": 1.0, + "content": ", long step parameter", + "type": "text" + }, + { + "bbox": [ + 315, + 97, + 340, + 108 + ], + "score": 0.89, + "content": "\\kappa \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 96, + 468, + 109 + ], + "score": 1.0, + "content": ", statistical advantage parameter", + "type": "text" + }, + { + "bbox": [ + 468, + 96, + 502, + 109 + ], + "score": 0.88, + "content": "\\xi \\le \\sqrt { \\kappa }", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 110, + 107, + 502, + 120 + ], + "spans": [ + { + "bbox": [ + 110, + 108, + 123, + 119 + ], + "score": 1.0, + "content": "1:", + "type": "text" + }, + { + "bbox": [ + 123, + 109, + 163, + 119 + ], + "score": 0.84, + "content": "\\bar { w } _ { 0 } w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 108, + 167, + 119 + ], + "score": 1.0, + "content": ";", + "type": "text" + }, + { + "bbox": [ + 167, + 108, + 193, + 118 + ], + "score": 0.78, + "content": "t \\gets 0", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 107, + 480, + 120 + ], + "score": 1.0, + "content": "/*Set running average to", + "type": "text" + }, + { + "bbox": [ + 481, + 109, + 502, + 119 + ], + "score": 0.85, + "content": "\\boldsymbol { w _ { 0 } } ^ { * } /", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 109, + 117, + 503, + 133 + ], + "spans": [ + { + "bbox": [ + 109, + 117, + 190, + 133 + ], + "score": 1.0, + "content": "2: α ← 1 − 0.72·ξ", + "type": "text" + }, + { + "bbox": [ + 398, + 120, + 503, + 133 + ], + "score": 1.0, + "content": "/*Set momentum value*/", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 109, + 132, + 234, + 145 + ], + "spans": [ + { + "bbox": [ + 109, + 132, + 148, + 145 + ], + "score": 1.0, + "content": "3: while", + "type": "text" + }, + { + "bbox": [ + 149, + 135, + 160, + 144 + ], + "score": 0.81, + "content": "w _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 132, + 234, + 145 + ], + "score": 1.0, + "content": "not converged do", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 110, + 141, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 110, + 141, + 132, + 164 + ], + "score": 1.0, + "content": "4:", + "type": "text" + }, + { + "bbox": [ + 133, + 143, + 337, + 163 + ], + "score": 0.71, + "content": "\\begin{array} { r } { \\bar { w } _ { t + 1 } \\alpha \\cdot \\bar { w } _ { t } + \\overline { { ( 1 - \\alpha ) \\cdot \\Big ( w _ { t } - \\frac { \\kappa \\cdot \\delta } { 0 . 7 } \\cdot \\widehat \\nabla f _ { t } ( w _ { t } ) \\Big ) } } } \\end{array}", + "type": "inline_equation", + "image_path": "a380f88ffdf107cfd28c041d6496862a76731ddbe7dff39150ae8d94d38f43ac.jpg" + }, + { + "bbox": [ + 366, + 145, + 506, + 161 + ], + "score": 1.0, + "content": "/*Update the running average as a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 132, + 159, + 427, + 174 + ], + "spans": [ + { + "bbox": [ + 132, + 159, + 417, + 174 + ], + "score": 1.0, + "content": "weighted average of previous running average and a long step gradient", + "type": "text" + }, + { + "bbox": [ + 417, + 162, + 427, + 172 + ], + "score": 0.29, + "content": "^ { * } /", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 110, + 172, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 110, + 176, + 123, + 188 + ], + "score": 1.0, + "content": "5:", + "type": "text" + }, + { + "bbox": [ + 133, + 172, + 382, + 192 + ], + "score": 0.8, + "content": "\\begin{array} { r } { \\overline { { w _ { t + 1 } } } \\frac { 0 . 7 } { 0 . 7 + ( 1 - \\alpha ) } \\cdot ( w _ { t } - \\delta \\cdot \\widehat { \\nabla } f _ { t } ( w _ { t } ) ) + \\frac { 1 - \\alpha } { 0 . 7 + ( 1 - \\alpha ) } \\cdot \\bar { w } _ { t + 1 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 174, + 506, + 189 + ], + "score": 1.0, + "content": "/*Update the iterate as", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 131, + 189, + 415, + 203 + ], + "spans": [ + { + "bbox": [ + 131, + 189, + 415, + 203 + ], + "score": 1.0, + "content": "weighted average of current running average and short step gradient*/", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 110, + 201, + 175, + 213 + ], + "spans": [ + { + "bbox": [ + 110, + 201, + 124, + 213 + ], + "score": 1.0, + "content": "6:", + "type": "text" + }, + { + "bbox": [ + 133, + 201, + 175, + 212 + ], + "score": 0.49, + "content": "t \\gets t + 1", + "type": "inline_equation" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 213, + 162, + 224 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 161, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 148, + 227 + ], + "score": 1.0, + "content": "Output:", + "type": "text" + }, + { + "bbox": [ + 149, + 214, + 161, + 224 + ], + "score": 0.63, + "content": "w _ { t }", + "type": "inline_equation" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 401, + 212, + 504, + 223 + ], + "lines": [ + { + "bbox": [ + 401, + 212, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 401, + 212, + 505, + 223 + ], + "score": 1.0, + "content": "/*Return the last iterate*/", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 246, + 504, + 273 + ], + "lines": [ + { + "bbox": [ + 102, + 243, + 509, + 264 + ], + "spans": [ + { + "bbox": [ + 102, + 243, + 323, + 264 + ], + "score": 1.0, + "content": "Corollary 1 (of Theorem 1 of Jain et al. (2016)). Let", + "type": "text" + }, + { + "bbox": [ + 323, + 246, + 350, + 260 + ], + "score": 0.92, + "content": "w _ { t } ^ { S G D }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 243, + 378, + 264 + ], + "score": 1.0, + "content": "be the", + "type": "text" + }, + { + "bbox": [ + 379, + 247, + 389, + 258 + ], + "score": 0.85, + "content": "t ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 243, + 509, + 264 + ], + "score": 1.0, + "content": "iterate of SGD on the above", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 254, + 464, + 274 + ], + "spans": [ + { + "bbox": [ + 104, + 258, + 218, + 272 + ], + "score": 1.0, + "content": "problem with starting point", + "type": "text" + }, + { + "bbox": [ + 218, + 261, + 231, + 270 + ], + "score": 0.85, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 258, + 283, + 272 + ], + "score": 1.0, + "content": "and stepsize", + "type": "text" + }, + { + "bbox": [ + 281, + 262, + 305, + 274 + ], + "score": 1.0, + "content": "cσ 2", + "type": "text" + }, + { + "bbox": [ + 299, + 254, + 354, + 273 + ], + "score": 1.0, + "content": "t . The error of", + "type": "text" + }, + { + "bbox": [ + 354, + 258, + 381, + 271 + ], + "score": 0.92, + "content": "w _ { t } ^ { S G D }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 254, + 464, + 273 + ], + "score": 1.0, + "content": "can be bounded as,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 278, + 426, + 306 + ], + "lines": [ + { + "bbox": [ + 184, + 278, + 426, + 306 + ], + "spans": [ + { + "bbox": [ + 184, + 278, + 426, + 306 + ], + "score": 0.92, + "content": "\\mathbb { E } \\left[ f \\left( w _ { t } ^ { S G D } \\right) \\right] - f \\left( w _ { * } \\right) \\leq \\exp \\left( \\frac { - t } { \\kappa } \\right) \\left( f \\left( w _ { 0 } \\right) - f \\left( w _ { * } \\right) \\right) .", + "type": "interline_equation", + "image_path": "b30294f1e21f3409bb3e144fb0eb94c356390ce17e81d7c46e0c66dbcecabc43.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 184, + 278, + 426, + 306 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 359, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 360, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 360, + 330 + ], + "score": 1.0, + "content": "On the other hand, ASGD achieves the following superior rate.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 329, + 505, + 364 + ], + "lines": [ + { + "bbox": [ + 104, + 327, + 507, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 327, + 331, + 344 + ], + "score": 1.0, + "content": "Corollary 2 (of Theorem 1 of Jain et al. (2017)). Let", + "type": "text" + }, + { + "bbox": [ + 331, + 329, + 364, + 343 + ], + "score": 0.92, + "content": "w _ { t } ^ { A S G D }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 327, + 394, + 344 + ], + "score": 1.0, + "content": "be the", + "type": "text" + }, + { + "bbox": [ + 395, + 330, + 406, + 340 + ], + "score": 0.86, + "content": "t ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 327, + 507, + 344 + ], + "score": 1.0, + "content": "iterate of ASGD on the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 103, + 338, + 507, + 355 + ], + "spans": [ + { + "bbox": [ + 103, + 338, + 248, + 355 + ], + "score": 1.0, + "content": "above problem with starting point", + "type": "text" + }, + { + "bbox": [ + 249, + 344, + 261, + 353 + ], + "score": 0.83, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 338, + 440, + 355 + ], + "score": 1.0, + "content": "and appropriate parameters. The error of", + "type": "text" + }, + { + "bbox": [ + 440, + 341, + 473, + 354 + ], + "score": 0.92, + "content": "\\dot { w } _ { t } ^ { A S G D }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 338, + 507, + 355 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 351, + 158, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 158, + 365 + ], + "score": 1.0, + "content": "bounded as,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 367, + 450, + 395 + ], + "lines": [ + { + "bbox": [ + 159, + 367, + 450, + 395 + ], + "spans": [ + { + "bbox": [ + 159, + 367, + 450, + 395 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left[ f \\left( w _ { t } ^ { A S G D } \\right) \\right] - f \\left( w _ { * } \\right) \\le \\mathrm { p o l y } ( \\kappa ) \\exp \\left( \\frac { - t } { \\sqrt { \\kappa \\tilde { \\kappa } } } \\right) \\left( f \\left( w _ { 0 } \\right) - f \\left( w _ { * } \\right) \\right) .", + "type": "interline_equation", + "image_path": "6dd8c8d023ad022cab755d475f6c6e8f6efdd3fcd5383300d1b87cfe80f81a20.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 159, + 367, + 450, + 395 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 297, + 427 + ], + "score": 1.0, + "content": "Note that for a given problem/input distribution", + "type": "text" + }, + { + "bbox": [ + 298, + 410, + 322, + 420 + ], + "score": 0.9, + "content": "\\tilde { \\kappa } = c", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 403, + 398, + 427 + ], + "score": 1.0, + "content": "is a constant while", + "type": "text" + }, + { + "bbox": [ + 399, + 406, + 433, + 425 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\kappa = \\frac { c \\sigma _ { 1 } ^ { 2 } } { \\sigma _ { 2 } ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 403, + 505, + 427 + ], + "score": 1.0, + "content": "can be arbitrarily", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 174, + 436 + ], + "score": 1.0, + "content": "large. Note that", + "type": "text" + }, + { + "bbox": [ + 174, + 425, + 223, + 435 + ], + "score": 0.9, + "content": "\\kappa > \\tilde { \\kappa } = c", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 424, + 465, + 436 + ], + "score": 1.0, + "content": ". Hence, ASGD improves upon rate of SGD by a factor of", + "type": "text" + }, + { + "bbox": [ + 466, + 424, + 480, + 436 + ], + "score": 0.9, + "content": "\\sqrt { \\kappa }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 424, + 505, + 436 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "following proposition, which is the main result of this section, establishes that HB (Algorithm 1)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 447, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 458 + ], + "score": 1.0, + "content": "cannot provide a similar improvement over SGD as what ASGD offers. In fact, we show no matter", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 458, + 504, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 504, + 469 + ], + "score": 1.0, + "content": "the choice of parameters of HB, its performance does not improve over SGD by more than a constant.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 104, + 469, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 186, + 486 + ], + "score": 1.0, + "content": "Proposition 3. Let", + "type": "text" + }, + { + "bbox": [ + 186, + 471, + 209, + 484 + ], + "score": 0.92, + "content": "w _ { t } ^ { H B }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 469, + 237, + 486 + ], + "score": 1.0, + "content": "be the", + "type": "text" + }, + { + "bbox": [ + 237, + 471, + 248, + 482 + ], + "score": 0.86, + "content": "t ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 469, + 351, + 486 + ], + "score": 1.0, + "content": "iterate of HB (Algorithm", + "type": "text" + }, + { + "bbox": [ + 352, + 473, + 357, + 482 + ], + "score": 0.4, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 469, + 506, + 486 + ], + "score": 1.0, + "content": ") on the above problem with starting", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 483, + 129, + 496 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 129, + 484, + 141, + 494 + ], + "score": 0.83, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 483, + 250, + 496 + ], + "score": 1.0, + "content": ". 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On the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 228, + 583 + ], + "score": 1.0, + "content": "other hand, ASGD can obtain", + "type": "text" + }, + { + "bbox": [ + 228, + 572, + 233, + 579 + ], + "score": 0.56, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 569, + 306, + 583 + ], + "score": 1.0, + "content": "-approximation to", + "type": "text" + }, + { + "bbox": [ + 307, + 570, + 320, + 580 + ], + "score": 0.87, + "content": "w ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 569, + 332, + 583 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 332, + 569, + 406, + 582 + ], + "score": 0.9, + "content": "\\mathcal { O } ( \\sqrt { \\kappa } \\log \\kappa \\log \\frac { 1 } { \\epsilon } )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "iterations. We note that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 581, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 309, + 593 + ], + "score": 1.0, + "content": "the gains offered by ASGD are meaningful when", + "type": "text" + }, + { + "bbox": [ + 310, + 581, + 353, + 593 + ], + "score": 0.92, + "content": "\\kappa > \\mathcal { O } ( c )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 581, + 506, + 593 + ], + "score": 1.0, + "content": "(Jain et al., 2017); otherwise, all the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 591, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 603 + ], + "score": 1.0, + "content": "algorithms including SGD achieve nearly the same rates (upto constant factors). While we do not", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 602, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 602, + 506, + 614 + ], + "score": 1.0, + "content": "prove it theoretically, we observe empirically that for the same problem instance, NAG also obtains", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "nearly same rate as HB and SGD. We conjecture that a lower bound for NAG can be established", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "using a similar proof technique as that of HB (i.e. Proposition 3). We also believe that the constant", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 636, + 476, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 450, + 648 + ], + "score": 1.0, + "content": "in the lower bound described in proposition 3 can be improved to some small number", + "type": "text" + }, + { + "bbox": [ + 450, + 636, + 472, + 647 + ], + "score": 0.78, + "content": "( \\leq 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 636, + 476, + 648 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 107, + 663, + 191, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 662, + 192, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 192, + 677 + ], + "score": 1.0, + "content": "4 ALGORITHM", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "We will now present and explain an intuitive version of ASGD (pseudo code in Algorithm 3). The", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 264, + 712 + ], + "score": 1.0, + "content": "algorithm takes three inputs: short step", + "type": "text" + }, + { + "bbox": [ + 264, + 700, + 270, + 709 + ], + "score": 0.72, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 698, + 354, + 712 + ], + "score": 1.0, + "content": ", long step parameter", + "type": "text" + }, + { + "bbox": [ + 355, + 701, + 362, + 709 + ], + "score": 0.76, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "and statistical advantage parameter", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 113, + 721 + ], + "score": 0.79, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 709, + 175, + 723 + ], + "score": 1.0, + "content": ". 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(2016)). Let", + "type": "text" + }, + { + "bbox": [ + 323, + 246, + 350, + 260 + ], + "score": 0.92, + "content": "w _ { t } ^ { S G D }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 243, + 378, + 264 + ], + "score": 1.0, + "content": "be the", + "type": "text" + }, + { + "bbox": [ + 379, + 247, + 389, + 258 + ], + "score": 0.85, + "content": "t ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 243, + 509, + 264 + ], + "score": 1.0, + "content": "iterate of SGD on the above", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 254, + 464, + 274 + ], + "spans": [ + { + "bbox": [ + 104, + 258, + 218, + 272 + ], + "score": 1.0, + "content": "problem with starting point", + "type": "text" + }, + { + "bbox": [ + 218, + 261, + 231, + 270 + ], + "score": 0.85, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 258, + 283, + 272 + ], + "score": 1.0, + "content": "and stepsize", + "type": "text" + }, + { + "bbox": [ + 281, + 262, + 305, + 274 + ], + "score": 1.0, + "content": "cσ 2", + "type": "text" + }, + { + "bbox": [ + 299, + 254, + 354, + 273 + ], + "score": 1.0, + "content": "t . The error of", + "type": "text" + }, + { + "bbox": [ + 354, + 258, + 381, + 271 + ], + "score": 0.92, + "content": "w _ { t } ^ { S G D }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 254, + 464, + 273 + ], + "score": 1.0, + "content": "can be bounded as,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 102, + 243, + 509, + 274 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 278, + 426, + 306 + ], + "lines": [ + { + "bbox": [ + 184, + 278, + 426, + 306 + ], + "spans": [ + { + "bbox": [ + 184, + 278, + 426, + 306 + ], + "score": 0.92, + "content": "\\mathbb { E } \\left[ f \\left( w _ { t } ^ { S G D } \\right) \\right] - f \\left( w _ { * } \\right) \\leq \\exp \\left( \\frac { - t } { \\kappa } \\right) \\left( f \\left( w _ { 0 } \\right) - f \\left( w _ { * } \\right) \\right) .", + "type": "interline_equation", + "image_path": "b30294f1e21f3409bb3e144fb0eb94c356390ce17e81d7c46e0c66dbcecabc43.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 184, + 278, + 426, + 306 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 359, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 360, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 360, + 330 + ], + "score": 1.0, + "content": "On the other hand, ASGD achieves the following superior rate.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 314, + 360, + 330 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 329, + 505, + 364 + ], + "lines": [ + { + "bbox": [ + 104, + 327, + 507, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 327, + 331, + 344 + ], + "score": 1.0, + "content": "Corollary 2 (of Theorem 1 of Jain et al. 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The error of", + "type": "text" + }, + { + "bbox": [ + 440, + 341, + 473, + 354 + ], + "score": 0.92, + "content": "\\dot { w } _ { t } ^ { A S G D }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 338, + 507, + 355 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 351, + 158, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 158, + 365 + ], + "score": 1.0, + "content": "bounded as,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 103, + 327, + 507, + 365 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 367, + 450, + 395 + ], + "lines": [ + { + "bbox": [ + 159, + 367, + 450, + 395 + ], + "spans": [ + { + "bbox": [ + 159, + 367, + 450, + 395 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left[ f \\left( w _ { t } ^ { A S G D } \\right) \\right] - f \\left( w _ { * } \\right) \\le \\mathrm { p o l y } ( \\kappa ) \\exp \\left( \\frac { - t } { \\sqrt { \\kappa \\tilde { \\kappa } } } \\right) \\left( f \\left( w _ { 0 } \\right) - f \\left( w _ { * } \\right) \\right) .", + "type": "interline_equation", + "image_path": "6dd8c8d023ad022cab755d475f6c6e8f6efdd3fcd5383300d1b87cfe80f81a20.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 159, + 367, + 450, + 395 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 407, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 297, + 427 + ], + "score": 1.0, + "content": "Note that for a given problem/input distribution", + "type": "text" + }, + { + "bbox": [ + 298, + 410, + 322, + 420 + ], + "score": 0.9, + "content": "\\tilde { \\kappa } = c", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 403, + 398, + 427 + ], + "score": 1.0, + "content": "is a constant while", + "type": "text" + }, + { + "bbox": [ + 399, + 406, + 433, + 425 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\kappa = \\frac { c \\sigma _ { 1 } ^ { 2 } } { \\sigma _ { 2 } ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 403, + 505, + 427 + ], + "score": 1.0, + "content": "can be arbitrarily", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 174, + 436 + ], + "score": 1.0, + "content": "large. Note that", + "type": "text" + }, + { + "bbox": [ + 174, + 425, + 223, + 435 + ], + "score": 0.9, + "content": "\\kappa > \\tilde { \\kappa } = c", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 424, + 465, + 436 + ], + "score": 1.0, + "content": ". Hence, ASGD improves upon rate of SGD by a factor of", + "type": "text" + }, + { + "bbox": [ + 466, + 424, + 480, + 436 + ], + "score": 0.9, + "content": "\\sqrt { \\kappa }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 424, + 505, + 436 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "following proposition, which is the main result of this section, establishes that HB (Algorithm 1)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 447, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 458 + ], + "score": 1.0, + "content": "cannot provide a similar improvement over SGD as what ASGD offers. In fact, we show no matter", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 458, + 504, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 504, + 469 + ], + "score": 1.0, + "content": "the choice of parameters of HB, its performance does not improve over SGD by more than a constant.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 403, + 506, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 471, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 104, + 469, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 186, + 486 + ], + "score": 1.0, + "content": "Proposition 3. Let", + "type": "text" + }, + { + "bbox": [ + 186, + 471, + 209, + 484 + ], + "score": 0.92, + "content": "w _ { t } ^ { H B }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 469, + 237, + 486 + ], + "score": 1.0, + "content": "be the", + "type": "text" + }, + { + "bbox": [ + 237, + 471, + 248, + 482 + ], + "score": 0.86, + "content": "t ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 469, + 351, + 486 + ], + "score": 1.0, + "content": "iterate of HB (Algorithm", + "type": "text" + }, + { + "bbox": [ + 352, + 473, + 357, + 482 + ], + "score": 0.4, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 469, + 506, + 486 + ], + "score": 1.0, + "content": ") on the above problem with starting", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 483, + 129, + 496 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 129, + 484, + 141, + 494 + ], + "score": 0.83, + "content": "w _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 483, + 250, + 496 + ], + "score": 1.0, + "content": ". For any choice of stepsize", + "type": "text" + }, + { + "bbox": [ + 250, + 483, + 256, + 493 + ], + "score": 0.59, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 483, + 320, + 496 + ], + "score": 1.0, + "content": "and momentum", + "type": "text" + }, + { + "bbox": [ + 320, + 483, + 359, + 495 + ], + "score": 0.84, + "content": "\\alpha \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 483, + 363, + 496 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 363, + 483, + 378, + 493 + ], + "score": 0.55, + "content": "\\exists T", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 483, + 470, + 496 + ], + "score": 1.0, + "content": "large enough such that", + "type": "text" + }, + { + "bbox": [ + 471, + 483, + 501, + 494 + ], + "score": 0.88, + "content": "\\forall t \\geq T", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 483, + 505, + 496 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 493, + 144, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 144, + 507 + ], + "score": 1.0, + "content": "we have,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 469, + 506, + 507 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 504, + 454, + 532 + ], + "lines": [ + { + "bbox": [ + 155, + 504, + 454, + 532 + ], + "spans": [ + { + "bbox": [ + 155, + 504, + 454, + 532 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left[ f \\left( w _ { t } ^ { H B } \\right) \\right] - f \\left( w _ { * } \\right) \\ge C ( \\kappa , \\delta , \\alpha ) \\cdot \\exp \\left( \\frac { - 5 0 0 t } { \\kappa } \\right) \\left( f \\left( w _ { 0 } \\right) - f \\left( w _ { * } \\right) \\right) ,", + "type": "interline_equation", + "image_path": "2bd934bd656cc5053bdfd788fb196e86a945c179b3f1da9d64e16654a2903dd3.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 155, + 504, + 454, + 532 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 324, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 325, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 133, + 550 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 536, + 175, + 548 + ], + "score": 0.93, + "content": "C ( \\kappa , \\delta , \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 534, + 224, + 550 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 224, + 536, + 240, + 547 + ], + "score": 0.88, + "content": "\\kappa , \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 534, + 259, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 259, + 538, + 267, + 546 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 534, + 313, + 550 + ], + "score": 1.0, + "content": "(but not on", + "type": "text" + }, + { + "bbox": [ + 314, + 537, + 319, + 546 + ], + "score": 0.48, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 534, + 325, + 550 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 534, + 325, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 173, + 570 + ], + "score": 1.0, + "content": "Thus, to obtain", + "type": "text" + }, + { + "bbox": [ + 173, + 557, + 182, + 567 + ], + "score": 0.81, + "content": "\\widehat { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 556, + 202, + 570 + ], + "score": 1.0, + "content": "s.t.", + "type": "text" + }, + { + "bbox": [ + 203, + 556, + 269, + 569 + ], + "score": 0.92, + "content": "\\| \\widehat { \\boldsymbol { w } } - \\boldsymbol { w } ^ { * } \\| \\le \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 556, + 327, + 570 + ], + "score": 1.0, + "content": ", HB requires", + "type": "text" + }, + { + "bbox": [ + 327, + 556, + 371, + 569 + ], + "score": 0.91, + "content": "\\Omega ( \\kappa \\log { \\frac { 1 } { \\epsilon } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "samples and iterations. On the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 228, + 583 + ], + "score": 1.0, + "content": "other hand, ASGD can obtain", + "type": "text" + }, + { + "bbox": [ + 228, + 572, + 233, + 579 + ], + "score": 0.56, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 569, + 306, + 583 + ], + "score": 1.0, + "content": "-approximation to", + "type": "text" + }, + { + "bbox": [ + 307, + 570, + 320, + 580 + ], + "score": 0.87, + "content": "w ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 569, + 332, + 583 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 332, + 569, + 406, + 582 + ], + "score": 0.9, + "content": "\\mathcal { O } ( \\sqrt { \\kappa } \\log \\kappa \\log \\frac { 1 } { \\epsilon } )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "iterations. We note that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 581, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 309, + 593 + ], + "score": 1.0, + "content": "the gains offered by ASGD are meaningful when", + "type": "text" + }, + { + "bbox": [ + 310, + 581, + 353, + 593 + ], + "score": 0.92, + "content": "\\kappa > \\mathcal { O } ( c )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 581, + 506, + 593 + ], + "score": 1.0, + "content": "(Jain et al., 2017); otherwise, all the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 591, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 603 + ], + "score": 1.0, + "content": "algorithms including SGD achieve nearly the same rates (upto constant factors). While we do not", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 602, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 602, + 506, + 614 + ], + "score": 1.0, + "content": "prove it theoretically, we observe empirically that for the same problem instance, NAG also obtains", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "nearly same rate as HB and SGD. We conjecture that a lower bound for NAG can be established", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "using a similar proof technique as that of HB (i.e. Proposition 3). We also believe that the constant", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 636, + 476, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 450, + 648 + ], + "score": 1.0, + "content": "in the lower bound described in proposition 3 can be improved to some small number", + "type": "text" + }, + { + "bbox": [ + 450, + 636, + 472, + 647 + ], + "score": 0.78, + "content": "( \\leq 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 636, + 476, + 648 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 556, + 506, + 648 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 663, + 191, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 662, + 192, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 192, + 677 + ], + "score": 1.0, + "content": "4 ALGORITHM", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "We will now present and explain an intuitive version of ASGD (pseudo code in Algorithm 3). The", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 264, + 712 + ], + "score": 1.0, + "content": "algorithm takes three inputs: short step", + "type": "text" + }, + { + "bbox": [ + 264, + 700, + 270, + 709 + ], + "score": 0.72, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 698, + 354, + 712 + ], + "score": 1.0, + "content": ", long step parameter", + "type": "text" + }, + { + "bbox": [ + 355, + 701, + 362, + 709 + ], + "score": 0.76, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "and statistical advantage parameter", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 113, + 721 + ], + "score": 0.79, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 709, + 175, + 723 + ], + "score": 1.0, + "content": ". The short step", + "type": "text" + }, + { + "bbox": [ + 176, + 710, + 182, + 720 + ], + "score": 0.76, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "is precisely the same as the step size in SGD, HB or NAG. For convex problems,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 448, + 733 + ], + "score": 1.0, + "content": "this scales inversely with the smoothness of the function. The long step parameter", + "type": "text" + }, + { + "bbox": [ + 449, + 722, + 456, + 730 + ], + "score": 0.75, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "is intended", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "to give an estimate of the ratio of the largest and smallest curvatures of the function; for convex", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 437, + 106 + ], + "score": 1.0, + "content": "functions, this is just the condition number. The statistical advantage parameter", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 437, + 94, + 444, + 105 + ], + "score": 0.83, + "content": "\\xi", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 444, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "captures trade√", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 467, + 117 + ], + "score": 1.0, + "content": "off between statistical and computational condition numbers – in the deterministic case,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 468, + 104, + 504, + 116 + ], + "score": 0.92, + "content": "\\xi = \\sqrt { \\kappa }", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 406, + 128 + ], + "score": 1.0, + "content": "and ASGD is equivalent to NAG, while in the high stochasticity regime,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 406, + 116, + 413, + 127 + ], + "score": 0.83, + "content": "\\xi", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 414, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "is much smaller. The", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 300, + 140 + ], + "score": 1.0, + "content": "algorithm maintains two iterates: descent iterate", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 301, + 128, + 313, + 137 + ], + "score": 0.85, + "content": "w _ { t }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 313, + 125, + 404, + 140 + ], + "score": 1.0, + "content": "and a running average", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 404, + 128, + 416, + 137 + ], + "score": 0.87, + "content": "\\bar { w } _ { t }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 416, + 125, + 506, + 140 + ], + "score": 1.0, + "content": ". The running average", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "is a weighted average of the previous average and a long gradient step from the descent iterate, while", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 504, + 160 + ], + "score": 1.0, + "content": "the descent iterate is updated as a convex combination of short gradient step from the descent iterate", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 173 + ], + "score": 1.0, + "content": "and the running average. The idea is that since the algorithm takes a long step as well as short step", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "and an appropriate average of both of them, it can make progress on different directions at a similar", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "pace. Appendix B shows the equivalence between Algorithm 3 and ASGD as proposed in Jain et al.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 191, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 505, + 205 + ], + "score": 1.0, + "content": "(2017). Note that the constant 0.7 appearing in Algorithm 3 has no special significance. Jain et al.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 503, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 246, + 218 + ], + "score": 1.0, + "content": "(2017) require it to be smaller than", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 246, + 203, + 272, + 217 + ], + "score": 0.92, + "content": "\\sqrt { 1 / 6 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 272, + 203, + 503, + 218 + ], + "score": 1.0, + "content": "but any constant smaller than 1 seems to work in practice.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 688, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "to give an estimate of the ratio of the largest and smallest curvatures of the function; for convex", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 437, + 106 + ], + "score": 1.0, + "content": "functions, this is just the condition number. The statistical advantage parameter", + "type": "text" + }, + { + "bbox": [ + 437, + 94, + 444, + 105 + ], + "score": 0.83, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "captures trade√", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 467, + 117 + ], + "score": 1.0, + "content": "off between statistical and computational condition numbers – in the deterministic case,", + "type": "text" + }, + { + "bbox": [ + 468, + 104, + 504, + 116 + ], + "score": 0.92, + "content": "\\xi = \\sqrt { \\kappa }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 406, + 128 + ], + "score": 1.0, + "content": "and ASGD is equivalent to NAG, while in the high stochasticity regime,", + "type": "text" + }, + { + "bbox": [ + 406, + 116, + 413, + 127 + ], + "score": 0.83, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "is much smaller. The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 300, + 140 + ], + "score": 1.0, + "content": "algorithm maintains two iterates: descent iterate", + "type": "text" + }, + { + "bbox": [ + 301, + 128, + 313, + 137 + ], + "score": 0.85, + "content": "w _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 125, + 404, + 140 + ], + "score": 1.0, + "content": "and a running average", + "type": "text" + }, + { + "bbox": [ + 404, + 128, + 416, + 137 + ], + "score": 0.87, + "content": "\\bar { w } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 125, + 506, + 140 + ], + "score": 1.0, + "content": ". The running average", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "is a weighted average of the previous average and a long gradient step from the descent iterate, while", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 504, + 160 + ], + "score": 1.0, + "content": "the descent iterate is updated as a convex combination of short gradient step from the descent iterate", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 173 + ], + "score": 1.0, + "content": "and the running average. The idea is that since the algorithm takes a long step as well as short step", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "and an appropriate average of both of them, it can make progress on different directions at a similar", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "pace. Appendix B shows the equivalence between Algorithm 3 and ASGD as proposed in Jain et al.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 191, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 505, + 205 + ], + "score": 1.0, + "content": "(2017). Note that the constant 0.7 appearing in Algorithm 3 has no special significance. Jain et al.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 503, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 246, + 218 + ], + "score": 1.0, + "content": "(2017) require it to be smaller than", + "type": "text" + }, + { + "bbox": [ + 246, + 203, + 272, + 217 + ], + "score": 0.92, + "content": "\\sqrt { 1 / 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 203, + 503, + 218 + ], + "score": 1.0, + "content": "but any constant smaller than 1 seems to work in practice.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 107, + 227, + 200, + 240 + ], + "lines": [ + { + "bbox": [ + 104, + 225, + 202, + 241 + ], + "spans": [ + { + "bbox": [ + 104, + 225, + 202, + 241 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 504, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "We now present our experimental results exploring performance of SGD, HB, NAG and ASGD. Our", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 263, + 377, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 377, + 277 + ], + "score": 1.0, + "content": "experiments are geared towards answering the following questions:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 132, + 284, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 132, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 132, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "• Even for linear regression, is the suboptimality of HB restricted to specific distributions in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 295, + 502, + 307 + ], + "spans": [ + { + "bbox": [ + 142, + 295, + 502, + 307 + ], + "score": 1.0, + "content": "Section 3 or does it hold for more general distributions as well? Is the same true of NAG?", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 137, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 137, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "What is the reason for the superiority of HB and NAG in practice? Is it because momen-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 142, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "tum methods have better performance that SGD for stochastic gradients or due to mini-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 332, + 404, + 343 + ], + "spans": [ + { + "bbox": [ + 142, + 332, + 404, + 343 + ], + "score": 1.0, + "content": "batching? Does this superiority hold even for small minibatches?", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 132, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 132, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "• How does the performance of ASGD compare to that of SGD, HB and NAG, when training", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 358, + 207, + 370 + ], + "spans": [ + { + "bbox": [ + 142, + 358, + 207, + 370 + ], + "score": 1.0, + "content": "deep networks?", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 378, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "Section 5.1 and parts of Section 5.2 address the first two questions. Section 5.2 and 5.3 address", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "Question 2 partially and the last question. We use Matlab to conduct experiments presented in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "Section 5.1 and use PyTorch (pytorch, 2017) for our deep networks related experiments. Pytorch", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 411, + 503, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 503, + 424 + ], + "score": 1.0, + "content": "code implementing the ASGD algorithm can be found at https://github.com/rahulkidambi/AccSGD.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 107, + 436, + 223, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 223, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 223, + 448 + ], + "score": 1.0, + "content": "5.1 LINEAR REGRESSION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "In this section, we will present results on performance of the four optimization methods (SGD,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "HB, NAG, and ASGD) for linear regression problems. We consider two different class of linear", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 352, + 491 + ], + "score": 1.0, + "content": "regression problems, both of them in two dimensions. Given", + "type": "text" + }, + { + "bbox": [ + 352, + 481, + 360, + 489 + ], + "score": 0.74, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "which stands for condition number,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 490, + 285, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 285, + 502 + ], + "score": 1.0, + "content": "we consider the following two distributions:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 505, + 446, + 519 + ], + "lines": [ + { + "bbox": [ + 147, + 500, + 358, + 527 + ], + "spans": [ + { + "bbox": [ + 147, + 509, + 176, + 518 + ], + "score": 0.89, + "content": "a = e _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 500, + 228, + 527 + ], + "score": 1.0, + "content": "w.p. 0.5 and", + "type": "text" + }, + { + "bbox": [ + 229, + 505, + 272, + 520 + ], + "score": 0.94, + "content": "\\textstyle a = { \\frac { 2 } { \\kappa } } \\cdot e _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 500, + 311, + 527 + ], + "score": 1.0, + "content": "with 0.5;", + "type": "text" + }, + { + "bbox": [ + 311, + 508, + 321, + 518 + ], + "score": 0.76, + "content": "e _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 500, + 345, + 527 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 346, + 506, + 358, + 516 + ], + "score": 0.89, + "content": "i ^ { t h }", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 488, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 486, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 154, + 546 + ], + "score": 1.0, + "content": "Gaussian :", + "type": "text" + }, + { + "bbox": [ + 154, + 531, + 185, + 542 + ], + "score": 0.86, + "content": "a \\in \\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 523, + 451, + 546 + ], + "score": 1.0, + "content": "is distributed as a Gaussian random vector with covariance matrix", + "type": "text" + }, + { + "bbox": [ + 451, + 523, + 486, + 551 + ], + "score": 0.86, + "content": "\\left[ { \\begin{array} { c c } { 1 } & { 0 } \\\\ { 0 } & { { \\frac { 1 } { \\kappa } } } \\end{array} } \\right] .", + "type": "inline_equation" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 504, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 228, + 569 + ], + "score": 1.0, + "content": "We fix a randomly generated", + "type": "text" + }, + { + "bbox": [ + 228, + 556, + 268, + 566 + ], + "score": 0.92, + "content": "w ^ { \\ast } \\in \\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 555, + 450, + 569 + ], + "score": 1.0, + "content": "and for both the distributions above, we let", + "type": "text" + }, + { + "bbox": [ + 450, + 556, + 501, + 568 + ], + "score": 0.93, + "content": "b = \\langle w ^ { * } , a \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 555, + 505, + 569 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 563, + 507, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 142, + 581 + ], + "score": 1.0, + "content": "We vary", + "type": "text" + }, + { + "bbox": [ + 142, + 569, + 150, + 577 + ], + "score": 0.74, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 563, + 173, + 581 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 173, + 567, + 238, + 579 + ], + "score": 0.92, + "content": "\\{ \\mathbf { \\bar { 2 } ^ { 4 } } , 2 ^ { 5 } , . . . , 2 ^ { 1 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 563, + 291, + 581 + ], + "score": 1.0, + "content": "and for each", + "type": "text" + }, + { + "bbox": [ + 291, + 569, + 299, + 577 + ], + "score": 0.78, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 563, + 507, + 581 + ], + "score": 1.0, + "content": "in this set, we run 100 independent runs of all four", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 219, + 591 + ], + "score": 1.0, + "content": "methods, each for a total of", + "type": "text" + }, + { + "bbox": [ + 219, + 579, + 249, + 588 + ], + "score": 0.89, + "content": "t = 5 \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "iterations. We define that the algorithm converges if there is no", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 245, + 601 + ], + "score": 1.0, + "content": "error in the second half (i.e. after", + "type": "text" + }, + { + "bbox": [ + 245, + 589, + 265, + 600 + ], + "score": 0.83, + "content": "2 . 5 \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "updates) that exceeds the starting error - this is reasonable", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 341, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 341, + 613 + ], + "score": 1.0, + "content": "since we expect geometric convergence of the initial error.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 616, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 617, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 629 + ], + "score": 1.0, + "content": "Unlike ASGD and SGD, we do not know optimal learning rate and momentum parameters for NAG", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "and HB in the stochastic gradient model. So, we perform a grid search over the values of the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 369, + 651 + ], + "score": 1.0, + "content": "learning rate and momentum parameters. In particular, we lay a", + "type": "text" + }, + { + "bbox": [ + 369, + 639, + 404, + 650 + ], + "score": 0.9, + "content": "1 0 \\times 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 639, + 435, + 651 + ], + "score": 1.0, + "content": "grid in", + "type": "text" + }, + { + "bbox": [ + 435, + 639, + 489, + 651 + ], + "score": 0.92, + "content": "[ 0 , 1 ] \\times [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 651, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 662 + ], + "score": 1.0, + "content": "learning rate and momentum and run NAG and HB. Then, for each grid point, we consider the subset", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "of 100 trials that converged and computed the final error using these. Finally, the parameters that", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "yield the minimal error are chosen for NAG and HB, and these numbers are reported. We measure", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 684, + 289, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 289, + 696 + ], + "score": 1.0, + "content": "convergence performance of a method using:", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41 + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 699, + 374, + 723 + ], + "lines": [ + { + "bbox": [ + 235, + 699, + 374, + 723 + ], + "spans": [ + { + "bbox": [ + 235, + 699, + 374, + 723 + ], + "score": 0.94, + "content": "{ \\mathrm { r a t e } } = { \\frac { \\log ( f ( w _ { 0 } ) ) - \\log ( f ( w _ { t } ) ) } { t } } ,", + "type": "interline_equation", + "image_path": "b9dc3438e9ba56ee3d2a40a0d96780a48908dc1caffc92e0d02e001399705b0d.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 235, + 699, + 374, + 723 + ], + "spans": [], + "index": 45 + } + ] + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 217 + ], + "lines": [], + "index": 5.5, + "bbox_fs": [ + 105, + 82, + 506, + 218 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 227, + 200, + 240 + ], + "lines": [ + { + "bbox": [ + 104, + 225, + 202, + 241 + ], + "spans": [ + { + "bbox": [ + 104, + 225, + 202, + 241 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 504, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "We now present our experimental results exploring performance of SGD, HB, NAG and ASGD. Our", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 263, + 377, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 377, + 277 + ], + "score": 1.0, + "content": "experiments are geared towards answering the following questions:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 252, + 505, + 277 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 284, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 132, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 132, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "• Even for linear regression, is the suboptimality of HB restricted to specific distributions in", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 295, + 502, + 307 + ], + "spans": [ + { + "bbox": [ + 142, + 295, + 502, + 307 + ], + "score": 1.0, + "content": "Section 3 or does it hold for more general distributions as well? Is the same true of NAG?", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 137, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 137, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "What is the reason for the superiority of HB and NAG in practice? Is it because momen-", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 142, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "tum methods have better performance that SGD for stochastic gradients or due to mini-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 332, + 404, + 343 + ], + "spans": [ + { + "bbox": [ + 142, + 332, + 404, + 343 + ], + "score": 1.0, + "content": "batching? Does this superiority hold even for small minibatches?", + "type": "text" + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 132, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "• How does the performance of ASGD compare to that of SGD, HB and NAG, when training", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 358, + 207, + 370 + ], + "spans": [ + { + "bbox": [ + 142, + 358, + 207, + 370 + ], + "score": 1.0, + "content": "deep networks?", + "type": "text" + } + ], + "index": 21, + "is_list_end_line": true + } + ], + "index": 18, + "bbox_fs": [ + 132, + 284, + 505, + 370 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 378, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "Section 5.1 and parts of Section 5.2 address the first two questions. Section 5.2 and 5.3 address", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "Question 2 partially and the last question. We use Matlab to conduct experiments presented in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "Section 5.1 and use PyTorch (pytorch, 2017) for our deep networks related experiments. Pytorch", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 411, + 503, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 503, + 424 + ], + "score": 1.0, + "content": "code implementing the ASGD algorithm can be found at https://github.com/rahulkidambi/AccSGD.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 378, + 506, + 424 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 436, + 223, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 223, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 223, + 448 + ], + "score": 1.0, + "content": "5.1 LINEAR REGRESSION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "In this section, we will present results on performance of the four optimization methods (SGD,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "HB, NAG, and ASGD) for linear regression problems. We consider two different class of linear", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 352, + 491 + ], + "score": 1.0, + "content": "regression problems, both of them in two dimensions. Given", + "type": "text" + }, + { + "bbox": [ + 352, + 481, + 360, + 489 + ], + "score": 0.74, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "which stands for condition number,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 490, + 285, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 285, + 502 + ], + "score": 1.0, + "content": "we consider the following two distributions:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 457, + 505, + 502 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 505, + 446, + 519 + ], + "lines": [ + { + "bbox": [ + 147, + 500, + 358, + 527 + ], + "spans": [ + { + "bbox": [ + 147, + 509, + 176, + 518 + ], + "score": 0.89, + "content": "a = e _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 500, + 228, + 527 + ], + "score": 1.0, + "content": "w.p. 0.5 and", + "type": "text" + }, + { + "bbox": [ + 229, + 505, + 272, + 520 + ], + "score": 0.94, + "content": "\\textstyle a = { \\frac { 2 } { \\kappa } } \\cdot e _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 500, + 311, + 527 + ], + "score": 1.0, + "content": "with 0.5;", + "type": "text" + }, + { + "bbox": [ + 311, + 508, + 321, + 518 + ], + "score": 0.76, + "content": "e _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 500, + 345, + 527 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 346, + 506, + 358, + 516 + ], + "score": 0.89, + "content": "i ^ { t h }", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 147, + 500, + 358, + 527 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 488, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 486, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 154, + 546 + ], + "score": 1.0, + "content": "Gaussian :", + "type": "text" + }, + { + "bbox": [ + 154, + 531, + 185, + 542 + ], + "score": 0.86, + "content": "a \\in \\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 523, + 451, + 546 + ], + "score": 1.0, + "content": "is distributed as a Gaussian random vector with covariance matrix", + "type": "text" + }, + { + "bbox": [ + 451, + 523, + 486, + 551 + ], + "score": 0.86, + "content": "\\left[ { \\begin{array} { c c } { 1 } & { 0 } \\\\ { 0 } & { { \\frac { 1 } { \\kappa } } } \\end{array} } \\right] .", + "type": "inline_equation" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 523, + 486, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 504, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 228, + 569 + ], + "score": 1.0, + "content": "We fix a randomly generated", + "type": "text" + }, + { + "bbox": [ + 228, + 556, + 268, + 566 + ], + "score": 0.92, + "content": "w ^ { \\ast } \\in \\mathbb { R } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 555, + 450, + 569 + ], + "score": 1.0, + "content": "and for both the distributions above, we let", + "type": "text" + }, + { + "bbox": [ + 450, + 556, + 501, + 568 + ], + "score": 0.93, + "content": "b = \\langle w ^ { * } , a \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 555, + 505, + 569 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 563, + 507, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 142, + 581 + ], + "score": 1.0, + "content": "We vary", + "type": "text" + }, + { + "bbox": [ + 142, + 569, + 150, + 577 + ], + "score": 0.74, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 563, + 173, + 581 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 173, + 567, + 238, + 579 + ], + "score": 0.92, + "content": "\\{ \\mathbf { \\bar { 2 } ^ { 4 } } , 2 ^ { 5 } , . . . , 2 ^ { 1 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 563, + 291, + 581 + ], + "score": 1.0, + "content": "and for each", + "type": "text" + }, + { + "bbox": [ + 291, + 569, + 299, + 577 + ], + "score": 0.78, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 563, + 507, + 581 + ], + "score": 1.0, + "content": "in this set, we run 100 independent runs of all four", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 219, + 591 + ], + "score": 1.0, + "content": "methods, each for a total of", + "type": "text" + }, + { + "bbox": [ + 219, + 579, + 249, + 588 + ], + "score": 0.89, + "content": "t = 5 \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "iterations. We define that the algorithm converges if there is no", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 245, + 601 + ], + "score": 1.0, + "content": "error in the second half (i.e. after", + "type": "text" + }, + { + "bbox": [ + 245, + 589, + 265, + 600 + ], + "score": 0.83, + "content": "2 . 5 \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "updates) that exceeds the starting error - this is reasonable", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 341, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 341, + 613 + ], + "score": 1.0, + "content": "since we expect geometric convergence of the initial error.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 555, + 507, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 616, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 617, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 629 + ], + "score": 1.0, + "content": "Unlike ASGD and SGD, we do not know optimal learning rate and momentum parameters for NAG", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "and HB in the stochastic gradient model. So, we perform a grid search over the values of the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 369, + 651 + ], + "score": 1.0, + "content": "learning rate and momentum parameters. 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A value of", + "type": "text" + }, + { + "bbox": [ + 458, + 334, + 466, + 344 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "indicates", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 341, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 250, + 357 + ], + "score": 1.0, + "content": "that the error decays at a rate of exp", + "type": "text" + }, + { + "bbox": [ + 251, + 342, + 271, + 356 + ], + "score": 0.88, + "content": "\\left( { \\frac { - t } { \\kappa ^ { \\gamma } } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 341, + 351, + 357 + ], + "score": 1.0, + "content": ". A smaller value of", + "type": "text" + }, + { + "bbox": [ + 351, + 344, + 359, + 354 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 341, + 506, + 357 + ], + "score": 1.0, + "content": "indicates a faster rate of error decay.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + } + ], + "index": 8.75 + }, + { + "type": "text", + "bbox": [ + 106, + 378, + 505, + 433 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 504, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 422, + 390 + ], + "score": 1.0, + "content": "We compute the rate (1) for all the algorithms with varying condition number", + "type": "text" + }, + { + "bbox": [ + 422, + 381, + 429, + 388 + ], + "score": 0.72, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 378, + 496, + 390 + ], + "score": 1.0, + "content": ". Given a rate vs", + "type": "text" + }, + { + "bbox": [ + 497, + 381, + 504, + 388 + ], + "score": 0.6, + "content": "\\kappa", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 316, + 402 + ], + "score": 1.0, + "content": "plot for a method, we compute it’s slope (denoted as", + "type": "text" + }, + { + "bbox": [ + 317, + 390, + 324, + 401 + ], + "score": 0.71, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 388, + 505, + 402 + ], + "score": 1.0, + "content": ") using linear regression. Table 1 presents the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 399, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 193, + 414 + ], + "score": 1.0, + "content": "estimated slopes (i.e.", + "type": "text" + }, + { + "bbox": [ + 193, + 402, + 201, + 412 + ], + "score": 0.67, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 399, + 505, + 414 + ], + "score": 1.0, + "content": ") for various methods for both the discrete and the Gaussian case. The slope", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 472, + 424 + ], + "score": 1.0, + "content": "values clearly show that the rate of SGD, HB and NAG have a nearly linear dependence on √", + "type": "text" + }, + { + "bbox": [ + 472, + 413, + 479, + 421 + ], + "score": 0.74, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "while", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 422, + 295, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 277, + 435 + ], + "score": 1.0, + "content": "that of ASGD seems to scale linearly with", + "type": "text" + }, + { + "bbox": [ + 277, + 422, + 292, + 434 + ], + "score": 0.91, + "content": "\\sqrt { \\kappa }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 422, + 295, + 435 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 441, + 283, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 285, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 285, + 454 + ], + "score": 1.0, + "content": "5.2 DEEP AUTOENCODERS FOR MNIST", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 462, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 504, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 504, + 475 + ], + "score": 1.0, + "content": "In this section, we present experimental results on training deep autoencoders for the mnist dataset,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "and we closely follow the setup of Hinton & Salakhutdinov (2006). 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The network architecture", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 417, + 518 + ], + "score": 1.0, + "content": "follows previous work (Hinton & Salakhutdinov, 2006) and is represented as", + "type": "text" + }, + { + "bbox": [ + 417, + 506, + 505, + 517 + ], + "score": 0.88, + "content": "7 8 4 - 1 0 0 0 - 5 0 0 -", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 245, + 528 + ], + "score": 0.88, + "content": "2 5 0 - 3 0 - 2 5 0 - 5 0 0 - 1 0 0 0 - 7 8 4", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "with the first and last 784 nodes representing the input and output", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "respectively. 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Note that the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "final loss values reported are suboptimal compared to those in published literature e.g., Sutskever", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "et al. (2013); while Sutskever et al. 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A value of", + "type": "text" + }, + { + "bbox": [ + 458, + 334, + 466, + 344 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "indicates", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 341, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 250, + 357 + ], + "score": 1.0, + "content": "that the error decays at a rate of exp", + "type": "text" + }, + { + "bbox": [ + 251, + 342, + 271, + 356 + ], + "score": 0.88, + "content": "\\left( { \\frac { - t } { \\kappa ^ { \\gamma } } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 341, + 351, + 357 + ], + "score": 1.0, + "content": ". 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Table 1 presents the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 399, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 193, + 414 + ], + "score": 1.0, + "content": "estimated slopes (i.e.", + "type": "text" + }, + { + "bbox": [ + 193, + 402, + 201, + 412 + ], + "score": 0.67, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 399, + 505, + 414 + ], + "score": 1.0, + "content": ") for various methods for both the discrete and the Gaussian case. 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For each of SGD, HB, NAG and ASGD,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "a grid search over learning rate, momentum and long step parameter (whichever is applicable) was", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "done and best parameters were chosen based on achieving the smallest training error in the same", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "protocol followed by Sutskever et al. (2013). The grid was extended whenever the best parameter", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "fell at the edge of a grid. For the parameters chosen by grid search, we perform 10 runs with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "different seeds and averaged the results. The results are presented in Figures 2 and 3. Note that the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "final loss values reported are suboptimal compared to those in published literature e.g., Sutskever", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "et al. (2013); while Sutskever et al. (2013) report results after 750000 updates with a large batch size", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 236, + 672 + ], + "score": 1.0, + "content": "of 200 (which implies a total of", + "type": "text" + }, + { + "bbox": [ + 237, + 660, + 333, + 671 + ], + "score": 0.9, + "content": "7 5 0 0 0 0 \\times 2 0 0 = 1 5 0 \\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "gradient evaluations), whereas, our results", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 670, + 479, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 479, + 683 + ], + "score": 1.0, + "content": "are after 1.8M updates of SGD with a batch size 1 (which is just 1.8M gradient evaluations).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 462, + 506, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 504, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 504, + 699 + ], + "score": 1.0, + "content": "Effect of minibatch sizes: While HB and NAG decay the loss faster compared to SGD for a mini-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 698, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 504, + 711 + ], + "score": 1.0, + "content": "batch size of 8 (Figure 2), this superior decay rate does not hold for a minibatch size of 1 (Figure 3).", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "This supports our intuitions from the stochastic linear regression setting, where we demonstrate", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 721, + 407, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 407, + 732 + ], + "score": 1.0, + "content": "that HB and NAG are suboptimal in the stochastic first order oracle model.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 687, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 132, + 92, + 473, + 220 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 132, + 92, + 473, + 220 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 92, + 473, + 220 + ], + "spans": [ + { + "bbox": [ + 132, + 92, + 473, + 220 + ], + "score": 0.791, + "type": "image", + "image_path": "e7e9ca2132cb1ddd893c549fd607c122fbf8832557608c7292e2d72b3058de61.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 132, + 92, + 473, + 134.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 132, + 134.66666666666666, + 473, + 177.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 132, + 177.33333333333331, + 473, + 219.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 224, + 504, + 257 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "Figure 2: Training loss (left) and test loss (right) while training deep autoencoder for mnist with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 235, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 247 + ], + "score": 1.0, + "content": "minibatch size 8. Clearly, ASGD matches performance of NAG and outperforms SGD on the test", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 246, + 240, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 240, + 258 + ], + "score": 1.0, + "content": "data. HB also outperforms SGD.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "image", + "bbox": [ + 132, + 271, + 473, + 399 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 132, + 271, + 473, + 399 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 132, + 271, + 473, + 399 + ], + "spans": [ + { + "bbox": [ + 132, + 271, + 473, + 399 + ], + "score": 0.971, + "type": "image", + "image_path": "1de51d3d68f2b8fcd7d7f8f74749949493d74864a33ffe1b0da66f947386a0f1.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 132, + 271, + 473, + 313.6666666666667 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 132, + 313.6666666666667, + 473, + 356.33333333333337 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 132, + 356.33333333333337, + 473, + 399.00000000000006 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 403, + 505, + 436 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "Figure 3: Training loss (left) and test loss (right) while training deep autoencoder for mnist with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "minibatch size 1. Interestingly, SGD, HB and NAG, all decrease the loss at a similar rate,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 425, + 252, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 252, + 437 + ], + "score": 1.0, + "content": "while ASGD decays at a faster rate.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 108, + 448, + 503, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "Comparison of ASGD with momentum methods: While ASGD performs slightly better than", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "score": 1.0, + "content": "NAG for batch size 8 in the training error (Figure 2), ASGD decays the error at a faster rate compared", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 469, + 351, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 351, + 482 + ], + "score": 1.0, + "content": "to all the three other methods for a batch size of 1 (Figure 3).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 107, + 489, + 323, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 323, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 323, + 502 + ], + "score": 1.0, + "content": "5.3 DEEP RESIDUAL NETWORKS FOR CIFAR-10", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "We will now present experimental results on training deep residual networks (He et al., 2016b) with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 521, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 536 + ], + "score": 1.0, + "content": "pre-activation blocks He et al. (2016a) for classifying images in cifar-10 (Krizhevsky & Hinton,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 533, + 504, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 504, + 545 + ], + "score": 1.0, + "content": "2009); the network we use has 44 layers (dubbed preresnet-44). The code for this section was down-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "score": 1.0, + "content": "loaded from preresnet (2017). One of the most distinct characteristics of this experiment compared", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "to our previous experiments is learning rate decay. We use a validation set based decay scheme,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "wherein, after every 3 epochs, we decay the learning rate by a certain factor (which we grid search", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 578, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 589 + ], + "score": 1.0, + "content": "on) if the validation zero one error does not decrease by at least a certain amount (precise numbers", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "are provided in the appendix since they vary across batch sizes). Due to space constraints, we present", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "only a subset of training error plots. Please see Appendix C.3 for some more plots on training errors.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 616, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "Effect of minibatch sizes: Our first experiment tries to understand how the performance", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "of HB and NAG compare with that of SGD and how it varies with minibatch sizes. Figure 4 presents", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "the test zero one error for minibatch sizes of 8 and 128. While training with batch size 8 was done", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 650, + 504, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 504, + 660 + ], + "score": 1.0, + "content": "for 40 epochs, with batch size 128, it was done for 120 epochs. We perform a grid search over all", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "parameters for each of these algorithms. See Appendix C.3 for details on the grid search parameters.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "We observe that final error achieved by SGD, HB and NAG are all very close for both batch sizes.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "While NAG exhibits a superior rate of convergence compared to SGD and HB for batch size 128,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 692, + 368, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 368, + 705 + ], + "score": 1.0, + "content": "this superior rate of convergence disappears for a batch size of 8.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Comparison of ASGD with momentum methods: The next experiment tries to understand", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "how ASGD compares with HB and NAG. The errors achieved by various methods when we do", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 132, + 92, + 473, + 220 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 132, + 92, + 473, + 220 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 92, + 473, + 220 + ], + "spans": [ + { + "bbox": [ + 132, + 92, + 473, + 220 + ], + "score": 0.791, + "type": "image", + "image_path": "e7e9ca2132cb1ddd893c549fd607c122fbf8832557608c7292e2d72b3058de61.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 132, + 92, + 473, + 134.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 132, + 134.66666666666666, + 473, + 177.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 132, + 177.33333333333331, + 473, + 219.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 224, + 504, + 257 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "Figure 2: Training loss (left) and test loss (right) while training deep autoencoder for mnist with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 235, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 247 + ], + "score": 1.0, + "content": "minibatch size 8. 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HB also outperforms SGD.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "image", + "bbox": [ + 132, + 271, + 473, + 399 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 132, + 271, + 473, + 399 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 132, + 271, + 473, + 399 + ], + "spans": [ + { + "bbox": [ + 132, + 271, + 473, + 399 + ], + "score": 0.971, + "type": "image", + "image_path": "1de51d3d68f2b8fcd7d7f8f74749949493d74864a33ffe1b0da66f947386a0f1.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 132, + 271, + 473, + 313.6666666666667 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 132, + 313.6666666666667, + 473, + 356.33333333333337 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 132, + 356.33333333333337, + 473, + 399.00000000000006 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 403, + 505, + 436 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "Figure 3: Training loss (left) and test loss (right) while training deep autoencoder for mnist with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "minibatch size 1. Interestingly, SGD, HB and NAG, all decrease the loss at a similar rate,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 425, + 252, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 252, + 437 + ], + "score": 1.0, + "content": "while ASGD decays at a faster rate.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 108, + 448, + 503, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "Comparison of ASGD with momentum methods: While ASGD performs slightly better than", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "score": 1.0, + "content": "NAG for batch size 8 in the training error (Figure 2), ASGD decays the error at a faster rate compared", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 469, + 351, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 351, + 482 + ], + "score": 1.0, + "content": "to all the three other methods for a batch size of 1 (Figure 3).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 448, + 505, + 482 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 489, + 323, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 323, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 323, + 502 + ], + "score": 1.0, + "content": "5.3 DEEP RESIDUAL NETWORKS FOR CIFAR-10", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "We will now present experimental results on training deep residual networks (He et al., 2016b) with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 521, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 536 + ], + "score": 1.0, + "content": "pre-activation blocks He et al. (2016a) for classifying images in cifar-10 (Krizhevsky & Hinton,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 533, + 504, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 504, + 545 + ], + "score": 1.0, + "content": "2009); the network we use has 44 layers (dubbed preresnet-44). The code for this section was down-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "score": 1.0, + "content": "loaded from preresnet (2017). One of the most distinct characteristics of this experiment compared", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "to our previous experiments is learning rate decay. We use a validation set based decay scheme,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "wherein, after every 3 epochs, we decay the learning rate by a certain factor (which we grid search", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 578, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 589 + ], + "score": 1.0, + "content": "on) if the validation zero one error does not decrease by at least a certain amount (precise numbers", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "are provided in the appendix since they vary across batch sizes). Due to space constraints, we present", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "only a subset of training error plots. Please see Appendix C.3 for some more plots on training errors.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 511, + 506, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 616, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "Effect of minibatch sizes: Our first experiment tries to understand how the performance", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "of HB and NAG compare with that of SGD and how it varies with minibatch sizes. Figure 4 presents", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "the test zero one error for minibatch sizes of 8 and 128. While training with batch size 8 was done", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 650, + 504, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 504, + 660 + ], + "score": 1.0, + "content": "for 40 epochs, with batch size 128, it was done for 120 epochs. We perform a grid search over all", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "parameters for each of these algorithms. See Appendix C.3 for details on the grid search parameters.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "We observe that final error achieved by SGD, HB and NAG are all very close for both batch sizes.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "While NAG exhibits a superior rate of convergence compared to SGD and HB for batch size 128,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 692, + 368, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 368, + 705 + ], + "score": 1.0, + "content": "this superior rate of convergence disappears for a batch size of 8.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 615, + 505, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Comparison of ASGD with momentum methods: The next experiment tries to understand", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "how ASGD compares with HB and NAG. The errors achieved by various methods when we do", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 116, + 90, + 490, + 182 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 116, + 90, + 490, + 182 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 90, + 490, + 182 + ], + "spans": [ + { + "bbox": [ + 116, + 90, + 490, + 182 + ], + "score": 0.967, + "type": "image", + "image_path": "db5f311c1e2dbe4b962051b725dbf673899c05a23640369f13b18c9e0f70e473.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 116, + 90, + 490, + 120.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 116, + 120.66666666666667, + 490, + 151.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 116, + 151.33333333333334, + 490, + 182.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 185, + 504, + 208 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 185, + 504, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 504, + 198 + ], + "score": 1.0, + "content": "Figure 4: Test zero one loss for batch size 128 (left), batch size 8 (center) and training function value", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 196, + 296, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 296, + 208 + ], + "score": 1.0, + "content": "for batch size 8 (right) for SGD, HB and NAG.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "image", + "bbox": [ + 117, + 221, + 489, + 314 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 221, + 489, + 314 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 117, + 221, + 489, + 314 + ], + "spans": [ + { + "bbox": [ + 117, + 221, + 489, + 314 + ], + "score": 0.966, + "type": "image", + "image_path": "9082f0ff9ebf68aa4e0752f19fd7b95ba1e73a84b488c3c29ff243cc33fb06fa.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 117, + 221, + 489, + 252.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 117, + 252.0, + 489, + 283.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 117, + 283.0, + 489, + 314.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 317, + 505, + 362 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "score": 1.0, + "content": "Figure 5: Test zero one loss for batch size 128 (left), batch size 8 (center) and training function value", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 328, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 341 + ], + "score": 1.0, + "content": "for batch size 8 (right) for ASGD compared to HB. In the above plots, both ASGD and ASGD-Hb-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "Params refer to ASGD run with the learning rate and decay schedule of HB. ASGD-Fully-Optimized", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 350, + 467, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 467, + 363 + ], + "score": 1.0, + "content": "refers to ASGD where learning rate and decay schedule were also selected by grid search.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + } + ], + "index": 7.75 + }, + { + "type": "table", + "bbox": [ + 145, + 417, + 466, + 482 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 371, + 504, + 404 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "grid search over all parameters are presented in Table 2. Note that the final test errors for batch size", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "128 are better than those for batch size 8 since the former was run for 120 epochs while the latter", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 393, + 319, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 319, + 405 + ], + "score": 1.0, + "content": "was run only for 40 epochs (due to time constraints).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "table_body", + "bbox": [ + 145, + 417, + 466, + 482 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 417, + 466, + 482 + ], + "spans": [ + { + "bbox": [ + 145, + 417, + 466, + 482 + ], + "score": 0.971, + "html": "
AlgorithmFinal test error-batch size 128Final test error-batch size 8
SGD8.32± 0.219.57±0.18
HB7.98 ± 0.199.28± 0.25
NAG7.63 ± 0.189.07 ±0.18
ASGD7.23 ± 0.228.52 ± 0.16
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The hyperpa-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 501, + 277, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 277, + 513 + ], + "score": 1.0, + "content": "rameters have been chosen by grid search.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 526, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "While the final error achieved by ASGD is similar/favorable compared to all other methods, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "are also interested in understanding whether ASGD has a superior convergence speed. For this", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 549, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 561 + ], + "score": 1.0, + "content": "experiment, we need to address the issue of differing learning rates used by various algorithms and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "different iterations where they decay learning rates. So, for each of HB and NAG, we choose the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "learning rate and decay factors by grid search, use these values for ASGD and do grid search only", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 211, + 594 + ], + "score": 1.0, + "content": "over long step parameter", + "type": "text" + }, + { + "bbox": [ + 212, + 583, + 219, + 591 + ], + "score": 0.69, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 582, + 288, + 594 + ], + "score": 1.0, + "content": "and momentum", + "type": "text" + }, + { + "bbox": [ + 288, + 583, + 296, + 591 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "for ASGD. The results are presented in Figures 5", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "and 6. For batch size 128, ASGD decays error at a faster rate compared to both HB and NAG.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 603, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 616 + ], + "score": 1.0, + "content": "For batch size 8, while we see a superior convergence of ASGD compared to NAG, we do not see", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "this superiority over HB. 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These results suggest that ASGD decays error at a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 659, + 369, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 369, + 670 + ], + "score": 1.0, + "content": "faster rate compared to HB and NAG across different batch sizes.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 108, + 682, + 211, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 681, + 213, + 697 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 213, + 697 + ], + "score": 1.0, + "content": "6 RELATED WORK", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 504, + 722 + ], + "score": 1.0, + "content": "First order oracle methods: The primary method in this family is Gradient Descent (GD) (Cauchy,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 718, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 506, + 734 + ], + "score": 1.0, + "content": "1847). 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Note that the final test errors for batch size", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "128 are better than those for batch size 8 since the former was run for 120 epochs while the latter", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 393, + 319, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 319, + 405 + ], + "score": 1.0, + "content": "was run only for 40 epochs (due to time constraints).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "table_body", + "bbox": [ + 145, + 417, + 466, + 482 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 417, + 466, + 482 + ], + "spans": [ + { + "bbox": [ + 145, + 417, + 466, + 482 + ], + "score": 0.971, + "html": "
AlgorithmFinal test error-batch size 128Final test error-batch size 8
SGD8.32± 0.219.57±0.18
HB7.98 ± 0.199.28± 0.25
NAG7.63 ± 0.189.07 ±0.18
ASGD7.23 ± 0.228.52 ± 0.16
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(2016), which indicate acceleration is possible with noisy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "gradients do not hold in the SFO model satisfied by algorithms that are run in practice (see Jain et al.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 384, + 207, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 207, + 397 + ], + "score": 1.0, + "content": "(2017) for more details).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 108, + 402, + 504, + 424 + ], + "lines": [ + { + "bbox": [ + 107, + 402, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 107, + 402, + 505, + 413 + ], + "score": 1.0, + "content": "While HB (Polyak, 1964) and NAG (Nesterov, 1983) are known to be effective in case of exact first", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 412, + 483, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 483, + 424 + ], + "score": 1.0, + "content": "order oracle, for the SFO, the theoretical performance of HB and NAG is not well understood.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 429, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "Understanding Stochastic Heavy Ball: Understanding HB’s performance with inexact gradients", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "has been considered in efforts spanning several decades, in many communities like controls, opti-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "mization and signal processing. Polyak (1987) considered HB with noisy gradients and concluded", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "that the improvements offered by HB with inexact gradients vanish unless strong assumptions on", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 474, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 485 + ], + "score": 1.0, + "content": "the inexactness was considered; an instance of this is when the variance of inexactness decreased", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "as the iterates approach the minimizer. Proakis (1974); Roy & Shynk (1990); Sharma et al. 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These methods have", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "been improved using accelerated variants (Shalev-Shwartz & Zhang, 2014; Frostig et al., 2015a; Lin", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "et al., 2015; Defazio, 2016; Allen-Zhu, 2016). Note that these methods require storing the entire", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "training set in memory and taking multiple passes over the same for guaranteed progress. 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A", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 341, + 504, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 504, + 352 + ], + "score": 1.0, + "content": "result of Jain et al. (2017) developed the first provably accelerated SGD method for linear regression", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 352, + 504, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 504, + 363 + ], + "score": 1.0, + "content": "which achieved minimax rates, inspired by a method of Nesterov (2012b). Schemes of Ghadimi", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "& Lan (2012; 2013); Dieuleveut et al. 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(1998)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 496, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 507 + ], + "score": 1.0, + "content": "suggest that the improved non-asymptotic rates offered by stochastic HB arose at the cost of worse", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "asymptotic behavior. We resolve these unquantified improvements on rates as being just constant", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "factors over SGD, in stark contrast to the gains offered by ASGD. Loizou & Richtarik ´ (2017) state", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "their method as Stochastic HB but require stochastic gradients that nearly behave as exact gradients;", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 539, + 504, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 504, + 551 + ], + "score": 1.0, + "content": "indeed, their rates match that of the standard HB method (Polyak, 1964). Such rates are not infor-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "mation theoretically possible (see Jain et al. (2017)), especially with a batch size of 1 or even with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 561, + 218, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 218, + 573 + ], + "score": 1.0, + "content": "constant sized minibatches.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 429, + 506, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "Accelerated and Fast Methods for finite-sums: There have been developments pertaining to faster", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 589, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 600 + ], + "score": 1.0, + "content": "methods for finite-sums (also known as offline stochastic optimization): amongst these are methods", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "such as SDCA (Shalev-Shwartz & Zhang, 2012), SAG (Roux et al., 2012), SVRG (Johnson &", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "Zhang, 2013), SAGA (Defazio et al., 2014), which offer linear convergence rates for strongly con-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "vex finite-sums, improving over SGD’s sub-linear rates (Rakhlin et al., 2012). These methods have", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "been improved using accelerated variants (Shalev-Shwartz & Zhang, 2014; Frostig et al., 2015a; Lin", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "et al., 2015; Defazio, 2016; Allen-Zhu, 2016). Note that these methods require storing the entire", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "training set in memory and taking multiple passes over the same for guaranteed progress. Further-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "more, these methods require computing a batch gradient or require memory requirements (typically", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 122, + 689 + ], + "score": 0.79, + "content": "\\Omega ( \\lfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "training data points|)). For deep learning problems, data augmentation is often deemed neces-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "sary for achieving good performance; this implies computing quantities such as batch gradient (or", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "storage necessities) over this augmented dataset is often infeasible. Such requirements are miti-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "gated by the use of simple streaming methods such as SGD, ASGD, HB, NAG. For other technical", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 476, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 476, + 732 + ], + "score": 1.0, + "content": "distinctions between the offline and online stochastic methods refer to Frostig et al. (2015b).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 577, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 181 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 504, + 94 + ], + "score": 1.0, + "content": "Practical methods for training deep networks: Momentum based methods employed with", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "stochastic gradients (Sutskever et al., 2013) have become standard and very popular in practice.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "These schemes tend to outperform standard SGD on several important practical problems. As previ-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "ously mentioned, we attribute this improvement to effect of mini-batching rather than improvement", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "offered by HB or NAG in the SFO model. Schemes such as Adagrad (Duchi et al., 2011), RM-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "SProp (Tieleman & Hinton, 2012), Adam (Kingma & Ba, 2014) represent an important and useful", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "score": 1.0, + "content": "class of algorithms. The advantages offered by these methods are orthogonal to the advantages of-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "fered by fast gradient methods; it is an important direction to explore augmenting these methods", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 414, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 414, + 183 + ], + "score": 1.0, + "content": "with ASGD as opposed to standard HB or NAG based acceleration schemes.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 507, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 507, + 201 + ], + "score": 1.0, + "content": "Chaudhari et al. (2017) proposed Entropy-SGD, which is an altered objective that adds a local strong", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "convexity term to the actual empirical risk objective, with an aim to improve generalization. How-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "ever, we do not understand convergence rates for convex problems or the generalization ability of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "this technique in a rigorous manner. Chaudhari et al. (2017) propose to use SGD in their procedure", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "but mention that they employ the HB/NAG method in their implementation for achieving better per-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 240, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 255 + ], + "score": 1.0, + "content": "formance. Naturally, we can use ASGD in this context. Path normalized SGD (Neyshabur et al.,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "2015) is a variant of SGD that alters the metric on which the weights are optimized. As noted in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 502, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 502, + 276 + ], + "score": 1.0, + "content": "their paper, path normalized SGD could be improved using HB/NAG (or even the ASGD method).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 292, + 339, + 305 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 341, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 341, + 308 + ], + "score": 1.0, + "content": "7 CONCLUSIONS AND FUTURE DIRECTIONS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "In this paper, we show that the performance gain of HB over SGD in stochastic setting is attributed to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "mini-batching rather than the algorithm’s ability to accelerate with stochastic gradients. Concretely,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "we provide a formal proof that for several easy problem instances, HB does not outperform SGD", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "despite large condition number of the problem; we observe this trend for NAG in our experiments.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "In contrast, ASGD (Jain et al., 2017) provides significant improvement over SGD for these problem", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "instances. We observe similar trends when training a resnet on cifar-10 and an autoencoder on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 385, + 504, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 504, + 396 + ], + "score": 1.0, + "content": "mnist. This work motivates several directions such as understanding the behavior of ASGD on", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 394, + 498, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 498, + 407 + ], + "score": 1.0, + "content": "domains such as NLP, and developing automatic momentum tuning schemes (Zhang et al., 2017).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 108, + 420, + 200, + 430 + ], + "lines": [ + { + "bbox": [ + 107, + 421, + 200, + 431 + ], + "spans": [ + { + "bbox": [ + 107, + 421, + 200, + 431 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 438, + 469, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 470, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 470, + 452 + ], + "score": 1.0, + "content": "Sham Kakade acknowledges funding from NSF Awards CCF-1703574 and CCF-1740551.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 468, + 175, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 176, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 176, + 481 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "Zeyuan Allen-Zhu. 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Mathematical Programming, 146:37–75, 2014.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 181 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 504, + 94 + ], + "score": 1.0, + "content": "Practical methods for training deep networks: Momentum based methods employed with", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "stochastic gradients (Sutskever et al., 2013) have become standard and very popular in practice.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "These schemes tend to outperform standard SGD on several important practical problems. As previ-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "ously mentioned, we attribute this improvement to effect of mini-batching rather than improvement", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "offered by HB or NAG in the SFO model. Schemes such as Adagrad (Duchi et al., 2011), RM-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "SProp (Tieleman & Hinton, 2012), Adam (Kingma & Ba, 2014) represent an important and useful", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "score": 1.0, + "content": "class of algorithms. The advantages offered by these methods are orthogonal to the advantages of-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "fered by fast gradient methods; it is an important direction to explore augmenting these methods", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 414, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 414, + 183 + ], + "score": 1.0, + "content": "with ASGD as opposed to standard HB or NAG based acceleration schemes.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 82, + 505, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 507, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 507, + 201 + ], + "score": 1.0, + "content": "Chaudhari et al. (2017) proposed Entropy-SGD, which is an altered objective that adds a local strong", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "convexity term to the actual empirical risk objective, with an aim to improve generalization. How-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "ever, we do not understand convergence rates for convex problems or the generalization ability of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "this technique in a rigorous manner. Chaudhari et al. (2017) propose to use SGD in their procedure", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "but mention that they employ the HB/NAG method in their implementation for achieving better per-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 240, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 255 + ], + "score": 1.0, + "content": "formance. Naturally, we can use ASGD in this context. Path normalized SGD (Neyshabur et al.,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "2015) is a variant of SGD that alters the metric on which the weights are optimized. As noted in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 502, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 502, + 276 + ], + "score": 1.0, + "content": "their paper, path normalized SGD could be improved using HB/NAG (or even the ASGD method).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 185, + 507, + 276 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 292, + 339, + 305 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 341, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 341, + 308 + ], + "score": 1.0, + "content": "7 CONCLUSIONS AND FUTURE DIRECTIONS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "In this paper, we show that the performance gain of HB over SGD in stochastic setting is attributed to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "mini-batching rather than the algorithm’s ability to accelerate with stochastic gradients. Concretely,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "we provide a formal proof that for several easy problem instances, HB does not outperform SGD", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "despite large condition number of the problem; we observe this trend for NAG in our experiments.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "In contrast, ASGD (Jain et al., 2017) provides significant improvement over SGD for these problem", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "instances. We observe similar trends when training a resnet on cifar-10 and an autoencoder on", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 385, + 504, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 504, + 396 + ], + "score": 1.0, + "content": "mnist. This work motivates several directions such as understanding the behavior of ASGD on", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 394, + 498, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 498, + 407 + ], + "score": 1.0, + "content": "domains such as NLP, and developing automatic momentum tuning schemes (Zhang et al., 2017).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 318, + 505, + 407 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 420, + 200, + 430 + ], + "lines": [ + { + "bbox": [ + 107, + 421, + 200, + 431 + ], + "spans": [ + { + "bbox": [ + 107, + 421, + 200, + 431 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 438, + 469, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 470, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 470, + 452 + ], + "score": 1.0, + "content": "Sham Kakade acknowledges funding from NSF Awards CCF-1703574 and CCF-1740551.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 438, + 470, + 452 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 468, + 175, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 176, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 176, + 481 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "Zeyuan Allen-Zhu. 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Let", + "type": "text" + }, + { + "bbox": [ + 417, + 104, + 438, + 121 + ], + "score": 0.93, + "content": "\\pmb { \\theta } _ { t + 1 } ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 99, + 505, + 127 + ], + "score": 1.0, + "content": "denote the con-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 119, + 366, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 264, + 133 + ], + "score": 1.0, + "content": "catenated and centered estimates in the", + "type": "text" + }, + { + "bbox": [ + 264, + 120, + 276, + 132 + ], + "score": 0.89, + "content": "j ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 119, + 329, + 133 + ], + "score": 1.0, + "content": "direction for", + "type": "text" + }, + { + "bbox": [ + 330, + 120, + 363, + 132 + ], + "score": 0.91, + "content": "j = 1 , 2", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 119, + 366, + 133 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 138, + 387, + 172 + ], + "lines": [ + { + "bbox": [ + 223, + 138, + 387, + 172 + ], + "spans": [ + { + "bbox": [ + 223, + 138, + 387, + 172 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\pmb { \\theta } _ { t + 1 } ^ { ( j ) } \\stackrel { \\mathrm { d e f } } { = } \\left[ \\mathbf { w } _ { t + 1 } ^ { ( j ) } - ( \\mathbf { w } ^ { * } ) ^ { ( j ) } \\right] , \\quad j = 1 , 2 . } \\end{array}", + "type": "interline_equation", + "image_path": "2d7f7d93663979170a21ec6947f999faf51d397b4be388611d75c371d13ec849.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 138, + 387, + 155.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 223, + 155.0, + 387, + 172.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 180, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 103, + 178, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 103, + 178, + 217, + 196 + ], + "score": 1.0, + "content": "Since the distribution over", + "type": "text" + }, + { + "bbox": [ + 217, + 184, + 225, + 192 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 178, + 453, + 196 + ], + "score": 1.0, + "content": "is such that the coordinates are decoupled, we see that", + "type": "text" + }, + { + "bbox": [ + 453, + 179, + 474, + 195 + ], + "score": 0.93, + "content": "\\pmb { \\theta } _ { t + 1 } ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 178, + 506, + 196 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 194, + 215, + 211 + ], + "spans": [ + { + "bbox": [ + 104, + 194, + 183, + 211 + ], + "score": 1.0, + "content": "written in terms of", + "type": "text" + }, + { + "bbox": [ + 183, + 194, + 200, + 209 + ], + "score": 0.94, + "content": "\\pmb { \\theta } _ { t } ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 194, + 215, + 211 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 216, + 428, + 244 + ], + "lines": [ + { + "bbox": [ + 181, + 216, + 428, + 244 + ], + "spans": [ + { + "bbox": [ + 181, + 216, + 428, + 244 + ], + "score": 0.91, + "content": "\\pmb { \\theta } _ { t + 1 } ^ { ( j ) } = \\widehat { \\mathbf { A } } _ { t + 1 } ^ { ( j ) } \\pmb { \\theta } _ { t } ^ { ( j ) } , \\mathrm { w i t h } \\widehat { \\mathbf { A } } _ { t + 1 } ^ { ( j ) } = \\left[ \\frac { 1 + \\alpha - \\delta ( a _ { t + 1 } ^ { ( j ) } ) ^ { 2 } } { 1 } - \\alpha \\right] .", + "type": "interline_equation", + "image_path": "1e9087439006256f226453054d397c1ad759b357bf7d62525d93d55832d52800.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 181, + 216, + 428, + 244 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 503, + 283 + ], + "lines": [ + { + "bbox": [ + 100, + 245, + 504, + 275 + ], + "spans": [ + { + "bbox": [ + 100, + 245, + 123, + 275 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 124, + 251, + 231, + 271 + ], + "score": 0.92, + "content": "\\Phi _ { t + 1 } ^ { ( j ) } \\ { \\stackrel { \\mathrm { d e f } } { = } } \\ \\mathbb { E } \\left[ \\pmb { \\theta } _ { t + 1 } ^ { ( j ) } \\otimes \\pmb { \\theta } _ { t + 1 } ^ { ( j ) } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 245, + 365, + 275 + ], + "score": 1.0, + "content": "denote the covariance matrix of", + "type": "text" + }, + { + "bbox": [ + 365, + 252, + 385, + 268 + ], + "score": 0.93, + "content": "\\pmb { \\theta } _ { t + 1 } ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 245, + 429, + 275 + ], + "score": 1.0, + "content": ". 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The characteristic polynomial of", + "type": "text" + }, + { + "bbox": [ + 287, + 698, + 295, + 707 + ], + "score": 0.74, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 696, + 308, + 711 + ], + "score": 1.0, + "content": "is:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 714, + 477, + 730 + ], + "lines": [ + { + "bbox": [ + 135, + 714, + 477, + 730 + ], + "spans": [ + { + "bbox": [ + 135, + 714, + 477, + 730 + ], + "score": 0.89, + "content": "D ( z ) = z ^ { 4 } - ( t ^ { 2 } + ( c - 1 ) x ^ { 2 } ) z ^ { 3 } + ( 2 \\alpha t ^ { 2 } - 2 \\alpha ^ { 2 } ) z ^ { 2 } + ( - t ^ { 2 } + ( c - 1 ) x ^ { 2 } ) \\alpha ^ { 2 } z + \\alpha ^ { 4 } .", + "type": "interline_equation", + "image_path": "f4144805b38ad8db88f573b95f9fd054cd912f90e52958624cb428369ce3c519.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 135, + 714, + 477, + 730 + ], + "spans": [], + "index": 30 + } + ] + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "13", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 80, + 393, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 394, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 394, + 96 + ], + "score": 1.0, + "content": "A SUBOPTIMALITY OF HB: PROOF OF PROPOSITION 3", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 503, + 132 + ], + "lines": [ + { + "bbox": [ + 102, + 99, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 102, + 99, + 417, + 127 + ], + "score": 1.0, + "content": "Before proceeding to the proof, we introduce some additional notation. 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If the stepsize", + "type": "text" + }, + { + "bbox": [ + 211, + 493, + 217, + 502 + ], + "score": 0.65, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 485, + 266, + 508 + ], + "score": 1.0, + "content": "is such that", + "type": "text" + }, + { + "bbox": [ + 266, + 487, + 332, + 507 + ], + "score": 0.96, + "content": "\\delta \\sigma _ { 1 } ^ { 2 } \\geq \\frac { 2 \\left( 1 - \\alpha ^ { 2 } \\right) } { c + ( c - 2 ) \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 485, + 356, + 508 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 356, + 491, + 375, + 503 + ], + "score": 0.88, + "content": "\\boldsymbol { B } ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 485, + 450, + 508 + ], + "score": 1.0, + "content": "has an eigenvalue", + "type": "text" + }, + { + "bbox": [ + 450, + 493, + 467, + 504 + ], + "score": 0.8, + "content": "\\geq 1", + "type": "inline_equation" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 103, + 485, + 467, + 508 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 104, + 509, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 210, + 532 + ], + "score": 1.0, + "content": "Lemma 5. If the stepsize", + "type": "text" + }, + { + "bbox": [ + 211, + 517, + 217, + 527 + ], + "score": 0.73, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 509, + 266, + 532 + ], + "score": 1.0, + "content": "is such that", + "type": "text" + }, + { + "bbox": [ + 266, + 511, + 332, + 531 + ], + "score": 0.95, + "content": "\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } < \\frac { 2 \\left( 1 - \\alpha ^ { 2 } \\right) } { c + ( c - 2 ) \\alpha } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 509, + 356, + 532 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 356, + 515, + 375, + 527 + ], + "score": 0.88, + "content": "B ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 509, + 506, + 532 + ], + "score": 1.0, + "content": "has an eigenvalue of magnitude", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 529, + 149, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 149, + 545 + ], + "score": 0.89, + "content": "\\textstyle { \\ge } 1 - { \\frac { 5 0 0 } { \\kappa } }", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 509, + 506, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 504, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 288, + 568 + ], + "score": 1.0, + "content": "Given this notation, we can now consider the", + "type": "text" + }, + { + "bbox": [ + 288, + 554, + 302, + 567 + ], + "score": 0.91, + "content": "j ^ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 552, + 506, + 568 + ], + "score": 1.0, + "content": "dimension without the superscripts; when needed,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 103, + 563, + 473, + 585 + ], + "spans": [ + { + "bbox": [ + 103, + 563, + 313, + 585 + ], + "score": 1.0, + "content": "they will be made clear in the exposition. Denoting", + "type": "text" + }, + { + "bbox": [ + 313, + 567, + 350, + 580 + ], + "score": 0.8, + "content": "x \\stackrel { \\mathrm { d e f } } { = } \\delta \\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 563, + 368, + 585 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 369, + 567, + 429, + 580 + ], + "score": 0.9, + "content": "t \\stackrel { \\mathrm { d e f } } { = } 1 + \\alpha - x", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 563, + 473, + 585 + ], + "score": 1.0, + "content": ", we have:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 103, + 552, + 506, + 585 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 587, + 392, + 635 + ], + "lines": [ + { + "bbox": [ + 219, + 587, + 392, + 635 + ], + "spans": [ + { + "bbox": [ + 219, + 587, + 392, + 635 + ], + "score": 0.93, + "content": "\\begin{array} { r } { B = \\left[ \\begin{array} { c c c c } { t ^ { 2 } + ( c - 1 ) x ^ { 2 } } & { - \\alpha t } & { - \\alpha t } & { \\alpha ^ { 2 } } \\\\ { t } & { 0 } & { - \\alpha } & { 0 } \\\\ { t } & { - \\alpha } & { 0 } & { 0 } \\\\ { 1 } & { 0 } & { 0 } & { 0 } \\end{array} \\right] } \\end{array}", + "type": "interline_equation", + "image_path": "7a3d15dc1dc2a465faf69d0d032b93a68286dfd83e1add214990fb4de90fc945.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 219, + 587, + 392, + 603.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 219, + 603.0, + 392, + 619.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 219, + 619.0, + 392, + 635.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 649, + 163, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 164, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 164, + 662 + ], + "score": 1.0, + "content": "A.1 PROOF", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 670, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 669, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 383, + 684 + ], + "score": 1.0, + "content": "The analysis goes via computation of the characteristic polynomial of", + "type": "text" + }, + { + "bbox": [ + 383, + 672, + 391, + 681 + ], + "score": 0.84, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 669, + 505, + 684 + ], + "score": 1.0, + "content": "and evaluating it at different", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 682, + 252, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 252, + 694 + ], + "score": 1.0, + "content": "values to obtain bounds on its roots.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 106, + 669, + 505, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 697, + 307, + 709 + ], + "lines": [ + { + "bbox": [ + 105, + 696, + 308, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 286, + 711 + ], + "score": 1.0, + "content": "Lemma 6. The characteristic polynomial of", + "type": "text" + }, + { + "bbox": [ + 287, + 698, + 295, + 707 + ], + "score": 0.74, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 696, + 308, + 711 + ], + "score": 1.0, + "content": "is:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 696, + 308, + 711 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 714, + 477, + 730 + ], + "lines": [ + { + "bbox": [ + 135, + 714, + 477, + 730 + ], + "spans": [ + { + "bbox": [ + 135, + 714, + 477, + 730 + ], + "score": 0.89, + "content": "D ( z ) = z ^ { 4 } - ( t ^ { 2 } + ( c - 1 ) x ^ { 2 } ) z ^ { 3 } + ( 2 \\alpha t ^ { 2 } - 2 \\alpha ^ { 2 } ) z ^ { 2 } + ( - t ^ { 2 } + ( c - 1 ) x ^ { 2 } ) \\alpha ^ { 2 } z + \\alpha ^ { 4 } .", + "type": "interline_equation", + "image_path": "f4144805b38ad8db88f573b95f9fd054cd912f90e52958624cb428369ce3c519.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 135, + 714, + 477, + 730 + ], + "spans": [], + "index": 30 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 394, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 394, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 394, + 96 + ], + "score": 1.0, + "content": "Proof. We first begin by writing out the expression for the determinant:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 98, + 424, + 146 + ], + "lines": [ + { + "bbox": [ + 186, + 98, + 424, + 146 + ], + "spans": [ + { + "bbox": [ + 186, + 98, + 424, + 146 + ], + "score": 0.94, + "content": "D e t ( B - z { \\mathcal { Z } } ) = \\left| \\begin{array} { c c c c } { t ^ { 2 } + ( c - 1 ) x ^ { 2 } - z } & { - \\alpha t } & { - \\alpha t } & { \\alpha ^ { 2 } } \\\\ { t } & { - z } & { - \\alpha } & { 0 } \\\\ { t } & { - \\alpha } & { - z } & { 0 } \\\\ { 1 } & { 0 } & { 0 } & { - z } \\end{array} \\right| .", + "type": "interline_equation", + "image_path": "ea76d69feaee8e0f22640391a543c329bef208829a17e3ee0b71a625e9d2db1f.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 186, + 98, + 424, + 114.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 186, + 114.0, + 424, + 130.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 186, + 130.0, + 424, + 146.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 151, + 280, + 163 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 281, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 281, + 164 + ], + "score": 1.0, + "content": "expanding along the first column, we have:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 167, + 514, + 199 + ], + "lines": [ + { + "bbox": [ + 111, + 167, + 514, + 199 + ], + "spans": [ + { + "bbox": [ + 111, + 167, + 514, + 199 + ], + "score": 0.88, + "content": "\\begin{array} { r l } & { \\gamma _ { e t } ( B - z \\mathcal { Z } ) = ( t ^ { 2 } + ( c - 1 ) x ^ { 2 } - z ) ( \\alpha ^ { 2 } z - z ^ { 3 } ) - t ( - \\alpha t z ^ { 2 } + \\alpha ^ { 2 } t z ) + t ( - \\alpha t ( \\alpha z ) + z \\cdot \\alpha t z ) - ( z \\cdot \\alpha ^ { 2 } z - \\alpha t \\mathcal { Z } ) } \\\\ & { \\qquad = ( t ^ { 2 } + ( c - 1 ) x ^ { 2 } - z ) ( \\alpha ^ { 2 } z - z ^ { 3 } ) - 2 t ( \\alpha ^ { 2 } t z - \\alpha t z ^ { 2 } ) - ( \\alpha ^ { 2 } z ^ { 2 } - \\alpha ^ { 4 } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "13db8664f5f384bd3705cf4d39cce332c87323864434569fa790e37f71bc0b1b.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 111, + 167, + 514, + 177.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 111, + 177.66666666666666, + 514, + 188.33333333333331 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 111, + 188.33333333333331, + 514, + 198.99999999999997 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 202, + 336, + 214 + ], + "lines": [ + { + "bbox": [ + 109, + 201, + 337, + 215 + ], + "spans": [ + { + "bbox": [ + 109, + 201, + 337, + 215 + ], + "score": 1.0, + "content": "Expanding the terms yields the expression in the lemma.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 105, + 225, + 387, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 385, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 385, + 240 + ], + "score": 1.0, + "content": "The next corollary follows by some simple arithmetic manipulations.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 105, + 240, + 461, + 252 + ], + "lines": [ + { + "bbox": [ + 106, + 239, + 464, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 213, + 254 + ], + "score": 1.0, + "content": "Corollary 7. Substituting", + "type": "text" + }, + { + "bbox": [ + 213, + 241, + 255, + 251 + ], + "score": 0.9, + "content": "z = 1 - \\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 239, + 418, + 254 + ], + "score": 1.0, + "content": "in the characteristic equation of Lemma", + "type": "text" + }, + { + "bbox": [ + 418, + 241, + 424, + 250 + ], + "score": 0.29, + "content": "6", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 239, + 464, + 254 + ], + "score": 1.0, + "content": ", we have:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 255, + 481, + 369 + ], + "lines": [ + { + "bbox": [ + 127, + 255, + 481, + 369 + ], + "spans": [ + { + "bbox": [ + 127, + 255, + 481, + 369 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { D ( 1 - \\tau ) = \\tau ^ { 4 } + \\tau ^ { 3 } ( - 4 + t ^ { 2 } + ( c - 1 ) x ^ { 2 } ) + \\tau ^ { 2 } ( 6 - 3 t ^ { 2 } - 3 ( c - 1 ) x ^ { 2 } - 2 \\alpha ^ { 2 } + 2 \\alpha t ^ { 2 } ) } \\\\ & { \\qquad + \\tau ( - 4 + 3 t ^ { 2 } + 3 ( c - 1 ) x ^ { 2 } + 4 \\alpha ^ { 2 } - 4 \\alpha t ^ { 2 } - ( c - 1 ) x ^ { 2 } \\alpha ^ { 2 } + t ^ { 2 } \\alpha ^ { 2 } ) } \\\\ & { \\qquad + ( 1 - t ^ { 2 } - ( c - 1 ) x ^ { 2 } - 2 \\alpha ^ { 2 } + 2 \\alpha t ^ { 2 } + ( c - 1 ) x ^ { 2 } \\alpha ^ { 2 } - t ^ { 2 } \\alpha ^ { 2 } + \\alpha ^ { 4 } ) } \\\\ & { \\qquad = \\tau ^ { 4 } + \\tau ^ { 3 } [ - ( 3 + \\alpha ) ( 1 - \\alpha ) - 2 x ( 1 + \\alpha ) + c x ^ { 2 } ] } \\\\ & { \\qquad + \\tau ^ { 2 } [ ( 3 - 4 \\alpha - \\alpha ^ { 2 } + 2 \\alpha ^ { 3 } ) - 2 x ( 1 + \\alpha ) ( 2 \\alpha - 3 ) + x ^ { 2 } ( 2 \\alpha - 3 c ) ] } \\\\ & { \\qquad + \\tau [ - ( 1 - \\alpha ) ^ { 2 } ( 1 - \\alpha ^ { 2 } ) - 2 x ( 3 - \\alpha ) ( 1 - \\alpha ^ { 2 } ) + x ^ { 2 } ( 3 c - 4 \\alpha + ( 2 - c ) \\alpha ^ { 2 } ) ] } \\\\ & { \\qquad + x ( 1 - \\alpha ) [ 2 ( 1 - \\alpha ^ { 2 } ) - x ( c + ( c - 2 ) \\alpha ) ] . } \\end{array}", + "type": "interline_equation", + "image_path": "1dcd7a6dadb1f017f50da83ae925f64de04e3cc31c884cb27b30f5e8c299e4a5.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 127, + 255, + 481, + 293.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 127, + 293.0, + 481, + 331.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 127, + 331.0, + 481, + 369.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 377, + 504, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 504, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 504, + 391 + ], + "score": 1.0, + "content": "Proof of Lemma 4. The first observation necessary to prove the lemma is that the characteristic poly-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 388, + 378, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 137, + 401 + ], + "score": 1.0, + "content": "nomial", + "type": "text" + }, + { + "bbox": [ + 137, + 389, + 159, + 401 + ], + "score": 0.92, + "content": "D ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 388, + 208, + 401 + ], + "score": 1.0, + "content": "approaches", + "type": "text" + }, + { + "bbox": [ + 208, + 390, + 219, + 398 + ], + "score": 0.75, + "content": "\\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 388, + 231, + 401 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 231, + 390, + 263, + 399 + ], + "score": 0.85, + "content": "z \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 388, + 284, + 401 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 284, + 388, + 373, + 401 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { z \\infty } D ( z ) = + \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 388, + 378, + 401 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 372, + 418 + ], + "score": 1.0, + "content": "Next, we evaluate the characteristic polynomial at 1, i.e. compute", + "type": "text" + }, + { + "bbox": [ + 372, + 405, + 394, + 417 + ], + "score": 0.9, + "content": "D ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 405, + 505, + 418 + ], + "score": 1.0, + "content": ". This follows in a straight-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 415, + 461, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 308, + 429 + ], + "score": 1.0, + "content": "forward manner from corollary (7) by substituting", + "type": "text" + }, + { + "bbox": [ + 309, + 417, + 334, + 426 + ], + "score": 0.9, + "content": "\\tau = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 415, + 461, + 429 + ], + "score": 1.0, + "content": "in equation (2), and this yields,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 432, + 439, + 460 + ], + "lines": [ + { + "bbox": [ + 172, + 432, + 439, + 460 + ], + "spans": [ + { + "bbox": [ + 172, + 432, + 439, + 460 + ], + "score": 0.93, + "content": "D ( 1 ) = ( 1 - \\alpha ) x \\cdot \\bigg ( 2 ( 1 - \\alpha ^ { 2 } ) - x ( 1 - \\alpha ) - ( c - 1 ) x ( 1 + \\alpha ) \\bigg ) .", + "type": "interline_equation", + "image_path": "e0bd88d6953eb39b22592e4d54f9d03958f770d27b69406b5c6350d5b00c653d.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 172, + 432, + 439, + 460 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 465, + 455, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 456, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 120, + 479 + ], + "score": 1.0, + "content": "As", + "type": "text" + }, + { + "bbox": [ + 120, + 466, + 146, + 477 + ], + "score": 0.85, + "content": "\\alpha < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 464, + 149, + 479 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 150, + 465, + 204, + 477 + ], + "score": 0.9, + "content": "x = \\delta \\sigma ^ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 464, + 339, + 479 + ], + "score": 1.0, + "content": ", we have the following by setting", + "type": "text" + }, + { + "bbox": [ + 340, + 466, + 380, + 478 + ], + "score": 0.93, + "content": "D ( 1 ) \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 464, + 444, + 479 + ], + "score": 1.0, + "content": "and solving for", + "type": "text" + }, + { + "bbox": [ + 445, + 469, + 451, + 476 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 464, + 456, + 479 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 266, + 483, + 344, + 510 + ], + "lines": [ + { + "bbox": [ + 266, + 483, + 344, + 510 + ], + "spans": [ + { + "bbox": [ + 266, + 483, + 344, + 510 + ], + "score": 0.95, + "content": "x \\geq \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } .", + "type": "interline_equation", + "image_path": "068e1807ff1d226a23c3f534bdac1e43bcbb26cf8aa58bf3a0da2d40f7fb882c.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 266, + 483, + 344, + 510 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 439, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 514, + 440, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 131, + 529 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 515, + 172, + 527 + ], + "score": 0.92, + "content": "D ( 1 ) \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 514, + 190, + 529 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 190, + 515, + 231, + 527 + ], + "score": 0.93, + "content": "D ( z ) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 514, + 243, + 529 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 243, + 517, + 275, + 525 + ], + "score": 0.86, + "content": "z \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 514, + 362, + 529 + ], + "score": 1.0, + "content": ", there exists a root of", + "type": "text" + }, + { + "bbox": [ + 362, + 515, + 382, + 527 + ], + "score": 0.91, + "content": "D ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 514, + 419, + 529 + ], + "score": 1.0, + "content": "which is", + "type": "text" + }, + { + "bbox": [ + 420, + 516, + 436, + 526 + ], + "score": 0.85, + "content": "\\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 514, + 440, + 529 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 506, + 608 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 415, + 545 + ], + "score": 1.0, + "content": "Remark 8. The above characterization is striking in the sense that for any", + "type": "text" + }, + { + "bbox": [ + 415, + 534, + 441, + 544 + ], + "score": 0.87, + "content": "c > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 533, + 505, + 545 + ], + "score": 1.0, + "content": ", increasing the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 545, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 196, + 557 + ], + "score": 1.0, + "content": "momentum parameter", + "type": "text" + }, + { + "bbox": [ + 196, + 547, + 204, + 554 + ], + "score": 0.28, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 545, + 393, + 557 + ], + "score": 1.0, + "content": "naturally requires the reduction in the step size", + "type": "text" + }, + { + "bbox": [ + 393, + 545, + 399, + 554 + ], + "score": 0.61, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 545, + 506, + 557 + ], + "score": 1.0, + "content": "to permit the convergence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 556, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 107, + 556, + 505, + 567 + ], + "score": 1.0, + "content": "of the algorithm, which is not observed when fast gradient methods are employed in deterministic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 419, + 579 + ], + "score": 1.0, + "content": "optimization. For instance, in the case of deterministic optimization, setting", + "type": "text" + }, + { + "bbox": [ + 420, + 567, + 447, + 577 + ], + "score": 0.89, + "content": "c = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 565, + 475, + 579 + ], + "score": 1.0, + "content": "yields", + "type": "text" + }, + { + "bbox": [ + 475, + 565, + 505, + 578 + ], + "score": 0.91, + "content": "\\delta \\sigma _ { 1 } ^ { 2 } <", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 576, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 107, + 578, + 144, + 591 + ], + "score": 0.9, + "content": "2 ( 1 + \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 576, + 453, + 592 + ], + "score": 1.0, + "content": ". On the other hand, when employing the stochastic heavy ball method with", + "type": "text" + }, + { + "bbox": [ + 453, + 578, + 501, + 592 + ], + "score": 0.94, + "content": "x ^ { ( j ) } = 2 \\sigma _ { j } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 576, + 505, + 592 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 591, + 417, + 610 + ], + "spans": [ + { + "bbox": [ + 104, + 591, + 214, + 610 + ], + "score": 1.0, + "content": "we have the condition that", + "type": "text" + }, + { + "bbox": [ + 214, + 595, + 237, + 604 + ], + "score": 0.88, + "content": "c = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 591, + 310, + 610 + ], + "score": 1.0, + "content": ", and this implies,", + "type": "text" + }, + { + "bbox": [ + 310, + 591, + 413, + 608 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } < \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { 2 } = 1 - \\alpha ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 591, + 417, + 610 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 396, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 397, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 397, + 631 + ], + "score": 1.0, + "content": "We now prove Lemma 5. We first consider the large momentum setting.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 630, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 290, + 642 + ], + "score": 1.0, + "content": "Lemma 9. When the momentum parameter", + "type": "text" + }, + { + "bbox": [ + 291, + 632, + 299, + 640 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 628, + 366, + 642 + ], + "score": 1.0, + "content": "is set such that", + "type": "text" + }, + { + "bbox": [ + 367, + 630, + 467, + 642 + ], + "score": 0.86, + "content": "1 - 4 5 0 / \\kappa \\leq \\alpha \\leq 1 , ~ ", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 630, + 473, + 640 + ], + "score": 0.33, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "has an", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 638, + 253, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 253, + 656 + ], + "score": 1.0, + "content": "eigenvalue of magnitude ≥ 1 − 450κ .", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 293, + 682 + ], + "score": 1.0, + "content": "Proof. This follows easily from the fact that", + "type": "text" + }, + { + "bbox": [ + 293, + 666, + 481, + 682 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\operatorname* { d e t } ( \\boldsymbol { B } ) = \\alpha ^ { 4 } = \\prod _ { j = 1 } ^ { 4 } \\lambda _ { j } ( \\boldsymbol { B } ) \\le ( \\lambda _ { \\operatorname* { m a x } } ( \\boldsymbol { B } ) ) ^ { 4 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 665, + 506, + 682 + ], + "score": 1.0, + "content": ", thus", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 679, + 502, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 145, + 695 + ], + "score": 1.0, + "content": "implying", + "type": "text" + }, + { + "bbox": [ + 145, + 680, + 263, + 693 + ], + "score": 0.93, + "content": "1 - 4 5 0 / \\kappa \\leq \\alpha \\leq | \\lambda _ { \\mathrm { m a x } } ( \\beta ) |", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 679, + 267, + 695 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 497, + 685, + 502, + 689 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 437, + 711 + ], + "score": 1.0, + "content": "Remark 10. Note that the above lemma holds for any value of the learning rate", + "type": "text" + }, + { + "bbox": [ + 438, + 699, + 443, + 709 + ], + "score": 0.62, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 698, + 505, + 711 + ], + "score": 1.0, + "content": ", and holds for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 202, + 723 + ], + "score": 1.0, + "content": "every eigen direction of", + "type": "text" + }, + { + "bbox": [ + 202, + 710, + 212, + 720 + ], + "score": 0.29, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 709, + 505, + 723 + ], + "score": 1.0, + "content": ". Thus, for “large” values of momentum, the behavior of stochastic heavy", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 377, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 377, + 733 + ], + "score": 1.0, + "content": "ball does degenerate to the behavior of stochastic gradient descent.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 515, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 496, + 517, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 496, + 517, + 505, + 526 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 202, + 505, + 213 + ], + "lines": [ + { + "bbox": [ + 496, + 204, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 496, + 204, + 505, + 214 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 394, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 394, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 394, + 96 + ], + "score": 1.0, + "content": "Proof. We first begin by writing out the expression for the determinant:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 394, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 98, + 424, + 146 + ], + "lines": [ + { + "bbox": [ + 186, + 98, + 424, + 146 + ], + "spans": [ + { + "bbox": [ + 186, + 98, + 424, + 146 + ], + "score": 0.94, + "content": "D e t ( B - z { \\mathcal { Z } } ) = \\left| \\begin{array} { c c c c } { t ^ { 2 } + ( c - 1 ) x ^ { 2 } - z } & { - \\alpha t } & { - \\alpha t } & { \\alpha ^ { 2 } } \\\\ { t } & { - z } & { - \\alpha } & { 0 } \\\\ { t } & { - \\alpha } & { - z } & { 0 } \\\\ { 1 } & { 0 } & { 0 } & { - z } \\end{array} \\right| .", + "type": "interline_equation", + "image_path": "ea76d69feaee8e0f22640391a543c329bef208829a17e3ee0b71a625e9d2db1f.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 186, + 98, + 424, + 114.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 186, + 114.0, + 424, + 130.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 186, + 130.0, + 424, + 146.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 151, + 280, + 163 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 281, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 281, + 164 + ], + "score": 1.0, + "content": "expanding along the first column, we have:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 150, + 281, + 164 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 167, + 514, + 199 + ], + "lines": [ + { + "bbox": [ + 111, + 167, + 514, + 199 + ], + "spans": [ + { + "bbox": [ + 111, + 167, + 514, + 199 + ], + "score": 0.88, + "content": "\\begin{array} { r l } & { \\gamma _ { e t } ( B - z \\mathcal { Z } ) = ( t ^ { 2 } + ( c - 1 ) x ^ { 2 } - z ) ( \\alpha ^ { 2 } z - z ^ { 3 } ) - t ( - \\alpha t z ^ { 2 } + \\alpha ^ { 2 } t z ) + t ( - \\alpha t ( \\alpha z ) + z \\cdot \\alpha t z ) - ( z \\cdot \\alpha ^ { 2 } z - \\alpha t \\mathcal { Z } ) } \\\\ & { \\qquad = ( t ^ { 2 } + ( c - 1 ) x ^ { 2 } - z ) ( \\alpha ^ { 2 } z - z ^ { 3 } ) - 2 t ( \\alpha ^ { 2 } t z - \\alpha t z ^ { 2 } ) - ( \\alpha ^ { 2 } z ^ { 2 } - \\alpha ^ { 4 } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "13db8664f5f384bd3705cf4d39cce332c87323864434569fa790e37f71bc0b1b.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 111, + 167, + 514, + 177.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 111, + 177.66666666666666, + 514, + 188.33333333333331 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 111, + 188.33333333333331, + 514, + 198.99999999999997 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 202, + 336, + 214 + ], + "lines": [ + { + "bbox": [ + 109, + 201, + 337, + 215 + ], + "spans": [ + { + "bbox": [ + 109, + 201, + 337, + 215 + ], + "score": 1.0, + "content": "Expanding the terms yields the expression in the lemma.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 109, + 201, + 337, + 215 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 225, + 387, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 385, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 385, + 240 + ], + "score": 1.0, + "content": "The next corollary follows by some simple arithmetic manipulations.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 225, + 385, + 240 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 240, + 461, + 252 + ], + "lines": [ + { + "bbox": [ + 106, + 239, + 464, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 213, + 254 + ], + "score": 1.0, + "content": "Corollary 7. Substituting", + "type": "text" + }, + { + "bbox": [ + 213, + 241, + 255, + 251 + ], + "score": 0.9, + "content": "z = 1 - \\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 239, + 418, + 254 + ], + "score": 1.0, + "content": "in the characteristic equation of Lemma", + "type": "text" + }, + { + "bbox": [ + 418, + 241, + 424, + 250 + ], + "score": 0.29, + "content": "6", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 239, + 464, + 254 + ], + "score": 1.0, + "content": ", we have:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 239, + 464, + 254 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 255, + 481, + 369 + ], + "lines": [ + { + "bbox": [ + 127, + 255, + 481, + 369 + ], + "spans": [ + { + "bbox": [ + 127, + 255, + 481, + 369 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { D ( 1 - \\tau ) = \\tau ^ { 4 } + \\tau ^ { 3 } ( - 4 + t ^ { 2 } + ( c - 1 ) x ^ { 2 } ) + \\tau ^ { 2 } ( 6 - 3 t ^ { 2 } - 3 ( c - 1 ) x ^ { 2 } - 2 \\alpha ^ { 2 } + 2 \\alpha t ^ { 2 } ) } \\\\ & { \\qquad + \\tau ( - 4 + 3 t ^ { 2 } + 3 ( c - 1 ) x ^ { 2 } + 4 \\alpha ^ { 2 } - 4 \\alpha t ^ { 2 } - ( c - 1 ) x ^ { 2 } \\alpha ^ { 2 } + t ^ { 2 } \\alpha ^ { 2 } ) } \\\\ & { \\qquad + ( 1 - t ^ { 2 } - ( c - 1 ) x ^ { 2 } - 2 \\alpha ^ { 2 } + 2 \\alpha t ^ { 2 } + ( c - 1 ) x ^ { 2 } \\alpha ^ { 2 } - t ^ { 2 } \\alpha ^ { 2 } + \\alpha ^ { 4 } ) } \\\\ & { \\qquad = \\tau ^ { 4 } + \\tau ^ { 3 } [ - ( 3 + \\alpha ) ( 1 - \\alpha ) - 2 x ( 1 + \\alpha ) + c x ^ { 2 } ] } \\\\ & { \\qquad + \\tau ^ { 2 } [ ( 3 - 4 \\alpha - \\alpha ^ { 2 } + 2 \\alpha ^ { 3 } ) - 2 x ( 1 + \\alpha ) ( 2 \\alpha - 3 ) + x ^ { 2 } ( 2 \\alpha - 3 c ) ] } \\\\ & { \\qquad + \\tau [ - ( 1 - \\alpha ) ^ { 2 } ( 1 - \\alpha ^ { 2 } ) - 2 x ( 3 - \\alpha ) ( 1 - \\alpha ^ { 2 } ) + x ^ { 2 } ( 3 c - 4 \\alpha + ( 2 - c ) \\alpha ^ { 2 } ) ] } \\\\ & { \\qquad + x ( 1 - \\alpha ) [ 2 ( 1 - \\alpha ^ { 2 } ) - x ( c + ( c - 2 ) \\alpha ) ] . } \\end{array}", + "type": "interline_equation", + "image_path": "1dcd7a6dadb1f017f50da83ae925f64de04e3cc31c884cb27b30f5e8c299e4a5.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 127, + 255, + 481, + 293.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 127, + 293.0, + 481, + 331.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 127, + 331.0, + 481, + 369.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 377, + 504, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 504, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 504, + 391 + ], + "score": 1.0, + "content": "Proof of Lemma 4. The first observation necessary to prove the lemma is that the characteristic poly-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 388, + 378, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 137, + 401 + ], + "score": 1.0, + "content": "nomial", + "type": "text" + }, + { + "bbox": [ + 137, + 389, + 159, + 401 + ], + "score": 0.92, + "content": "D ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 388, + 208, + 401 + ], + "score": 1.0, + "content": "approaches", + "type": "text" + }, + { + "bbox": [ + 208, + 390, + 219, + 398 + ], + "score": 0.75, + "content": "\\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 388, + 231, + 401 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 231, + 390, + 263, + 399 + ], + "score": 0.85, + "content": "z \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 388, + 284, + 401 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 284, + 388, + 373, + 401 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { z \\infty } D ( z ) = + \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 388, + 378, + 401 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 376, + 504, + 401 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 372, + 418 + ], + "score": 1.0, + "content": "Next, we evaluate the characteristic polynomial at 1, i.e. compute", + "type": "text" + }, + { + "bbox": [ + 372, + 405, + 394, + 417 + ], + "score": 0.9, + "content": "D ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 405, + 505, + 418 + ], + "score": 1.0, + "content": ". This follows in a straight-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 415, + 461, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 308, + 429 + ], + "score": 1.0, + "content": "forward manner from corollary (7) by substituting", + "type": "text" + }, + { + "bbox": [ + 309, + 417, + 334, + 426 + ], + "score": 0.9, + "content": "\\tau = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 415, + 461, + 429 + ], + "score": 1.0, + "content": "in equation (2), and this yields,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 405, + 505, + 429 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 432, + 439, + 460 + ], + "lines": [ + { + "bbox": [ + 172, + 432, + 439, + 460 + ], + "spans": [ + { + "bbox": [ + 172, + 432, + 439, + 460 + ], + "score": 0.93, + "content": "D ( 1 ) = ( 1 - \\alpha ) x \\cdot \\bigg ( 2 ( 1 - \\alpha ^ { 2 } ) - x ( 1 - \\alpha ) - ( c - 1 ) x ( 1 + \\alpha ) \\bigg ) .", + "type": "interline_equation", + "image_path": "e0bd88d6953eb39b22592e4d54f9d03958f770d27b69406b5c6350d5b00c653d.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 172, + 432, + 439, + 460 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 465, + 455, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 456, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 120, + 479 + ], + "score": 1.0, + "content": "As", + "type": "text" + }, + { + "bbox": [ + 120, + 466, + 146, + 477 + ], + "score": 0.85, + "content": "\\alpha < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 464, + 149, + 479 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 150, + 465, + 204, + 477 + ], + "score": 0.9, + "content": "x = \\delta \\sigma ^ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 464, + 339, + 479 + ], + "score": 1.0, + "content": ", we have the following by setting", + "type": "text" + }, + { + "bbox": [ + 340, + 466, + 380, + 478 + ], + "score": 0.93, + "content": "D ( 1 ) \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 464, + 444, + 479 + ], + "score": 1.0, + "content": "and solving for", + "type": "text" + }, + { + "bbox": [ + 445, + 469, + 451, + 476 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 464, + 456, + 479 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 464, + 456, + 479 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 266, + 483, + 344, + 510 + ], + "lines": [ + { + "bbox": [ + 266, + 483, + 344, + 510 + ], + "spans": [ + { + "bbox": [ + 266, + 483, + 344, + 510 + ], + "score": 0.95, + "content": "x \\geq \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } .", + "type": "interline_equation", + "image_path": "068e1807ff1d226a23c3f534bdac1e43bcbb26cf8aa58bf3a0da2d40f7fb882c.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 266, + 483, + 344, + 510 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 439, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 514, + 440, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 131, + 529 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 515, + 172, + 527 + ], + "score": 0.92, + "content": "D ( 1 ) \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 514, + 190, + 529 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 190, + 515, + 231, + 527 + ], + "score": 0.93, + "content": "D ( z ) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 514, + 243, + 529 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 243, + 517, + 275, + 525 + ], + "score": 0.86, + "content": "z \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 514, + 362, + 529 + ], + "score": 1.0, + "content": ", there exists a root of", + "type": "text" + }, + { + "bbox": [ + 362, + 515, + 382, + 527 + ], + "score": 0.91, + "content": "D ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 514, + 419, + 529 + ], + "score": 1.0, + "content": "which is", + "type": "text" + }, + { + "bbox": [ + 420, + 516, + 436, + 526 + ], + "score": 0.85, + "content": "\\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 514, + 440, + 529 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 514, + 440, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 506, + 608 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 415, + 545 + ], + "score": 1.0, + "content": "Remark 8. The above characterization is striking in the sense that for any", + "type": "text" + }, + { + "bbox": [ + 415, + 534, + 441, + 544 + ], + "score": 0.87, + "content": "c > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 533, + 505, + 545 + ], + "score": 1.0, + "content": ", increasing the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 545, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 196, + 557 + ], + "score": 1.0, + "content": "momentum parameter", + "type": "text" + }, + { + "bbox": [ + 196, + 547, + 204, + 554 + ], + "score": 0.28, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 545, + 393, + 557 + ], + "score": 1.0, + "content": "naturally requires the reduction in the step size", + "type": "text" + }, + { + "bbox": [ + 393, + 545, + 399, + 554 + ], + "score": 0.61, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 545, + 506, + 557 + ], + "score": 1.0, + "content": "to permit the convergence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 556, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 107, + 556, + 505, + 567 + ], + "score": 1.0, + "content": "of the algorithm, which is not observed when fast gradient methods are employed in deterministic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 419, + 579 + ], + "score": 1.0, + "content": "optimization. For instance, in the case of deterministic optimization, setting", + "type": "text" + }, + { + "bbox": [ + 420, + 567, + 447, + 577 + ], + "score": 0.89, + "content": "c = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 565, + 475, + 579 + ], + "score": 1.0, + "content": "yields", + "type": "text" + }, + { + "bbox": [ + 475, + 565, + 505, + 578 + ], + "score": 0.91, + "content": "\\delta \\sigma _ { 1 } ^ { 2 } <", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 576, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 107, + 578, + 144, + 591 + ], + "score": 0.9, + "content": "2 ( 1 + \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 576, + 453, + 592 + ], + "score": 1.0, + "content": ". On the other hand, when employing the stochastic heavy ball method with", + "type": "text" + }, + { + "bbox": [ + 453, + 578, + 501, + 592 + ], + "score": 0.94, + "content": "x ^ { ( j ) } = 2 \\sigma _ { j } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 576, + 505, + 592 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 591, + 417, + 610 + ], + "spans": [ + { + "bbox": [ + 104, + 591, + 214, + 610 + ], + "score": 1.0, + "content": "we have the condition that", + "type": "text" + }, + { + "bbox": [ + 214, + 595, + 237, + 604 + ], + "score": 0.88, + "content": "c = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 591, + 310, + 610 + ], + "score": 1.0, + "content": ", and this implies,", + "type": "text" + }, + { + "bbox": [ + 310, + 591, + 413, + 608 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } < \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { 2 } = 1 - \\alpha ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 591, + 417, + 610 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 533, + 506, + 610 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 396, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 397, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 397, + 631 + ], + "score": 1.0, + "content": "We now prove Lemma 5. We first consider the large momentum setting.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 612, + 397, + 631 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 630, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 290, + 642 + ], + "score": 1.0, + "content": "Lemma 9. When the momentum parameter", + "type": "text" + }, + { + "bbox": [ + 291, + 632, + 299, + 640 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 628, + 366, + 642 + ], + "score": 1.0, + "content": "is set such that", + "type": "text" + }, + { + "bbox": [ + 367, + 630, + 467, + 642 + ], + "score": 0.86, + "content": "1 - 4 5 0 / \\kappa \\leq \\alpha \\leq 1 , ~ ", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 630, + 473, + 640 + ], + "score": 0.33, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "has an", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 638, + 253, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 253, + 656 + ], + "score": 1.0, + "content": "eigenvalue of magnitude ≥ 1 − 450κ .", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 628, + 505, + 656 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 293, + 682 + ], + "score": 1.0, + "content": "Proof. This follows easily from the fact that", + "type": "text" + }, + { + "bbox": [ + 293, + 666, + 481, + 682 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\operatorname* { d e t } ( \\boldsymbol { B } ) = \\alpha ^ { 4 } = \\prod _ { j = 1 } ^ { 4 } \\lambda _ { j } ( \\boldsymbol { B } ) \\le ( \\lambda _ { \\operatorname* { m a x } } ( \\boldsymbol { B } ) ) ^ { 4 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 665, + 506, + 682 + ], + "score": 1.0, + "content": ", thus", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 679, + 502, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 145, + 695 + ], + "score": 1.0, + "content": "implying", + "type": "text" + }, + { + "bbox": [ + 145, + 680, + 263, + 693 + ], + "score": 0.93, + "content": "1 - 4 5 0 / \\kappa \\leq \\alpha \\leq | \\lambda _ { \\mathrm { m a x } } ( \\beta ) |", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 679, + 267, + 695 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 497, + 685, + 502, + 689 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 665, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 437, + 711 + ], + "score": 1.0, + "content": "Remark 10. Note that the above lemma holds for any value of the learning rate", + "type": "text" + }, + { + "bbox": [ + 438, + 699, + 443, + 709 + ], + "score": 0.62, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 698, + 505, + 711 + ], + "score": 1.0, + "content": ", and holds for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 202, + 723 + ], + "score": 1.0, + "content": "every eigen direction of", + "type": "text" + }, + { + "bbox": [ + 202, + 710, + 212, + 720 + ], + "score": 0.29, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 709, + 505, + 723 + ], + "score": 1.0, + "content": ". Thus, for “large” values of momentum, the behavior of stochastic heavy", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 377, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 377, + 733 + ], + "score": 1.0, + "content": "ball does degenerate to the behavior of stochastic gradient descent.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 698, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 396, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 397, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 397, + 96 + ], + "score": 1.0, + "content": "We now consider the setting where momentum is bounded away from 1.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 97, + 505, + 122 + ], + "lines": [ + { + "bbox": [ + 105, + 96, + 506, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 207, + 112 + ], + "score": 1.0, + "content": "Corollary 11. Consider", + "type": "text" + }, + { + "bbox": [ + 207, + 97, + 225, + 109 + ], + "score": 0.87, + "content": "B ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 96, + 290, + 112 + ], + "score": 1.0, + "content": ", by substituting", + "type": "text" + }, + { + "bbox": [ + 298, + 97, + 424, + 110 + ], + "score": 0.88, + "content": "= l / \\kappa , x = \\delta \\lambda _ { \\mathrm { m i n } } = c ( \\delta \\sigma _ { 1 } ^ { 2 } ) / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 96, + 506, + 112 + ], + "score": 1.0, + "content": "in equation (2) and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 109, + 338, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 272, + 123 + ], + "score": 1.0, + "content": "accumulating terms in varying powers of", + "type": "text" + }, + { + "bbox": [ + 273, + 110, + 289, + 122 + ], + "score": 0.89, + "content": "1 / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 109, + 338, + 123 + ], + "score": 1.0, + "content": ", we obtain:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 124, + 496, + 228 + ], + "lines": [ + { + "bbox": [ + 112, + 124, + 496, + 228 + ], + "spans": [ + { + "bbox": [ + 112, + 124, + 496, + 228 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { G ( l ) \\stackrel { d e f } { = } \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \\kappa ^ { 5 } } + l ^ { 4 } - 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ^ { 3 } ( 1 + \\alpha ) + ( 2 \\alpha - 3 c ) c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 2 } } } \\\\ { { \\ + \\ \\frac { - ( 3 + \\alpha ) ( 1 - \\alpha ) l ^ { 3 } - 2 ( 1 + \\alpha ) ( 2 \\alpha - 3 ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ^ { 2 } + ( 3 c - 4 \\alpha + ( 2 - c ) \\alpha ^ { 2 } ) c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \\kappa ^ { 3 } } } } \\\\ { { \\ + \\ \\frac { ( 3 - 4 \\alpha - \\alpha ^ { 2 } + 2 \\alpha ^ { 3 } ) l ^ { 2 } - 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ( 3 - \\alpha ) ( 1 - \\alpha ^ { 2 } ) - c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } ( 1 - \\alpha ) ( c + ( c - 2 ) \\alpha ) } { \\kappa ^ { 2 } } } } \\\\ { { \\ + \\ \\frac { - ( 1 - \\alpha ) ^ { 2 } ( 1 - \\alpha ^ { 2 } ) l + 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) ( 1 - \\alpha ) ( 1 - \\alpha ^ { 2 } ) } { \\kappa } } } \\end{array}", + "type": "interline_equation", + "image_path": "2e49ae20187a91b888fee3c37c47a6bfa117379d22ff04dc6574ea64371c31ed.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 112, + 124, + 496, + 158.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 112, + 158.66666666666666, + 496, + 193.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 112, + 193.33333333333331, + 496, + 227.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 443, + 252 + ], + "lines": [ + { + "bbox": [ + 104, + 233, + 444, + 252 + ], + "spans": [ + { + "bbox": [ + 104, + 233, + 174, + 252 + ], + "score": 1.0, + "content": "Lemma 12. Let", + "type": "text" + }, + { + "bbox": [ + 174, + 237, + 231, + 249 + ], + "score": 0.53, + "content": "2 < c < 3 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 233, + 235, + 252 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 235, + 235, + 305, + 250 + ], + "score": 0.67, + "content": "\\textstyle 0 \\leq \\alpha \\leq 1 - { \\frac { 4 5 0 } { \\kappa } }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 233, + 309, + 252 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 309, + 234, + 372, + 251 + ], + "score": 0.91, + "content": "\\begin{array} { r } { l = 1 + \\frac { 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) } { 1 - \\alpha } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 233, + 444, + 250 + ], + "score": 1.0, + "content": "2c(δσ21) . Then, G(l) ≤ 0.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 105, + 262, + 488, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 260, + 486, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 161, + 281 + ], + "score": 1.0, + "content": "Proof. Since", + "type": "text" + }, + { + "bbox": [ + 161, + 263, + 236, + 281 + ], + "score": 0.94, + "content": "\\begin{array} { r } { ( \\delta \\sigma _ { 1 } ^ { 2 } ) \\le \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 260, + 289, + 280 + ], + "score": 1.0, + "content": ", this implies", + "type": "text" + }, + { + "bbox": [ + 289, + 263, + 380, + 281 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { ( \\delta \\sigma _ { 1 } ^ { 2 } ) } { 1 - \\alpha } \\leq \\frac { 2 ( 1 + \\alpha ) } { c + ( c - 2 ) \\alpha } \\leq \\frac { 4 } { c } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 261, + 444, + 285 + ], + "score": 1.0, + "content": ", thus implying,", + "type": "text" + }, + { + "bbox": [ + 444, + 267, + 486, + 278 + ], + "score": 0.91, + "content": "1 \\leq l \\leq 9", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 108, + 286, + 471, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 468, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 206, + 300 + ], + "score": 1.0, + "content": "Substituting the value of", + "type": "text" + }, + { + "bbox": [ + 207, + 288, + 211, + 297 + ], + "score": 0.67, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 286, + 345, + 300 + ], + "score": 1.0, + "content": "in equation (3), the coefficient of", + "type": "text" + }, + { + "bbox": [ + 346, + 286, + 378, + 299 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / \\kappa )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 286, + 389, + 300 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 389, + 286, + 465, + 299 + ], + "score": 0.91, + "content": "- ( 1 - \\alpha ) ^ { 3 } ( 1 + \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 286, + 468, + 300 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 109, + 302, + 503, + 316 + ], + "lines": [ + { + "bbox": [ + 107, + 302, + 504, + 316 + ], + "spans": [ + { + "bbox": [ + 107, + 302, + 249, + 316 + ], + "score": 1.0, + "content": "We will bound this term along with", + "type": "text" + }, + { + "bbox": [ + 249, + 303, + 463, + 316 + ], + "score": 0.91, + "content": "( 3 - 4 \\alpha - \\alpha ^ { 2 } + 2 \\alpha ^ { 3 } ) l ^ { 2 } / \\kappa ^ { 2 } = ( 1 - \\alpha ) ^ { 2 } ( 3 + 2 \\alpha ) l ^ { 2 } / \\kappa ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 302, + 504, + 316 + ], + "score": 1.0, + "content": "to obtain:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 321, + 488, + 404 + ], + "lines": [ + { + "bbox": [ + 123, + 321, + 488, + 404 + ], + "spans": [ + { + "bbox": [ + 123, + 321, + 488, + 404 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\frac { - ( 1 - \\alpha ) ^ { 3 } ( 1 + \\alpha ) } { \\kappa } + \\frac { ( 1 - \\alpha ) ^ { 2 } ( 3 + 2 \\alpha ) l ^ { 2 } } { \\kappa ^ { 2 } } \\leq \\frac { - ( 1 - \\alpha ) ^ { 3 } ( 1 + \\alpha ) } { \\kappa } + \\frac { 4 0 5 ( 1 - \\alpha ) ^ { 2 } } { \\kappa ^ { 2 } } } \\\\ & { \\qquad \\leq \\frac { ( 1 - \\alpha ) ^ { 2 } } { \\kappa } \\bigg ( \\frac { 4 0 5 } { \\kappa } - ( 1 - \\alpha ^ { 2 } ) \\bigg ) } \\\\ & { \\qquad \\leq \\frac { ( 1 - \\alpha ) ^ { 2 } } { \\kappa } \\bigg ( \\frac { 4 0 5 } { \\kappa } - ( 1 - \\alpha ) \\bigg ) \\leq - \\frac { 4 5 \\cdot 4 5 0 ^ { 2 } } { \\kappa ^ { 4 } } , } \\end{array}", + "type": "interline_equation", + "image_path": "c227990b778998dcd360275c899bb2dfc87c9724d75f1bfe40777d978027adb7.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 123, + 321, + 488, + 348.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 123, + 348.6666666666667, + 488, + 376.33333333333337 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 123, + 376.33333333333337, + 488, + 404.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 406, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 214, + 418 + ], + "score": 1.0, + "content": "where, we use the fact that", + "type": "text" + }, + { + "bbox": [ + 214, + 407, + 267, + 418 + ], + "score": 0.31, + "content": "\\alpha < 1 , l \\le 9", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 406, + 505, + 418 + ], + "score": 1.0, + "content": ". The natural implication of this bound is that the terms that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 204, + 430 + ], + "score": 1.0, + "content": "are lower order, such as", + "type": "text" + }, + { + "bbox": [ + 205, + 418, + 241, + 430 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / \\kappa ^ { 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 417, + 261, + 430 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 261, + 417, + 297, + 430 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / \\kappa ^ { 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "will be negative owing to the large constant above.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 428, + 449, + 441 + ], + "score": 1.0, + "content": "Let us verify that this is indeed the case by considering the terms having powers of", + "type": "text" + }, + { + "bbox": [ + 449, + 428, + 486, + 441 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / \\kappa ^ { 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 438, + 221, + 452 + ], + "spans": [ + { + "bbox": [ + 107, + 439, + 144, + 452 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / \\kappa ^ { 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 438, + 221, + 452 + ], + "score": 1.0, + "content": "from equation (3):", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 455, + 457, + 535 + ], + "lines": [ + { + "bbox": [ + 154, + 455, + 457, + 535 + ], + "spans": [ + { + "bbox": [ + 154, + 455, + 457, + 535 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \\kappa ^ { 5 } } + \\frac { l ^ { 4 } - 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ^ { 3 } ( 1 + \\alpha ) + ( 2 \\alpha - 3 c ) c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 2 } } { \\kappa ^ { 4 } } - \\frac { 4 5 \\cdot 4 5 0 ^ { 2 } } { \\kappa ^ { 4 } } } \\\\ & { \\leq \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \\kappa ^ { 5 } } + \\frac { l ^ { 4 } } { \\kappa ^ { 4 } } - \\frac { 4 5 \\cdot 4 5 0 ^ { 2 } } { \\kappa ^ { 4 } } } \\\\ & { \\leq \\frac { c l ^ { 3 } } { \\kappa ^ { 5 } } + \\frac { ( 9 ^ { 4 } - ( 4 5 \\cdot 4 5 0 ^ { 2 } ) ) } { \\kappa ^ { 4 } } \\leq \\frac { 9 ^ { 3 } c + 9 ^ { 4 } - ( 4 5 \\cdot 4 5 0 ^ { 2 } ) } { \\kappa ^ { 4 } } } \\end{array}", + "type": "interline_equation", + "image_path": "d9c4930d752d40ea7b9abf18ee1e20a7394ed6064f0d4d22b6da3dd963b47530.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 154, + 455, + 457, + 481.6666666666667 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 154, + 481.6666666666667, + 457, + 508.33333333333337 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 154, + 508.33333333333337, + 457, + 535.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 243, + 550 + ], + "score": 1.0, + "content": "The expression above evaluates to", + "type": "text" + }, + { + "bbox": [ + 243, + 538, + 260, + 549 + ], + "score": 0.88, + "content": "\\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 537, + 407, + 550 + ], + "score": 1.0, + "content": "given an upperbound on the value of", + "type": "text" + }, + { + "bbox": [ + 407, + 540, + 412, + 547 + ], + "score": 0.73, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 537, + 505, + 550 + ], + "score": 1.0, + "content": ". The expression above", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 549, + 265, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 210, + 560 + ], + "score": 1.0, + "content": "follows from the fact that", + "type": "text" + }, + { + "bbox": [ + 210, + 549, + 261, + 560 + ], + "score": 0.92, + "content": "l \\leq 9 , \\kappa \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 549, + 265, + 560 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 397, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 397, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 246, + 579 + ], + "score": 1.0, + "content": "Next, consider the terms involving", + "type": "text" + }, + { + "bbox": [ + 246, + 564, + 283, + 578 + ], + "score": 0.94, + "content": "\\mathcal { O } ( 1 / \\kappa ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 562, + 302, + 579 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 302, + 565, + 339, + 578 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / \\kappa ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 562, + 397, + 579 + ], + "score": 1.0, + "content": ", in particular,", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 581, + 445, + 739 + ], + "lines": [ + { + "bbox": [ + 165, + 581, + 445, + 739 + ], + "spans": [ + { + "bbox": [ + 165, + 581, + 445, + 739 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\frac { ( 3 c - 4 \\alpha + ( 2 - c ) \\alpha ^ { 2 } ) c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l } { \\kappa ^ { 3 } } - \\frac { c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } ( 1 - \\alpha ) ( c + ( c - 2 ) \\alpha ) } { \\kappa ^ { 2 } } } \\\\ & { \\leq \\frac { c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { l ( 3 c + 2 ) } { \\kappa } - ( 1 - \\alpha ) ( c + ( c - 2 ) \\alpha ) ) } \\\\ & { \\leq \\frac { c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { 5 \\epsilon } { \\kappa } - ( 1 - \\alpha ) ( c + ( c - 2 ) \\alpha ) ) } \\\\ & { \\leq \\frac { c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { 5 \\epsilon l } { \\kappa } - ( 1 - \\alpha ) c ) } \\\\ & { \\leq \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { 5 l } { \\kappa } - \\frac { 4 5 0 } { \\kappa } ) } \\\\ & { \\leq \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { 1 - \\delta ^ { 2 } } { \\kappa } ) } \\\\ & { \\leq \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } \\cdot \\frac { - 4 4 5 0 } { \\kappa } \\leq 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "4ca8e3746ecda1b419bb3c5576e6b565c92c10c25b1ae667713f3542cc658b64.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 165, + 581, + 445, + 633.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 165, + 633.6666666666666, + 445, + 686.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 165, + 686.3333333333333, + 445, + 738.9999999999999 + ], + "spans": [], + "index": 25 + } + ] + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "15", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 396, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 397, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 397, + 96 + ], + "score": 1.0, + "content": "We now consider the setting where momentum is bounded away from 1.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 82, + 397, + 96 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 97, + 505, + 122 + ], + "lines": [ + { + "bbox": [ + 105, + 96, + 506, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 207, + 112 + ], + "score": 1.0, + "content": "Corollary 11. Consider", + "type": "text" + }, + { + "bbox": [ + 207, + 97, + 225, + 109 + ], + "score": 0.87, + "content": "B ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 96, + 290, + 112 + ], + "score": 1.0, + "content": ", by substituting", + "type": "text" + }, + { + "bbox": [ + 298, + 97, + 424, + 110 + ], + "score": 0.88, + "content": "= l / \\kappa , x = \\delta \\lambda _ { \\mathrm { m i n } } = c ( \\delta \\sigma _ { 1 } ^ { 2 } ) / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 96, + 506, + 112 + ], + "score": 1.0, + "content": "in equation (2) and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 109, + 338, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 272, + 123 + ], + "score": 1.0, + "content": "accumulating terms in varying powers of", + "type": "text" + }, + { + "bbox": [ + 273, + 110, + 289, + 122 + ], + "score": 0.89, + "content": "1 / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 109, + 338, + 123 + ], + "score": 1.0, + "content": ", we obtain:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 96, + 506, + 123 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 124, + 496, + 228 + ], + "lines": [ + { + "bbox": [ + 112, + 124, + 496, + 228 + ], + "spans": [ + { + "bbox": [ + 112, + 124, + 496, + 228 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { G ( l ) \\stackrel { d e f } { = } \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \\kappa ^ { 5 } } + l ^ { 4 } - 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ^ { 3 } ( 1 + \\alpha ) + ( 2 \\alpha - 3 c ) c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 2 } } } \\\\ { { \\ + \\ \\frac { - ( 3 + \\alpha ) ( 1 - \\alpha ) l ^ { 3 } - 2 ( 1 + \\alpha ) ( 2 \\alpha - 3 ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ^ { 2 } + ( 3 c - 4 \\alpha + ( 2 - c ) \\alpha ^ { 2 } ) c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \\kappa ^ { 3 } } } } \\\\ { { \\ + \\ \\frac { ( 3 - 4 \\alpha - \\alpha ^ { 2 } + 2 \\alpha ^ { 3 } ) l ^ { 2 } - 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ( 3 - \\alpha ) ( 1 - \\alpha ^ { 2 } ) - c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } ( 1 - \\alpha ) ( c + ( c - 2 ) \\alpha ) } { \\kappa ^ { 2 } } } } \\\\ { { \\ + \\ \\frac { - ( 1 - \\alpha ) ^ { 2 } ( 1 - \\alpha ^ { 2 } ) l + 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) ( 1 - \\alpha ) ( 1 - \\alpha ^ { 2 } ) } { \\kappa } } } \\end{array}", + "type": "interline_equation", + "image_path": "2e49ae20187a91b888fee3c37c47a6bfa117379d22ff04dc6574ea64371c31ed.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 112, + 124, + 496, + 158.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 112, + 158.66666666666666, + 496, + 193.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 112, + 193.33333333333331, + 496, + 227.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 443, + 252 + ], + "lines": [ + { + "bbox": [ + 104, + 233, + 444, + 252 + ], + "spans": [ + { + "bbox": [ + 104, + 233, + 174, + 252 + ], + "score": 1.0, + "content": "Lemma 12. Let", + "type": "text" + }, + { + "bbox": [ + 174, + 237, + 231, + 249 + ], + "score": 0.53, + "content": "2 < c < 3 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 233, + 235, + 252 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 235, + 235, + 305, + 250 + ], + "score": 0.67, + "content": "\\textstyle 0 \\leq \\alpha \\leq 1 - { \\frac { 4 5 0 } { \\kappa } }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 233, + 309, + 252 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 309, + 234, + 372, + 251 + ], + "score": 0.91, + "content": "\\begin{array} { r } { l = 1 + \\frac { 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) } { 1 - \\alpha } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 233, + 444, + 250 + ], + "score": 1.0, + "content": "2c(δσ21) . Then, G(l) ≤ 0.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 233, + 444, + 252 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 262, + 488, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 260, + 486, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 161, + 281 + ], + "score": 1.0, + "content": "Proof. Since", + "type": "text" + }, + { + "bbox": [ + 161, + 263, + 236, + 281 + ], + "score": 0.94, + "content": "\\begin{array} { r } { ( \\delta \\sigma _ { 1 } ^ { 2 } ) \\le \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 260, + 289, + 280 + ], + "score": 1.0, + "content": ", this implies", + "type": "text" + }, + { + "bbox": [ + 289, + 263, + 380, + 281 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { ( \\delta \\sigma _ { 1 } ^ { 2 } ) } { 1 - \\alpha } \\leq \\frac { 2 ( 1 + \\alpha ) } { c + ( c - 2 ) \\alpha } \\leq \\frac { 4 } { c } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 261, + 444, + 285 + ], + "score": 1.0, + "content": ", thus implying,", + "type": "text" + }, + { + "bbox": [ + 444, + 267, + 486, + 278 + ], + "score": 0.91, + "content": "1 \\leq l \\leq 9", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 260, + 486, + 285 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 286, + 471, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 468, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 206, + 300 + ], + "score": 1.0, + "content": "Substituting the value of", + "type": "text" + }, + { + "bbox": [ + 207, + 288, + 211, + 297 + ], + "score": 0.67, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 286, + 345, + 300 + ], + "score": 1.0, + "content": "in equation (3), the coefficient of", + "type": "text" + }, + { + "bbox": [ + 346, + 286, + 378, + 299 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / \\kappa )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 286, + 389, + 300 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 389, + 286, + 465, + 299 + ], + "score": 0.91, + "content": "- ( 1 - \\alpha ) ^ { 3 } ( 1 + \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 286, + 468, + 300 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 286, + 468, + 300 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 302, + 503, + 316 + ], + "lines": [ + { + "bbox": [ + 107, + 302, + 504, + 316 + ], + "spans": [ + { + "bbox": [ + 107, + 302, + 249, + 316 + ], + "score": 1.0, + "content": "We will bound this term along with", + "type": "text" + }, + { + "bbox": [ + 249, + 303, + 463, + 316 + ], + "score": 0.91, + "content": "( 3 - 4 \\alpha - \\alpha ^ { 2 } + 2 \\alpha ^ { 3 } ) l ^ { 2 } / \\kappa ^ { 2 } = ( 1 - \\alpha ) ^ { 2 } ( 3 + 2 \\alpha ) l ^ { 2 } / \\kappa ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 302, + 504, + 316 + ], + "score": 1.0, + "content": "to obtain:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 107, + 302, + 504, + 316 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 321, + 488, + 404 + ], + "lines": [ + { + "bbox": [ + 123, + 321, + 488, + 404 + ], + "spans": [ + { + "bbox": [ + 123, + 321, + 488, + 404 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\frac { - ( 1 - \\alpha ) ^ { 3 } ( 1 + \\alpha ) } { \\kappa } + \\frac { ( 1 - \\alpha ) ^ { 2 } ( 3 + 2 \\alpha ) l ^ { 2 } } { \\kappa ^ { 2 } } \\leq \\frac { - ( 1 - \\alpha ) ^ { 3 } ( 1 + \\alpha ) } { \\kappa } + \\frac { 4 0 5 ( 1 - \\alpha ) ^ { 2 } } { \\kappa ^ { 2 } } } \\\\ & { \\qquad \\leq \\frac { ( 1 - \\alpha ) ^ { 2 } } { \\kappa } \\bigg ( \\frac { 4 0 5 } { \\kappa } - ( 1 - \\alpha ^ { 2 } ) \\bigg ) } \\\\ & { \\qquad \\leq \\frac { ( 1 - \\alpha ) ^ { 2 } } { \\kappa } \\bigg ( \\frac { 4 0 5 } { \\kappa } - ( 1 - \\alpha ) \\bigg ) \\leq - \\frac { 4 5 \\cdot 4 5 0 ^ { 2 } } { \\kappa ^ { 4 } } , } \\end{array}", + "type": "interline_equation", + "image_path": "c227990b778998dcd360275c899bb2dfc87c9724d75f1bfe40777d978027adb7.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 123, + 321, + 488, + 348.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 123, + 348.6666666666667, + 488, + 376.33333333333337 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 123, + 376.33333333333337, + 488, + 404.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 406, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 214, + 418 + ], + "score": 1.0, + "content": "where, we use the fact that", + "type": "text" + }, + { + "bbox": [ + 214, + 407, + 267, + 418 + ], + "score": 0.31, + "content": "\\alpha < 1 , l \\le 9", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 406, + 505, + 418 + ], + "score": 1.0, + "content": ". The natural implication of this bound is that the terms that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 204, + 430 + ], + "score": 1.0, + "content": "are lower order, such as", + "type": "text" + }, + { + "bbox": [ + 205, + 418, + 241, + 430 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / \\kappa ^ { 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 417, + 261, + 430 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 261, + 417, + 297, + 430 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / \\kappa ^ { 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "will be negative owing to the large constant above.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 428, + 449, + 441 + ], + "score": 1.0, + "content": "Let us verify that this is indeed the case by considering the terms having powers of", + "type": "text" + }, + { + "bbox": [ + 449, + 428, + 486, + 441 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / \\kappa ^ { 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 438, + 221, + 452 + ], + "spans": [ + { + "bbox": [ + 107, + 439, + 144, + 452 + ], + "score": 0.92, + "content": "\\mathcal { O } ( 1 / \\kappa ^ { 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 438, + 221, + 452 + ], + "score": 1.0, + "content": "from equation (3):", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 406, + 505, + 452 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 455, + 457, + 535 + ], + "lines": [ + { + "bbox": [ + 154, + 455, + 457, + 535 + ], + "spans": [ + { + "bbox": [ + 154, + 455, + 457, + 535 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \\kappa ^ { 5 } } + \\frac { l ^ { 4 } - 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ^ { 3 } ( 1 + \\alpha ) + ( 2 \\alpha - 3 c ) c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 2 } } { \\kappa ^ { 4 } } - \\frac { 4 5 \\cdot 4 5 0 ^ { 2 } } { \\kappa ^ { 4 } } } \\\\ & { \\leq \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l ^ { 3 } } { \\kappa ^ { 5 } } + \\frac { l ^ { 4 } } { \\kappa ^ { 4 } } - \\frac { 4 5 \\cdot 4 5 0 ^ { 2 } } { \\kappa ^ { 4 } } } \\\\ & { \\leq \\frac { c l ^ { 3 } } { \\kappa ^ { 5 } } + \\frac { ( 9 ^ { 4 } - ( 4 5 \\cdot 4 5 0 ^ { 2 } ) ) } { \\kappa ^ { 4 } } \\leq \\frac { 9 ^ { 3 } c + 9 ^ { 4 } - ( 4 5 \\cdot 4 5 0 ^ { 2 } ) } { \\kappa ^ { 4 } } } \\end{array}", + "type": "interline_equation", + "image_path": "d9c4930d752d40ea7b9abf18ee1e20a7394ed6064f0d4d22b6da3dd963b47530.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 154, + 455, + 457, + 481.6666666666667 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 154, + 481.6666666666667, + 457, + 508.33333333333337 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 154, + 508.33333333333337, + 457, + 535.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 243, + 550 + ], + "score": 1.0, + "content": "The expression above evaluates to", + "type": "text" + }, + { + "bbox": [ + 243, + 538, + 260, + 549 + ], + "score": 0.88, + "content": "\\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 537, + 407, + 550 + ], + "score": 1.0, + "content": "given an upperbound on the value of", + "type": "text" + }, + { + "bbox": [ + 407, + 540, + 412, + 547 + ], + "score": 0.73, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 537, + 505, + 550 + ], + "score": 1.0, + "content": ". The expression above", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 549, + 265, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 210, + 560 + ], + "score": 1.0, + "content": "follows from the fact that", + "type": "text" + }, + { + "bbox": [ + 210, + 549, + 261, + 560 + ], + "score": 0.92, + "content": "l \\leq 9 , \\kappa \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 549, + 265, + 560 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 537, + 505, + 560 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 397, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 397, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 246, + 579 + ], + "score": 1.0, + "content": "Next, consider the terms involving", + "type": "text" + }, + { + "bbox": [ + 246, + 564, + 283, + 578 + ], + "score": 0.94, + "content": "\\mathcal { O } ( 1 / \\kappa ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 562, + 302, + 579 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 302, + 565, + 339, + 578 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / \\kappa ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 562, + 397, + 579 + ], + "score": 1.0, + "content": ", in particular,", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 562, + 397, + 579 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 581, + 445, + 739 + ], + "lines": [ + { + "bbox": [ + 165, + 581, + 445, + 739 + ], + "spans": [ + { + "bbox": [ + 165, + 581, + 445, + 739 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\frac { ( 3 c - 4 \\alpha + ( 2 - c ) \\alpha ^ { 2 } ) c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } l } { \\kappa ^ { 3 } } - \\frac { c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } ( 1 - \\alpha ) ( c + ( c - 2 ) \\alpha ) } { \\kappa ^ { 2 } } } \\\\ & { \\leq \\frac { c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { l ( 3 c + 2 ) } { \\kappa } - ( 1 - \\alpha ) ( c + ( c - 2 ) \\alpha ) ) } \\\\ & { \\leq \\frac { c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { 5 \\epsilon } { \\kappa } - ( 1 - \\alpha ) ( c + ( c - 2 ) \\alpha ) ) } \\\\ & { \\leq \\frac { c ^ { 2 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { 5 \\epsilon l } { \\kappa } - ( 1 - \\alpha ) c ) } \\\\ & { \\leq \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { 5 l } { \\kappa } - \\frac { 4 5 0 } { \\kappa } ) } \\\\ & { \\leq \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } ( \\frac { 1 - \\delta ^ { 2 } } { \\kappa } ) } \\\\ & { \\leq \\frac { c ^ { 3 } ( \\delta \\sigma _ { 1 } ^ { 2 } ) ^ { 2 } } { \\kappa ^ { 2 } } \\cdot \\frac { - 4 4 5 0 } { \\kappa } \\leq 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "4ca8e3746ecda1b419bb3c5576e6b565c92c10c25b1ae667713f3542cc658b64.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 165, + 581, + 445, + 633.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 165, + 633.6666666666666, + 445, + 686.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 165, + 686.3333333333333, + 445, + 738.9999999999999 + ], + "spans": [], + "index": 25 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 130, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 132, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 132, + 97 + ], + "score": 1.0, + "content": "Next,", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 97, + 424, + 237 + ], + "lines": [ + { + "bbox": [ + 186, + 97, + 424, + 237 + ], + "spans": [ + { + "bbox": [ + 186, + 97, + 424, + 237 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\frac { - 2 ( 1 + \\alpha ) ( 2 \\alpha - 3 ) e ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ^ { 2 } } { \\kappa ^ { 3 } } - \\frac { 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ( 3 - \\alpha ) ( 1 - \\alpha ^ { 2 } ) } { \\kappa ^ { 2 } } } \\\\ & { \\leq \\frac { 2 ( 1 + \\alpha ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l } { \\kappa ^ { 2 } } \\Big ( \\frac { - ( 2 \\alpha - 3 ) l } { \\kappa } - ( 3 - \\alpha ) ( 1 - \\alpha ) \\Big ) } \\\\ & { \\leq \\frac { 2 ( 1 + \\alpha ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l } { \\kappa ^ { 2 } } \\Big ( \\frac { 3 l } { \\kappa } - 2 ( 1 - \\alpha ) \\Big ) } \\\\ & { \\leq \\frac { 2 ( 1 + \\alpha ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l } { \\kappa ^ { 2 } } \\Big ( \\frac { 3 l } { \\kappa } - \\frac { 2 \\cdot 4 5 0 } { \\kappa } \\Big ) } \\\\ & { \\leq \\frac { 2 ( 1 + \\alpha ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l } { \\kappa ^ { 2 } } \\Big ( \\frac { 3 \\cdot 2 7 } { \\kappa } - \\frac { 2 \\cdot 4 5 0 } { \\kappa } \\Big ) \\leq 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "c6d3a6cbc597db87d0d5141e5bf9a7de29a1c946fa9e13405ca548749641087c.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 97, + 424, + 114.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 186, + 114.5, + 424, + 132.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 186, + 132.0, + 424, + 149.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 186, + 149.5, + 424, + 167.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 186, + 167.0, + 424, + 184.5 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 186, + 184.5, + 424, + 202.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 186, + 202.0, + 424, + 219.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 186, + 219.5, + 424, + 237.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 240, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 101, + 234, + 510, + 264 + ], + "spans": [ + { + "bbox": [ + 101, + 234, + 273, + 264 + ], + "score": 1.0, + "content": "In both these cases, we used the fact that remaining terms are negative.", + "type": "text" + }, + { + "bbox": [ + 274, + 240, + 327, + 254 + ], + "score": 0.92, + "content": "\\textstyle \\alpha \\leq 1 - { \\frac { 4 5 0 } { \\kappa } }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 234, + 368, + 264 + ], + "score": 1.0, + "content": "implying", + "type": "text" + }, + { + "bbox": [ + 368, + 240, + 443, + 254 + ], + "score": 0.92, + "content": "\\begin{array} { r } { - ( 1 - \\alpha ) \\le \\frac { - 4 5 0 } { \\kappa } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 234, + 510, + 264 + ], + "score": 1.0, + "content": ". Finally, other", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 289 + ], + "score": 1.0, + "content": "Before rounding up the proof of the proposition, we need the following lemma to ensure that our", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 287, + 504, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 280, + 299 + ], + "score": 1.0, + "content": "lower bounds on the largest eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 280, + 288, + 289, + 297 + ], + "score": 0.8, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 287, + 504, + 299 + ], + "score": 1.0, + "content": "indeed affect the algorithm’s rates and are true irre-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 104, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "spective of where the algorithm is begun. Note that this allows our result to be much stronger than", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 308, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 324 + ], + "score": 1.0, + "content": "typical optimization lowerbounds that rely on specific initializations to ensure a component along", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 319, + 421, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 421, + 333 + ], + "score": 1.0, + "content": "the largest eigendirection of the update operator, for which bounds are proven.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 334, + 504, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 332, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 259, + 348 + ], + "score": 1.0, + "content": "Lemma 13. For any starting iterate", + "type": "text" + }, + { + "bbox": [ + 259, + 334, + 302, + 346 + ], + "score": 0.9, + "content": "\\mathbf { w } _ { 0 } \\neq \\mathbf { w } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 332, + 505, + 348 + ], + "score": 1.0, + "content": ", the HB method produces a non-zero component", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 345, + 260, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 248, + 357 + ], + "score": 1.0, + "content": "along the largest eigen direction of", + "type": "text" + }, + { + "bbox": [ + 249, + 346, + 256, + 355 + ], + "score": 0.68, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 345, + 260, + 357 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 368, + 506, + 468 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "Proof. We note that in a similar manner as other proofs, it suffices to argue for each dimension of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "the problem separately. But before we start looking at each dimension separately, let us consider", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 389, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 122, + 405 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 390, + 134, + 403 + ], + "score": 0.89, + "content": "\\bar { j } ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 389, + 506, + 405 + ], + "score": 1.0, + "content": "dimension, and detail the approach we use to prove the claim: the idea is to examine the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 102, + 399, + 509, + 423 + ], + "spans": [ + { + "bbox": [ + 102, + 399, + 238, + 423 + ], + "score": 1.0, + "content": "subspace spanned by covariance", + "type": "text" + }, + { + "bbox": [ + 238, + 401, + 301, + 422 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left[ \\pmb { \\theta } _ { . } ^ { ( j ) } \\otimes \\pmb { \\theta } _ { . } ^ { ( j ) } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 399, + 359, + 423 + ], + "score": 1.0, + "content": "of the iterates", + "type": "text" + }, + { + "bbox": [ + 360, + 402, + 432, + 418 + ], + "score": 0.93, + "content": "\\pmb { \\theta } _ { 0 } ^ { ( j ) } , \\pmb { \\theta } _ { 1 } ^ { ( j ) } , \\pmb { \\theta } _ { 2 } ^ { ( j ) } , . . . ,", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 399, + 509, + 423 + ], + "score": 1.0, + "content": "for every starting", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 419, + 507, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 135, + 438 + ], + "score": 1.0, + "content": "iterate", + "type": "text" + }, + { + "bbox": [ + 135, + 420, + 196, + 436 + ], + "score": 0.94, + "content": "\\pmb { \\theta } _ { 0 } ^ { ( j ) } \\neq \\left[ 0 , 0 \\right] ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 419, + 459, + 438 + ], + "score": 1.0, + "content": "and prove that the largest eigenvector of the expected operator", + "type": "text" + }, + { + "bbox": [ + 459, + 422, + 477, + 433 + ], + "score": 0.9, + "content": "B ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 419, + 507, + 438 + ], + "score": 1.0, + "content": "is not", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 434, + 503, + 455 + ], + "spans": [ + { + "bbox": [ + 104, + 434, + 440, + 455 + ], + "score": 1.0, + "content": "orthogonal to this subspace. This implies that there exists a non-zero component of", + "type": "text" + }, + { + "bbox": [ + 440, + 435, + 503, + 455 + ], + "score": 0.89, + "content": "\\mathbb { E } \\left[ \\pmb { \\theta } _ { \\cdot } ^ { ( j ) } \\otimes \\pmb { \\theta } _ { \\cdot } ^ { ( j ) } \\right]", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 452, + 460, + 469 + ], + "spans": [ + { + "bbox": [ + 104, + 452, + 233, + 469 + ], + "score": 1.0, + "content": "in the largest eigen direction of", + "type": "text" + }, + { + "bbox": [ + 233, + 453, + 251, + 465 + ], + "score": 0.9, + "content": "B ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 452, + 407, + 469 + ], + "score": 1.0, + "content": ", and this decays at a rate that is at best", + "type": "text" + }, + { + "bbox": [ + 408, + 454, + 454, + 467 + ], + "score": 0.91, + "content": "\\lambda _ { \\operatorname* { m a x } } ( B ^ { ( j ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 452, + 460, + 469 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 472, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 104, + 469, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 133, + 487 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 133, + 472, + 192, + 484 + ], + "score": 0.94, + "content": "\\boldsymbol { B } ^ { ( j ) } ~ \\in ~ \\mathbb { R } ^ { 4 \\times 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 469, + 506, + 487 + ], + "score": 1.0, + "content": ", we begin by examining the expected covariance spanned by the iterates", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 479, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 107, + 484, + 186, + 499 + ], + "score": 0.91, + "content": "\\pmb { \\theta } _ { 0 } ^ { ( j ) } , \\pmb { \\theta } _ { 1 } ^ { ( j ) } , \\pmb { \\theta } _ { 2 } ^ { ( j ) } , \\pmb { \\theta } _ { 3 } ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 479, + 211, + 507 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 211, + 484, + 390, + 500 + ], + "score": 0.93, + "content": "\\mathbf { w } _ { 0 } ^ { ( j ) } - ( \\mathbf { w } ^ { * } ) ^ { ( j ) } = \\mathbf { \\bar { w } } _ { - 1 } ^ { ( j ) } - ( \\mathbf { w } ^ { * } ) ^ { ( j ) } = k ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 479, + 473, + 507 + ], + "score": 1.0, + "content": ". Now, this implies", + "type": "text" + }, + { + "bbox": [ + 474, + 484, + 505, + 500 + ], + "score": 0.91, + "content": "\\theta _ { 0 } ^ { ( j ) } =", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 497, + 189, + 516 + ], + "spans": [ + { + "bbox": [ + 107, + 500, + 157, + 514 + ], + "score": 0.92, + "content": "\\boldsymbol { k } ^ { ( j ) } \\cdot \\left[ 1 , 1 \\right] ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 497, + 189, + 516 + ], + "score": 1.0, + "content": ". 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This in turn implies that in order to analyze", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 100, + 559, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 100, + 559, + 313, + 583 + ], + "score": 1.0, + "content": "the subspace spanned by the covariance of iterates", + "type": "text" + }, + { + "bbox": [ + 314, + 561, + 366, + 576 + ], + "score": 0.9, + "content": "\\theta _ { 0 } ^ { ( j ) } , \\theta _ { 1 } ^ { ( j ) } , . . . ,", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 559, + 433, + 583 + ], + "score": 1.0, + "content": ", we can assume", + "type": "text" + }, + { + "bbox": [ + 433, + 562, + 470, + 574 + ], + "score": 0.91, + "content": "k ^ { ( j ) } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 559, + 506, + 583 + ], + "score": 1.0, + "content": "without", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 574, + 507, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 257, + 591 + ], + "score": 1.0, + "content": "any loss in generality. This implies,", + "type": "text" + }, + { + "bbox": [ + 257, + 575, + 317, + 590 + ], + "score": 0.93, + "content": "\\pmb { \\theta } _ { 0 } ^ { ( j ) } = \\left[ 1 , 1 \\right] ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 574, + 507, + 591 + ], + "score": 1.0, + "content": ". 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2 ( 1 + \\alpha ) ( 2 \\alpha - 3 ) e ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ^ { 2 } } { \\kappa ^ { 3 } } - \\frac { 2 c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l ( 3 - \\alpha ) ( 1 - \\alpha ^ { 2 } ) } { \\kappa ^ { 2 } } } \\\\ & { \\leq \\frac { 2 ( 1 + \\alpha ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l } { \\kappa ^ { 2 } } \\Big ( \\frac { - ( 2 \\alpha - 3 ) l } { \\kappa } - ( 3 - \\alpha ) ( 1 - \\alpha ) \\Big ) } \\\\ & { \\leq \\frac { 2 ( 1 + \\alpha ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l } { \\kappa ^ { 2 } } \\Big ( \\frac { 3 l } { \\kappa } - 2 ( 1 - \\alpha ) \\Big ) } \\\\ & { \\leq \\frac { 2 ( 1 + \\alpha ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l } { \\kappa ^ { 2 } } \\Big ( \\frac { 3 l } { \\kappa } - \\frac { 2 \\cdot 4 5 0 } { \\kappa } \\Big ) } \\\\ & { \\leq \\frac { 2 ( 1 + \\alpha ) c ( \\delta \\sigma _ { 1 } ^ { 2 } ) l } { \\kappa ^ { 2 } } \\Big ( \\frac { 3 \\cdot 2 7 } { \\kappa } - \\frac { 2 \\cdot 4 5 0 } { \\kappa } \\Big ) \\leq 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "c6d3a6cbc597db87d0d5141e5bf9a7de29a1c946fa9e13405ca548749641087c.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 97, + 424, + 114.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 186, + 114.5, + 424, + 132.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 186, + 132.0, + 424, + 149.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 186, + 149.5, + 424, + 167.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 186, + 167.0, + 424, + 184.5 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 186, + 184.5, + 424, + 202.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 186, + 202.0, + 424, + 219.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 186, + 219.5, + 424, + 237.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 240, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 101, + 234, + 510, + 264 + ], + "spans": [ + { + "bbox": [ + 101, + 234, + 273, + 264 + ], + "score": 1.0, + "content": "In both these cases, we used the fact that remaining terms are negative.", + "type": "text" + }, + { + "bbox": [ + 274, + 240, + 327, + 254 + ], + "score": 0.92, + "content": "\\textstyle \\alpha \\leq 1 - { \\frac { 4 5 0 } { \\kappa } }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 234, + 368, + 264 + ], + "score": 1.0, + "content": "implying", + "type": "text" + }, + { + "bbox": [ + 368, + 240, + 443, + 254 + ], + "score": 0.92, + "content": "\\begin{array} { r } { - ( 1 - \\alpha ) \\le \\frac { - 4 5 0 } { \\kappa } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 234, + 510, + 264 + ], + "score": 1.0, + "content": ". Finally, other", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 101, + 234, + 510, + 264 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 505, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 289 + ], + "score": 1.0, + "content": "Before rounding up the proof of the proposition, we need the following lemma to ensure that our", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 287, + 504, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 280, + 299 + ], + "score": 1.0, + "content": "lower bounds on the largest eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 280, + 288, + 289, + 297 + ], + "score": 0.8, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 287, + 504, + 299 + ], + "score": 1.0, + "content": "indeed affect the algorithm’s rates and are true irre-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 104, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "spective of where the algorithm is begun. Note that this allows our result to be much stronger than", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 308, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 324 + ], + "score": 1.0, + "content": "typical optimization lowerbounds that rely on specific initializations to ensure a component along", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 319, + 421, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 421, + 333 + ], + "score": 1.0, + "content": "the largest eigendirection of the update operator, for which bounds are proven.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 275, + 506, + 333 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 334, + 504, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 332, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 259, + 348 + ], + "score": 1.0, + "content": "Lemma 13. For any starting iterate", + "type": "text" + }, + { + "bbox": [ + 259, + 334, + 302, + 346 + ], + "score": 0.9, + "content": "\\mathbf { w } _ { 0 } \\neq \\mathbf { w } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 332, + 505, + 348 + ], + "score": 1.0, + "content": ", the HB method produces a non-zero component", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 345, + 260, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 248, + 357 + ], + "score": 1.0, + "content": "along the largest eigen direction of", + "type": "text" + }, + { + "bbox": [ + 249, + 346, + 256, + 355 + ], + "score": 0.68, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 345, + 260, + 357 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 332, + 505, + 357 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 368, + 506, + 468 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "Proof. We note that in a similar manner as other proofs, it suffices to argue for each dimension of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 506, + 392 + ], + "score": 1.0, + "content": "the problem separately. But before we start looking at each dimension separately, let us consider", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 389, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 122, + 405 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 390, + 134, + 403 + ], + "score": 0.89, + "content": "\\bar { j } ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 389, + 506, + 405 + ], + "score": 1.0, + "content": "dimension, and detail the approach we use to prove the claim: the idea is to examine the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 102, + 399, + 509, + 423 + ], + "spans": [ + { + "bbox": [ + 102, + 399, + 238, + 423 + ], + "score": 1.0, + "content": "subspace spanned by covariance", + "type": "text" + }, + { + "bbox": [ + 238, + 401, + 301, + 422 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left[ \\pmb { \\theta } _ { . } ^ { ( j ) } \\otimes \\pmb { \\theta } _ { . } ^ { ( j ) } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 399, + 359, + 423 + ], + "score": 1.0, + "content": "of the iterates", + "type": "text" + }, + { + "bbox": [ + 360, + 402, + 432, + 418 + ], + "score": 0.93, + "content": "\\pmb { \\theta } _ { 0 } ^ { ( j ) } , \\pmb { \\theta } _ { 1 } ^ { ( j ) } , \\pmb { \\theta } _ { 2 } ^ { ( j ) } , . . . ,", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 399, + 509, + 423 + ], + "score": 1.0, + "content": "for every starting", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 419, + 507, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 135, + 438 + ], + "score": 1.0, + "content": "iterate", + "type": "text" + }, + { + "bbox": [ + 135, + 420, + 196, + 436 + ], + "score": 0.94, + "content": "\\pmb { \\theta } _ { 0 } ^ { ( j ) } \\neq \\left[ 0 , 0 \\right] ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 419, + 459, + 438 + ], + "score": 1.0, + "content": "and prove that the largest eigenvector of the expected operator", + "type": "text" + }, + { + "bbox": [ + 459, + 422, + 477, + 433 + ], + "score": 0.9, + "content": "B ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 419, + 507, + 438 + ], + "score": 1.0, + "content": "is not", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 434, + 503, + 455 + ], + "spans": [ + { + "bbox": [ + 104, + 434, + 440, + 455 + ], + "score": 1.0, + "content": "orthogonal to this subspace. This implies that there exists a non-zero component of", + "type": "text" + }, + { + "bbox": [ + 440, + 435, + 503, + 455 + ], + "score": 0.89, + "content": "\\mathbb { E } \\left[ \\pmb { \\theta } _ { \\cdot } ^ { ( j ) } \\otimes \\pmb { \\theta } _ { \\cdot } ^ { ( j ) } \\right]", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 452, + 460, + 469 + ], + "spans": [ + { + "bbox": [ + 104, + 452, + 233, + 469 + ], + "score": 1.0, + "content": "in the largest eigen direction of", + "type": "text" + }, + { + "bbox": [ + 233, + 453, + 251, + 465 + ], + "score": 0.9, + "content": "B ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 452, + 407, + 469 + ], + "score": 1.0, + "content": ", and this decays at a rate that is at best", + "type": "text" + }, + { + "bbox": [ + 408, + 454, + 454, + 467 + ], + "score": 0.91, + "content": "\\lambda _ { \\operatorname* { m a x } } ( B ^ { ( j ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 452, + 460, + 469 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 102, + 368, + 509, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 472, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 104, + 469, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 133, + 487 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 133, + 472, + 192, + 484 + ], + "score": 0.94, + "content": "\\boldsymbol { B } ^ { ( j ) } ~ \\in ~ \\mathbb { R } ^ { 4 \\times 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 469, + 506, + 487 + ], + "score": 1.0, + "content": ", we begin by examining the expected covariance spanned by the iterates", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 479, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 107, + 484, + 186, + 499 + ], + "score": 0.91, + "content": "\\pmb { \\theta } _ { 0 } ^ { ( j ) } , \\pmb { \\theta } _ { 1 } ^ { ( j ) } , \\pmb { \\theta } _ { 2 } ^ { ( j ) } , \\pmb { \\theta } _ { 3 } ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 479, + 211, + 507 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 211, + 484, + 390, + 500 + ], + "score": 0.93, + "content": "\\mathbf { w } _ { 0 } ^ { ( j ) } - ( \\mathbf { w } ^ { * } ) ^ { ( j ) } = \\mathbf { \\bar { w } } _ { - 1 } ^ { ( j ) } - ( \\mathbf { w } ^ { * } ) ^ { ( j ) } = k ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 479, + 473, + 507 + ], + "score": 1.0, + "content": ". Now, this implies", + "type": "text" + }, + { + "bbox": [ + 474, + 484, + 505, + 500 + ], + "score": 0.91, + "content": "\\theta _ { 0 } ^ { ( j ) } =", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 497, + 189, + 516 + ], + "spans": [ + { + "bbox": [ + 107, + 500, + 157, + 514 + ], + "score": 0.92, + "content": "\\boldsymbol { k } ^ { ( j ) } \\cdot \\left[ 1 , 1 \\right] ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 497, + 189, + 516 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 469, + 506, + 516 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 518, + 473, + 546 + ], + "lines": [ + { + "bbox": [ + 137, + 518, + 473, + 546 + ], + "spans": [ + { + "bbox": [ + 137, + 518, + 473, + 546 + ], + "score": 0.86, + "content": "\\pmb { \\theta } _ { 1 } ^ { ( j ) } = k ^ { ( j ) } \\widehat { \\mathbf { A } } _ { 1 } ^ { ( j ) } \\left[ 1 \\right] , \\mathrm { w i t h } \\widehat { \\mathbf { A } } _ { 1 } ^ { ( j ) } = \\left[ 1 + \\alpha - \\delta \\widehat { \\mathbf { H } } _ { 1 } ^ { ( j ) } \\quad - \\alpha \\right] , \\mathrm { w h e r e } \\widehat { \\mathbf { H } } _ { 1 } ^ { ( j ) } = \\big ( a _ { 1 } ^ { ( j ) } \\big ) ^ { 2 } .", + "type": "interline_equation", + "image_path": "075e58cbc0117e80b6adfaa51e1cf1a4b122387aa3cfe9a35517173163888eb2.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 137, + 518, + 473, + 546 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 506, + 630 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 180, + 563 + ], + "score": 1.0, + "content": "This implies that", + "type": "text" + }, + { + "bbox": [ + 180, + 551, + 187, + 561 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "just appears as a scale factor. This in turn implies that in order to analyze", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 100, + 559, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 100, + 559, + 313, + 583 + ], + "score": 1.0, + "content": "the subspace spanned by the covariance of iterates", + "type": "text" + }, + { + "bbox": [ + 314, + 561, + 366, + 576 + ], + "score": 0.9, + "content": "\\theta _ { 0 } ^ { ( j ) } , \\theta _ { 1 } ^ { ( j ) } , . . . ,", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 559, + 433, + 583 + ], + "score": 1.0, + "content": ", we can assume", + "type": "text" + }, + { + "bbox": [ + 433, + 562, + 470, + 574 + ], + "score": 0.91, + "content": "k ^ { ( j ) } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 559, + 506, + 583 + ], + "score": 1.0, + "content": "without", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 574, + 507, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 257, + 591 + ], + "score": 1.0, + "content": "any loss in generality. This implies,", + "type": "text" + }, + { + "bbox": [ + 257, + 575, + 317, + 590 + ], + "score": 0.93, + "content": "\\pmb { \\theta } _ { 0 } ^ { ( j ) } = \\left[ 1 , 1 \\right] ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 574, + 507, + 591 + ], + "score": 1.0, + "content": ". Note that with this in place, we see that we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 227, + 602 + ], + "score": 1.0, + "content": "can now drop the superscript", + "type": "text" + }, + { + "bbox": [ + 227, + 590, + 234, + 601 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "that represents the dimension, since the analysis decouples across", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 601, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 171, + 616 + ], + "score": 1.0, + "content": "the dimensions", + "type": "text" + }, + { + "bbox": [ + 171, + 603, + 216, + 615 + ], + "score": 0.93, + "content": "j \\in \\{ 1 , 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 601, + 391, + 616 + ], + "score": 1.0, + "content": ". Furthermore, let the entries of the vector", + "type": "text" + }, + { + "bbox": [ + 391, + 604, + 403, + 615 + ], + "score": 0.88, + "content": "\\pmb { \\theta } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 601, + 478, + 616 + ], + "score": 1.0, + "content": "be represented as", + "type": "text" + }, + { + "bbox": [ + 479, + 601, + 491, + 615 + ], + "score": 0.74, + "content": "\\pmb { \\theta } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 601, + 505, + 616 + ], + "score": 1.0, + "content": "def =", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 615, + 348, + 631 + ], + "spans": [ + { + "bbox": [ + 107, + 615, + 157, + 630 + ], + "score": 0.85, + "content": "\\left[ \\theta _ { k 1 } \\quad \\theta _ { k 2 } \\right] ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 616, + 212, + 631 + ], + "score": 1.0, + "content": "Next, denote", + "type": "text" + }, + { + "bbox": [ + 212, + 615, + 289, + 629 + ], + "score": 0.92, + "content": "1 + \\alpha - \\delta \\widehat { \\mathbf { H } } _ { k } = \\widehat { t } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 616, + 348, + 631 + ], + "score": 1.0, + "content": ". This implies,", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 100, + 549, + 507, + 631 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 267, + 633, + 344, + 662 + ], + "lines": [ + { + "bbox": [ + 267, + 633, + 344, + 662 + ], + "spans": [ + { + "bbox": [ + 267, + 633, + 344, + 662 + ], + "score": 0.95, + "content": "\\begin{array} { r } \\widehat { \\mathbf { A } } _ { k } = \\left[ \\begin{array} { c c } { \\widehat { t } _ { k } } & { - \\alpha \\right] . } \\end{array} \\end{array}", + "type": "interline_equation", + "image_path": "563194c085e318134ba1dcd4c84395874345a02b1d6ed4fcb6126b19077e91b2.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 267, + 633, + 344, + 662 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 160, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 162, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 162, + 677 + ], + "score": 1.0, + "content": "Furthermore,", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 664, + 162, + 677 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 680, + 428, + 738 + ], + "lines": [ + { + "bbox": [ + 182, + 680, + 428, + 738 + ], + "spans": [ + { + "bbox": [ + 182, + 680, + 428, + 738 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\pmb \\theta _ { 1 } = \\widehat { \\mathbf A } _ { 1 } \\pmb \\theta _ { 0 } = \\left[ \\hat { t } _ { 1 } - \\alpha \\right] , \\pmb \\theta _ { 2 } = \\widehat { \\mathbf A } _ { 2 } \\pmb \\theta _ { 1 } = \\left[ \\hat { t } _ { 2 } ( \\hat { t } _ { 1 } - \\alpha ) - \\alpha \\right] , } \\\\ { \\pmb \\theta _ { 3 } = \\widehat { \\mathbf A } _ { 3 } \\pmb \\theta _ { 2 } = \\left[ \\hat { t } _ { 3 } ( \\hat { t } _ { 2 } ( \\hat { t } _ { 1 } - \\alpha ) - \\alpha ) - \\alpha ( \\hat { t } _ { 1 } - \\alpha ) \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "0614d54ea272ff955e32aeeab22d751a11862b3257157be460f37fd70aae708d.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 182, + 680, + 428, + 699.3333333333334 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 182, + 699.3333333333334, + 428, + 718.6666666666667 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 182, + 718.6666666666667, + 428, + 738.0000000000001 + ], + "spans": [], + "index": 38 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 261, + 95 + ], + "score": 1.0, + "content": "Let us consider the vectorized form of", + "type": "text" + }, + { + "bbox": [ + 262, + 82, + 334, + 95 + ], + "score": 0.92, + "content": "\\Phi _ { j } = \\mathbb { E } \\left[ \\pmb { \\theta } _ { j } \\otimes \\pmb { \\theta } _ { j } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 82, + 441, + 95 + ], + "score": 1.0, + "content": ", and we denote this as vec", + "type": "text" + }, + { + "bbox": [ + 441, + 82, + 461, + 95 + ], + "score": 0.68, + "content": "( \\Phi _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 141, + 106 + ], + "score": 0.85, + "content": "\\mathrm { v e c } ( \\Phi _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 93, + 171, + 106 + ], + "score": 1.0, + "content": "makes", + "type": "text" + }, + { + "bbox": [ + 171, + 94, + 184, + 106 + ], + "score": 0.89, + "content": "\\Phi _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 93, + 314, + 106 + ], + "score": 1.0, + "content": "become a column vector of size", + "type": "text" + }, + { + "bbox": [ + 314, + 95, + 337, + 104 + ], + "score": 0.87, + "content": "4 \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 93, + 415, + 106 + ], + "score": 1.0, + "content": ". Now, consider vec", + "type": "text" + }, + { + "bbox": [ + 416, + 94, + 436, + 106 + ], + "score": 0.61, + "content": "( \\Phi _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 93, + 452, + 106 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 452, + 94, + 504, + 105 + ], + "score": 0.91, + "content": "\\bar { \\boldsymbol { j } } = 0 , 1 , 2 , 3", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 363, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 334, + 117 + ], + "score": 1.0, + "content": "and concatenate these to form a matrix that we denote as", + "type": "text" + }, + { + "bbox": [ + 335, + 105, + 344, + 114 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 104, + 363, + 117 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 120, + 406, + 134 + ], + "lines": [ + { + "bbox": [ + 205, + 120, + 406, + 134 + ], + "spans": [ + { + "bbox": [ + 205, + 120, + 406, + 134 + ], + "score": 0.87, + "content": "\\mathcal { D } = \\left[ \\mathrm { v e c } ( \\Phi _ { 0 } ) \\mathrm { v e c } ( \\Phi _ { 1 } ) \\mathrm { v e c } ( \\Phi _ { 2 } ) \\mathrm { v e c } ( \\Phi _ { 3 } ) \\right] .", + "type": "interline_equation", + "image_path": "d69f4ec225f119d9ce46cad40495b635c267c73ffe69a3ea0e4729eb7c48c06f.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 205, + 120, + 406, + 134 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 506, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 217, + 151 + ], + "score": 1.0, + "content": "Now, since we note that", + "type": "text" + }, + { + "bbox": [ + 218, + 139, + 232, + 151 + ], + "score": 0.89, + "content": "\\Phi _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 138, + 303, + 151 + ], + "score": 1.0, + "content": "is a symmetric", + "type": "text" + }, + { + "bbox": [ + 304, + 139, + 332, + 149 + ], + "score": 0.9, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 138, + 369, + 151 + ], + "score": 1.0, + "content": "matrix,", + "type": "text" + }, + { + "bbox": [ + 369, + 139, + 379, + 149 + ], + "score": 0.81, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 138, + 506, + 151 + ], + "score": 1.0, + "content": "should contain two identical", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "score": 1.0, + "content": "rows implying that it has an eigenvalue that is zero and a corresponding eigenvector that is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 160, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 107, + 160, + 226, + 177 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\left[ { 0 \\mathrm { ~ \\ t ~ { ~ - } 1 / { \\sqrt { 2 } } ~ } } { \\hat { 1 } } / { \\sqrt { ( 2 \\tau ) } } \\mathrm { ~ \\ t ~ { ~ } } \\right] ^ { \\top } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 163, + 416, + 177 + ], + "score": 1.0, + "content": ". It turns out that this is also an eigenvector of", + "type": "text" + }, + { + "bbox": [ + 417, + 164, + 425, + 174 + ], + "score": 0.8, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 163, + 506, + 177 + ], + "score": 1.0, + "content": "with an eigenvalue", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 176, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 107, + 179, + 114, + 187 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 176, + 159, + 190 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 160, + 177, + 213, + 189 + ], + "score": 0.9, + "content": "\\operatorname* { d e t } ( B ) = \\alpha ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 176, + 505, + 190 + ], + "score": 1.0, + "content": ". This implies there are two cases that we need to consider: (i) when all", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 187, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 167, + 201 + ], + "score": 1.0, + "content": "eigenvalues of", + "type": "text" + }, + { + "bbox": [ + 167, + 189, + 175, + 198 + ], + "score": 0.8, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 187, + 281, + 201 + ], + "score": 1.0, + "content": "have the same magnitude", + "type": "text" + }, + { + "bbox": [ + 281, + 188, + 305, + 199 + ], + "score": 0.85, + "content": "( = \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 187, + 506, + 201 + ], + "score": 1.0, + "content": ". In this case, we are already done, because there", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 199, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 279, + 211 + ], + "score": 1.0, + "content": "exists at least one non zero eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 280, + 199, + 289, + 209 + ], + "score": 0.83, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 199, + 506, + 211 + ], + "score": 1.0, + "content": "and this should have some component along one of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 184, + 223 + ], + "score": 1.0, + "content": "the eigenvectors of", + "type": "text" + }, + { + "bbox": [ + 184, + 210, + 192, + 220 + ], + "score": 0.8, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "and we know that all eigenvectors have eigenvalues with a magnitude equal to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 143, + 232 + ], + "score": 0.92, + "content": "\\lambda _ { \\mathrm { m a x } } ( B )", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 221, + 505, + 234 + ], + "score": 1.0, + "content": ". Thus, there exists an iterate which has a non-zero component along the largest eigendirec-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 135, + 244 + ], + "score": 1.0, + "content": "tion of", + "type": "text" + }, + { + "bbox": [ + 136, + 232, + 144, + 241 + ], + "score": 0.81, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 231, + 505, + 244 + ], + "score": 1.0, + "content": ". (ii) the second case is the situation when we have eigenvalues with different magnitudes.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 241, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 196, + 256 + ], + "score": 1.0, + "content": "In this case, note that", + "type": "text" + }, + { + "bbox": [ + 196, + 242, + 313, + 254 + ], + "score": 0.91, + "content": "\\operatorname* { d e t } ( \\mathcal B ) = \\alpha ^ { 4 } < ( \\lambda _ { \\operatorname* { m a x } } ( \\mathcal B ) ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 241, + 353, + 256 + ], + "score": 1.0, + "content": "implying", + "type": "text" + }, + { + "bbox": [ + 353, + 242, + 412, + 254 + ], + "score": 0.93, + "content": "\\bar { \\lambda } _ { \\mathrm { m a x } } ( B ) > \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 241, + 506, + 256 + ], + "score": 1.0, + "content": ". In this case, we need", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 161, + 265 + ], + "score": 1.0, + "content": "to prove that", + "type": "text" + }, + { + "bbox": [ + 161, + 254, + 171, + 263 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "spans a three-dimensional subspace; if it does, it contains a component along the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 504, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 206, + 276 + ], + "score": 1.0, + "content": "largest eigendirection of", + "type": "text" + }, + { + "bbox": [ + 206, + 265, + 215, + 275 + ], + "score": 0.81, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 264, + 494, + 276 + ], + "score": 1.0, + "content": "which will round up the proof. Since we need to understand whether", + "type": "text" + }, + { + "bbox": [ + 495, + 265, + 504, + 274 + ], + "score": 0.79, + "content": "\\mathcal { D }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "spans a three dimensional subspace, we can consider a different (yet related) matrix, which we call", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 286, + 205, + 298 + ], + "spans": [ + { + "bbox": [ + 107, + 287, + 116, + 296 + ], + "score": 0.85, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 286, + 205, + 298 + ], + "score": 1.0, + "content": "and this is defined as:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 299, + 385, + 340 + ], + "lines": [ + { + "bbox": [ + 226, + 299, + 385, + 340 + ], + "spans": [ + { + "bbox": [ + 226, + 299, + 385, + 340 + ], + "score": 0.94, + "content": "\\mathcal { R } \\stackrel { \\mathrm { d e f } } { = } \\mathbb { E } \\left( \\begin{array} { c c c } { \\theta _ { 0 1 } ^ { 2 } } & { \\theta _ { 1 1 } ^ { 2 } } & { \\theta _ { 2 1 } ^ { 2 } } \\\\ { \\theta _ { 0 1 } \\theta _ { 0 2 } } & { \\theta _ { 1 1 } \\theta _ { 1 2 } } & { \\theta _ { 2 1 } \\theta _ { 2 2 } } \\\\ { \\theta _ { 0 2 } ^ { 2 } } & { \\theta _ { 1 2 } ^ { 2 } } & { \\theta _ { 2 2 } ^ { 2 } } \\end{array} \\right)", + "type": "interline_equation", + "image_path": "3c20cce8c2c0d6134445db18bfd15fcba74adcbd9ac0bdca4df8f04fef5db123.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 226, + 299, + 385, + 312.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 226, + 312.6666666666667, + 385, + 326.33333333333337 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 226, + 326.33333333333337, + 385, + 340.00000000000006 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 344, + 504, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 212, + 358 + ], + "score": 1.0, + "content": "Given the expressions for", + "type": "text" + }, + { + "bbox": [ + 212, + 344, + 248, + 358 + ], + "score": 0.93, + "content": "\\{ \\pmb { \\theta } _ { j } \\} _ { j = 0 } ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 344, + 317, + 358 + ], + "score": 1.0, + "content": "(by definition of", + "type": "text" + }, + { + "bbox": [ + 317, + 345, + 329, + 356 + ], + "score": 0.88, + "content": "\\pmb { \\theta } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "and using equation 4), we can substitute to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 355, + 270, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 139, + 368 + ], + "score": 1.0, + "content": "see that", + "type": "text" + }, + { + "bbox": [ + 140, + 356, + 149, + 366 + ], + "score": 0.85, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 355, + 270, + 368 + ], + "score": 1.0, + "content": "has the following expression:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 371, + 430, + 412 + ], + "lines": [ + { + "bbox": [ + 180, + 371, + 430, + 412 + ], + "spans": [ + { + "bbox": [ + 180, + 371, + 430, + 412 + ], + "score": 0.95, + "content": "\\mathcal { R } = \\left[ { 1 \\atop 1 } \\begin{array} { c c } { { \\mathbb { E } \\left[ ( \\hat { t } _ { 1 } - \\alpha ) ^ { 2 } \\right] } } & { { \\mathbb { E } \\left[ ( \\hat { t } _ { 2 } ( \\hat { t } _ { 1 } - \\alpha ) - \\alpha ) ^ { 2 } \\right] } } \\\\ { { \\mathbb { E } \\left[ \\hat { t } _ { 1 } - \\alpha \\right] } } & { { \\mathbb { E } \\left[ ( ( \\hat { t } _ { 2 } ( \\hat { t } _ { 1 } - \\alpha ) - \\alpha ) ) ( \\hat { t } _ { 1 } - \\alpha ) \\right] } } \\\\ { { 1 } } & { { \\mathbb { E } \\left[ ( \\hat { t } _ { 1 } - \\alpha ) ^ { 2 } \\right] } } \\end{array} \\right] .", + "type": "interline_equation", + "image_path": "fff99e0533348a2104f79eb84a2054e261225711c13efbd3cf425173bcd8318a.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 180, + 371, + 430, + 384.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 180, + 384.6666666666667, + 430, + 398.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 180, + 398.33333333333337, + 430, + 412.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 504, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 226, + 429 + ], + "score": 1.0, + "content": "If we compute and prove that", + "type": "text" + }, + { + "bbox": [ + 227, + 416, + 273, + 428 + ], + "score": 0.86, + "content": "\\operatorname* { d e t } ( \\mathcal { R } ) \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 415, + 418, + 429 + ], + "score": 1.0, + "content": ", we are done since that implies that", + "type": "text" + }, + { + "bbox": [ + 418, + 417, + 428, + 426 + ], + "score": 0.85, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "has three non-zero", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 426, + 158, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 158, + 440 + ], + "score": 1.0, + "content": "eigenvalues.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 443, + 504, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 290, + 458 + ], + "score": 1.0, + "content": "This implies, we first define the following: let", + "type": "text" + }, + { + "bbox": [ + 290, + 443, + 395, + 456 + ], + "score": 0.92, + "content": "q _ { \\gamma } = ( t - 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Then,", + "type": "text" + }, + { + "bbox": [ + 425, + 444, + 435, + 454 + ], + "score": 0.85, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 442, + 506, + 458 + ], + "score": 1.0, + "content": "can be expressed", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 456, + 121, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 121, + 468 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 468, + 446, + 627 + ], + "lines": [ + { + "bbox": [ + 164, + 468, + 446, + 627 + ], + "spans": [ + { + "bbox": [ + 164, + 468, + 446, + 627 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathrm { d e t } ( \\mathcal { R } ) = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 1 } & { q _ { \\alpha } } & { 2 q _ { \\alpha } - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 1 } & { t - \\alpha } & { t q _ { \\alpha } - \\alpha ( t - \\alpha ) } \\end{array} \\right] \\right) } } \\\\ & { \\qquad = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 1 } & { q _ { \\alpha } } & { 2 \\alpha } \\\\ { 1 } & { 1 } & { \\phi _ { \\alpha } } \\\\ { 1 } & { 1 } & { \\phi _ { \\alpha } } \\\\ { 1 } & { t - \\alpha } & { q _ { \\alpha } - \\alpha ( t - \\alpha ) - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 1 } & { 1 } & { \\theta _ { \\alpha } } \\end{array} \\right] \\right) } } \\\\ & { \\qquad = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 1 } & { q _ { \\alpha } - 1 } & { q _ { \\alpha } ( q _ { \\alpha } - q _ { \\alpha } ) - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 1 } & { t - \\alpha - 1 } & { 1 } & { 0 } \\\\ { 1 } & { 0 } & { t q _ { \\alpha } - \\alpha ( t - \\alpha ) - 0 } & { ( t - \\alpha ) q _ { \\alpha } } \\end{array} \\right] \\right) } } \\\\ & { \\qquad = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 0 } & { q _ { \\alpha } - 1 } & { q _ { \\alpha } ( q _ { \\alpha } - q _ { \\alpha } ) - ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 0 } & { 1 } & { \\theta _ { \\alpha } } \\end{array} \\right] \\right) } } \\\\ & { \\qquad = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 0 } & { q _ { \\alpha } - 1 } & { q _ { \\alpha } ( q _ { \\alpha } - q _ { \\alpha } ) - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 0 } & { t - \\alpha - 1 } & { 1 } \\end{array} \\right] \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "5e6b21a6080326b5942c4da37a2a03ce47ec3a1c714e266fd9d1fcec5fed47b3.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 164, + 468, + 446, + 521.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 164, + 521.0, + 446, + 574.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 164, + 574.0, + 446, + 627.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 504, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 142, + 645 + ], + "score": 1.0, + "content": "Note: (i)", + "type": "text" + }, + { + "bbox": [ + 142, + 633, + 505, + 645 + ], + "score": 0.77, + "content": "q _ { \\alpha } - 1 = ( t - \\alpha ) ^ { 2 } - 1 + ( c - 1 ) x ^ { 2 } = ( 1 - x ) ^ { 2 } - 1 + ( c - 1 ) x ^ { 2 } = - 2 x + x ^ { 2 } + ( c - 1 ) x ^ { 2 } = 0", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 641, + 157, + 657 + ], + "spans": [ + { + "bbox": [ + 107, + 644, + 152, + 654 + ], + "score": 0.9, + "content": "- 2 x + c x ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 641, + 157, + 657 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 656, + 504, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 189, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 121, + 667 + ], + "score": 1.0, + "content": "(ii)", + "type": "text" + }, + { + "bbox": [ + 121, + 656, + 189, + 665 + ], + "score": 0.87, + "content": "t - \\alpha - 1 = - x", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 664, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 122, + 688 + ], + "score": 1.0, + "content": "(iv)", + "type": "text" + }, + { + "bbox": [ + 105, + 664, + 122, + 679 + ], + "score": 1.0, + "content": "(iii)", + "type": "text" + }, + { + "bbox": [ + 125, + 666, + 504, + 689 + ], + "score": 0.42, + "content": "\\begin{array} { r l } & { \\alpha ( q _ { \\alpha } - ( t - \\alpha ) ) = \\alpha ( ( t - \\alpha ) ^ { 2 } - ( t - \\alpha ) + ( c - 1 ) x ^ { 2 } ) = \\alpha ( ( 1 - x ) ( - x ) + ( c - 1 ) x ^ { 2 } ) = \\alpha x ( - 1 + c x ) } \\\\ & { q _ { 0 } - q _ { \\alpha } = t ^ { 2 } - ( t - \\alpha ) ^ { 2 } = \\alpha ( 2 t - \\alpha ) = 2 t \\alpha - \\alpha ^ { 2 } . } \\end{array}", + "type": "inline_equation", + "image_path": "47f026b02c742640634acb9d908bfff54014c48400179953ea7be45bbe0af47d.jpg" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 686, + 135, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 135, + 701 + ], + "score": 1.0, + "content": "Then,", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 702, + 439, + 735 + ], + "lines": [ + { + "bbox": [ + 149, + 702, + 439, + 735 + ], + "spans": [ + { + "bbox": [ + 149, + 702, + 439, + 735 + ], + "score": 0.86, + "content": "\\begin{array} { r } { ( 2 \\alpha t - \\alpha ^ { 2 } ) q _ { \\alpha } - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } = 2 t \\alpha ( q _ { \\alpha } - ( t - \\alpha ) ) + \\alpha ^ { 2 } ( 1 - q _ { \\alpha } ) } \\\\ { = 2 t \\alpha ( - x + c x ^ { 2 } ) - \\alpha ^ { 2 } ( - 2 x + c x ^ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "20fbd1362c636fb537b4b2079712bb5c9e76d381bbdf6e4400c3cc2cb4011a96.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 149, + 702, + 439, + 713.0 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 149, + 713.0, + 439, + 724.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 149, + 724.0, + 439, + 735.0 + ], + "spans": [], + "index": 40 + } + ] + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 261, + 95 + ], + "score": 1.0, + "content": "Let us consider the vectorized form of", + "type": "text" + }, + { + "bbox": [ + 262, + 82, + 334, + 95 + ], + "score": 0.92, + "content": "\\Phi _ { j } = \\mathbb { E } \\left[ \\pmb { \\theta } _ { j } \\otimes \\pmb { \\theta } _ { j } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 82, + 441, + 95 + ], + "score": 1.0, + "content": ", and we denote this as vec", + "type": "text" + }, + { + "bbox": [ + 441, + 82, + 461, + 95 + ], + "score": 0.68, + "content": "( \\Phi _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 141, + 106 + ], + "score": 0.85, + "content": "\\mathrm { v e c } ( \\Phi _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 93, + 171, + 106 + ], + "score": 1.0, + "content": "makes", + "type": "text" + }, + { + "bbox": [ + 171, + 94, + 184, + 106 + ], + "score": 0.89, + "content": "\\Phi _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 93, + 314, + 106 + ], + "score": 1.0, + "content": "become a column vector of size", + "type": "text" + }, + { + "bbox": [ + 314, + 95, + 337, + 104 + ], + "score": 0.87, + "content": "4 \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 93, + 415, + 106 + ], + "score": 1.0, + "content": ". Now, consider vec", + "type": "text" + }, + { + "bbox": [ + 416, + 94, + 436, + 106 + ], + "score": 0.61, + "content": "( \\Phi _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 93, + 452, + 106 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 452, + 94, + 504, + 105 + ], + "score": 0.91, + "content": "\\bar { \\boldsymbol { j } } = 0 , 1 , 2 , 3", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 363, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 334, + 117 + ], + "score": 1.0, + "content": "and concatenate these to form a matrix that we denote as", + "type": "text" + }, + { + "bbox": [ + 335, + 105, + 344, + 114 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 104, + 363, + 117 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 505, + 117 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 120, + 406, + 134 + ], + "lines": [ + { + "bbox": [ + 205, + 120, + 406, + 134 + ], + "spans": [ + { + "bbox": [ + 205, + 120, + 406, + 134 + ], + "score": 0.87, + "content": "\\mathcal { D } = \\left[ \\mathrm { v e c } ( \\Phi _ { 0 } ) \\mathrm { v e c } ( \\Phi _ { 1 } ) \\mathrm { v e c } ( \\Phi _ { 2 } ) \\mathrm { v e c } ( \\Phi _ { 3 } ) \\right] .", + "type": "interline_equation", + "image_path": "d69f4ec225f119d9ce46cad40495b635c267c73ffe69a3ea0e4729eb7c48c06f.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 205, + 120, + 406, + 134 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 506, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 217, + 151 + ], + "score": 1.0, + "content": "Now, since we note that", + "type": "text" + }, + { + "bbox": [ + 218, + 139, + 232, + 151 + ], + "score": 0.89, + "content": "\\Phi _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 138, + 303, + 151 + ], + "score": 1.0, + "content": "is a symmetric", + "type": "text" + }, + { + "bbox": [ + 304, + 139, + 332, + 149 + ], + "score": 0.9, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 138, + 369, + 151 + ], + "score": 1.0, + "content": "matrix,", + "type": "text" + }, + { + "bbox": [ + 369, + 139, + 379, + 149 + ], + "score": 0.81, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 138, + 506, + 151 + ], + "score": 1.0, + "content": "should contain two identical", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "score": 1.0, + "content": "rows implying that it has an eigenvalue that is zero and a corresponding eigenvector that is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 160, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 107, + 160, + 226, + 177 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\left[ { 0 \\mathrm { ~ \\ t ~ { ~ - } 1 / { \\sqrt { 2 } } ~ } } { \\hat { 1 } } / { \\sqrt { ( 2 \\tau ) } } \\mathrm { ~ \\ t ~ { ~ } } \\right] ^ { \\top } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 163, + 416, + 177 + ], + "score": 1.0, + "content": ". 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This implies there are two cases that we need to consider: (i) when all", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 187, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 167, + 201 + ], + "score": 1.0, + "content": "eigenvalues of", + "type": "text" + }, + { + "bbox": [ + 167, + 189, + 175, + 198 + ], + "score": 0.8, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 187, + 281, + 201 + ], + "score": 1.0, + "content": "have the same magnitude", + "type": "text" + }, + { + "bbox": [ + 281, + 188, + 305, + 199 + ], + "score": 0.85, + "content": "( = \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 187, + 506, + 201 + ], + "score": 1.0, + "content": ". In this case, we are already done, because there", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 199, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 279, + 211 + ], + "score": 1.0, + "content": "exists at least one non zero eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 280, + 199, + 289, + 209 + ], + "score": 0.83, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 199, + 506, + 211 + ], + "score": 1.0, + "content": "and this should have some component along one of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 184, + 223 + ], + "score": 1.0, + "content": "the eigenvectors of", + "type": "text" + }, + { + "bbox": [ + 184, + 210, + 192, + 220 + ], + "score": 0.8, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "and we know that all eigenvectors have eigenvalues with a magnitude equal to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 143, + 232 + ], + "score": 0.92, + "content": "\\lambda _ { \\mathrm { m a x } } ( B )", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 221, + 505, + 234 + ], + "score": 1.0, + "content": ". Thus, there exists an iterate which has a non-zero component along the largest eigendirec-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 135, + 244 + ], + "score": 1.0, + "content": "tion of", + "type": "text" + }, + { + "bbox": [ + 136, + 232, + 144, + 241 + ], + "score": 0.81, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 231, + 505, + 244 + ], + "score": 1.0, + "content": ". (ii) the second case is the situation when we have eigenvalues with different magnitudes.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 241, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 196, + 256 + ], + "score": 1.0, + "content": "In this case, note that", + "type": "text" + }, + { + "bbox": [ + 196, + 242, + 313, + 254 + ], + "score": 0.91, + "content": "\\operatorname* { d e t } ( \\mathcal B ) = \\alpha ^ { 4 } < ( \\lambda _ { \\operatorname* { m a x } } ( \\mathcal B ) ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 241, + 353, + 256 + ], + "score": 1.0, + "content": "implying", + "type": "text" + }, + { + "bbox": [ + 353, + 242, + 412, + 254 + ], + "score": 0.93, + "content": "\\bar { \\lambda } _ { \\mathrm { m a x } } ( B ) > \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 241, + 506, + 256 + ], + "score": 1.0, + "content": ". In this case, we need", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 161, + 265 + ], + "score": 1.0, + "content": "to prove that", + "type": "text" + }, + { + "bbox": [ + 161, + 254, + 171, + 263 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "spans a three-dimensional subspace; if it does, it contains a component along the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 504, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 206, + 276 + ], + "score": 1.0, + "content": "largest eigendirection of", + "type": "text" + }, + { + "bbox": [ + 206, + 265, + 215, + 275 + ], + "score": 0.81, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 264, + 494, + 276 + ], + "score": 1.0, + "content": "which will round up the proof. Since we need to understand whether", + "type": "text" + }, + { + "bbox": [ + 495, + 265, + 504, + 274 + ], + "score": 0.79, + "content": "\\mathcal { D }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "spans a three dimensional subspace, we can consider a different (yet related) matrix, which we call", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 286, + 205, + 298 + ], + "spans": [ + { + "bbox": [ + 107, + 287, + 116, + 296 + ], + "score": 0.85, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 286, + 205, + 298 + ], + "score": 1.0, + "content": "and this is defined as:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 10.5, + "bbox_fs": [ + 104, + 138, + 506, + 298 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 299, + 385, + 340 + ], + "lines": [ + { + "bbox": [ + 226, + 299, + 385, + 340 + ], + "spans": [ + { + "bbox": [ + 226, + 299, + 385, + 340 + ], + "score": 0.94, + "content": "\\mathcal { R } \\stackrel { \\mathrm { d e f } } { = } \\mathbb { E } \\left( \\begin{array} { c c c } { \\theta _ { 0 1 } ^ { 2 } } & { \\theta _ { 1 1 } ^ { 2 } } & { \\theta _ { 2 1 } ^ { 2 } } \\\\ { \\theta _ { 0 1 } \\theta _ { 0 2 } } & { \\theta _ { 1 1 } \\theta _ { 1 2 } } & { \\theta _ { 2 1 } \\theta _ { 2 2 } } \\\\ { \\theta _ { 0 2 } ^ { 2 } } & { \\theta _ { 1 2 } ^ { 2 } } & { \\theta _ { 2 2 } ^ { 2 } } \\end{array} \\right)", + "type": "interline_equation", + "image_path": "3c20cce8c2c0d6134445db18bfd15fcba74adcbd9ac0bdca4df8f04fef5db123.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 226, + 299, + 385, + 312.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 226, + 312.6666666666667, + 385, + 326.33333333333337 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 226, + 326.33333333333337, + 385, + 340.00000000000006 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 344, + 504, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 212, + 358 + ], + "score": 1.0, + "content": "Given the expressions for", + "type": "text" + }, + { + "bbox": [ + 212, + 344, + 248, + 358 + ], + "score": 0.93, + "content": "\\{ \\pmb { \\theta } _ { j } \\} _ { j = 0 } ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 344, + 317, + 358 + ], + "score": 1.0, + "content": "(by definition of", + "type": "text" + }, + { + "bbox": [ + 317, + 345, + 329, + 356 + ], + "score": 0.88, + "content": "\\pmb { \\theta } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "and using equation 4), we can substitute to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 355, + 270, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 139, + 368 + ], + "score": 1.0, + "content": "see that", + "type": "text" + }, + { + "bbox": [ + 140, + 356, + 149, + 366 + ], + "score": 0.85, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 355, + 270, + 368 + ], + "score": 1.0, + "content": "has the following expression:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 344, + 506, + 368 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 371, + 430, + 412 + ], + "lines": [ + { + "bbox": [ + 180, + 371, + 430, + 412 + ], + "spans": [ + { + "bbox": [ + 180, + 371, + 430, + 412 + ], + "score": 0.95, + "content": "\\mathcal { R } = \\left[ { 1 \\atop 1 } \\begin{array} { c c } { { \\mathbb { E } \\left[ ( \\hat { t } _ { 1 } - \\alpha ) ^ { 2 } \\right] } } & { { \\mathbb { E } \\left[ ( \\hat { t } _ { 2 } ( \\hat { t } _ { 1 } - \\alpha ) - \\alpha ) ^ { 2 } \\right] } } \\\\ { { \\mathbb { E } \\left[ \\hat { t } _ { 1 } - \\alpha \\right] } } & { { \\mathbb { E } \\left[ ( ( \\hat { t } _ { 2 } ( \\hat { t } _ { 1 } - \\alpha ) - \\alpha ) ) ( \\hat { t } _ { 1 } - \\alpha ) \\right] } } \\\\ { { 1 } } & { { \\mathbb { E } \\left[ ( \\hat { t } _ { 1 } - \\alpha ) ^ { 2 } \\right] } } \\end{array} \\right] .", + "type": "interline_equation", + "image_path": "fff99e0533348a2104f79eb84a2054e261225711c13efbd3cf425173bcd8318a.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 180, + 371, + 430, + 384.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 180, + 384.6666666666667, + 430, + 398.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 180, + 398.33333333333337, + 430, + 412.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 504, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 226, + 429 + ], + "score": 1.0, + "content": "If we compute and prove that", + "type": "text" + }, + { + "bbox": [ + 227, + 416, + 273, + 428 + ], + "score": 0.86, + "content": "\\operatorname* { d e t } ( \\mathcal { R } ) \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 415, + 418, + 429 + ], + "score": 1.0, + "content": ", we are done since that implies that", + "type": "text" + }, + { + "bbox": [ + 418, + 417, + 428, + 426 + ], + "score": 0.85, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "has three non-zero", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 426, + 158, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 158, + 440 + ], + "score": 1.0, + "content": "eigenvalues.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 415, + 506, + 440 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 443, + 504, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 290, + 458 + ], + "score": 1.0, + "content": "This implies, we first define the following: let", + "type": "text" + }, + { + "bbox": [ + 290, + 443, + 395, + 456 + ], + "score": 0.92, + "content": "q _ { \\gamma } = ( t - \\gamma ) ^ { 2 } + ( c - 1 ) x ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 442, + 425, + 458 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 425, + 444, + 435, + 454 + ], + "score": 0.85, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 442, + 506, + 458 + ], + "score": 1.0, + "content": "can be expressed", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 456, + 121, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 121, + 468 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 442, + 506, + 468 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 468, + 446, + 627 + ], + "lines": [ + { + "bbox": [ + 164, + 468, + 446, + 627 + ], + "spans": [ + { + "bbox": [ + 164, + 468, + 446, + 627 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathrm { d e t } ( \\mathcal { R } ) = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 1 } & { q _ { \\alpha } } & { 2 q _ { \\alpha } - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 1 } & { t - \\alpha } & { t q _ { \\alpha } - \\alpha ( t - \\alpha ) } \\end{array} \\right] \\right) } } \\\\ & { \\qquad = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 1 } & { q _ { \\alpha } } & { 2 \\alpha } \\\\ { 1 } & { 1 } & { \\phi _ { \\alpha } } \\\\ { 1 } & { 1 } & { \\phi _ { \\alpha } } \\\\ { 1 } & { t - \\alpha } & { q _ { \\alpha } - \\alpha ( t - \\alpha ) - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 1 } & { 1 } & { \\theta _ { \\alpha } } \\end{array} \\right] \\right) } } \\\\ & { \\qquad = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 1 } & { q _ { \\alpha } - 1 } & { q _ { \\alpha } ( q _ { \\alpha } - q _ { \\alpha } ) - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 1 } & { t - \\alpha - 1 } & { 1 } & { 0 } \\\\ { 1 } & { 0 } & { t q _ { \\alpha } - \\alpha ( t - \\alpha ) - 0 } & { ( t - \\alpha ) q _ { \\alpha } } \\end{array} \\right] \\right) } } \\\\ & { \\qquad = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 0 } & { q _ { \\alpha } - 1 } & { q _ { \\alpha } ( q _ { \\alpha } - q _ { \\alpha } ) - ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 0 } & { 1 } & { \\theta _ { \\alpha } } \\end{array} \\right] \\right) } } \\\\ & { \\qquad = \\mathrm { d e t } { \\left( \\left[ \\begin{array} { l l l } { 0 } & { q _ { \\alpha } - 1 } & { q _ { \\alpha } ( q _ { \\alpha } - q _ { \\alpha } ) - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } } \\\\ { 0 } & { t - \\alpha - 1 } & { 1 } \\end{array} \\right] \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "5e6b21a6080326b5942c4da37a2a03ce47ec3a1c714e266fd9d1fcec5fed47b3.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 164, + 468, + 446, + 521.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 164, + 521.0, + 446, + 574.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 164, + 574.0, + 446, + 627.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 504, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 142, + 645 + ], + "score": 1.0, + "content": "Note: (i)", + "type": "text" + }, + { + "bbox": [ + 142, + 633, + 505, + 645 + ], + "score": 0.77, + "content": "q _ { \\alpha } - 1 = ( t - \\alpha ) ^ { 2 } - 1 + ( c - 1 ) x ^ { 2 } = ( 1 - x ) ^ { 2 } - 1 + ( c - 1 ) x ^ { 2 } = - 2 x + x ^ { 2 } + ( c - 1 ) x ^ { 2 } = 0", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 641, + 157, + 657 + ], + "spans": [ + { + "bbox": [ + 107, + 644, + 152, + 654 + ], + "score": 0.9, + "content": "- 2 x + c x ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 641, + 157, + 657 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 631, + 505, + 657 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 656, + 504, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 189, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 121, + 667 + ], + "score": 1.0, + "content": "(ii)", + "type": "text" + }, + { + "bbox": [ + 121, + 656, + 189, + 665 + ], + "score": 0.87, + "content": "t - \\alpha - 1 = - x", + "type": "inline_equation" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 104, + 664, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 122, + 688 + ], + "score": 1.0, + "content": "(iv)", + "type": "text" + }, + { + "bbox": [ + 105, + 664, + 122, + 679 + ], + "score": 1.0, + "content": "(iii)", + "type": "text" + }, + { + "bbox": [ + 125, + 666, + 504, + 689 + ], + "score": 0.42, + "content": "\\begin{array} { r l } & { \\alpha ( q _ { \\alpha } - ( t - \\alpha ) ) = \\alpha ( ( t - \\alpha ) ^ { 2 } - ( t - \\alpha ) + ( c - 1 ) x ^ { 2 } ) = \\alpha ( ( 1 - x ) ( - x ) + ( c - 1 ) x ^ { 2 } ) = \\alpha x ( - 1 + c x ) } \\\\ & { q _ { 0 } - q _ { \\alpha } = t ^ { 2 } - ( t - \\alpha ) ^ { 2 } = \\alpha ( 2 t - \\alpha ) = 2 t \\alpha - \\alpha ^ { 2 } . } \\end{array}", + "type": "inline_equation", + "image_path": "47f026b02c742640634acb9d908bfff54014c48400179953ea7be45bbe0af47d.jpg" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 686, + 135, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 135, + 701 + ], + "score": 1.0, + "content": "Then,", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + } + ], + "index": 36, + "bbox_fs": [ + 104, + 654, + 504, + 701 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 702, + 439, + 735 + ], + "lines": [ + { + "bbox": [ + 149, + 702, + 439, + 735 + ], + "spans": [ + { + "bbox": [ + 149, + 702, + 439, + 735 + ], + "score": 0.86, + "content": "\\begin{array} { r } { ( 2 \\alpha t - \\alpha ^ { 2 } ) q _ { \\alpha } - 2 \\alpha t ( t - \\alpha ) + \\alpha ^ { 2 } = 2 t \\alpha ( q _ { \\alpha } - ( t - \\alpha ) ) + \\alpha ^ { 2 } ( 1 - q _ { \\alpha } ) } \\\\ { = 2 t \\alpha ( - x + c x ^ { 2 } ) - \\alpha ^ { 2 } ( - 2 x + c x ^ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "20fbd1362c636fb537b4b2079712bb5c9e76d381bbdf6e4400c3cc2cb4011a96.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 149, + 702, + 439, + 713.0 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 149, + 713.0, + 439, + 724.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 149, + 724.0, + 439, + 735.0 + ], + "spans": [], + "index": 40 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "interline_equation", + "bbox": [ + 286, + 81, + 461, + 144 + ], + "lines": [ + { + "bbox": [ + 286, + 81, + 461, + 144 + ], + "spans": [ + { + "bbox": [ + 286, + 81, + 461, + 144 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { = - 2 t \\alpha x + 2 x \\alpha ^ { 2 } + 2 t \\alpha c x ^ { 2 } - c \\alpha ^ { 2 } x ^ { 2 } } \\\\ & { = 2 \\alpha x ( - t + \\alpha ) + c \\alpha x ^ { 2 } ( 2 t - \\alpha ) } \\\\ & { = - 2 \\alpha x ( 1 - x ) + 2 c \\alpha x ^ { 2 } ( 1 - x ) + c \\alpha ^ { 2 } x ^ { 2 } } \\\\ & { = 2 \\alpha x ( 1 - x ) ( - 1 + c x ) + c \\alpha ^ { 2 } x ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "c2d87a6adadba871e5f064c44116e20243172461ca53c598ba5613e87d00c556.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 286, + 81, + 461, + 102.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 286, + 102.0, + 461, + 123.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 286, + 123.0, + 461, + 144.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 147, + 132, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 146, + 133, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 133, + 161 + ], + "score": 1.0, + "content": "Then,", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 163, + 446, + 276 + ], + "lines": [ + { + "bbox": [ + 164, + 163, + 446, + 276 + ], + "spans": [ + { + "bbox": [ + 164, + 163, + 446, + 276 + ], + "score": 0.95, + "content": "{ \\begin{array} { r l } & { \\operatorname* { d e t } ( \\mathcal { R } ) = \\operatorname* { d e t } { \\left( \\begin{array} { l l l } { 0 } & { x ( c x - 2 ) } & { 2 \\alpha x ( 1 - x ) ( - 1 + c x ) + c \\alpha ^ { 2 } x ^ { 2 } } \\\\ { 0 } & { - x } & { \\alpha x ( c x - 1 ) } \\\\ { 1 } & { 0 } & { 0 } \\end{array} \\right) } } \\\\ & { \\qquad = x ^ { 2 } \\alpha \\operatorname* { d e t } \\left( { \\left[ \\begin{array} { l l l } { 0 } & { ( c x - 2 ) } & { c \\alpha x + 2 ( 1 - x ) ( c x - 1 ) } \\\\ { 0 } & { - 1 } & { c x - 1 } \\\\ { 1 } & { 0 } & { 0 } \\end{array} \\right] } \\right) } \\\\ & { \\qquad = x ^ { 3 } \\alpha \\operatorname* { d e t } \\left( { \\left[ \\begin{array} { l l l } { 0 } & { c } & { c \\alpha - 2 ( c x - 1 ) } \\\\ { 0 } & { - 1 } & { c x - 1 } \\\\ { 1 } & { 0 } & { 0 } \\end{array} \\right] } \\right) } \\end{array} }", + "type": "interline_equation", + "image_path": "a16691d3c7ee37009e76b6c2fee6d01aeac1460ff850eb443b092b9570c3a0ef.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 164, + 163, + 446, + 200.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 164, + 200.66666666666666, + 446, + 238.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 164, + 238.33333333333331, + 446, + 276.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 132, + 291 + ], + "lines": [ + { + "bbox": [ + 106, + 278, + 134, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 134, + 294 + ], + "score": 1.0, + "content": "Then,", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 295, + 408, + 352 + ], + "lines": [ + { + "bbox": [ + 202, + 295, + 408, + 352 + ], + "spans": [ + { + "bbox": [ + 202, + 295, + 408, + 352 + ], + "score": 0.93, + "content": "\\begin{array} { c } { { \\operatorname * { d e t } ( \\mathcal { R } ) = x ^ { 3 } \\alpha \\bigg ( c ( - 1 + c x ) - 2 ( - 1 + c x ) + c \\alpha \\bigg ) } } \\\\ { { = \\alpha x ^ { 3 } \\bigg ( ( c - 2 ) ( - 1 + c x ) + c \\alpha \\bigg ) } } \\end{array}", + "type": "interline_equation", + "image_path": "0eadf542bbb122afce66a44b53d459f6100df1faae43af7754b8b176cf312392.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 202, + 295, + 408, + 314.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 202, + 314.0, + 408, + 333.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 202, + 333.0, + 408, + 352.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 285, + 366 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 285, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 285, + 367 + ], + "score": 1.0, + "content": "Note that this determinant can be zero when", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 260, + 368, + 351, + 393 + ], + "lines": [ + { + "bbox": [ + 260, + 368, + 351, + 393 + ], + "spans": [ + { + "bbox": [ + 260, + 368, + 351, + 393 + ], + "score": 0.95, + "content": "\\alpha = \\frac { ( c - 2 ) ( 1 - c x ) } { c } .", + "type": "interline_equation", + "image_path": "3eb8355657aeb44dfcbd395c2082fbf6fd540dce6e01864969c1289e7447458a.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 260, + 368, + 351, + 393 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "We show this is not possible by splitting our argument into two parts, one about the convergent", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 406, + 474, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 236, + 425 + ], + "score": 1.0, + "content": "regime of the algorithm (where,", + "type": "text" + }, + { + "bbox": [ + 236, + 408, + 306, + 426 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } < \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 406, + 474, + 425 + ], + "score": 1.0, + "content": "and the other about the divergent regime.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 429, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "Let us first provide a proof for the convergent regime of the algorithm. For this regime, let the chosen", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 440, + 504, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 113, + 451 + ], + "score": 0.72, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 440, + 185, + 454 + ], + "score": 1.0, + "content": "be represented as", + "type": "text" + }, + { + "bbox": [ + 186, + 441, + 198, + 451 + ], + "score": 0.88, + "content": "\\delta ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 440, + 353, + 454 + ], + "score": 1.0, + "content": ". Now, for the smaller eigen direction,", + "type": "text" + }, + { + "bbox": [ + 353, + 441, + 455, + 453 + ], + "score": 0.93, + "content": "x = \\delta ^ { + } \\lambda _ { \\mathrm { m i n } } = c \\bar { \\delta ^ { + } } \\sigma _ { 1 } ^ { 2 } / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 440, + 496, + 454 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 496, + 443, + 504, + 451 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 452, + 228, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 228, + 464 + ], + "score": 1.0, + "content": "was chosen as per equation 5,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 468, + 367, + 519 + ], + "lines": [ + { + "bbox": [ + 245, + 468, + 367, + 519 + ], + "spans": [ + { + "bbox": [ + 245, + 468, + 367, + 519 + ], + "score": 0.93, + "content": "\\begin{array} { c } { { \\displaystyle \\frac { c \\alpha } { c - 2 } = 1 - \\frac { c ^ { 2 } \\delta ^ { + } \\sigma _ { 1 } ^ { 2 } } { \\kappa } } } \\\\ { { \\implies \\delta ^ { + } \\sigma _ { 1 } ^ { 2 } = \\displaystyle \\frac { \\kappa } { c ^ { 2 } } - \\frac { \\kappa \\alpha } { c ( c - 2 ) } . } } \\end{array}", + "type": "interline_equation", + "image_path": "de1df55d734357c1c9e66322d4865ff165cc66c9c2614aecd7423dde5b5f49cf.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 245, + 468, + 367, + 493.5 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 245, + 493.5, + 367, + 519.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 103, + 517, + 508, + 541 + ], + "spans": [ + { + "bbox": [ + 103, + 517, + 265, + 541 + ], + "score": 1.0, + "content": "We will now prove that δ+σ21 = κc ( 1c", + "type": "text" + }, + { + "bbox": [ + 291, + 520, + 508, + 540 + ], + "score": 1.0, + "content": "is much larger than one allowed by the convergence", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 532, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 202, + 556 + ], + "score": 1.0, + "content": "of the HB updates, i.e.,", + "type": "text" + }, + { + "bbox": [ + 203, + 537, + 316, + 555 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } < \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } \\le \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 532, + 442, + 558 + ], + "score": 1.0, + "content": ". In particular, if we prove that", + "type": "text" + }, + { + "bbox": [ + 442, + 538, + 505, + 554 + ], + "score": 0.92, + "content": "\\textstyle { \\frac { \\kappa } { c } } { \\bigl ( } { \\frac { 1 } { c } } - { \\frac { \\alpha } { c - 2 } } { \\bigr ) } >", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 553, + 318, + 571 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 318, + 571 + ], + "score": 1.0, + "content": "2(1−α2) for any admissible value of α, we are done.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 574, + 396, + 678 + ], + "lines": [ + { + "bbox": [ + 211, + 574, + 396, + 678 + ], + "spans": [ + { + "bbox": [ + 211, + 574, + 396, + 678 + ], + "score": 0.94, + "content": "\\begin{array} { c } { { \\displaystyle \\frac \\kappa c ( \\frac 1 c - \\frac \\alpha { c - 2 } ) > \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c } } } \\\\ { { \\Leftrightarrow \\displaystyle \\frac \\kappa c - \\frac { \\kappa \\alpha } { c - 2 } > 2 - 2 \\alpha ^ { 2 } } } \\\\ { { \\Leftrightarrow \\displaystyle \\frac \\kappa c - \\frac { \\kappa \\alpha } { c - 2 } > \\frac \\kappa c - \\frac { \\kappa \\alpha } { c } > 2 - 2 \\alpha ^ { 2 } } } \\\\ { { \\Leftrightarrow \\kappa - \\kappa \\alpha > 2 c - 2 c \\alpha ^ { 2 } } } \\\\ { { \\Leftrightarrow 2 c \\alpha ^ { 2 } - \\kappa \\alpha + ( \\kappa - 2 c ) > 0 . } } \\end{array}", + "type": "interline_equation", + "image_path": "4209fa860481162363cfcba17cb63e9b25abe1e5f76a87c55d10cbe7223d24b4.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 211, + 574, + 396, + 591.3333333333334 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 211, + 591.3333333333334, + 396, + 608.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 211, + 608.6666666666667, + 396, + 626.0000000000001 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 211, + 626.0000000000001, + 396, + 643.3333333333335 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 211, + 643.3333333333335, + 396, + 660.6666666666669 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 211, + 660.6666666666669, + 396, + 678.0000000000002 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 682, + 506, + 735 + ], + "lines": [ + { + "bbox": [ + 104, + 679, + 507, + 697 + ], + "spans": [ + { + "bbox": [ + 104, + 679, + 289, + 697 + ], + "score": 1.0, + "content": "The two roots of this quadratic equation are", + "type": "text" + }, + { + "bbox": [ + 290, + 682, + 348, + 695 + ], + "score": 0.89, + "content": "\\alpha ^ { + } = \\textstyle { \\frac { \\kappa } { 2 c } } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 679, + 368, + 697 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 368, + 683, + 404, + 693 + ], + "score": 0.9, + "content": "\\alpha ^ { - } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 679, + 451, + 697 + ], + "score": 1.0, + "content": ". 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And, for any", + "type": "text" + }, + { + "bbox": [ + 447, + 695, + 478, + 705 + ], + "score": 0.89, + "content": "\\kappa \\geq 4 c", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 692, + 506, + 707 + ], + "score": 1.0, + "content": ", note,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 703, + 505, + 719 + ], + "spans": [ + { + "bbox": [ + 106, + 705, + 147, + 716 + ], + "score": 0.91, + "content": "\\alpha ^ { + } > \\alpha ^ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 703, + 338, + 719 + ], + "score": 1.0, + "content": ", indicating that the above equation holds true if", + "type": "text" + }, + { + "bbox": [ + 338, + 705, + 410, + 718 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\alpha > \\alpha ^ { + } = \\frac { \\kappa } { 2 c } - 1 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 703, + 430, + 719 + ], + "score": 1.0, + "content": "or if", + "type": "text" + }, + { + "bbox": [ + 431, + 705, + 483, + 715 + ], + "score": 0.91, + "content": "\\alpha < \\alpha ^ { - } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 703, + 505, + 719 + ], + "score": 1.0, + "content": ". 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2 ) } & { 2 \\alpha x ( 1 - x ) ( - 1 + c x ) + c \\alpha ^ { 2 } x ^ { 2 } } \\\\ { 0 } & { - x } & { \\alpha x ( c x - 1 ) } \\\\ { 1 } & { 0 } & { 0 } \\end{array} \\right) } } \\\\ & { \\qquad = x ^ { 2 } \\alpha \\operatorname* { d e t } \\left( { \\left[ \\begin{array} { l l l } { 0 } & { ( c x - 2 ) } & { c \\alpha x + 2 ( 1 - x ) ( c x - 1 ) } \\\\ { 0 } & { - 1 } & { c x - 1 } \\\\ { 1 } & { 0 } & { 0 } \\end{array} \\right] } \\right) } \\\\ & { \\qquad = x ^ { 3 } \\alpha \\operatorname* { d e t } \\left( { \\left[ \\begin{array} { l l l } { 0 } & { c } & { c \\alpha - 2 ( c x - 1 ) } \\\\ { 0 } & { - 1 } & { c x - 1 } \\\\ { 1 } & { 0 } & { 0 } \\end{array} \\right] } \\right) } \\end{array} }", + "type": "interline_equation", + "image_path": "a16691d3c7ee37009e76b6c2fee6d01aeac1460ff850eb443b092b9570c3a0ef.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 164, + 163, + 446, + 200.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 164, + 200.66666666666666, + 446, + 238.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 164, + 238.33333333333331, + 446, + 276.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 132, + 291 + ], + "lines": [ + { + "bbox": [ + 106, + 278, + 134, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 134, + 294 + ], + "score": 1.0, + "content": "Then,", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 278, + 134, + 294 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 295, + 408, + 352 + ], + "lines": [ + { + "bbox": [ + 202, + 295, + 408, + 352 + ], + "spans": [ + { + "bbox": [ + 202, + 295, + 408, + 352 + ], + "score": 0.93, + "content": "\\begin{array} { c } { { \\operatorname * { d e t } ( \\mathcal { R } ) = x ^ { 3 } \\alpha \\bigg ( c ( - 1 + c x ) - 2 ( - 1 + c x ) + c \\alpha \\bigg ) } } \\\\ { { = \\alpha x ^ { 3 } \\bigg ( ( c - 2 ) ( - 1 + c x ) + c \\alpha \\bigg ) } } \\end{array}", + "type": "interline_equation", + "image_path": "0eadf542bbb122afce66a44b53d459f6100df1faae43af7754b8b176cf312392.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 202, + 295, + 408, + 314.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 202, + 314.0, + 408, + 333.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 202, + 333.0, + 408, + 352.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 285, + 366 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 285, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 285, + 367 + ], + "score": 1.0, + "content": "Note that this determinant can be zero when", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 353, + 285, + 367 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 260, + 368, + 351, + 393 + ], + "lines": [ + { + "bbox": [ + 260, + 368, + 351, + 393 + ], + "spans": [ + { + "bbox": [ + 260, + 368, + 351, + 393 + ], + "score": 0.95, + "content": "\\alpha = \\frac { ( c - 2 ) ( 1 - c x ) } { c } .", + "type": "interline_equation", + "image_path": "3eb8355657aeb44dfcbd395c2082fbf6fd540dce6e01864969c1289e7447458a.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 260, + 368, + 351, + 393 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "We show this is not possible by splitting our argument into two parts, one about the convergent", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 406, + 474, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 236, + 425 + ], + "score": 1.0, + "content": "regime of the algorithm (where,", + "type": "text" + }, + { + "bbox": [ + 236, + 408, + 306, + 426 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } < \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 406, + 474, + 425 + ], + "score": 1.0, + "content": "and the other about the divergent regime.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 396, + 505, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 429, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "Let us first provide a proof for the convergent regime of the algorithm. For this regime, let the chosen", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 440, + 504, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 113, + 451 + ], + "score": 0.72, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 440, + 185, + 454 + ], + "score": 1.0, + "content": "be represented as", + "type": "text" + }, + { + "bbox": [ + 186, + 441, + 198, + 451 + ], + "score": 0.88, + "content": "\\delta ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 440, + 353, + 454 + ], + "score": 1.0, + "content": ". Now, for the smaller eigen direction,", + "type": "text" + }, + { + "bbox": [ + 353, + 441, + 455, + 453 + ], + "score": 0.93, + "content": "x = \\delta ^ { + } \\lambda _ { \\mathrm { m i n } } = c \\bar { \\delta ^ { + } } \\sigma _ { 1 } ^ { 2 } / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 440, + 496, + 454 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 496, + 443, + 504, + 451 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 452, + 228, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 228, + 464 + ], + "score": 1.0, + "content": "was chosen as per equation 5,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 429, + 505, + 464 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 468, + 367, + 519 + ], + "lines": [ + { + "bbox": [ + 245, + 468, + 367, + 519 + ], + "spans": [ + { + "bbox": [ + 245, + 468, + 367, + 519 + ], + "score": 0.93, + "content": "\\begin{array} { c } { { \\displaystyle \\frac { c \\alpha } { c - 2 } = 1 - \\frac { c ^ { 2 } \\delta ^ { + } \\sigma _ { 1 } ^ { 2 } } { \\kappa } } } \\\\ { { \\implies \\delta ^ { + } \\sigma _ { 1 } ^ { 2 } = \\displaystyle \\frac { \\kappa } { c ^ { 2 } } - \\frac { \\kappa \\alpha } { c ( c - 2 ) } . } } \\end{array}", + "type": "interline_equation", + "image_path": "de1df55d734357c1c9e66322d4865ff165cc66c9c2614aecd7423dde5b5f49cf.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 245, + 468, + 367, + 493.5 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 245, + 493.5, + 367, + 519.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 103, + 517, + 508, + 541 + ], + "spans": [ + { + "bbox": [ + 103, + 517, + 265, + 541 + ], + "score": 1.0, + "content": "We will now prove that δ+σ21 = κc ( 1c", + "type": "text" + }, + { + "bbox": [ + 291, + 520, + 508, + 540 + ], + "score": 1.0, + "content": "is much larger than one allowed by the convergence", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 532, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 202, + 556 + ], + "score": 1.0, + "content": "of the HB updates, i.e.,", + "type": "text" + }, + { + "bbox": [ + 203, + 537, + 316, + 555 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } < \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } \\le \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 532, + 442, + 558 + ], + "score": 1.0, + "content": ". In particular, if we prove that", + "type": "text" + }, + { + "bbox": [ + 442, + 538, + 505, + 554 + ], + "score": 0.92, + "content": "\\textstyle { \\frac { \\kappa } { c } } { \\bigl ( } { \\frac { 1 } { c } } - { \\frac { \\alpha } { c - 2 } } { \\bigr ) } >", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 553, + 318, + 571 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 318, + 571 + ], + "score": 1.0, + "content": "2(1−α2) for any admissible value of α, we are done.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 103, + 517, + 508, + 571 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 574, + 396, + 678 + ], + "lines": [ + { + "bbox": [ + 211, + 574, + 396, + 678 + ], + "spans": [ + { + "bbox": [ + 211, + 574, + 396, + 678 + ], + "score": 0.94, + "content": "\\begin{array} { c } { { \\displaystyle \\frac \\kappa c ( \\frac 1 c - \\frac \\alpha { c - 2 } ) > \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c } } } \\\\ { { \\Leftrightarrow \\displaystyle \\frac \\kappa c - \\frac { \\kappa \\alpha } { c - 2 } > 2 - 2 \\alpha ^ { 2 } } } \\\\ { { \\Leftrightarrow \\displaystyle \\frac \\kappa c - \\frac { \\kappa \\alpha } { c - 2 } > \\frac \\kappa c - \\frac { \\kappa \\alpha } { c } > 2 - 2 \\alpha ^ { 2 } } } \\\\ { { \\Leftrightarrow \\kappa - \\kappa \\alpha > 2 c - 2 c \\alpha ^ { 2 } } } \\\\ { { \\Leftrightarrow 2 c \\alpha ^ { 2 } - \\kappa \\alpha + ( \\kappa - 2 c ) > 0 . } } \\end{array}", + "type": "interline_equation", + "image_path": "4209fa860481162363cfcba17cb63e9b25abe1e5f76a87c55d10cbe7223d24b4.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 211, + 574, + 396, + 591.3333333333334 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 211, + 591.3333333333334, + 396, + 608.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 211, + 608.6666666666667, + 396, + 626.0000000000001 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 211, + 626.0000000000001, + 396, + 643.3333333333335 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 211, + 643.3333333333335, + 396, + 660.6666666666669 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 211, + 660.6666666666669, + 396, + 678.0000000000002 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 682, + 506, + 735 + ], + "lines": [ + { + "bbox": [ + 104, + 679, + 507, + 697 + ], + "spans": [ + { + "bbox": [ + 104, + 679, + 289, + 697 + ], + "score": 1.0, + "content": "The two roots of this quadratic equation are", + "type": "text" + }, + { + "bbox": [ + 290, + 682, + 348, + 695 + ], + "score": 0.89, + "content": "\\alpha ^ { + } = \\textstyle { \\frac { \\kappa } { 2 c } } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 679, + 368, + 697 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 368, + 683, + 404, + 693 + ], + "score": 0.9, + "content": "\\alpha ^ { - } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 679, + 451, + 697 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 452, + 683, + 502, + 694 + ], + "score": 0.9, + "content": "\\kappa \\geq \\widetilde { \\kappa } = c", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 679, + 507, + 697 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 692, + 506, + 707 + ], + "spans": [ + { + "bbox": [ + 104, + 692, + 344, + 707 + ], + "score": 1.0, + "content": "note that there is not much any method gains over SGD if", + "type": "text" + }, + { + "bbox": [ + 345, + 694, + 386, + 705 + ], + "score": 0.9, + "content": "\\kappa = \\mathcal { O } ( c )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 692, + 447, + 707 + ], + "score": 1.0, + "content": ". And, for any", + "type": "text" + }, + { + "bbox": [ + 447, + 695, + 478, + 705 + ], + "score": 0.89, + "content": "\\kappa \\geq 4 c", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 692, + 506, + 707 + ], + "score": 1.0, + "content": ", note,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 703, + 505, + 719 + ], + "spans": [ + { + "bbox": [ + 106, + 705, + 147, + 716 + ], + "score": 0.91, + "content": "\\alpha ^ { + } > \\alpha ^ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 703, + 338, + 719 + ], + "score": 1.0, + "content": ", indicating that the above equation holds true if", + "type": "text" + }, + { + "bbox": [ + 338, + 705, + 410, + 718 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\alpha > \\alpha ^ { + } = \\frac { \\kappa } { 2 c } - 1 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 703, + 430, + 719 + ], + "score": 1.0, + "content": "or if", + "type": "text" + }, + { + "bbox": [ + 431, + 705, + 483, + 715 + ], + "score": 0.91, + "content": "\\alpha < \\alpha ^ { - } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 703, + 505, + 719 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 103, + 715, + 424, + 736 + ], + "spans": [ + { + "bbox": [ + 103, + 715, + 320, + 736 + ], + "score": 1.0, + "content": "latter condition is true and hence the proposition that", + "type": "text" + }, + { + "bbox": [ + 320, + 718, + 393, + 735 + ], + "score": 0.95, + "content": "\\delta ^ { + } \\sigma _ { 1 } ^ { 2 } > \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 715, + 424, + 736 + ], + "score": 1.0, + "content": "is true.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 103, + 679, + 507, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "We need to prove that the determinant does not vanish in the divergent regime for rounding up the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 188, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 188, + 105 + ], + "score": 1.0, + "content": "proof to the lemma.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 506, + 172 + ], + "lines": [ + { + "bbox": [ + 104, + 109, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 111, + 395, + 126 + ], + "score": 1.0, + "content": "Now, let us consider the divergent regime of the algorithm, i.e., when,", + "type": "text" + }, + { + "bbox": [ + 395, + 110, + 464, + 128 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } > \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 109, + 506, + 129 + ], + "score": 1.0, + "content": ". Further-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 362, + 143 + ], + "score": 1.0, + "content": "more, for the larger eigendirection, the determinant is zero when", + "type": "text" + }, + { + "bbox": [ + 362, + 128, + 464, + 145 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } = \\frac { 1 - \\frac { c \\alpha } { c - 2 } } { c } = \\frac { 1 } { c } - \\frac { \\alpha } { c - 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 131, + 505, + 144 + ], + "score": 1.0, + "content": "(obtained", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 169, + 164 + ], + "score": 1.0, + "content": "by substituting", + "type": "text" + }, + { + "bbox": [ + 169, + 146, + 207, + 159 + ], + "score": 0.93, + "content": "x = \\delta \\sigma _ { 1 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 142, + 341, + 164 + ], + "score": 1.0, + "content": "in equation 5). If we show that", + "type": "text" + }, + { + "bbox": [ + 341, + 144, + 429, + 162 + ], + "score": 0.93, + "content": "\\textstyle { \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } } > { \\frac { 1 } { c } } - { \\frac { \\alpha } { c - 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 147, + 506, + 159 + ], + "score": 1.0, + "content": "for all admissible", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 160, + 359, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 145, + 172 + ], + "score": 1.0, + "content": "values of", + "type": "text" + }, + { + "bbox": [ + 145, + 163, + 150, + 170 + ], + "score": 0.67, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 160, + 359, + 172 + ], + "score": 1.0, + "content": ", we are done. We will explore this in greater detail:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 175, + 451, + 288 + ], + "lines": [ + { + "bbox": [ + 158, + 175, + 451, + 288 + ], + "spans": [ + { + "bbox": [ + 158, + 175, + 451, + 288 + ], + "score": 0.96, + "content": "\\begin{array} { c } { { \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } > \\displaystyle \\frac { 1 } { c } - \\frac { \\alpha } { c - 2 } } } \\\\ { { \\Leftrightarrow 2 ( 1 - \\alpha ^ { 2 } ) \\geq 1 + \\displaystyle \\frac { c - 2 } { c } \\alpha - \\displaystyle \\frac { c } { c - 2 } \\alpha - \\alpha ^ { 2 } } } \\\\ { { \\Leftrightarrow 1 - \\alpha ^ { 2 } \\geq \\displaystyle \\frac { - 4 ( c - 1 ) } { c ( c - 2 ) } \\alpha } } \\\\ { { \\Leftrightarrow c ^ { 2 } - 2 c - \\alpha ^ { 2 } c ^ { 2 } + 2 c \\alpha ^ { 2 } \\geq - 4 c \\alpha + 4 \\alpha } } \\\\ { { \\Leftrightarrow c ^ { 2 } ( 1 - \\alpha ^ { 2 } ) - 2 c ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) - 4 \\alpha \\geq 0 . } } \\end{array}", + "type": "interline_equation", + "image_path": "401b2fa244d3b195ef2a5a6b2f13fcfb4f6525d18fb05b869c3d40152ea3178a.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 158, + 175, + 451, + 212.66666666666666 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 158, + 212.66666666666666, + 451, + 250.33333333333331 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 158, + 250.33333333333331, + 451, + 288.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 408, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 408, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 361, + 305 + ], + "score": 1.0, + "content": "considering the quadratic in the left hand size and solving it for", + "type": "text" + }, + { + "bbox": [ + 361, + 294, + 367, + 301 + ], + "score": 0.73, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 290, + 408, + 305 + ], + "score": 1.0, + "content": ", we have:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 307, + 458, + 424 + ], + "lines": [ + { + "bbox": [ + 152, + 307, + 458, + 424 + ], + "spans": [ + { + "bbox": [ + 152, + 307, + 458, + 424 + ], + "score": 0.96, + "content": "\\begin{array} { l } { { c ^ { \\pm } = \\frac { 2 ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) \\pm \\sqrt { 4 ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) ^ { 2 } + 1 6 \\alpha ( 1 - \\alpha ^ { 2 } ) } } { 2 ( 1 - \\alpha ^ { 2 } ) } } } \\\\ { { { } ~ = \\frac { ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) \\pm \\sqrt { ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) ^ { 2 } + 4 \\alpha ( 1 - \\alpha ^ { 2 } ) } } { ( 1 - \\alpha ^ { 2 } ) } } } \\\\ { { { } ~ = \\frac { ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) \\pm \\sqrt { 1 + \\alpha ^ { 4 } + 4 \\alpha ^ { 2 } - 2 \\alpha ^ { 2 } - 4 \\alpha + 4 \\alpha ^ { 3 } + 4 \\alpha ( 1 - \\alpha ^ { 2 } ) } } { ( 1 - \\alpha ^ { 2 } ) } } } \\\\ { { { } ~ = \\frac { ( 1 - \\alpha ^ { 2 } - 2 \\alpha ) \\pm ( 1 + \\alpha ^ { 2 } ) } { ( 1 - \\alpha ^ { 2 } ) } } } \\end{array}", + "type": "interline_equation", + "image_path": "ae8558b43deba1c0f65d2bf390e9ee9196ec81de566ff02ddc856a88fe2909fd.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 152, + 307, + 458, + 346.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 152, + 346.0, + 458, + 385.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 152, + 385.0, + 458, + 424.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 181, + 439 + ], + "lines": [ + { + "bbox": [ + 107, + 427, + 181, + 439 + ], + "spans": [ + { + "bbox": [ + 107, + 427, + 181, + 439 + ], + "score": 1.0, + "content": "This holds true iff", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 442, + 374, + 466 + ], + "lines": [ + { + "bbox": [ + 236, + 442, + 374, + 466 + ], + "spans": [ + { + "bbox": [ + 236, + 442, + 374, + 466 + ], + "score": 0.94, + "content": "c \\leq c ^ { - } = \\frac { - 2 \\alpha ( 1 + \\alpha ) } { 1 - \\alpha ^ { 2 } } = \\frac { - 2 \\alpha } { 1 - \\alpha } ,", + "type": "interline_equation", + "image_path": "562c8cb38fb814a43502622bc28811015cf0482d08d354218f24c09b304f3432.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 236, + 442, + 374, + 466 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 131, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 133, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 133, + 483 + ], + "score": 1.0, + "content": "or iff,", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 485, + 367, + 511 + ], + "lines": [ + { + "bbox": [ + 243, + 485, + 367, + 511 + ], + "spans": [ + { + "bbox": [ + 243, + 485, + 367, + 511 + ], + "score": 0.94, + "content": "c \\geq c ^ { + } = { \\frac { 2 ( 1 - \\alpha ) } { 1 - \\alpha ^ { 2 } } } = { \\frac { 2 } { 1 + \\alpha } } .", + "type": "interline_equation", + "image_path": "5183796f3ff4c4c7c38f9ea246d689041c470c4cd14b7849b8936f32cfdc455e.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 243, + 485, + 367, + 511 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 429, + 527 + ], + "lines": [ + { + "bbox": [ + 106, + 514, + 430, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 242, + 528 + ], + "score": 1.0, + "content": "Which is true automatically since", + "type": "text" + }, + { + "bbox": [ + 242, + 515, + 266, + 525 + ], + "score": 0.89, + "content": "c > 2", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 514, + 430, + 528 + ], + "score": 1.0, + "content": ". This completes the proof of the lemma.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 108, + 538, + 257, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 536, + 259, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 259, + 551 + ], + "score": 1.0, + "content": "We are now ready to prove Lemma 5.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "score": 1.0, + "content": "Proof of Lemma 5. Combining Lemmas 9 and 12, we see that no matter what stepsize and momen-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 570, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 104, + 570, + 173, + 588 + ], + "score": 1.0, + "content": "tum we choose,", + "type": "text" + }, + { + "bbox": [ + 174, + 573, + 192, + 584 + ], + "score": 0.89, + "content": "\\boldsymbol { B } ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 570, + 358, + 588 + ], + "score": 1.0, + "content": "has an eigenvalue of magnitude at least", + "type": "text" + }, + { + "bbox": [ + 358, + 573, + 392, + 587 + ], + "score": 0.91, + "content": "1 - { \\frac { 5 0 0 } { \\kappa } }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 570, + 433, + 588 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 433, + 573, + 478, + 586 + ], + "score": 0.93, + "content": "j \\in \\{ 1 , 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 570, + 506, + 588 + ], + "score": 1.0, + "content": ". This", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 182, + 597 + ], + "score": 1.0, + "content": "proves the lemma.", + "type": "text" + }, + { + "bbox": [ + 493, + 585, + 505, + 596 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 107, + 612, + 360, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 360, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 360, + 627 + ], + "score": 1.0, + "content": "B EQUIVALENCE OF ALGORITHM 3 AND ASGD", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 636, + 503, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "We begin by writing out the updates of ASGD as written out in Jain et al. (2017), which starts with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 648, + 469, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 154, + 663 + ], + "score": 1.0, + "content": "two iterates", + "type": "text" + }, + { + "bbox": [ + 155, + 650, + 166, + 661 + ], + "score": 0.88, + "content": "\\widehat { a } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 649, + 184, + 663 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 184, + 648, + 195, + 661 + ], + "score": 0.89, + "content": "\\widehat { d } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 649, + 258, + 663 + ], + "score": 1.0, + "content": ", and from time", + "type": "text" + }, + { + "bbox": [ + 258, + 650, + 327, + 662 + ], + "score": 0.92, + "content": "t = 0 , 1 , . . . T - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 649, + 469, + 663 + ], + "score": 1.0, + "content": "implements the following updates:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 666, + 361, + 735 + ], + "lines": [ + { + "bbox": [ + 249, + 666, + 361, + 735 + ], + "spans": [ + { + "bbox": [ + 249, + 666, + 361, + 735 + ], + "score": 0.93, + "content": "\\begin{array} { r l r } & { } & { \\widehat { b } _ { t } = \\alpha _ { 1 } \\widehat { a } _ { t } + ( 1 - \\alpha _ { 1 } ) \\widehat { d } _ { t } } \\\\ & { } & { \\widehat { a } _ { t + 1 } = \\widehat { b } _ { t } - \\delta _ { 1 } \\widehat { \\nabla } f _ { t + 1 } ( \\widehat { b } _ { t } ) } \\\\ & { } & { \\widehat { c } _ { t } = \\beta _ { 1 } \\widehat { b } _ { t } + ( 1 - \\beta _ { 1 } ) \\widehat { d } _ { t } } \\\\ & { } & { \\widehat { d } _ { t + 1 } = \\widehat { c } _ { t } - \\gamma _ { 1 } \\widehat { \\nabla } f _ { t + 1 } ( \\widehat { b } _ { t } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "ddeca56d9e7b854d0dc0e34d661bc923e33374f1f00c93564287c1a65170c120.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 249, + 666, + 361, + 700.5 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 249, + 700.5, + 361, + 735.0 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 514, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 496, + 516, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 496, + 516, + 505, + 526 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "We need to prove that the determinant does not vanish in the divergent regime for rounding up the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 188, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 188, + 105 + ], + "score": 1.0, + "content": "proof to the lemma.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 505, + 105 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 506, + 172 + ], + "lines": [ + { + "bbox": [ + 104, + 109, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 111, + 395, + 126 + ], + "score": 1.0, + "content": "Now, let us consider the divergent regime of the algorithm, i.e., when,", + "type": "text" + }, + { + "bbox": [ + 395, + 110, + 464, + 128 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\delta \\sigma _ { 1 } ^ { 2 } > \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 109, + 506, + 129 + ], + "score": 1.0, + "content": ". 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If we show that", + "type": "text" + }, + { + "bbox": [ + 341, + 144, + 429, + 162 + ], + "score": 0.93, + "content": "\\textstyle { \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } } > { \\frac { 1 } { c } } - { \\frac { \\alpha } { c - 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 147, + 506, + 159 + ], + "score": 1.0, + "content": "for all admissible", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 160, + 359, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 145, + 172 + ], + "score": 1.0, + "content": "values of", + "type": "text" + }, + { + "bbox": [ + 145, + 163, + 150, + 170 + ], + "score": 0.67, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 160, + 359, + 172 + ], + "score": 1.0, + "content": ", we are done. We will explore this in greater detail:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 104, + 109, + 506, + 172 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 175, + 451, + 288 + ], + "lines": [ + { + "bbox": [ + 158, + 175, + 451, + 288 + ], + "spans": [ + { + "bbox": [ + 158, + 175, + 451, + 288 + ], + "score": 0.96, + "content": "\\begin{array} { c } { { \\frac { 2 ( 1 - \\alpha ^ { 2 } ) } { c + ( c - 2 ) \\alpha } > \\displaystyle \\frac { 1 } { c } - \\frac { \\alpha } { c - 2 } } } \\\\ { { \\Leftrightarrow 2 ( 1 - \\alpha ^ { 2 } ) \\geq 1 + \\displaystyle \\frac { c - 2 } { c } \\alpha - \\displaystyle \\frac { c } { c - 2 } \\alpha - \\alpha ^ { 2 } } } \\\\ { { \\Leftrightarrow 1 - \\alpha ^ { 2 } \\geq \\displaystyle \\frac { - 4 ( c - 1 ) } { c ( c - 2 ) } \\alpha } } \\\\ { { \\Leftrightarrow c ^ { 2 } - 2 c - \\alpha ^ { 2 } c ^ { 2 } + 2 c \\alpha ^ { 2 } \\geq - 4 c \\alpha + 4 \\alpha } } \\\\ { { \\Leftrightarrow c ^ { 2 } ( 1 - 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This completes the proof of the lemma.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 514, + 430, + 528 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 538, + 257, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 536, + 259, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 259, + 551 + ], + "score": 1.0, + "content": "We are now ready to prove Lemma 5.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 536, + 259, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 575 + ], + "score": 1.0, + "content": "Proof of Lemma 5. Combining Lemmas 9 and 12, we see that no matter what stepsize and momen-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 570, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 104, + 570, + 173, + 588 + ], + "score": 1.0, + "content": "tum we choose,", + "type": "text" + }, + { + "bbox": [ + 174, + 573, + 192, + 584 + ], + "score": 0.89, + "content": "\\boldsymbol { B } ^ { ( j ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 570, + 358, + 588 + ], + "score": 1.0, + "content": "has an eigenvalue of magnitude at least", + "type": "text" + }, + { + "bbox": [ + 358, + 573, + 392, + 587 + ], + "score": 0.91, + "content": "1 - { \\frac { 5 0 0 } { \\kappa } }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 570, + 433, + 588 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 433, + 573, + 478, + 586 + ], + "score": 0.93, + "content": "j \\in \\{ 1 , 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 570, + 506, + 588 + ], + "score": 1.0, + "content": ". This", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 182, + 597 + ], + "score": 1.0, + "content": "proves the lemma.", + "type": "text" + }, + { + "bbox": [ + 493, + 585, + 505, + 596 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 561, + 506, + 597 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 612, + 360, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 360, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 360, + 627 + ], + "score": 1.0, + "content": "B EQUIVALENCE OF ALGORITHM 3 AND ASGD", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 636, + 503, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "We begin by writing out the updates of ASGD as written out in Jain et al. 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\\alpha _ { 1 } ) \\widehat { d } _ { t + 1 } } \\\\ & { \\qquad = \\alpha _ { 1 } \\left( \\widehat { b } _ { t } - \\delta _ { 1 } \\widehat { \\nabla } f _ { t + 1 } ( \\widehat { b } _ { t } ) \\right) + ( 1 - \\alpha _ { 1 } ) \\widehat { d } _ { t + 1 } . } \\end{array}", + "type": "interline_equation", + "image_path": "265d66dacb4c6ae7532fa2778a10f860acbc96d81447052b4f3e15acc9ee3207.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 205, + 238, + 405, + 251.33333333333334 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 205, + 251.33333333333334, + 405, + 264.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 205, + 264.6666666666667, + 405, + 278.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 225, + 293 + ], + "score": 1.0, + "content": "By substituting the value of", + "type": "text" + }, + { + "bbox": [ + 225, + 283, + 236, + 292 + ], + "score": 0.87, + "content": "\\alpha _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "we note that this is indeed the update of the iterate as a convex", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "combination of the current running average and a short gradient step as written in this paper. In this", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 303, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 162, + 315 + ], + "score": 1.0, + "content": "paper, we set", + "type": "text" + }, + { + "bbox": [ + 163, + 304, + 173, + 314 + ], + "score": 0.83, + "content": "c _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 303, + 506, + 315 + ], + "score": 1.0, + "content": "to be equal to 0.7, and any constant less than 1 works. 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Just as in Appendix A, it is indeed possible to compute the expected error of all the algorithms", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 433, + 504, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 504, + 445 + ], + "score": 1.0, + "content": "among SGD, HB, NAG and ASGD, by tracking certain covariance matrices which evolve as lin-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 102, + 442, + 509, + 464 + ], + "spans": [ + { + "bbox": [ + 102, + 442, + 292, + 464 + ], + "score": 1.0, + "content": "ear systems. For SGD, for instance, denoting", + "type": "text" + }, + { + "bbox": [ + 293, + 445, + 486, + 461 + ], + "score": 0.92, + "content": "\\Phi _ { t } ^ { S G D } \\stackrel { \\mathrm { d e f } } { = } \\mathbb { E } \\left[ \\left( \\mathbf { w } _ { t } ^ { S G D } - w ^ { * } \\right) \\otimes \\left( \\mathbf { w } _ { t } ^ { S G D } - w ^ { * } \\right) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 442, + 509, + 464 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 103, + 456, + 507, + 478 + ], + "spans": [ + { + "bbox": [ + 103, + 457, + 174, + 477 + ], + "score": 1.0, + "content": "see that ΦSGDt+1", + "type": "text" + }, + { + "bbox": [ + 142, + 459, + 232, + 474 + ], + "score": 0.92, + "content": "\\Phi _ { t + 1 } ^ { S G D } \\ : = \\ : B \\circ \\Phi _ { t } ^ { S G D }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 456, + 266, + 478 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 266, + 461, + 275, + 471 + ], + "score": 0.81, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 456, + 400, + 478 + ], + "score": 1.0, + "content": "is a linear operator acting on", + "type": "text" + }, + { + "bbox": [ + 400, + 462, + 426, + 472 + ], + "score": 0.89, + "content": "d \\times d", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 456, + 507, + 478 + ], + "score": 1.0, + "content": "matrices such that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 471, + 507, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 327, + 489 + ], + "score": 0.76, + "content": "{ \\mathcal { B } } \\circ M { \\stackrel { \\mathrm { d e f } } { = } } M - \\delta H M - \\delta M H + \\delta ^ { 2 } \\mathbb { E } \\left[ \\left. x , M x \\right. x x ^ { \\top } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 471, + 507, + 492 + ], + "score": 1.0, + "content": ". Similarly, HB, NAG and ASGD also have", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "corresponding operators (see Appendix A for more details on the operator corresponding to HB).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 497, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 513 + ], + "score": 1.0, + "content": "The largest magnitude of the eigenvalues of these matrices indicate the rate of decay achieved by", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 509, + 389, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 389, + 523 + ], + "score": 1.0, + "content": "the particular algorithm – smaller it is compared to 1, faster the decay.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 477, + 540 + ], + "score": 1.0, + "content": "We now detail the range of parameters explored for these results: the condition number", + "type": "text" + }, + { + "bbox": [ + 477, + 529, + 485, + 537 + ], + "score": 0.72, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 525, + 506, + 540 + ], + "score": 1.0, + "content": "was", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 535, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 156, + 552 + ], + "score": 1.0, + "content": "varied from", + "type": "text" + }, + { + "bbox": [ + 156, + 537, + 218, + 550 + ], + "score": 0.94, + "content": "\\{ 2 ^ { 4 } , 2 ^ { 5 } , . . , \\bar { 2 } ^ { 2 8 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 535, + 506, + 552 + ], + "score": 1.0, + "content": "for all the optimization methods and for both the discrete and gaussian", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "problem. 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For NAG and HB, we did a very fine grid search by sampling 50", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 195, + 583 + ], + "score": 1.0, + "content": "values in the interval", + "type": "text" + }, + { + "bbox": [ + 195, + 570, + 216, + 582 + ], + "score": 0.74, + "content": "( 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "for both the learning rate and the momentum parameter and chose the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 280, + 595 + ], + "score": 1.0, + "content": "parameter setting that yielded the smallest", + "type": "text" + }, + { + "bbox": [ + 280, + 582, + 317, + 593 + ], + "score": 0.93, + "content": "\\lambda _ { \\mathrm { m a x } } ( B )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "that is less than 1 (so that it falls in the range", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 592, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 473, + 604 + ], + "score": 1.0, + "content": "of convergence of the algorithm). As for SGD and ASGD, we employed a learning rate of", + "type": "text" + }, + { + "bbox": [ + 473, + 592, + 489, + 604 + ], + "score": 0.45, + "content": "1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 593, + 506, + 604 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 604, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 506, + 616 + ], + "score": 1.0, + "content": "the Gaussian case and a step size of 0.9 for the discrete case. The statistical advantage parameter of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 614, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 209, + 628 + ], + "score": 1.0, + "content": "ASGD was chosen to be", + "type": "text" + }, + { + "bbox": [ + 209, + 614, + 241, + 628 + ], + "score": 0.93, + "content": "\\sqrt { 3 \\kappa / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 615, + 350, + 628 + ], + "score": 1.0, + "content": "for the Gaussian case and", + "type": "text" + }, + { + "bbox": [ + 351, + 614, + 382, + 628 + ], + "score": 0.93, + "content": "\\sqrt { 2 \\kappa / 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "for the Discrete case, and the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 212, + 640 + ], + "score": 1.0, + "content": "a long step parameters of", + "type": "text" + }, + { + "bbox": [ + 212, + 628, + 224, + 637 + ], + "score": 0.74, + "content": "3 \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 627, + 243, + 640 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 243, + 627, + 255, + 637 + ], + "score": 0.82, + "content": "2 \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "were chosen for the Gaussian and Discrete case respectively.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "The reason it appears as if we choose a parameter above the theoretically maximal allowed value", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 330, + 661 + ], + "score": 1.0, + "content": "of the advantage parameter is because the definition of", + "type": "text" + }, + { + "bbox": [ + 330, + 650, + 338, + 659 + ], + "score": 0.79, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 649, + 456, + 661 + ], + "score": 1.0, + "content": "is different in this case. The", + "type": "text" + }, + { + "bbox": [ + 456, + 650, + 464, + 659 + ], + "score": 0.65, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "we speak", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 218, + 673 + ], + "score": 1.0, + "content": "about for this experiment is", + "type": "text" + }, + { + "bbox": [ + 219, + 660, + 265, + 672 + ], + "score": 0.93, + "content": "\\lambda _ { \\operatorname* { m a x } } / \\lambda _ { \\operatorname* { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "unlike the condition number for the stochastic optimization", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "problem. In a manner similar to actually running the algorithms (the results of whose are presented", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "in the main paper), we also note that we can compute the rate as in equation 1 and join all these rates", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 693, + 458, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 458, + 704 + ], + "score": 1.0, + "content": "using a curve and estimate its slope (in the log scale). This result is indicated in table 3.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Figure 7 presents these results, where for each method, we did grid search over all parameters", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 270, + 734 + ], + "score": 1.0, + "content": "and chose parameters that give smallest", + "type": "text" + }, + { + "bbox": [ + 270, + 721, + 289, + 732 + ], + "score": 0.93, + "content": "\\lambda _ { \\operatorname* { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 720, + 505, + 734 + ], + "score": 1.0, + "content": ". We see the same pattern as in Figure 1 from actual", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 230, + 95 + ], + "score": 1.0, + "content": "Next, we specify the step sizes", + "type": "text" + }, + { + "bbox": [ + 230, + 81, + 288, + 95 + ], + "score": 0.9, + "content": "\\beta _ { 1 } = c _ { 3 } ^ { 2 } / \\sqrt { \\kappa \\widetilde { \\kappa } }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 81, + 291, + 95 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 291, + 81, + 363, + 95 + ], + "score": 0.77, + "content": "\\alpha _ { 1 } = c _ { 3 } / ( c _ { 3 } + \\beta )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 81, + 367, + 95 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 367, + 82, + 438, + 95 + ], + "score": 0.86, + "content": "\\gamma _ { 1 } = \\beta / ( c _ { 3 } \\lambda _ { \\operatorname* { m i n } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 81, + 456, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 456, + 82, + 501, + 95 + ], + "score": 0.93, + "content": "\\delta _ { 1 } = 1 / R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 81, + 505, + 95 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 133, + 106 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 93, + 190, + 106 + ], + "score": 0.93, + "content": "\\kappa = R ^ { 2 } / \\lambda _ { \\operatorname* { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 92, + 438, + 106 + ], + "score": 1.0, + "content": "e. Note that the step sizes in the paper of Jain et al. (2017) with", + "type": "text" + }, + { + "bbox": [ + 438, + 96, + 448, + 105 + ], + "score": 0.85, + "content": "c _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "in their paper", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "score": 1.0, + "content": "set to 1 yields the step sizes above. 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\\beta _ { 1 } ) \\widehat { d } _ { t } } \\\\ & { \\qquad = \\beta _ { 1 } \\left( \\widehat { b } _ { t } - \\frac { \\delta \\kappa } { c _ { 3 } } \\hat { \\nabla } f _ { t + 1 } ( \\widehat { b } _ { t } ) \\right) + ( 1 - \\beta _ { 1 } ) \\widehat { d } _ { t } . } \\end{array}", + "type": "interline_equation", + "image_path": "436d2b99f7356ae7befe7bc769245732cd69a9850d47a86cfd7e5b20851b9a81.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 130, + 410, + 144.5 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 199, + 144.5, + 410, + 159.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 199, + 159.0, + 410, + 173.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 199, + 173.5, + 410, + 188.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 192, + 505, + 216 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 154, + 207 + ], + "score": 1.0, + "content": "We see that", + "type": "text" + }, + { + "bbox": [ + 154, + 191, + 173, + 205 + ], + "score": 0.93, + "content": "\\widehat { d } _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 191, + 355, + 207 + ], + "score": 1.0, + "content": "is precisely the update of the running average", + "type": "text" + }, + { + "bbox": [ + 355, + 194, + 377, + 205 + ], + "score": 0.9, + "content": "\\bar { w } _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 191, + 506, + 207 + ], + "score": 1.0, + "content": "in the ASGD method employed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 202, + 161, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 161, + 218 + ], + "score": 1.0, + "content": "in this paper.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 191, + 506, + 218 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 221, + 482, + 235 + ], + "lines": [ + { + "bbox": [ + 105, + 220, + 484, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 170, + 237 + ], + "score": 1.0, + "content": "We now update", + "type": "text" + }, + { + "bbox": [ + 171, + 221, + 179, + 234 + ], + "score": 0.88, + "content": "\\widehat { b } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 220, + 224, + 237 + ], + "score": 1.0, + "content": "to become", + "type": "text" + }, + { + "bbox": [ + 225, + 220, + 243, + 235 + ], + "score": 0.92, + "content": "\\widehat { b } _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 220, + 442, + 237 + ], + "score": 1.0, + "content": "and this can be done by writing out equation 6 at", + "type": "text" + }, + { + "bbox": [ + 442, + 223, + 463, + 234 + ], + "score": 0.86, + "content": "t + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 220, + 484, + 237 + ], + "score": 1.0, + "content": ", i.e:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 220, + 484, + 237 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 238, + 405, + 278 + ], + "lines": [ + { + "bbox": [ + 205, + 238, + 405, + 278 + ], + "spans": [ + { + "bbox": [ + 205, + 238, + 405, + 278 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\widehat { b } _ { t + 1 } = \\alpha _ { 1 } \\widehat { a } _ { t + 1 } + ( 1 - 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In this", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 303, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 162, + 315 + ], + "score": 1.0, + "content": "paper, we set", + "type": "text" + }, + { + "bbox": [ + 163, + 304, + 173, + 314 + ], + "score": 0.83, + "content": "c _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 303, + 506, + 315 + ], + "score": 1.0, + "content": "to be equal to 0.7, and any constant less than 1 works. In terms of variables, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 314, + 366, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 144, + 326 + ], + "score": 1.0, + "content": "note that", + "type": "text" + }, + { + "bbox": [ + 144, + 315, + 151, + 324 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 315, + 333, + 326 + ], + "score": 1.0, + "content": "in this paper’s algorithm description maps to", + "type": "text" + }, + { + "bbox": [ + 333, + 314, + 361, + 325 + ], + "score": 0.91, + "content": "1 - \\beta _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 315, + 366, + 326 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 281, + 506, + 326 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 341, + 300, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 301, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 301, + 356 + ], + "score": 1.0, + "content": "C MORE DETAILS ON EXPERIMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 390, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 391, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 391, + 381 + ], + "score": 1.0, + "content": "In this section, we will present more details on our experimental setup.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 362, + 391, + 381 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 390, + 225, + 402 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 226, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 226, + 402 + ], + "score": 1.0, + "content": "C.1 LINEAR REGRESSION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 411, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "In this section, we will present some more results on our experiments on the linear regression prob-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "lem. Just as in Appendix A, it is indeed possible to compute the expected error of all the algorithms", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 433, + 504, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 504, + 445 + ], + "score": 1.0, + "content": "among SGD, HB, NAG and ASGD, by tracking certain covariance matrices which evolve as lin-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 102, + 442, + 509, + 464 + ], + "spans": [ + { + "bbox": [ + 102, + 442, + 292, + 464 + ], + "score": 1.0, + "content": "ear systems. For SGD, for instance, denoting", + "type": "text" + }, + { + "bbox": [ + 293, + 445, + 486, + 461 + ], + "score": 0.92, + "content": "\\Phi _ { t } ^ { S G D } \\stackrel { \\mathrm { d e f } } { = } \\mathbb { E } \\left[ \\left( \\mathbf { w } _ { t } ^ { S G D } - w ^ { * } \\right) \\otimes \\left( \\mathbf { w } _ { t } ^ { S G D } - w ^ { * } \\right) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 442, + 509, + 464 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 103, + 456, + 507, + 478 + ], + "spans": [ + { + "bbox": [ + 103, + 457, + 174, + 477 + ], + "score": 1.0, + "content": "see that ΦSGDt+1", + "type": "text" + }, + { + "bbox": [ + 142, + 459, + 232, + 474 + ], + "score": 0.92, + "content": "\\Phi _ { t + 1 } ^ { S G D } \\ : = \\ : B \\circ \\Phi _ { t } ^ { S G D }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 456, + 266, + 478 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 266, + 461, + 275, + 471 + ], + "score": 0.81, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 456, + 400, + 478 + ], + "score": 1.0, + "content": "is a linear operator acting on", + "type": "text" + }, + { + "bbox": [ + 400, + 462, + 426, + 472 + ], + "score": 0.89, + "content": "d \\times d", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 456, + 507, + 478 + ], + "score": 1.0, + "content": "matrices such that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 471, + 507, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 327, + 489 + ], + "score": 0.76, + "content": "{ \\mathcal { B } } \\circ M { \\stackrel { \\mathrm { d e f } } { = } } M - \\delta H M - \\delta M H + \\delta ^ { 2 } \\mathbb { E } \\left[ \\left. x , M x \\right. x x ^ { \\top } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 471, + 507, + 492 + ], + "score": 1.0, + "content": ". Similarly, HB, NAG and ASGD also have", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "corresponding operators (see Appendix A for more details on the operator corresponding to HB).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 497, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 513 + ], + "score": 1.0, + "content": "The largest magnitude of the eigenvalues of these matrices indicate the rate of decay achieved by", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 509, + 389, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 389, + 523 + ], + "score": 1.0, + "content": "the particular algorithm – smaller it is compared to 1, faster the decay.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25, + "bbox_fs": [ + 102, + 411, + 509, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 477, + 540 + ], + "score": 1.0, + "content": "We now detail the range of parameters explored for these results: the condition number", + "type": "text" + }, + { + "bbox": [ + 477, + 529, + 485, + 537 + ], + "score": 0.72, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 525, + 506, + 540 + ], + "score": 1.0, + "content": "was", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 535, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 156, + 552 + ], + "score": 1.0, + "content": "varied from", + "type": "text" + }, + { + "bbox": [ + 156, + 537, + 218, + 550 + ], + "score": 0.94, + "content": "\\{ 2 ^ { 4 } , 2 ^ { 5 } , . . , \\bar { 2 } ^ { 2 8 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 535, + 506, + 552 + ], + "score": 1.0, + "content": "for all the optimization methods and for both the discrete and gaussian", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "problem. For each of these experiments, we draw 1000 samples and compute the empirical estimate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "of the fourth moment tensor. For NAG and HB, we did a very fine grid search by sampling 50", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 195, + 583 + ], + "score": 1.0, + "content": "values in the interval", + "type": "text" + }, + { + "bbox": [ + 195, + 570, + 216, + 582 + ], + "score": 0.74, + "content": "( 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "for both the learning rate and the momentum parameter and chose the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 280, + 595 + ], + "score": 1.0, + "content": "parameter setting that yielded the smallest", + "type": "text" + }, + { + "bbox": [ + 280, + 582, + 317, + 593 + ], + "score": 0.93, + "content": "\\lambda _ { \\mathrm { m a x } } ( B )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "that is less than 1 (so that it falls in the range", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 592, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 473, + 604 + ], + "score": 1.0, + "content": "of convergence of the algorithm). As for SGD and ASGD, we employed a learning rate of", + "type": "text" + }, + { + "bbox": [ + 473, + 592, + 489, + 604 + ], + "score": 0.45, + "content": "1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 593, + 506, + 604 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 604, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 506, + 616 + ], + "score": 1.0, + "content": "the Gaussian case and a step size of 0.9 for the discrete case. The statistical advantage parameter of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 614, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 209, + 628 + ], + "score": 1.0, + "content": "ASGD was chosen to be", + "type": "text" + }, + { + "bbox": [ + 209, + 614, + 241, + 628 + ], + "score": 0.93, + "content": "\\sqrt { 3 \\kappa / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 615, + 350, + 628 + ], + "score": 1.0, + "content": "for the Gaussian case and", + "type": "text" + }, + { + "bbox": [ + 351, + 614, + 382, + 628 + ], + "score": 0.93, + "content": "\\sqrt { 2 \\kappa / 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "for the Discrete case, and the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 212, + 640 + ], + "score": 1.0, + "content": "a long step parameters of", + "type": "text" + }, + { + "bbox": [ + 212, + 628, + 224, + 637 + ], + "score": 0.74, + "content": "3 \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 627, + 243, + 640 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 243, + 627, + 255, + 637 + ], + "score": 0.82, + "content": "2 \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "were chosen for the Gaussian and Discrete case respectively.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "The reason it appears as if we choose a parameter above the theoretically maximal allowed value", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 330, + 661 + ], + "score": 1.0, + "content": "of the advantage parameter is because the definition of", + "type": "text" + }, + { + "bbox": [ + 330, + 650, + 338, + 659 + ], + "score": 0.79, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 649, + 456, + 661 + ], + "score": 1.0, + "content": "is different in this case. The", + "type": "text" + }, + { + "bbox": [ + 456, + 650, + 464, + 659 + ], + "score": 0.65, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "we speak", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 218, + 673 + ], + "score": 1.0, + "content": "about for this experiment is", + "type": "text" + }, + { + "bbox": [ + 219, + 660, + 265, + 672 + ], + "score": 0.93, + "content": "\\lambda _ { \\operatorname* { m a x } } / \\lambda _ { \\operatorname* { m i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "unlike the condition number for the stochastic optimization", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "problem. In a manner similar to actually running the algorithms (the results of whose are presented", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "in the main paper), we also note that we can compute the rate as in equation 1 and join all these rates", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 693, + 458, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 458, + 704 + ], + "score": 1.0, + "content": "using a curve and estimate its slope (in the log scale). This result is indicated in table 3.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 525, + 506, + 704 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Figure 7 presents these results, where for each method, we did grid search over all parameters", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 270, + 734 + ], + "score": 1.0, + "content": "and chose parameters that give smallest", + "type": "text" + }, + { + "bbox": [ + 270, + 721, + 289, + 732 + ], + "score": 0.93, + "content": "\\lambda _ { \\operatorname* { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 720, + 505, + 734 + ], + "score": 1.0, + "content": ". We see the same pattern as in Figure 1 from actual", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 414, + 95 + ], + "score": 1.0, + "content": "runs – SGD,HB and NAG all have linear dependence on condition number", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 415, + 85, + 421, + 92 + ], + "score": 0.68, + "content": "\\kappa", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 422, + 81, + 506, + 95 + ], + "score": 1.0, + "content": ", while ASGD has a", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 186, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 167, + 106 + ], + "score": 1.0, + "content": "dependence of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 167, + 93, + 182, + 106 + ], + "score": 0.9, + "content": "\\sqrt { \\kappa }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 182, + 93, + 186, + 106 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 709, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 414, + 95 + ], + "score": 1.0, + "content": "runs – SGD,HB and NAG all have linear dependence on condition number", + "type": "text" + }, + { + "bbox": [ + 415, + 85, + 421, + 92 + ], + "score": 0.68, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 81, + 506, + 95 + ], + "score": 1.0, + "content": ", while ASGD has a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 186, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 167, + 106 + ], + "score": 1.0, + "content": "dependence of", + "type": "text" + }, + { + "bbox": [ + 167, + 93, + 182, + 106 + ], + "score": 0.9, + "content": "\\sqrt { \\kappa }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 93, + 186, + 106 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "image", + "bbox": [ + 115, + 127, + 484, + 267 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 127, + 484, + 267 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 127, + 484, + 267 + ], + "spans": [ + { + "bbox": [ + 115, + 127, + 484, + 267 + ], + "score": 0.967, + "type": "image", + "image_path": "2ce4c7a58a43a49aac0ae0911e2b09d770a20e8605b3023da8d2c7e23eef036f.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 115, + 127, + 484, + 173.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 115, + 173.66666666666666, + 484, + 220.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 115, + 220.33333333333331, + 484, + 267.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 104, + 279, + 506, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "Figure 7: Expected rate of error decay (equation 1) vs condition number for various methods for the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 291, + 488, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 488, + 303 + ], + "score": 1.0, + "content": "linear regression problem. Left is for discrete distribution and right is for Gaussian distribution.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "image_caption", + "bbox": [ + 106, + 396, + 504, + 420 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 396, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 192, + 409 + ], + "score": 1.0, + "content": "Table 3: Slopes (i.e.", + "type": "text" + }, + { + "bbox": [ + 192, + 398, + 200, + 408 + ], + "score": 0.64, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 396, + 458, + 409 + ], + "score": 1.0, + "content": ") obtained by fitting a line to the curves in Figure 7. A value of", + "type": "text" + }, + { + "bbox": [ + 458, + 398, + 466, + 408 + ], + "score": 0.8, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 396, + 506, + 409 + ], + "score": 1.0, + "content": "indicates", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 406, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 250, + 421 + ], + "score": 1.0, + "content": "that the error decays at a rate of exp", + "type": "text" + }, + { + "bbox": [ + 251, + 407, + 271, + 421 + ], + "score": 0.9, + "content": "\\left( { \\frac { - t } { \\kappa ^ { \\gamma } } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 406, + 351, + 421 + ], + "score": 1.0, + "content": ". A smaller value of", + "type": "text" + }, + { + "bbox": [ + 351, + 409, + 359, + 419 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 406, + 506, + 421 + ], + "score": 1.0, + "content": "indicates a faster rate of error decay.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + } + ], + "index": 5.5 + }, + { + "type": "table", + "bbox": [ + 197, + 321, + 414, + 388 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 197, + 321, + 414, + 388 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 197, + 321, + 414, + 388 + ], + "spans": [ + { + "bbox": [ + 197, + 321, + 414, + 388 + ], + "score": 0.935, + "html": "
AlgorithmSlope - discreteSlope- Gaussian
SGD0.99900.9995
HB1.03400.9989
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1.0, + "content": ", learning rate decay factor", + "type": "text" + }, + { + "bbox": [ + 329, + 494, + 376, + 507 + ], + "score": 0.93, + "content": "\\{ 2 , { \\sqrt { 1 0 } } , 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 494, + 380, + 507 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 505, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "As a final remark, for any comparison across algorithms, such as, (i) ASGD vs. NAG, (ii) ASGD", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "vs HB, we fix the starting learning rate, learning rate decay factor and decay schedule chosen by", + "type": "text" + } + ], + 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In a similar manner, when we compare (iii) SGD vs NAG or, (iv)", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "SGD vs. HB, we choose the learning rate, learning rate decay factor and decay schedule of SGD and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "simply sweep over the momentum parameter of NAG or HB and choose the momentum that offers", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 582, + 205, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 205, + 594 + ], + "score": 1.0, + "content": "the best validation error.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 459, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 460, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 460, + 612 + ], + "score": 1.0, + "content": "We now present plots of training function value for different algorithms and batch sizes.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "Effect of minibatch sizes: Figure 8 plots training function value for batch sizes of 128 and 8", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "for SGD, HB and NAG. We notice that in the initial stages of training, NAG obtains substantial", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "improvements compared to SGD and HB for batch size 128 but not for batch size 8. Towards the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "end of training however, NAG starts decreasing the training function value rapidly for both the batch", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "sizes. The reason for this phenomenon is not clear. Note however, that at this point, the test error", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 671, + 393, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 393, + 682 + ], + "score": 1.0, + "content": "has already stabilized and the algorithms are just overfitting to the data.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 107, + 688, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Comparison of ASGD with momentum methods: We now present the training error plots", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "for ASGD compared to HB and NAG in Figures 9 and 10 respectively. As mentioned earlier, in", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "order to see a clear trend, we constrain the learning rate and decay schedule of ASGD to be the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "same as that of HB and NAG respectively, which themselves were learned using grid search. We see", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5 + } + ], + "page_idx": 21, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "22", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 109, + 82, + 322, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 325, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 325, + 95 + ], + "score": 1.0, + "content": "C.3 DEEP RESIDUAL NETWORKS FOR CIFAR-10", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 116 + ], + "score": 1.0, + "content": "In this section, we will provide more details on our experiments on cifar-10, as well as present", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "some additional results. We used a weight decay of 0.0005 in all our experiments. The grid search", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 125, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 104, + 125, + 506, + 139 + ], + "score": 1.0, + "content": "parameters we used for various algorithms are as follows. Note that the ranges in which parameters", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 135, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 104, + 135, + 506, + 150 + ], + "score": 1.0, + "content": "such as learning rate need to be searched differ based on batch size (Jain et al., 2016). Furthermore,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 160 + ], + "score": 1.0, + "content": "we tend to extrapolate the grid search whenever a parameter (except for the learning rate decay", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "score": 1.0, + "content": "factor) at the edge of the grid has been chosen; this is done so that we always tend to lie in the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 182 + ], + "score": 1.0, + "content": "interior of the grid that we have searched on. Note that for the purposes of the grid search, we", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 179, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 506, + 194 + ], + "score": 1.0, + "content": "choose a hold out set from the training data and add it in to the training data after the parameters are", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 191, + 206, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 206, + 203 + ], + "score": 1.0, + "content": "chosen, for the final run.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 104, + 506, + 203 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 208, + 505, + 274 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 221 + ], + "score": 1.0, + "content": "Batch Size 8: Note: (i) parameters chosen by running for 40 epochs and picking the grid search", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 104, + 219, + 284, + 232 + ], + "score": 1.0, + "content": "parameter that yields the smallest validation", + "type": "text" + }, + { + "bbox": [ + 285, + 219, + 301, + 231 + ], + "score": 0.81, + "content": "0 / 1", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "error. 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105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "vs HB, we fix the starting learning rate, learning rate decay factor and decay schedule chosen by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "the best grid search run of NAG/HB respectively and perform a grid search over the long step and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "advantage parameter of ASGD. In a similar manner, when we compare (iii) SGD vs NAG or, (iv)", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "SGD vs. HB, we choose the learning rate, learning rate decay factor and decay schedule of SGD and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "simply sweep over the momentum parameter of NAG or HB and choose the momentum that offers", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 582, + 205, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 205, + 594 + ], + "score": 1.0, + "content": "the best validation error.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 516, + 505, + 594 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 459, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 460, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 460, + 612 + ], + "score": 1.0, + "content": "We now present plots of training function value for different algorithms and batch sizes.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 106, + 598, + 460, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "Effect of minibatch sizes: Figure 8 plots training function value for batch sizes of 128 and 8", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "for SGD, HB and NAG. We notice that in the initial stages of training, NAG obtains substantial", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "improvements compared to SGD and HB for batch size 128 but not for batch size 8. Towards the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "end of training however, NAG starts decreasing the training function value rapidly for both the batch", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "sizes. The reason for this phenomenon is not clear. Note however, that at this point, the test error", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 671, + 393, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 393, + 682 + ], + "score": 1.0, + "content": "has already stabilized and the algorithms are just overfitting to the data.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 615, + 506, + 682 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 688, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Comparison of ASGD with momentum methods: We now present the training error plots", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "for ASGD compared to HB and NAG in Figures 9 and 10 respectively. As mentioned earlier, in", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "order to see a clear trend, we constrain the learning rate and decay schedule of ASGD to be the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "same as that of HB and NAG respectively, which themselves were learned using grid search. We see", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "similar trends as in the validation error plots from Figures 5 and 6. Please see the figures and their", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 286, + 209, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 209, + 298 + ], + "score": 1.0, + "content": "captions for more details.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 113, + 92, + 484, + 230 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 92, + 484, + 230 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 92, + 484, + 230 + ], + "spans": [ + { + "bbox": [ + 113, + 92, + 484, + 230 + ], + "score": 0.967, + "type": "image", + "image_path": "d4f9589382a35d59086f9e033e6141ac77583966f826f72c83135bd3dea92e69.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 113, + 92, + 484, + 138.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 138.0, + 484, + 184.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 113, + 184.0, + 484, + 230.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 131, + 242, + 477, + 255 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 242, + 478, + 256 + ], + "spans": [ + { + "bbox": [ + 132, + 242, + 478, + 256 + ], + "score": 1.0, + "content": "Figure 8: Training loss for batch sizes 128 and 8 respectively for SGD, HB and NAG.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 274, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "similar trends as in the validation error plots from Figures 5 and 6. 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0000000000000000000000000000000000000000..fa4dd2f423929a4ce892f07cb6f3170085158e7f --- /dev/null +++ b/parse/train/rkZzY-lCb/rkZzY-lCb.md @@ -0,0 +1,401 @@ +# FEAT2VEC: DENSE VECTOR REPRESENTATION OF DATA WITH ARBITRARY FEATURES + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Methods that calculate dense vector representations for features in unstructured data—such as words in a document—have proven to be very successful for knowledge representation. We study how to estimate dense representations when multiple feature types exist within a dataset for supervised learning where explicit labels are available, as well as for unsupervised learning where there are no labels. Feat2Vec calculates embeddings for data with multiple feature types enforcing that all different feature types exist in a common space. In the supervised case, we show that our method has advantages over recently proposed methods; such as enabling higher prediction accuracy, and providing a way to avoid the cold-start problem. In the unsupervised case, our experiments suggest that Feat2Vec significantly outperforms existing algorithms that do not leverage the structure of the data. We believe that we are the first to propose a method for learning unsupervised embeddings that leverage the structure of multiple feature types. + +# 1 INTRODUCTION + +Informally, in machine learning a dense representation, or embedding of a vector $\vec { x } \in \mathbb { R } ^ { n }$ is another vector $\vec { y } \in \mathbb { R } ^ { r }$ that has much lower dimensionality $( r \ll n )$ than the original representation, and can be used to replace the original vector in downstream prediction tasks. Embeddings have multiple advantages, as they enable more efficient training (Mikolov et al., 2013), and unsupervised learning (Schnabel et al., 2015). For example, when applied to text, semantically similar words are mapped to nearby points. + +We consider two kind of algorithms that use embeddings: + +1. Unsupervised methods (sometimes referred as self-supervised methods) like Word2Vec (Mikolov et al., 2013), are designed to provide embeddings that are useful for a wide-array of predictions tasks. For example, the loss function of the continuous bag of words (CBOW) algorithm of Word2Vec is tuned to predict the next word of a sequence; however, in practice, the embeddings produced are mostly used for other tasks, such as analogy solving (Mikolov et al., 2013), or sentiment analysis (Le & Mikolov, 2014). In the context of this paper, we refer to the embeddings of an unsupervised method that can be used for a variety of auxiliary prediction tasks as general-purpose. +2. Supervised methods, like matrix factorization, produce embeddings that are highly tuned to a prediction task. These embeddings may be interpretable but do not usually generalize to other tasks. We refer to these embeddings as task-specific. Matrix factorization and Word2Vec are unable to calculate embeddings for items that are not available during training (“cold-start” problem). While recent work using n-gram features (Bojanowski et al., 2016) have addressed this limitation for supervised and unsupervised tasks, it can only be used for a single feature type—words. + +In this paper we propose Feat2Vec as a novel method that allows calculating embeddings of arbitrary feature types from both supervised and unsupervised data. Our main contributions are: + +• Unsupervised Feat2Vec. Existing general-purpose dense representation methods are largely restricted to one or two feature types. For example, the Word2Vec methods can only calculate embeddings for words, while follow-up work has enabled embeddings for both words and documents (Le & Mikolov, 2014). To our knowledge, Feat2Vec is the first algorithm that is able to calculate general-purpose embeddings that are not tuned for a single specific prediction task for arbitrary feature types. + +• Supervised Feat2Vec. Task-specific methods can use arbitrary feature types, but are restricted in that embeddings must be calculated for each individual feature, while sometimes higher-level of abstractions may be desirable—for example, we may want to have embeddings of documents instead of simply words. This capability makes Supervised Feat2Vec extremely flexible. We demonstrate that our method can be used to calculate embeddings of unseen (cold-start) items when there is an alternative textual description. + +# 2 PRELIMINARIES + +Factorization Machine (Rendle, 2010) is one of the most successful methods for general-purpose factorization. Rendle (2010) formulated it as an extension to polynomial regression. Consider a degree-2 polynomial (quadratic) regression, where we want to predict a target variable $y$ from a vector of inputs ${ \vec { x } } \in \mathbb { R } ^ { n }$ : + +$$ +\hat { y } ( \vec { x } ; \vec { b } , \vec { w } ) = \omega \big ( b _ { 0 } + \sum _ { i } b _ { i } x _ { i } + \sum _ { i = 1 } ^ { n } \sum _ { j = i + 1 } ^ { n } w _ { i , j } \ x _ { i } x _ { j } \big ) +$$ + +In words, $n$ is the total number of features, the term $b _ { 0 }$ is an intercept, $b _ { i }$ is the strength of the $i$ -th feature, and $w _ { i , j }$ is the interaction coefficient between the $i$ -th and $j$ -th feature. The function $\omega$ is an activation. Choices for $\omega$ include a linear link $\omega ( x ) = x ,$ ) for continuous outputs, or a logistic link $\begin{array} { r } { ( \omega ( x ) = \frac { \exp ( x ) } { \exp ( x ) + 1 } ) } \end{array}$ for binary outputs. + +Factorization Machine replaces the two-way individual pairwise parameters $w _ { i , j }$ for each interaction with a vector of parameters $\vec { w } _ { i }$ for each feature. This is a rank- $r$ vector of latent factors—embeddings in the neural literature—that encode the interaction between features and replaces the quadratic regression model with the following: + +$$ +{ \hat { y } } ( { \vec { x } } ; { \vec { b } } , { \vec { w } } ) = \omega { \big ( } b _ { 0 } + \sum _ { i } b _ { i } x _ { i } + \sum _ { i = 1 } ^ { n } \sum _ { j = i + 1 } ^ { n } ( x _ { i } { \vec { w _ { i } } } ) \cdot ( x _ { j } { \vec { w _ { j } } } ) { \big ) } +$$ + +Intuitively, the dot product $( \cdot )$ returns a scalar that measures the (dis)similarity between the latent factors of features $x _ { i }$ and $x _ { j }$ . Polynomial regression has $n ^ { 2 }$ interaction parameters, and Factorization Machine has $n \times r$ . While setting $r \ll n$ makes the model less expressive, factorization will typically exploit features having some shared latent structure. Factorization Machine may dramatically reduce the number of parameters to estimate. Rendle (2010) shows that when the feature vector x consists only of two categorical features in one-hot encoding, Factorization Machine is equivalent to the popular Matrix Factorization algorithm (Koren et al., 2009). + +# 3 FEAT2VEC + +We now describe how Feat2Vec extends the Factorization Machine model by allowing grouping of features, and enabling arbitrary feature extraction functions $( \ S \ 3 . 1 )$ . We also report a supervised method to learning Feat2Vec (§ 3.2), as well as a novel unsupervised training procedure (§ 3.3). + +# 3.1 MODEL + +We propose a framework for extending factorization machine with neural methods, by introducing structure into the feature interactions. Specifically, we do this by defining feature groups, \~κ, where each group contains features of a particular type. Explicitly, $\dot { \vec { \kappa } }$ is a partition of the set of feature columns in a dataset and each set within the partition is a feature group. The embeddings of a feature group are then learned via a feature extraction function, $\phi _ { i }$ , defined for each feature group. Feat2Vec will then extract features from each feature group, and build $r$ latent factors from them. In Factorization Machine, all the feature embeddings interact with each other, while in Feat2Vec, the interactions only occur between different feature groups. + +Formally, the addition of deep extraction methods yields the following statistical model: + +$$ +{ \hat { y } } ( { \vec { x } } , { \vec { b } } , { \vec { \phi } } ) = \omega \bigg ( b _ { 0 } + \sum _ { i = 1 } ^ { n } b _ { i } x _ { i } ~ + ~ \sum _ { i = 1 } ^ { | { \vec { \kappa } } | } \sum _ { j = i } ^ { | { \vec { \kappa } } | } \phi _ { i } ( { \vec { x } } _ { { \vec { \kappa } } _ { i } } ) \cdot \phi _ { j } ( { \vec { x } } _ { { \vec { \kappa } } _ { j } } ) \bigg ) +$$ + +In this notation, $\vec { x } _ { \vec { \kappa } _ { i } }$ is a subvector that contains all of the features that belong to the group ${ \vec { \kappa } } _ { i }$ . Thus, $x _ { \vec { \kappa } _ { i } } = [ x _ { j } : j \in \vec { \kappa } _ { i } ]$ . The intuition is that by grouping (sub-)features as a single entity, we can can reason on a higher level of abstraction. Instead of individual sub-features interacting among each other, the embeddings of feature groups interact with those of other groups. $\phi _ { i }$ is a feature extraction that inputs the $i$ -th feature group of the instance, and returns an $r$ -dimensional embedding. The feature extraction function $\phi _ { i }$ can allow for an arbitrary processing of its subfeatures. Across groups, entities interact with each other via the output of $\phi$ only. + +As a concrete example of an application of this grouping/feature extraction, we might group the individual words of a document into a “document” feature group, and allow this document embedding to then interact with learned embeddings of other document metadata (such as author id). We might expect the extraction function $\phi$ for the words in a document to extract features that characterize the attributes of the document taken as a whole, rather than simply the sum of its individual words. + +Figure 1 compares existing factorization methods with our novel model. In this example, Feat2Vec is using two feature groups: the first group only has a single feature which is projected to an embedding (just like a regular Factorization Machine); the second group has multiple features, which are together projected to a single embedding. + +![](images/88cf493d4cacf4a87c6bcf9e8dec89a2e33aab3199b3a3b0c39e17d487871d57.jpg) +Figure 1: Network architectures for factorization models. The white clouds $( \bigcirc )$ represent deep layers, for example a convolutional network for text features. + +The simplest implementation for $\phi _ { i }$ is a linear fully-connected layer, where the output of the $r$ -th entry is: + +$$ +\phi _ { i } \big ( \vec { x } _ { i } ; \vec { w } \big ) _ { r } = \sum _ { a = 1 } ^ { d _ { i } } w _ { r _ { a } } x _ { i _ { a } } +$$ + +Note that without loss of generality, we could define a model that is equivalent to a shallow Factorization Machine by allowing each feature group to be a singleton : $\vec { \kappa } \overset { \cdot } { = } \{ \{ x _ { 1 } \} , \{ x _ { 2 } \} \ldots \{ x _ { n } \} \}$ and the linear extraction function presented in Equation 4. + +We can use Feat2Vec to both use large feature sets and overcome the cold-start problem. This is only possible when there is an alternative description of the item available (for example an image or a passage of text). In Figure 2, we show how we address this problem by treating the words as indexed features, but placed within a structured feature group $\kappa _ { w }$ , the group of word features. A feature extraction function $\phi$ acts on the features in $\kappa _ { w }$ , and the other features interact with the words only via the output of $\phi$ . Notice that this implies we can precompute and store the latent factors of the target task seen during training, so that predictions during inference can be sped-up. For example if we have two feature groups (e.g, a label and an item), first we compute the feature extraction function to the unseen items and their embeddings, and then we simply apply a dot product over the stored vectors of the labels. + +![](images/cc57b1066f59d22b82314167436fc6f203035c20cf16e8f70ec5112b69514f54.jpg) +Figure 2: Comparison of how factorization may use item descriptions features. + +Figure 1c shows an approach of using neural networks within factorization machines that has been proposed multiple times (Dziugaite & Roy, 2015; Guo et al., 2017). It replaces the dot product of factors with a learned neural function, which has been shown to improve predictive accuracy for various tasks. In this case, fast inference for cold-start documents using pre-computed label embeddings is no longer possible. It needs to store the entire neural function that takes the embeddings as inputs. Another shortcoming of replacing the dot product with a neural function is that it would no longer be possible to interpret the embeddings as containing latent factors related to the target task; There may be highly complex mappings from the embeddings to the final output via this neural function. However, it would be straightforward to combine this approach with Feat2Vec. This is not explored in this work. + +# 3.2 SUPERVISED LEARNING FROM DATA + +We can learn the the parameters of a deep factorization model θ using training data by minimizing a loss function $\mathcal { L }$ : + +$$ +\arg \operatorname* { m i n } _ { \vec { \Theta } } \sum _ { x } \mathcal { L } \big ( y ( x ) , \hat { y } ( x ; \vec { \theta } ) \big ) + \gamma | | \Theta | | ^ { w } +$$ + +Here, $y ( x )$ is the true target value for $x$ obtained from training data, and ${ \hat { y } } ( x )$ is the one estimated by the model; the hyperparameter $\gamma$ controls the amount of regularization. For the labeling and classification tasks, we optimize the binary cross-entropy for $y \in \{ 0 , 1 \}$ : + +$$ +\mathcal { L } ( y , \hat { y } ) = - \big ( y \log ( \hat { y } ) \big ) - ( 1 - y ) \log ( 1 - \hat { y } ) \big ) +$$ + +For the regression tasks where the target value is continuous, we optimize the mean squared error (MSE): + +$$ +\mathcal { L } ( y , \hat { y } ) = ( y - \hat { y } ) ^ { 2 } +$$ + +Neural models are typically learned using mini-batch updates, where the incremental descent is performed with respect to several instances at a time. For the implementation of this paper, we built our models using the Keras programming toolkit (Chollet et al., 2015), that is now part of Tensorflow (Abadi et al., 2015). It enables automatic differentiation, and is bundled with a generalpurpose optimization algorithm called ADAM (Kingma & Ba, 2014) that needs no tuning of gradient step sizes. + +It is straightforward to optimize Equation 5 directly for multiclass or binary classification. However, when the number of labels is very large, it is common practice use a binary classifier and sample the negative examples (Dyer, 2014). For the multi-label classification tasks, we use Feat2Vec with a binary output. In this case we would have at least two feature groups—one of the feature groups is the label that we want to predict, and the other group(s) is the input from which we want to make the prediction. The output indicates whether the label is associated with the input $( y = + 1 )$ ), or not $( y = 0 )$ ). The datasets we use for our labeling experiments only contains positive labels, thus for each training example we sample a set of negative labels equal to the number of positive labels. It is typical to use one of the following sampling strategies according to the best validation error, in each case excluding the actual positive labels for each training example – (i) uniformly from all possible labels, or (ii) from the empirical distributions of positive labels. Other sampling strategies have been proposed (Rendle et al., 2009; Rendle & Freudenthaler, 2014). + +# 3.3 UNSUPERVISED LEARNING FROM DATA + +We now discuss how Feat2Vec can be used to learn embeddings in an unsupervised setting with no explicit target for prediction. + +The training dataset for a Feat2Vec model consists of only the observed data. In natural language, these would be documents written by humans. Since Feat2Vec (Equation 3) requires positive and negative examples, we also need to supply unobserved data as negative examples. Consider a feature group $\vec { \mathsf { K } } _ { i }$ , that exists in very high dimensional space. For example, this could happen because we are modeling with one-hot encoding a categorical variable with large number of possible values. In such scenario, it is overwhelmingly costly to feed the model all negative labels, particularly if the model is fairly sparse. + +A shortcut around this is a concept known as implicit sampling, where instead of using all of the possible negative labels, one simply samples a fixed number $( k )$ from the set of possible negative labels for each positively labelled record. Word2Vec makes use of an algorithm called Negative Sampling, that has little theoretical guarantees (Dyer, 2014). In short, their approach samples a negative observation from a noise distribution $\mathcal { Q } _ { w 2 v }$ , that is proportional to the empirical frequency of a word in the training data. + +We introduce a new implicit sampling method that enables learning unsupervised embeddings for structured feature sets. We can learn the correlation of features within a dataset by imputing negative labels, simply by generating unobserved records as our negative samples. Unlike Word2Vec, we do not constraint features types to be words. Features groups can be individual columns in a data matrix, but they need not to be. By grouping subfeatures using the parameter $\boldsymbol { \mathsf { K } }$ in Equation 3, the model can reason on more abstract entities in the data. By entity, we mean a particular feature group value. For example, in our experiments on a movie dataset, we use a “genre” feature group, where we group non-mutually exclusive indicators for movie genres including comedy, action, and drama films. + +We start with a dataset $S ^ { + }$ of records with |\~κ| feature groups. We then mark all observed records in the training set as positive examples. For each positive record, we generate $k$ negative labels using the following 2-step algorithm: + +# Algorithm 1 Implicit sampling algorithm for unsupervised Feat2Vec: $\mathcal { Q }$ + +1: function FEAT2VEC SAMPLE(S+, k, α1, α2) +2: S − ← ∅ +3: for \~x + ∈ S+ do +4: Draw a random feature group $\kappa _ { i } \sim \mathcal { Q } _ { 1 } ( \{ \mathrm { p a r a m s } ( \phi _ { i } ) \} _ { i = 1 } ^ { | \vec { \kappa } | } , \alpha _ { 1 } )$ +5: for $j \in \{ 1 , \ldots , k \}$ do +6: \~x − ← \~x + $\triangleright$ set initially to be equal to the positive sample +7: Draw a random feature group value $\tilde { x } \sim \mathcal { Q } _ { 2 } ( \mathrm { X } _ { \kappa _ { i } , \alpha _ { 2 } } )$ +8: $\begin{array} { l } { { { \vec { x } _ { \kappa _ { i } } ^ { - } \tilde { x } } } } \\ { { { S ^ { - } S ^ { - } + \{ \vec { x } ^ { - } \} } } } \end{array}$ $\triangleright$ substitute the $i$ -th feature type with the sampled one +9: +10: end for +11: end for +12: return $S ^ { - }$ +13: end function + +Explained in words, our negative sampling method for unsupervised learning iterates over all of the observations of the training dataset. For each observation ${ \bar { x } } ^ { + }$ , it randomly selects the $i$ -th feature group from a noise distribution $\mathcal { Q } _ { 1 } ( \cdot )$ . Then, it creates a negative observation that is identical to ${ \vec { x } } ^ { + }$ , except that its $i$ -th feature group is replaced by a value sampled from a noise distribution $\mathcal { Q } _ { 2 } ( \cdot )$ . In our application, we use the same class of noise distributions (flattened multinomial) for both levels of sampling, but this need not necessarily be the case. + +We now describe the two noise distributions that we use. We use $P _ { \mathcal { Q } } ( x )$ to denote the probability of $x$ under a distribution $\mathcal { Q }$ . + +Sampling Feature Groups. The function params calculates the complexity of a feature extraction function $\phi _ { i }$ . To sample a feature group, we choose a feature group $\kappa _ { i }$ from a multinomial distribution with probabilities proportional a feature’s complexity. By complexity, we mean the number of parameters we need to learn that are associated with a particular feature group. This choice places more weight on features that have more parameters and thus are going to require more training iterations to properly learn. The sampling probabilities of each feature group are: + +$$ +P _ { \mathcal { Q } _ { 1 } } ( \kappa _ { i } | \operatorname { p a r a m s } ( \phi _ { i } ) \} _ { i = 1 } ^ { | \sharp | } , \alpha _ { 1 } ) = \frac { \operatorname { p a r a m s } ( \phi _ { i } ) ^ { \alpha _ { 1 } } } { \sum _ { j = 1 } ^ { | \sharp | } \operatorname { p a r a m s } ( \phi _ { j } ) ^ { \alpha _ { 1 } } } , \quad \alpha _ { 1 } \in [ 0 , 1 ] +$$ + +For categorical variables using a linear fully-connected layer, the complexity is simply proportional to the number of categories in the feature group. However, if we have multiple intermediate layers for some feature extraction functions (e.g., convolutional layers), these parameters should also be counted towards a feature group’s complexity. The hyper-parameter $\alpha _ { 1 }$ helps flatten the distribution. When $\alpha _ { 1 } = 0$ , the feature groups are sampled uniformly, and when $\alpha _ { 1 } = 1$ , they are sampled proportional to their complexity. Figure A.1 in the Appendix provides a visualization of how the feature sampling rate varies with the hyperparameter for features with differing levels of complexity. + +Sampling Feature Group Values. To sample a value from within a feature groups $\kappa _ { i }$ , we use a similar strategy to Word2Vec and use the empirical distribution of values: + +$$ +P _ { \mathcal Q _ { 2 } } ( x | \mathrm X _ { \kappa _ { i } } , \alpha _ { 2 } ) = \frac { \mathrm { c o u n t } ( x ) ^ { \alpha _ { 2 } } } { \sum _ { x _ { \kappa _ { i } } ^ { \prime } \in S ^ { + } } \mathrm { c o u n t } ( x _ { \kappa _ { i } } ^ { \prime } ) ^ { \alpha _ { 2 } } } , \quad \alpha _ { 2 } \in [ 0 , 1 ] +$$ + +Here, $\operatorname { c o u n t } ( x )$ is the number of times a feature group value $x$ appeared in the training dataset $S ^ { + }$ , and $\alpha _ { 2 }$ is again a flattening hyperparameter. + +This method will sometimes by chance generate negatively labeled samples that do exist in our sample of observed records. The literature offers two possibilities: in the Negative Sampling that Word2Vec follows, the duplicate negative samples are simply ignored (Dyer, 2014). Alternatively, it is possible to account for the probability of random negative labels that are identical to positively labeled data using Noise Contrastive Estimation (NCE) (Gutmann & Hyvarinen, 2010). ¨ + +# 3.3.1 THE LOSS FUNCTION FOR UNSUPERVISED LEARNING + +For our unsupervised learning of embeddings, we optimize a NCE loss function, to adjust the structural statistical model $\hat { y } = p ( y = 1 | \vec { x } , \vec { \phi } , \Theta )$ , expressed in Equation 3 to account for the possibility of random negative labels that appear identical to positively labeled data. θ here represents the parameters learned in during training (i.e. the $b _ { i }$ terms and parameters associated with the extraction functions $\phi _ { i }$ in Equation 3). Since we only deal with a dichotomous label, indicating a positive or negative sample, for unsupervised learning, we restrict our attention to usage of Equation 3 with $\omega$ as a logistic link function. + +An additional burden of NCE is that we need to calculate a partition function $Z _ { \vec { x } }$ for each unique record type $\vec { x }$ in the data that transforms the probability $\hat { y }$ of a positive or negative label into a wellbehaved distribution that integrates to 1. Normally, this would introduce an astronomical amount of computation and greatly increase the complexity of the model. As a work-around, we appeal to the work of Mnih & Teh (2012), who showed that in the context of language models that setting the $Z _ { \vec { x } } = 1$ in advance effectively does not change the performance of the model. The intuition is that if the underlying model has enough free parameters that it will effectively learn the probabilities itself. Thus, it does not over/under predict the probabilities on average (since that will result in penalties on the loss function). + +Written explicitly, the new structural probability model is: + +$$ +\tilde { p } ( Y = 1 | \vec { x } , \vec { \phi } , \mathbf { \Theta } \Theta ) = \frac { \exp \bigl ( s ( \vec { x } , \vec { \phi } , \mathbf { \Theta } \Theta ) \bigr ) } { \exp ( s ( \vec { x } , \vec { \phi } , \mathbf { \Theta } \Theta ) \bigr ) + P _ { \mathcal { Q } } ( \vec { x } | \alpha _ { 1 } , \alpha _ { 2 } ) } +$$ + +where $s ( . )$ denotes the score of a record $\vec { x }$ given parameter values/extraction functions: + +$$ +s ( \vec { x } , \vec { \phi } , \Theta ) = b _ { 0 } + \sum _ { i = 1 } ^ { n } b _ { i } x _ { i } + \sum _ { i = 1 } ^ { | \vec { \kappa } | } \sum _ { j = i } ^ { | \vec { \kappa } | } \phi _ { i } ( \vec { x } _ { \vec { \kappa } _ { i } } ) \cdot \phi _ { j } ( \vec { x } _ { \vec { \kappa } _ { j } } ) +$$ + +and $P _ { \mathcal { Q } } ( . )$ denotes the total probability of a record $\vec { x _ { i } }$ being drawn from our negative sampling algorithm, conditional on the positively labeled record ${ \vec { x } } ^ { + }$ the negative sample is drawn for: + +$$ +P _ { \mathcal { Q } } ( \vec { x } | \alpha _ { 1 } , \alpha _ { 2 } , \mathrm { X } , \vec { x } ^ { + } ) = P _ { \mathcal { Q } _ { 2 } } ( \vec { x } _ { \bf { \kappa } _ { i } } | \mathrm { X } _ { \bf { \kappa } _ { i } } , \alpha _ { 2 } ) P _ { \mathcal { Q } _ { 1 } } ( \kappa _ { i } | \mathrm { p a r a m s } ( \phi _ { i } ) \rbrace _ { i = 1 } ^ { n } , \alpha _ { 1 } ) +$$ + +Our loss function $L$ optimizes $\theta$ , the parameters of the feature extraction functions $\vec { \phi }$ , while accounting for the probability of negative samples. + +$$ +L ( S ) = \arg \operatorname* { m i n } _ { \Theta } \frac { 1 } { | S ^ { + } | } \sum _ { \vec { x } ^ { + } \in S ^ { + } } \Big ( \log ( \tilde { p } ( y = 1 | \vec { x } ^ { + } , \vec { \phi } , \Theta ) ) \ + \sum _ { \vec { x } ^ { - } \sim \mathcal { Q } ( \cdot | \vec { x } ^ { + } ) } ^ { k } \log ( \tilde { p } ( y = 0 | \vec { x } ^ { - } , \vec { \phi } , \Theta ) ) \Big ) +$$ + +Feat2Vec has interesting theoretical properties. For example, it is well known that Factorization Machines can be used as a multi-label classifier: with at least two features, one can use one of the feature as the target label, and the other as the input feature to make a prediction. In such setting, the output indicates whether the label is associated with the input $( y = + 1 )$ ), or not $( y = 0 )$ ), and therefore the input can be associated with more than one label. With $n$ feature types, Feat2Vec is equivalent to optimizing a convex combination of the loss functions from $n$ individual Factorization Machines. In other words, it optimizes $n$ multi-label classifiers, where each classifier is optimized for a different target (i.e.,a specific feature group). We show the proof of this in the Appendix 1. + +# 4 EMPIRICAL RESULTS + +# 4.1 SUPERVISED EMBEDDINGS + +We now address our working hypotheses for evaluating supervised embeddings. For all our experiments we define a development set and a single test set which is $10 \%$ of the dataset, and a part of the development set is used for early stopping or validating hyper-parameters. Since these datasets are large and require significant time to train on an Nvidia K80 GPU cluster, we report results on only a single training-test split. For the multi-label classification task in 4.1.1 we predict a probability for each document-label pair and use an evaluation metric called Area Under the Curve (AUC) of the Receiver Operating Characteristic (ROC). Since we only observe positive labels, for each positive label in the test set we sample negative labels according to the label frequency. This ensures that if a model merely predicts the labels according to their popularity, it would have an AUC of 0.5. A caveat of our evaluation strategy is that we could be underestimating the performance of our models—there is a small probability that the sampled negatives labels are false negatives. However, since we apply the same evaluation strategy consistently across our methods and baselines, the relative difference of the AUC is meaningful. We choose the AUC as a metric because it is popular for both classification and ranking problems. For the regression task in 4.1.2, we use mean squared error (MSE) as the evaluation metric. In preliminary experiments we noticed that regularization slows down convergence with no gains in prediction accuracy, so we avoid overfitting only by using early stopping. We share most of the code for the experiments online1 for reproducibility. + +For our feature extraction function $\phi$ for text, we use a Convolutional Neural Network (CNN) that has been shown to be effective for natural language tasks (Kalchbrenner et al., 2014; Weston et al., 2014). In Appendix B we describe this network and its hyper-parameters. Instead of tuning the hyper-parameters, we follow previously published guidelines (Zhang & Wallace, 2015). + +# 4.1.1 IS Feat2Vec EFFECTIVE FOR COLD-START PREDICTIONS? + +We compare Feat2Vec with an extension of matrix factorization that can generalize to unseen items for text documents, Collaborative Topic Regression (CTR– Wang & Blei (2011)), a method with an open-source Python implementation2. We evaluate them on the CiteULike dataset which consists of pairs of scientific articles and the users who have added them to their personal libraries, and it contains 16,980 unique articles and 5,551 unique users. We use the models to predict users who may have added a given article to their library. We compare the performance of Feat2Vec with CTR using pre-defined cross-validation splits3. We use $1 \%$ of the training set for early stopping. + +Table 1: Yelp rating prediction + +
MSEImprovement over Matrix Factorization
Matrix Factorization1.561=
Feat2Vec0.48069.2 %
DeepCoNN1.44119.6 %
+ +For CTR we use the hyper-parameters reported by the authors as best, except for $r$ which we found had a significant impact on training time . We only consider $r \in \{ 5 , 1 0 , 1 5 \}$ and choose the value which gives the best performance for CTR (details in Appendix A.2). On the warm-start condition, CTR has an AUC of 0.9356; however, it shows significant degradation in performance for unseen documents and it only performs slightly better than random chance with an AUC of 0.5047. On the other hand, Feat2Vec achieves AUC of 0.9401 on the warm-start condition, and it only degrades to 0.9124 on unseen documents. Feat2Vec can also be trained over ten times faster, since it can leverage GPUs.4 We also note that we have not tuned the architecture or hyper-parameters of the feature extraction function $\phi$ and greater improvements are possible by optimizing them. + +# 4.1.2 COMPARISON WITH ALTERNATIVE CNN-BASED TEXT FACTORIZATION + +We now compare with a method called DeepCoNN, a deep network specifically designed for incorporating text into matrix factorization (Zheng et al., 2017)—which reportedly, is the state of the art for predicting customer ratings when textual reviews are available. For Feat2Vec we use the same feature extraction function (see Appendix B.1 for details) used by DeepCoNN. We evaluate on the Yelp dataset5, which consists of 4.7 million reviews of restaurants. For each user-item pair, DeepCoNN concatenates the text from all reviews for that item and all reviews by that user. The concatenated text is fed into a feature extraction function followed by a factorization machine. In contrast, for Feat2Vec, we build 3 feature groups: item identifiers (in this case, restaurants), users and review text. + +Table 1 compares our methods to DeepCoNN’s published results because a public implementation is not available. We see that Feat2Vec provides a large performance increase when comparing the reported improvement, over Matrix Factorization, of the mean squared error. Our approach is more general, and we claim that it is also more efficient. Since DeepCoNN concatenates text, when the average reviews per user is $\bar { n _ { u } }$ and reviews per item is $\bar { n _ { i } }$ , each text is duplicated on average $\bar { n _ { i } } \times \bar { n _ { u } }$ times per training epoch. In contrast, for Feat2Vec each review is seen only once per epoch. Thus it can be 1-2 orders of magnitude more efficient for datasets where $\bar { n _ { i } } \times \bar { n _ { u } }$ is large. + +# 4.2 GENERAL-PURPOSE EMBEDDINGS + +# 4.2.1 DOES Feat2Vec ENABLE BETTER EMBEDDINGS? + +Ex ante, it is unclear to us how to evaluate the performance of an unsupervised embedding algorithm, but we felt that a reasonable task would be a ranking task one might practically attempt using our datasets. This task will assess the similarity of trained embeddings using unseen records in a left-out dataset. In order to test the relative performance of our learned embeddings, we train our unsupervised Feat2Vec algorithm and compare its performance in a targeted ranking task to Word2Vec’s CBOW algorithm for learning embeddings. In our evaluation approach, we compare the cosine similarity of the embeddings of two entities where these entities are known to be associated with each other since they appear in the same observation in a test dataset. In particular, in the movie dataset we compare the similarity of movie directors to those of actors who were cast in the same film for a left-out set of films. For our educational dataset, we compare rankings of textbooks by evaluating the similarity of textbook and user embeddings. We evaluate the rankings according to their mean percentile rank (MPR): + +$$ +M P R = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \frac { R _ { i } } { \operatorname* { m a x } R } +$$ + +where $R _ { i }$ is the rank of the entity under our evaluation procedure for observation $i$ . This measures on average how well we rank actual entities. A score of 0 would indicate perfect performance (i.e. top rank every test sample given), so a lower value is better under this metric. See the appendix $\ S \mathrm { A } . 1$ for further details on the experimental setup. + +# 4.2.2 DATASETS + +Movies The Internet Movie Database (IMDB) is a publicly available dataset6 of information related to films, television programs and video games. Though in this paper, we focus only on data on its 465,136 movies. Table A.1 in the appendix $( \ S \mathrm { A } . 1 )$ summarizes the feature types we use. It contains information on writers, directors, and principal cast members attached to each film, along with metadata. + +Education We use a dataset from an anonymized leading technology company that provides educational services. In this proprietary dataset, we have 57 million observations and 9 categorical feature types which include textbook identifier, user identifier, school identifier, and course the book is typically used with, along with other proprietary features. Here, each observation is an “interaction” a user had with a textbook. + +# 4.2.3 RESULTS + +After training, we use the cast members associated with the movies of the test set and attempt to predict the actual director the film was directed. We take the sum of the cast member embeddings, and rank the directors by cosine similarity of their embeddings to the summed cast member vector. If there is a cast member in the test dataset who did not appear in the training data, we exclude them from the summation. For the educational dataset, we simply use the user embedding directly to get the most similar textbooks. + +Table 2 presents the results from our evaluation. Feat2Vec sizably outperforms CBOW in the MPR metric. In fact, Feat2Vec predicts the actual director $2 . 4 3 \%$ of the times, while CBOW only does so $1 . 2 6 \%$ of the time, making our approach almost 2 times better in terms of Top-1 Precision metric. We explore in greater detail the distribution of the rankings in the appendix in $\ S \mathrm { A } . 2$ . + +Table 2: Mean percentile rank + +
DatasetFeat2VecCBOW
IMDB19.36%24.15%
Educational25.2%29.2%
+ +# 4.3 UNSUPERVISED Feat2Vec PERFORMANCE WITH CONTINUOUS INPUTS + +We now focus on how well Feat2Vec performs on a real-valued feature with a complex feature extraction function. We expect this task to highlight Feat2Vec’s advantage over token-based embedding learning algorithms, such as Word2Vec, since our rating embedding extraction function will require embeddings of numerically similar ratings to be close , while Word2Vec will treat two differing ratings tokens as completely different entities. We evaluate the prediction of the real-valued rating of movies in the test dataset by choosing the IMDB rating embedding most similar7 to the embedding of the movie’s director, and compute the Root Mean Squared Error (RMSE) of the predicted rating in the test dataset. We also vary $\alpha _ { 1 }$ , the flattening hyperparameter for feature group sampling, to see what effect this hyperparameter has on our performance. Intuitively, a low $\alpha _ { 1 }$ will greatly improve the quality of the ratings embeddings learned, since it has relatively few parameters and is otherwise sampled infrequently. At the same time, with low $\alpha _ { 1 }$ the director feature will be sampled less since it is one of the most complex features to learn, so the learned director embeddings may be of poorer quality. Figure 3 displays the results of our experiment, benchmarked against the performance of Word2Vec’s CBOW algorithm in the prediction task. We also show as a baseline the RMSE of a random uniform variable over the range of possible ratings (0 to 10). As is evident from the plot, CBOW performs a bit better than a random prediction, but is also handily outperformed by Feat2Vec across all hyper-parameter settings. The algorithm’s performance does not seem very sensitive to the hyperparameter choice. + +![](images/617543a6a2a807e0a383fe6a61bcbf4ec4eb3fbaf9fd269a94493f8b5ea3637d.jpg) +Figure 3: RMSE in Ratings Task as a Function of $\alpha _ { 1 }$ + +# 5 RELATION TO PRIOR WORK + +The original Factorization Machine formulation has been extended for multiple contexts. For example, Field-Aware Factrorization Machine (Juan et al., 2016) allows different weights for some feature interactions, but does not allow feature groups or feature extraction functions like Feat2Vec does. + +Algorithms that calculate continuous representations of entities other than words have been proposed for biological sequences (Abrahamsson & Plotkin, 2009), of vertices in network graphs (Perozzi et al., 2014) or in machine translation for embeddings of complete sentences (Kiros et al., 2015). Generative Adversarial Networks (Goodfellow et al., 2014)(GANs) have been used to produce unsupervised embeddings of images effective for classification (Radford et al., 2015) and for generating natural language (Press et al., 2017). To our knowledge, GANs have not been used for jointly embedding multiple feature types. Adversarial training could be an alternative to NCE for unsupervised learning, but we leave this for future study. + +We recently discovered a promising direction for an algorithm still in development called StarSpace (Wu et al., 2017) with similar goals from ours. Even though they intend to be able to embed all types of features, at the time of the writing of this paper, their pre-print method was limited to only work for bag of words. While Feat2Vec can jointly learn embeddings for all feature values in a dataset, StarSpace samples a single arbitrary feature. Our preliminary experiments suggest that sampling a single feature does not produce embeddings that generalize well. Nonetheless, a limitation of our work is that we do not compare with StarSpace, which future work may decide to do. + +# 6 CONCLUSION + +Embeddings have proven useful in a wide variety of contexts, but they are typically built from datasets with a single feature type as in the case of Word2Vec, or tuned for a single prediction task as in the case of Factorization Machine. We believe Feat2Vec is an important step towards generalpurpose methods, because it decouples feature extraction from prediction for datasets with multiple feature types, it is general-purpose, and its embeddings are easily interpretable. + +In the supervised setting, Feat2Vec is able to calculate embeddings for whole passages of texts, and we show experimental results outperforming an algorithm specifically designed for text—even when using the same feature extraction CNN. This suggests that the need for ad-hoc networks should be situated in relationship to the improvements over a general-purpose method. + +In the unsupervised setting, Feat2Vec’s embeddings are able to capture relationships across features that can be twice as better as Word2Vec’s CBOW algorithm on some evaluation metrics. Feat2Vec exploits the structure of a datasets to learn embeddings in a way that is structurally more sensible than existing methods. The sampling method, and loss function that we use have interesting theoretical properties. To the extent of our knowledge, Unsupervised Feat2Vec is the first method able to calculate continuous representations of data with arbitrary feature types. + +Future work could study how to reduce the amount of human knowledge our approach requires; for example by automatically grouping features into entities, or by automatically choosing a feature extraction function. These ideas can extend to our codebase that we make available 8. 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In Proceedings of the Tenth ACM International Conference on Web Search and Data Mining, WSDM ’17, pp. 425–434, New York, NY, USA, 2017. ACM. ISBN 978-1-4503-4675-7. doi: 10.1145/3018661.3018665. URL http://doi.acm.org/10.1145/3018661.3018665. + +![](images/bf34d4eebcb9f890edf4a4cbf4846f7ce55c933140f8d2fbbd7a590e3701c8c8.jpg) +Figure A.1: Feature Sampling Probabilities as a Function of $\alpha _ { 1 }$ + +Table A.1: IMDB dataset features + +
Feature Type NameType# of feats.Example for an instance
Runtime (minutes)Real-valued1116
IMDB rating (0-10)Real-valued17.8
# of IMDB rating votesReal-valued1435,682
Is adult film?Boolean2False
Movie releaes yearCategorical2712001
Movie titleText165,471“Ocean's”,“Eleven”
DirectorsBag of categories174,382‘Steven Soderbergh”
GenresBag of categories28“Crime”,“Thriller”
WritersBag of categories244,241“George Johnson”,“Jack Russell"
Principal cast members (actors)Bag of categories1,104,280“George Clooney”,“Brad Pitt”,“Julia Roberts”
+ +# A APPENDIXES + +# A.1 UNSUPERVISED RANKING EXPERIMENT DETAILS + +For our evaluation, we define a testing set that was not used to tune the parameters of the model. For the IMDB dataset, we randomly select a $10 \%$ sample of the observations that contain a director that appears at least twice in the database 9. We do this to guarantee that the set of directors in the left-out dataset appear during training at least once, so that each respective algorithm can learn something about the characteristics of these directors. For the educational dataset, our testing set only has observations of textbooks and users that appear at least 10 times in training. + +For both Feat2Vec and CBOW, we perform cross-validation on the loss function, by splitting the $10 \%$ of the training data randomly into a validation set, to determine the number of epochs to train, and then train the full training dataset with this number of epochs. 10 While regularization of the embeddings during training is possible, this did not dramatically change results, so we ignore this dimension of hyperparameters. + +We rank left-out entity pairs in the test dataset using the ordinal ranking of the cosine similarity of target and input embeddings. For the IMDB dataset, the target is the director embedding, and the input embedding is the sum of the cast member embeddings. For the educational dataset, the target is the textbook embedding, and the input embedding is the user embedding. + +For training Feat2Vec we set $\alpha _ { 1 } = \alpha _ { 2 } = 3 / 4$ in the IMDB dataset; and $\alpha _ { 1 } = 0$ and $\alpha _ { 2 } = 0 . 5$ for the educational. In each setting, $\alpha _ { 2 }$ is set to the same flattening hyperparameter we use for CBOW to negatively sample words in a document. We learn $r = 5 0$ dimensional embeddings under both algorithms. + +Below we describe how CBOW is implemented on our datasets for unsupervised experiments and what extraction functions are used to represent features in the IMDB dataset. + +Word2Vec For every observation in each of the datasets, we create a document that tokenizes the same information that we feed into Feat2Vec. We prepend each feature value by its feature name, and we remove spaces from within features. In Figure A.2 we show an example document. Some features may allow multiple values (e.g., multiple writers, directors). To feed these features into the models, for convenience, we constraint the number of values, by truncating each feature to no more than 10 levels (and sometimes less if reasonable). This results in retaining the full set of information for well over $9 5 \%$ of the values. We pad the sequences with a “null” category whenever necessary to maintain a fixed length. We do this consistently for both Word2Vec and Feat2Vec. We use the CBOW Word2Vec algorithm and set the context window to encompass all other tokens in a document during training, since the text in this application is unordered. + +![](images/330946bb7dcbf02917794a98c168e5aef01780c17fee4188796103aa1971d0ff.jpg) +Figure A.2: Sample document for Word2Vec for the Ocean’s Eleven movie + +Feat2Vec Feature representation in Feat2Vec requires a feature extraction function for each feature type. Here, we explain how we build these functions: + +• Bag of categories, categorical, and boolean: For all of the categorical variables, we learn a unique $r$ -dimensional embedding for each entity using a linear fully-connected layer (Equation 4). We do not require one-hot encodings, and thus we allow multiple categories to be active; resulting in a single embedding for the group that is the sum of the embeddings of the subfeatures. This is ordering-invariant: the embedding of “Brad Pitt” would be the same when he appears in a movie as a principal cast member, regardless whether he was 1st or 2nd star. Though, if he were listed as a director it may result in a different embedding. + +• Text: We preprocess the text by removing non alpha-numeric characters, stopwords, and stemming the remaining words. We then follow the same approach that we did for categorical variables, summing learned word embeddings to a “title embedding” before interacting. It would be easy to use more sophisticated methods (e.g, convolutions), but we felt this would not extract further information. + +• Real-valued: For all real-valued features, we pass these features through a 3-layer feedforward fully connected neural network that outputs a vector of dimension $r$ , which we treat as the feature’s embedding. Each intermediate layer has $r$ units with $\mathtt { r e l u }$ activation functions. These real-valued features highlight one of the advantages of the Feat2Vec algorithm: using a numeric value as an input, Feat2Vec can learn a highly nonlinear relation mapping a real number to our high-dimensional embedding space. In contrast, Word2Vec would be unable to know ex ante that an IMDB rating of 5.5 is similar to 5.6. + +# A.2 DISTRIBUTION OF IMDB DIRECTOR RANKINGS + +Figure A.3 shows the full distribution of rankings of the IMDB dataset, rather than summary statistics, in the form of a Cumulative Distribution Function (CDF) of all rankings calculated in the test dataset. The graphic makes it apparent for the vast majority of the ranking space, the rank CDF of Feat2Vec is to the left of CBOW, indicating a greater probability of a lower ranking under Feat2Vec. This is not, however, the case at the upper tail of ranking space, where it appears CBOW is superior. + +However, when we zoom-in on the absolute upper region of rankings (1 to 25), which might be a sensible length of ranks one might give as actual recommendatiosn, it is the case that up until rank 8 or so, Feat2Vec outperforms CBOW still. Intermediate rankings are still strong signals that our Feat2Vec algorithm is doing a better job of extracting information into embeddings, particularly those entities that appear sparsely in the training data and so are especially difficult to learn. + +![](images/fe8ebcf2f1729c2c77fccf48acc7e972098c21fdd9cc58b0c0592dc3fc7497e5.jpg) +Figure A.3: Cumulative Distribution Function of Director Rankings (With Zoom-in to Top 25 Ranks) + +# A.3 PROOF TO THEOREM 1 + +Theorem 1. The gradient for learning embeddings with Feat2Vec is a convex combination of the gradient from n targeted Factorization Machines for each feature in the data when each feature group is a singleton, where n is the total number of features in the dataset. + +Proof. Let $S _ { \kappa _ { i } } ^ { + }$ denote the positively labeled records whose corresponding negative samples resample feature $\kappa _ { i }$ . For convenience, suppress the inclusion of learned parameters θ in the notation in this section while understanding the feature extraction functions $\vec { \phi }$ implicitly include these parameters. We can express the loss function $L ( . )$ , the binary cross-entropy of the data given the Feat2Vec model, as follows: + +$$ +\begin{array} { r l } { L ( S ^ { + } | \vec { \phi } ) = \displaystyle \frac { 1 } { | S ^ { + } | } \sum _ { \tau ^ { \prime } \in S ^ { + } } \Big ( \log ( \tilde { \rho } ( y | \vec { \phi } - 1 | \vec { \phi } , \vec { x } ^ { + } ) ) + \underbrace { \sum _ { \tau ^ { \prime } \in S ^ { + } } ^ { \vec { K } } \log ( | \vec { \rho } ( y = 0 | | \vec { \phi } , \vec { x } ^ { - } ) ) } _ { \tau ^ { \prime } - \tau < 2 ( | \vec { x } ^ { + } - \vec { x } ^ { + } | ) ^ { \tilde { \phi } } ( \vec { x } ^ { + } ) } \Big . } & { } \\ { = \frac { 1 } { | S ^ { + } | } \sum _ { \tau ^ { \prime } \in S ^ { + } } \Big ( \log ( \tilde { \rho } ( y | \vec { \phi } - 1 | \vec { \phi } , \vec { x } ^ { + } ) , \vec { x } ^ { + } + S _ { \tau , y } ^ { + } ) \rho ( \vec { x } ^ { + } \in S _ { \tau , x } ^ { + } ) ) } \\ { + \underbrace { \sum _ { \tau ^ { \prime } \in S ^ { + } } ^ { \vec { K } } \log ( \tilde { \rho } ( y = 0 | \vec { \phi } , \vec { x } ^ { - } , \vec { x } ^ { + } \in S _ { \tau , y } ^ { + } ) \rho ( \vec { x } ^ { + } \in S _ { \tau , x } ^ { + } ) ) } _ { \tau ^ { \prime } - \tau ^ { \prime } ( | \vec { x } ^ { + } | ) ^ { \tilde { \phi } } ( \vec { x } ^ { + } \in S _ { \tau , x } ^ { + } ) } \Big ) } & { } \\ - \displaystyle \frac { 1 } { | S ^ { + } | } \sum _ { \tau ^ { \prime } \in S ^ { + } } ^ { \vec { K } } \log ( \log ( \frac { e ^ { - i ( \tau ^ { + } \cdot \vec { \phi } ) } p | \vec { x } ^ { \top } \in S _ { \tau , y } ^ { + } } { e ^ { i ( \tau ^ { + } \cdot \vec { \phi } ) } + P _ { 0 } ( | \vec { x } ^ { + } | \vec { x } ^ { + } , \vec { x } ^ { + } \in S _ { \tau , x } ^ { + } ) } \\ + \underbrace \sum _ { \tau ^ { \prime } \in S ^ { + } } ^ \vec \end{array} +$$ + +Note now that $P _ { \mathcal { Q } } ( \vec { x } | \vec { x } ^ { + } , \vec { x } ^ { + } \in S _ { \kappa _ { i } } ^ { + } )$ is simply the probability of the record’s feature value $\vec { x } _ { f }$ under the second step noise distribution $\mathcal { Q } _ { 2 } ( \mathrm { X } _ { \mathrm { f } } , \alpha _ { 2 } )$ : $P _ { \mathcal { Q } } ( \vec { x } | \vec { x } ^ { + } , \vec { x } ^ { + } \in S _ { \kappa _ { i } } ^ { + } ) = P _ { \mathcal { Q } _ { 2 } } ( \vec { x } _ { f } )$ + +$$ +\begin{array} { r l } & { = \displaystyle \frac { 1 } { | S ^ { + } | } \displaystyle \sum _ { i = 1 } ^ { n } \sum _ { \bar { x } ^ { + } \in S _ { \star _ { i } } ^ { + } } \Big ( \log ( \frac { e ^ { s ( \bar { x } ^ { + } , \bar { \phi } ) } p ( \bar { x } ^ { + } \in S _ { \kappa _ { i } } ^ { + } ) } { e ^ { s ( \bar { x } ^ { + } , \bar { \phi } ) } + P _ { Q _ { 2 } } ( \vec { x } _ { \kappa _ { i } } ^ { + } ) } ) + \frac { k } { \bar { x } ^ { - } \sim Q ( \cdot | \vec { x } ^ { + } , i \in S _ { \star _ { i } } ^ { + } ) } \log ( \frac { P _ { Q _ { 2 } } ( \vec { x } _ { f } ^ { - } ) p ( \bar { x } ^ { + } \in S _ { \kappa _ { i } } ^ { + } ) } { e ^ { s ( \bar { x } ^ { - } , \bar { \phi } ) } + P _ { Q _ { 2 } } ( \vec { x } _ { f } ^ { - } ) ) } \Big ) } \\ & { = \displaystyle \frac { 1 } { | S ^ { + } | } \displaystyle \sum _ { i = 1 } ^ { n } \sum _ { \bar { x } ^ { + } \in S _ { \star _ { i } } ^ { + } } \Big ( \log ( \frac { e ^ { s ( \bar { x } ^ { + } , \bar { \phi } ) } } { e ^ { s ( \bar { x } ^ { + } , \bar { \phi } ) } + P _ { Q _ { 2 } } ( \vec { x } _ { \kappa _ { i } } ^ { + } ) } ) + \log ( p ( \bar { x } ^ { + } \in S _ { \kappa _ { i } } ^ { + } ) ^ { k + 1 } ) } \\ & { \quad + \left. \frac { k } { \bar { x } ^ { - } \sim Q ( \cdot | \vec { x } ^ { + } , \vec { x } ^ { + } \in S _ { \star _ { i } } ^ { + } ) } \log ( \frac { P _ { Q _ { 2 } } ( \vec { x } _ { f } ^ { - } ) } { e ^ { s ( \bar { x } ^ { - } , \bar { \phi } ) } + P _ { Q _ { 2 } } ( \vec { x } _ { f } ^ { - } ) } ) \right) } \end{array} +$$ + +We now drop the term containing the probability of assignment to a feature group $p ( \vec { x } ^ { + } \in S _ { \kappa _ { i } } ^ { + }$ ) since it is outside of the learned model parameters $\vec { \phi }$ and fixed in advance: + +$$ +\begin{array} { c } \displaystyle \propto \displaystyle \frac { 1 } { | S ^ { + } | } \displaystyle \sum _ { i = 1 } ^ { n } \sum _ { \vec { x } ^ { + } \in S _ { \mathbf { * } _ { i } } ^ { + } } \left( \log ( \frac { e ^ { s ( \vec { x } ^ { + } , \vec { \phi } ) } } { e ^ { s ( \vec { x } ^ { + } , \vec { \phi } ) } + P _ { Q _ { 2 } } ( \vec { x } _ { \mathbf { * } _ { i } } ^ { + } ) } ) + \displaystyle \sum _ { \vec { x } ^ { - } \sim Q ( \cdot , | \vec { x } ^ { + } , \vec { x } ^ { + } \in S _ { \mathbf { * } _ { i } } ^ { + } ) } ^ { k } \log ( \frac { P _ { Q _ { 2 } } ( \vec { x } _ { f } ^ { - } ) } { e ^ { s ( \vec { x } ^ { - } , \vec { \phi } ) } + P _ { Q _ { 2 } } ( \vec { x } _ { f } ^ { - } ) } ) \right) \nonumber _ { \vec { x } ^ { + } \sim \mathbb { S } _ { \mathbf { \ * } ^ { + } \sim \mathbb { S } _ { \mathbf { \ * } ^ { + } \sim \mathbb { S } _ { \mathbf { \ * } ^ { + } \sim \mathbb { S } _ { \mathbf { \ * } ^ { + } \sim \mathbb { S } _ { \mathbf { \ * } ^ { + } \sim \mathbb { S } _ { \mathbf { \ * } ^ { + } \sim \mathbb { S } _ { \mathbf { \ * } ^ { + } \sim \mathbb { S } _ { \mathbf { \ * } ^ { - } \mathbb { S } _ { \mathbf { \ * } ^ { - } \mathbb { S } _ { \mathbf { \ * } ^ { - } \mathbb { S } _ { \mathcal \delta } } } } } } } } } } } } \\ \displaystyle \xrightarrow [ { \vec { x } ^ { + } | \to \infty } ] { n } p ( \vec { x } ^ { + } \in S _ { \mathbf { \star } _ { i } } ^ { + } ) E \Big [ \log ( \frac { e ^ { s ( \vec { x } ^ { + } , \vec { \phi } ) } } { e ^ { s ( \vec { x } ^ { + } , \vec { \phi } ) } + P _ { Q _ { 2 } } ( \vec { x } _ { \mathbf { \star } _ { i } } ^ { + } ) } ) + \sum _ { \vec { x } ^ { - } \sim Q ( \cdot , | \vec { x } ^ { + } , \vec { x } ^ { + } \in S _ { \mathbf { \star } _ { i } } ^ { + } ) } ^ { k } \ \end{array} +$$ + +Thus, the loss function is just a convex combination of the loss functions of the targeted classifiers for each of the $p$ features, and by extension so is the gradient since: + +$$ +\frac { \partial } { \partial \phi } \sum _ { i = 1 } ^ { n } p ( \vec { x } ^ { + } \in S _ { \mathrm { \bf { k } } _ { i } } ^ { + } ) E \Big [ L ( \vec { x } | \vec { \phi } , \mathrm { t a r g e t } = f ) \Big ] = \sum _ { i = 1 } ^ { n } p ( \vec { x } ^ { + } \in S _ { \mathrm { \bf { k } } _ { i } } ^ { + } ) \frac { \partial } { \partial \phi } E \Big [ L ( \vec { x } | \vec { \phi } , \mathrm { t a r g e t } = f ) \Big ] +$$ + +Thus the algorithm will, at each step, learn a convex combination of the gradient for a targeted classifier on feature $f$ , with weights proportional to the feature group sampling probabilities in step 1 of + +the sampling algorithm. Note that if feature groups are not singletons, the gradient from unsupervised Feat2Vec will analogously be a convex combination of $n$ gradients learned from supervised learning tasks on each of the $n$ feature groups. □ + +# B FEATURE EXTRACTION NETWORK FOR NATURAL LANGUAGE + +![](images/71766fd860ac992bd9f8f7e92e427018a951fd40442b3983d9ec580cd9dd0aa9.jpg) +Figure A.4: Feature extraction network used for labelling tasks. We use $\mathrm { f } { = } 1 0 0 0$ convolutional filters each of width 3 (words) + +Here we describe the details of the feature extraction function $\phi$ used in our experiments for supervised tasks in $\ S 4 . 1$ . An overview of the network is given in Fig. A.4. We choose the most common words of each dataset to build a vocabulary of size $n$ , and convert the words of each document to a sequence of length $t$ of one-hot encodings of the input words. If the input text is shorter than $t$ , then we pad it with zeros; if the text is longer, we truncate it by discarding the trailing words. Therefore, for a vocabulary size $n$ , the input has dimensions $t \times n$ . These $t \times$ dimensional matrix is then passed through the following layers: + +1. We use an embedding layer to assign a $d$ -dimensional vector to each word in the input passage of text. This is done through a $d \times n$ -dimensional lookup table, which results in an $t \times d$ matrix. +2. We extract features from the embeddings with functions called convolutional filters (LeCun et al., 1998) (also called feature maps). A convolutional filter is simply a matrix learned from an input. We learn $f$ filters that are applied on groups of $m$ adjacent word embeddings, thus each of our filters is a $d \times m$ matrix of learned parameters. Filters are applied by computing the element-wise dot product of the filter along a sliding window of the entire input. The resulting output for each filter is a vector of length $t - m + 1$ . We also apply a ReLU activation to the output of each filter. +3. Consider the case of inputs of different lengths. For very short texts, the output of the filters will be mostly zero since the input is zero-padded. To enforce learning from the features of the text, and not just its length we apply a function called 1-max pooling to the output of the filters: from the $t - m + 1$ output vector of each filter, we select the maximum value. This yields a vector of length $F$ , a representation of the passage which is independent of its length. +4. We learn higher-level features from the convolutional filters. For this, we use a fully connected layer with $p$ units and a ReLU activation, +5. During training (not in inference), we prevent the units from co-adapting too much with a dropout layer (Srivastava et al., 2014). Dropout is a form of regularization that for each mini-batch randomly drops a specified percentage of units. +6. the final embedding for $x _ { j }$ (that is used in the factorization) is computed by a dense layer with $r$ output units and an activation function, where $r$ is the embedding size of our indexable items. + +We set the maximum vocabulary size $n$ to 100,000 words, and input embedding size $d$ to 50 for all experiments. We initialize the input word embeddings and the label embeddings using + +Word2Vec(Mikolov et al., 2013) We have have not evaluated multiple architectures or hyperparameter settings and obtain good results on diverse datasets with the same architecture, which was designed followed recommendations from a large scale evaluation of CNN hyper parameters(Zhang & Wallace, 2015). We set the number of convolutional filters $f$ to 1,000, and the dropout rate to 0.1. The maximum sequence length $t$ was chosen according to the typical document length (350 words for CiteULike and 250 for Yelp). For the CTR dataset, because we use very small values of $r$ , due to the tendency of the ReLU units to‘die’ during training (output zero for all examples), which can have a significant impact, we used instead PReLU activations (He et al., 2015) for the final layer, since they do not suffer from this issue. + +# B.1 FEATURE EXTRACTION FOR DEEPCONN COMPARISON + +The CNN architecture used for DeepCoNN (Zheng et al., 2017) is similar to the previous section. It consists of a word embedding lookup table, convolutional layer, 1-max pooling and a fully connected layer. We use the hyper-parameters that the authors report as best - 100 convolution filters and 50 units for the fully connected layer. We set the word embedding size to 100, the vocabulary size to 100,000 and the maximum document length to 250. + +# C HYPER-PARAMETERS FOR CTR + +To compare Feat2Vec with Collaborative Topic Regression, we choose the embedding size $r \in$ $\{ 5 , 1 0 , { \bar { 1 } } 5 \}$ for which CTR performs best. The results are show in Table A.2. + +Table A.2: Tuning embedding size for CTR + +
r=5r=10r=15Time (mins.)
Matrix Fact.0.87230.89110.90461
Feat2Vec0.90810.93030.9401133
C.T.R0.87630.92340.93561425
\ No newline at end of file diff --git a/parse/train/rkZzY-lCb/rkZzY-lCb_content_list.json b/parse/train/rkZzY-lCb/rkZzY-lCb_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..4bfeb32f2499e6ac0b58aa9a5611da0c48652b56 --- /dev/null +++ b/parse/train/rkZzY-lCb/rkZzY-lCb_content_list.json @@ -0,0 +1,1864 @@ +[ + { + "type": "text", + "text": "FEAT2VEC: DENSE VECTOR REPRESENTATION OF DATA WITH ARBITRARY FEATURES ", + "text_level": 1, + "bbox": [ + 176, + 98, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 236, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Methods that calculate dense vector representations for features in unstructured data—such as words in a document—have proven to be very successful for knowledge representation. We study how to estimate dense representations when multiple feature types exist within a dataset for supervised learning where explicit labels are available, as well as for unsupervised learning where there are no labels. Feat2Vec calculates embeddings for data with multiple feature types enforcing that all different feature types exist in a common space. In the supervised case, we show that our method has advantages over recently proposed methods; such as enabling higher prediction accuracy, and providing a way to avoid the cold-start problem. In the unsupervised case, our experiments suggest that Feat2Vec significantly outperforms existing algorithms that do not leverage the structure of the data. We believe that we are the first to propose a method for learning unsupervised embeddings that leverage the structure of multiple feature types. ", + "bbox": [ + 233, + 265, + 764, + 445 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 470, + 336, + 487 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Informally, in machine learning a dense representation, or embedding of a vector $\\vec { x } \\in \\mathbb { R } ^ { n }$ is another vector $\\vec { y } \\in \\mathbb { R } ^ { r }$ that has much lower dimensionality $( r \\ll n )$ than the original representation, and can be used to replace the original vector in downstream prediction tasks. Embeddings have multiple advantages, as they enable more efficient training (Mikolov et al., 2013), and unsupervised learning (Schnabel et al., 2015). For example, when applied to text, semantically similar words are mapped to nearby points. ", + "bbox": [ + 174, + 502, + 825, + 585 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We consider two kind of algorithms that use embeddings: ", + "bbox": [ + 176, + 593, + 550, + 607 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1. Unsupervised methods (sometimes referred as self-supervised methods) like Word2Vec (Mikolov et al., 2013), are designed to provide embeddings that are useful for a wide-array of predictions tasks. For example, the loss function of the continuous bag of words (CBOW) algorithm of Word2Vec is tuned to predict the next word of a sequence; however, in practice, the embeddings produced are mostly used for other tasks, such as analogy solving (Mikolov et al., 2013), or sentiment analysis (Le & Mikolov, 2014). In the context of this paper, we refer to the embeddings of an unsupervised method that can be used for a variety of auxiliary prediction tasks as general-purpose. \n2. Supervised methods, like matrix factorization, produce embeddings that are highly tuned to a prediction task. These embeddings may be interpretable but do not usually generalize to other tasks. We refer to these embeddings as task-specific. Matrix factorization and Word2Vec are unable to calculate embeddings for items that are not available during training (“cold-start” problem). While recent work using n-gram features (Bojanowski et al., 2016) have addressed this limitation for supervised and unsupervised tasks, it can only be used for a single feature type—words. ", + "bbox": [ + 212, + 618, + 825, + 830 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper we propose Feat2Vec as a novel method that allows calculating embeddings of arbitrary feature types from both supervised and unsupervised data. Our main contributions are: ", + "bbox": [ + 176, + 842, + 823, + 869 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "• Unsupervised Feat2Vec. Existing general-purpose dense representation methods are largely restricted to one or two feature types. For example, the Word2Vec methods can only calculate embeddings for words, while follow-up work has enabled embeddings for both words and documents (Le & Mikolov, 2014). To our knowledge, Feat2Vec is the first algorithm that is able to calculate general-purpose embeddings that are not tuned for a single specific prediction task for arbitrary feature types. ", + "bbox": [ + 217, + 882, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 230, + 103, + 823, + 146 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• Supervised Feat2Vec. Task-specific methods can use arbitrary feature types, but are restricted in that embeddings must be calculated for each individual feature, while sometimes higher-level of abstractions may be desirable—for example, we may want to have embeddings of documents instead of simply words. This capability makes Supervised Feat2Vec extremely flexible. We demonstrate that our method can be used to calculate embeddings of unseen (cold-start) items when there is an alternative textual description. ", + "bbox": [ + 217, + 150, + 823, + 234 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 253, + 339, + 270 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Factorization Machine (Rendle, 2010) is one of the most successful methods for general-purpose factorization. Rendle (2010) formulated it as an extension to polynomial regression. Consider a degree-2 polynomial (quadratic) regression, where we want to predict a target variable $y$ from a vector of inputs ${ \\vec { x } } \\in \\mathbb { R } ^ { n }$ : ", + "bbox": [ + 174, + 285, + 825, + 342 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/5d449542272befb568a06a6ca5956851cf689454caa8c562b817e3a274ede6b1.jpg", + "text": "$$\n\\hat { y } ( \\vec { x } ; \\vec { b } , \\vec { w } ) = \\omega \\big ( b _ { 0 } + \\sum _ { i } b _ { i } x _ { i } + \\sum _ { i = 1 } ^ { n } \\sum _ { j = i + 1 } ^ { n } w _ { i , j } \\ x _ { i } x _ { j } \\big )\n$$", + "text_format": "latex", + "bbox": [ + 323, + 348, + 674, + 391 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In words, $n$ is the total number of features, the term $b _ { 0 }$ is an intercept, $b _ { i }$ is the strength of the $i$ -th feature, and $w _ { i , j }$ is the interaction coefficient between the $i$ -th and $j$ -th feature. The function $\\omega$ is an activation. Choices for $\\omega$ include a linear link $\\omega ( x ) = x ,$ ) for continuous outputs, or a logistic link $\\begin{array} { r } { ( \\omega ( x ) = \\frac { \\exp ( x ) } { \\exp ( x ) + 1 } ) } \\end{array}$ for binary outputs. ", + "bbox": [ + 173, + 397, + 825, + 458 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Factorization Machine replaces the two-way individual pairwise parameters $w _ { i , j }$ for each interaction with a vector of parameters $\\vec { w } _ { i }$ for each feature. This is a rank- $r$ vector of latent factors—embeddings in the neural literature—that encode the interaction between features and replaces the quadratic regression model with the following: ", + "bbox": [ + 173, + 465, + 825, + 522 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/85e8e6df74ac2041ee0db844f4cc34f6bfbafe47ded21eb97d35582c8fd0bd42.jpg", + "text": "$$\n{ \\hat { y } } ( { \\vec { x } } ; { \\vec { b } } , { \\vec { w } } ) = \\omega { \\big ( } b _ { 0 } + \\sum _ { i } b _ { i } x _ { i } + \\sum _ { i = 1 } ^ { n } \\sum _ { j = i + 1 } ^ { n } ( x _ { i } { \\vec { w _ { i } } } ) \\cdot ( x _ { j } { \\vec { w _ { j } } } ) { \\big ) }\n$$", + "text_format": "latex", + "bbox": [ + 303, + 529, + 694, + 571 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Intuitively, the dot product $( \\cdot )$ returns a scalar that measures the (dis)similarity between the latent factors of features $x _ { i }$ and $x _ { j }$ . Polynomial regression has $n ^ { 2 }$ interaction parameters, and Factorization Machine has $n \\times r$ . While setting $r \\ll n$ makes the model less expressive, factorization will typically exploit features having some shared latent structure. Factorization Machine may dramatically reduce the number of parameters to estimate. Rendle (2010) shows that when the feature vector x consists only of two categorical features in one-hot encoding, Factorization Machine is equivalent to the popular Matrix Factorization algorithm (Koren et al., 2009). ", + "bbox": [ + 173, + 577, + 825, + 676 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 FEAT2VEC ", + "text_level": 1, + "bbox": [ + 176, + 695, + 297, + 712 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We now describe how Feat2Vec extends the Factorization Machine model by allowing grouping of features, and enabling arbitrary feature extraction functions $( \\ S \\ 3 . 1 )$ . We also report a supervised method to learning Feat2Vec (§ 3.2), as well as a novel unsupervised training procedure (§ 3.3). ", + "bbox": [ + 176, + 727, + 825, + 770 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3.1 MODEL ", + "text_level": 1, + "bbox": [ + 174, + 786, + 266, + 800 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We propose a framework for extending factorization machine with neural methods, by introducing structure into the feature interactions. Specifically, we do this by defining feature groups, \\~κ, where each group contains features of a particular type. Explicitly, $\\dot { \\vec { \\kappa } }$ is a partition of the set of feature columns in a dataset and each set within the partition is a feature group. The embeddings of a feature group are then learned via a feature extraction function, $\\phi _ { i }$ , defined for each feature group. Feat2Vec will then extract features from each feature group, and build $r$ latent factors from them. In Factorization Machine, all the feature embeddings interact with each other, while in Feat2Vec, the interactions only occur between different feature groups. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Formally, the addition of deep extraction methods yields the following statistical model: ", + "bbox": [ + 174, + 103, + 750, + 119 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/0e8e8a72a7cad74c4a1ab4e9f65f22867eacfe8dedfda3618ea358bcc780a61b.jpg", + "text": "$$\n{ \\hat { y } } ( { \\vec { x } } , { \\vec { b } } , { \\vec { \\phi } } ) = \\omega \\bigg ( b _ { 0 } + \\sum _ { i = 1 } ^ { n } b _ { i } x _ { i } ~ + ~ \\sum _ { i = 1 } ^ { | { \\vec { \\kappa } } | } \\sum _ { j = i } ^ { | { \\vec { \\kappa } } | } \\phi _ { i } ( { \\vec { x } } _ { { \\vec { \\kappa } } _ { i } } ) \\cdot \\phi _ { j } ( { \\vec { x } } _ { { \\vec { \\kappa } } _ { j } } ) \\bigg )\n$$", + "text_format": "latex", + "bbox": [ + 294, + 143, + 702, + 190 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this notation, $\\vec { x } _ { \\vec { \\kappa } _ { i } }$ is a subvector that contains all of the features that belong to the group ${ \\vec { \\kappa } } _ { i }$ . Thus, $x _ { \\vec { \\kappa } _ { i } } = [ x _ { j } : j \\in \\vec { \\kappa } _ { i } ]$ . The intuition is that by grouping (sub-)features as a single entity, we can can reason on a higher level of abstraction. Instead of individual sub-features interacting among each other, the embeddings of feature groups interact with those of other groups. $\\phi _ { i }$ is a feature extraction that inputs the $i$ -th feature group of the instance, and returns an $r$ -dimensional embedding. The feature extraction function $\\phi _ { i }$ can allow for an arbitrary processing of its subfeatures. Across groups, entities interact with each other via the output of $\\phi$ only. ", + "bbox": [ + 173, + 196, + 825, + 296 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As a concrete example of an application of this grouping/feature extraction, we might group the individual words of a document into a “document” feature group, and allow this document embedding to then interact with learned embeddings of other document metadata (such as author id). We might expect the extraction function $\\phi$ for the words in a document to extract features that characterize the attributes of the document taken as a whole, rather than simply the sum of its individual words. ", + "bbox": [ + 174, + 301, + 825, + 372 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Figure 1 compares existing factorization methods with our novel model. In this example, Feat2Vec is using two feature groups: the first group only has a single feature which is projected to an embedding (just like a regular Factorization Machine); the second group has multiple features, which are together projected to a single embedding. ", + "bbox": [ + 174, + 378, + 825, + 435 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/88cf493d4cacf4a87c6bcf9e8dec89a2e33aab3199b3a3b0c39e17d487871d57.jpg", + "image_caption": [ + "Figure 1: Network architectures for factorization models. The white clouds $( \\bigcirc )$ represent deep layers, for example a convolutional network for text features. " + ], + "image_footnote": [], + "bbox": [ + 250, + 448, + 823, + 628 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The simplest implementation for $\\phi _ { i }$ is a linear fully-connected layer, where the output of the $r$ -th entry is: ", + "bbox": [ + 174, + 684, + 821, + 712 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/9fbe22c7691890b6da1c28c35c459f75d3c6dd6cd2fada21fc0776d27ef137b1.jpg", + "text": "$$\n\\phi _ { i } \\big ( \\vec { x } _ { i } ; \\vec { w } \\big ) _ { r } = \\sum _ { a = 1 } ^ { d _ { i } } w _ { r _ { a } } x _ { i _ { a } }\n$$", + "text_format": "latex", + "bbox": [ + 413, + 710, + 583, + 753 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Note that without loss of generality, we could define a model that is equivalent to a shallow Factorization Machine by allowing each feature group to be a singleton : $\\vec { \\kappa } \\overset { \\cdot } { = } \\{ \\{ x _ { 1 } \\} , \\{ x _ { 2 } \\} \\ldots \\{ x _ { n } \\} \\}$ and the linear extraction function presented in Equation 4. ", + "bbox": [ + 174, + 762, + 825, + 806 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We can use Feat2Vec to both use large feature sets and overcome the cold-start problem. This is only possible when there is an alternative description of the item available (for example an image or a passage of text). In Figure 2, we show how we address this problem by treating the words as indexed features, but placed within a structured feature group $\\kappa _ { w }$ , the group of word features. A feature extraction function $\\phi$ acts on the features in $\\kappa _ { w }$ , and the other features interact with the words only via the output of $\\phi$ . Notice that this implies we can precompute and store the latent factors of the target task seen during training, so that predictions during inference can be sped-up. For example if we have two feature groups (e.g, a label and an item), first we compute the feature extraction function to the unseen items and their embeddings, and then we simply apply a dot product over the stored vectors of the labels. ", + "bbox": [ + 173, + 811, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/cc57b1066f59d22b82314167436fc6f203035c20cf16e8f70ec5112b69514f54.jpg", + "image_caption": [ + "Figure 2: Comparison of how factorization may use item descriptions features. " + ], + "image_footnote": [], + "bbox": [ + 359, + 147, + 642, + 234 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Figure 1c shows an approach of using neural networks within factorization machines that has been proposed multiple times (Dziugaite & Roy, 2015; Guo et al., 2017). It replaces the dot product of factors with a learned neural function, which has been shown to improve predictive accuracy for various tasks. In this case, fast inference for cold-start documents using pre-computed label embeddings is no longer possible. It needs to store the entire neural function that takes the embeddings as inputs. Another shortcoming of replacing the dot product with a neural function is that it would no longer be possible to interpret the embeddings as containing latent factors related to the target task; There may be highly complex mappings from the embeddings to the final output via this neural function. However, it would be straightforward to combine this approach with Feat2Vec. This is not explored in this work. ", + "bbox": [ + 173, + 280, + 825, + 419 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 SUPERVISED LEARNING FROM DATA ", + "text_level": 1, + "bbox": [ + 178, + 438, + 467, + 452 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We can learn the the parameters of a deep factorization model θ using training data by minimizing a loss function $\\mathcal { L }$ : ", + "bbox": [ + 173, + 464, + 823, + 491 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/406c04da6215b8c7fc0065eccd99ccfd7f79e398bc375c2f6333f01ccf72edfe.jpg", + "text": "$$\n\\arg \\operatorname* { m i n } _ { \\vec { \\Theta } } \\sum _ { x } \\mathcal { L } \\big ( y ( x ) , \\hat { y } ( x ; \\vec { \\theta } ) \\big ) + \\gamma | | \\Theta | | ^ { w }\n$$", + "text_format": "latex", + "bbox": [ + 366, + 489, + 632, + 523 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here, $y ( x )$ is the true target value for $x$ obtained from training data, and ${ \\hat { y } } ( x )$ is the one estimated by the model; the hyperparameter $\\gamma$ controls the amount of regularization. For the labeling and classification tasks, we optimize the binary cross-entropy for $y \\in \\{ 0 , 1 \\}$ : ", + "bbox": [ + 174, + 529, + 825, + 571 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/04399e73ffe3e26293c5a76d183a6510348dfdc3f5fcabdbf46132396ce3d7c6.jpg", + "text": "$$\n\\mathcal { L } ( y , \\hat { y } ) = - \\big ( y \\log ( \\hat { y } ) \\big ) - ( 1 - y ) \\log ( 1 - \\hat { y } ) \\big )\n$$", + "text_format": "latex", + "bbox": [ + 344, + 579, + 653, + 599 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For the regression tasks where the target value is continuous, we optimize the mean squared error (MSE): ", + "bbox": [ + 171, + 606, + 821, + 633 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f188263c38345d7fdaf07f6772407658afffd9e4f4edd29834257467b918a1bb.jpg", + "text": "$$\n\\mathcal { L } ( y , \\hat { y } ) = ( y - \\hat { y } ) ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 433, + 635, + 563, + 654 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Neural models are typically learned using mini-batch updates, where the incremental descent is performed with respect to several instances at a time. For the implementation of this paper, we built our models using the Keras programming toolkit (Chollet et al., 2015), that is now part of Tensorflow (Abadi et al., 2015). It enables automatic differentiation, and is bundled with a generalpurpose optimization algorithm called ADAM (Kingma & Ba, 2014) that needs no tuning of gradient step sizes. ", + "bbox": [ + 173, + 665, + 825, + 750 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "It is straightforward to optimize Equation 5 directly for multiclass or binary classification. However, when the number of labels is very large, it is common practice use a binary classifier and sample the negative examples (Dyer, 2014). For the multi-label classification tasks, we use Feat2Vec with a binary output. In this case we would have at least two feature groups—one of the feature groups is the label that we want to predict, and the other group(s) is the input from which we want to make the prediction. The output indicates whether the label is associated with the input $( y = + 1 )$ ), or not $( y = 0 )$ ). The datasets we use for our labeling experiments only contains positive labels, thus for each training example we sample a set of negative labels equal to the number of positive labels. It is typical to use one of the following sampling strategies according to the best validation error, in each case excluding the actual positive labels for each training example – (i) uniformly from all possible labels, or (ii) from the empirical distributions of positive labels. Other sampling strategies have been proposed (Rendle et al., 2009; Rendle & Freudenthaler, 2014). ", + "bbox": [ + 173, + 756, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 UNSUPERVISED LEARNING FROM DATA ", + "text_level": 1, + "bbox": [ + 176, + 103, + 490, + 117 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now discuss how Feat2Vec can be used to learn embeddings in an unsupervised setting with no explicit target for prediction. ", + "bbox": [ + 173, + 130, + 823, + 159 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The training dataset for a Feat2Vec model consists of only the observed data. In natural language, these would be documents written by humans. Since Feat2Vec (Equation 3) requires positive and negative examples, we also need to supply unobserved data as negative examples. Consider a feature group $\\vec { \\mathsf { K } } _ { i }$ , that exists in very high dimensional space. For example, this could happen because we are modeling with one-hot encoding a categorical variable with large number of possible values. In such scenario, it is overwhelmingly costly to feed the model all negative labels, particularly if the model is fairly sparse. ", + "bbox": [ + 174, + 165, + 825, + 262 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "A shortcut around this is a concept known as implicit sampling, where instead of using all of the possible negative labels, one simply samples a fixed number $( k )$ from the set of possible negative labels for each positively labelled record. Word2Vec makes use of an algorithm called Negative Sampling, that has little theoretical guarantees (Dyer, 2014). In short, their approach samples a negative observation from a noise distribution $\\mathcal { Q } _ { w 2 v }$ , that is proportional to the empirical frequency of a word in the training data. ", + "bbox": [ + 174, + 270, + 825, + 353 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We introduce a new implicit sampling method that enables learning unsupervised embeddings for structured feature sets. We can learn the correlation of features within a dataset by imputing negative labels, simply by generating unobserved records as our negative samples. Unlike Word2Vec, we do not constraint features types to be words. Features groups can be individual columns in a data matrix, but they need not to be. By grouping subfeatures using the parameter $\\boldsymbol { \\mathsf { K } }$ in Equation 3, the model can reason on more abstract entities in the data. By entity, we mean a particular feature group value. For example, in our experiments on a movie dataset, we use a “genre” feature group, where we group non-mutually exclusive indicators for movie genres including comedy, action, and drama films. ", + "bbox": [ + 174, + 359, + 825, + 484 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We start with a dataset $S ^ { + }$ of records with |\\~κ| feature groups. We then mark all observed records in the training set as positive examples. For each positive record, we generate $k$ negative labels using the following 2-step algorithm: ", + "bbox": [ + 176, + 492, + 825, + 534 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1 Implicit sampling algorithm for unsupervised Feat2Vec: $\\mathcal { Q }$ ", + "text_level": 1, + "bbox": [ + 176, + 549, + 653, + 565 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1: function FEAT2VEC SAMPLE(S+, k, α1, α2) \n2: S − ← ∅ \n3: for \\~x + ∈ S+ do \n4: Draw a random feature group $\\kappa _ { i } \\sim \\mathcal { Q } _ { 1 } ( \\{ \\mathrm { p a r a m s } ( \\phi _ { i } ) \\} _ { i = 1 } ^ { | \\vec { \\kappa } | } , \\alpha _ { 1 } )$ \n5: for $j \\in \\{ 1 , \\ldots , k \\}$ do \n6: \\~x − ← \\~x + $\\triangleright$ set initially to be equal to the positive sample \n7: Draw a random feature group value $\\tilde { x } \\sim \\mathcal { Q } _ { 2 } ( \\mathrm { X } _ { \\kappa _ { i } , \\alpha _ { 2 } } )$ \n8: $\\begin{array} { l } { { { \\vec { x } _ { \\kappa _ { i } } ^ { - } \\tilde { x } } } } \\\\ { { { S ^ { - } S ^ { - } + \\{ \\vec { x } ^ { - } \\} } } } \\end{array}$ $\\triangleright$ substitute the $i$ -th feature type with the sampled one \n9: \n10: end for \n11: end for \n12: return $S ^ { - }$ \n13: end function ", + "bbox": [ + 178, + 570, + 825, + 741 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Explained in words, our negative sampling method for unsupervised learning iterates over all of the observations of the training dataset. For each observation ${ \\bar { x } } ^ { + }$ , it randomly selects the $i$ -th feature group from a noise distribution $\\mathcal { Q } _ { 1 } ( \\cdot )$ . Then, it creates a negative observation that is identical to ${ \\vec { x } } ^ { + }$ , except that its $i$ -th feature group is replaced by a value sampled from a noise distribution $\\mathcal { Q } _ { 2 } ( \\cdot )$ . In our application, we use the same class of noise distributions (flattened multinomial) for both levels of sampling, but this need not necessarily be the case. ", + "bbox": [ + 174, + 756, + 825, + 840 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now describe the two noise distributions that we use. We use $P _ { \\mathcal { Q } } ( x )$ to denote the probability of $x$ under a distribution $\\mathcal { Q }$ . ", + "bbox": [ + 174, + 845, + 823, + 875 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Sampling Feature Groups. The function params calculates the complexity of a feature extraction function $\\phi _ { i }$ . To sample a feature group, we choose a feature group $\\kappa _ { i }$ from a multinomial distribution with probabilities proportional a feature’s complexity. By complexity, we mean the number of parameters we need to learn that are associated with a particular feature group. This choice places more weight on features that have more parameters and thus are going to require more training iterations to properly learn. The sampling probabilities of each feature group are: ", + "bbox": [ + 176, + 881, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 147 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/7295902368ddd593c25b39729ce1c7042346c7966c038b67f7f65c47147c14eb.jpg", + "text": "$$\nP _ { \\mathcal { Q } _ { 1 } } ( \\kappa _ { i } | \\operatorname { p a r a m s } ( \\phi _ { i } ) \\} _ { i = 1 } ^ { | \\sharp | } , \\alpha _ { 1 } ) = \\frac { \\operatorname { p a r a m s } ( \\phi _ { i } ) ^ { \\alpha _ { 1 } } } { \\sum _ { j = 1 } ^ { | \\sharp | } \\operatorname { p a r a m s } ( \\phi _ { j } ) ^ { \\alpha _ { 1 } } } , \\quad \\alpha _ { 1 } \\in [ 0 , 1 ]\n$$", + "text_format": "latex", + "bbox": [ + 266, + 165, + 732, + 204 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For categorical variables using a linear fully-connected layer, the complexity is simply proportional to the number of categories in the feature group. However, if we have multiple intermediate layers for some feature extraction functions (e.g., convolutional layers), these parameters should also be counted towards a feature group’s complexity. The hyper-parameter $\\alpha _ { 1 }$ helps flatten the distribution. When $\\alpha _ { 1 } = 0$ , the feature groups are sampled uniformly, and when $\\alpha _ { 1 } = 1$ , they are sampled proportional to their complexity. Figure A.1 in the Appendix provides a visualization of how the feature sampling rate varies with the hyperparameter for features with differing levels of complexity. ", + "bbox": [ + 173, + 208, + 825, + 308 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Sampling Feature Group Values. To sample a value from within a feature groups $\\kappa _ { i }$ , we use a similar strategy to Word2Vec and use the empirical distribution of values: ", + "bbox": [ + 173, + 313, + 823, + 342 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/f68f5ae3e7bc11909d0f84398bdb7ef9d95bbc5d429d34dc3c591343ac9a05c9.jpg", + "text": "$$\nP _ { \\mathcal Q _ { 2 } } ( x | \\mathrm X _ { \\kappa _ { i } } , \\alpha _ { 2 } ) = \\frac { \\mathrm { c o u n t } ( x ) ^ { \\alpha _ { 2 } } } { \\sum _ { x _ { \\kappa _ { i } } ^ { \\prime } \\in S ^ { + } } \\mathrm { c o u n t } ( x _ { \\kappa _ { i } } ^ { \\prime } ) ^ { \\alpha _ { 2 } } } , \\quad \\alpha _ { 2 } \\in [ 0 , 1 ]\n$$", + "text_format": "latex", + "bbox": [ + 302, + 349, + 696, + 388 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Here, $\\operatorname { c o u n t } ( x )$ is the number of times a feature group value $x$ appeared in the training dataset $S ^ { + }$ , and $\\alpha _ { 2 }$ is again a flattening hyperparameter. ", + "bbox": [ + 174, + 398, + 825, + 426 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This method will sometimes by chance generate negatively labeled samples that do exist in our sample of observed records. The literature offers two possibilities: in the Negative Sampling that Word2Vec follows, the duplicate negative samples are simply ignored (Dyer, 2014). Alternatively, it is possible to account for the probability of random negative labels that are identical to positively labeled data using Noise Contrastive Estimation (NCE) (Gutmann & Hyvarinen, 2010). ¨ ", + "bbox": [ + 173, + 433, + 825, + 503 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3.1 THE LOSS FUNCTION FOR UNSUPERVISED LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 520, + 602, + 535 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For our unsupervised learning of embeddings, we optimize a NCE loss function, to adjust the structural statistical model $\\hat { y } = p ( y = 1 | \\vec { x } , \\vec { \\phi } , \\Theta )$ , expressed in Equation 3 to account for the possibility of random negative labels that appear identical to positively labeled data. θ here represents the parameters learned in during training (i.e. the $b _ { i }$ terms and parameters associated with the extraction functions $\\phi _ { i }$ in Equation 3). Since we only deal with a dichotomous label, indicating a positive or negative sample, for unsupervised learning, we restrict our attention to usage of Equation 3 with $\\omega$ as a logistic link function. ", + "bbox": [ + 173, + 545, + 825, + 645 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "An additional burden of NCE is that we need to calculate a partition function $Z _ { \\vec { x } }$ for each unique record type $\\vec { x }$ in the data that transforms the probability $\\hat { y }$ of a positive or negative label into a wellbehaved distribution that integrates to 1. Normally, this would introduce an astronomical amount of computation and greatly increase the complexity of the model. As a work-around, we appeal to the work of Mnih & Teh (2012), who showed that in the context of language models that setting the $Z _ { \\vec { x } } = 1$ in advance effectively does not change the performance of the model. The intuition is that if the underlying model has enough free parameters that it will effectively learn the probabilities itself. Thus, it does not over/under predict the probabilities on average (since that will result in penalties on the loss function). ", + "bbox": [ + 173, + 651, + 825, + 779 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Written explicitly, the new structural probability model is: ", + "bbox": [ + 176, + 784, + 555, + 799 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/709090e493f8a8e10784f32fcaccee8c8d8a3f4f5065964f9d698d881bd0cc09.jpg", + "text": "$$\n\\tilde { p } ( Y = 1 | \\vec { x } , \\vec { \\phi } , \\mathbf { \\Theta } \\Theta ) = \\frac { \\exp \\bigl ( s ( \\vec { x } , \\vec { \\phi } , \\mathbf { \\Theta } \\Theta ) \\bigr ) } { \\exp ( s ( \\vec { x } , \\vec { \\phi } , \\mathbf { \\Theta } \\Theta ) \\bigr ) + P _ { \\mathcal { Q } } ( \\vec { x } | \\alpha _ { 1 } , \\alpha _ { 2 } ) }\n$$", + "text_format": "latex", + "bbox": [ + 318, + 806, + 679, + 848 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $s ( . )$ denotes the score of a record $\\vec { x }$ given parameter values/extraction functions: ", + "bbox": [ + 173, + 856, + 743, + 872 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/2fc29c9e7cd3d6acf17d19eb6b8d6ac4cc6ae573a92b49f97976bbabe0e60937.jpg", + "text": "$$\ns ( \\vec { x } , \\vec { \\phi } , \\Theta ) = b _ { 0 } + \\sum _ { i = 1 } ^ { n } b _ { i } x _ { i } + \\sum _ { i = 1 } ^ { | \\vec { \\kappa } | } \\sum _ { j = i } ^ { | \\vec { \\kappa } | } \\phi _ { i } ( \\vec { x } _ { \\vec { \\kappa } _ { i } } ) \\cdot \\phi _ { j } ( \\vec { x } _ { \\vec { \\kappa } _ { j } } )\n$$", + "text_format": "latex", + "bbox": [ + 310, + 881, + 687, + 926 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "and $P _ { \\mathcal { Q } } ( . )$ denotes the total probability of a record $\\vec { x _ { i } }$ being drawn from our negative sampling algorithm, conditional on the positively labeled record ${ \\vec { x } } ^ { + }$ the negative sample is drawn for: ", + "bbox": [ + 169, + 103, + 823, + 132 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/a54fa3367147a9a92842f28b770d10bc2b45c7003f3c41882d8b50211391d00b.jpg", + "text": "$$\nP _ { \\mathcal { Q } } ( \\vec { x } | \\alpha _ { 1 } , \\alpha _ { 2 } , \\mathrm { X } , \\vec { x } ^ { + } ) = P _ { \\mathcal { Q } _ { 2 } } ( \\vec { x } _ { \\bf { \\kappa } _ { i } } | \\mathrm { X } _ { \\bf { \\kappa } _ { i } } , \\alpha _ { 2 } ) P _ { \\mathcal { Q } _ { 1 } } ( \\kappa _ { i } | \\mathrm { p a r a m s } ( \\phi _ { i } ) \\rbrace _ { i = 1 } ^ { n } , \\alpha _ { 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 258, + 140, + 740, + 159 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Our loss function $L$ optimizes $\\theta$ , the parameters of the feature extraction functions $\\vec { \\phi }$ , while accounting for the probability of negative samples. ", + "bbox": [ + 173, + 176, + 825, + 205 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/2f663c4b47c3a3aca685b1724182b3712b6b8926249923ea3d7c65b83d4aeae5.jpg", + "text": "$$\nL ( S ) = \\arg \\operatorname* { m i n } _ { \\Theta } \\frac { 1 } { | S ^ { + } | } \\sum _ { \\vec { x } ^ { + } \\in S ^ { + } } \\Big ( \\log ( \\tilde { p } ( y = 1 | \\vec { x } ^ { + } , \\vec { \\phi } , \\Theta ) ) \\ + \\sum _ { \\vec { x } ^ { - } \\sim \\mathcal { Q } ( \\cdot | \\vec { x } ^ { + } ) } ^ { k } \\log ( \\tilde { p } ( y = 0 | \\vec { x } ^ { - } , \\vec { \\phi } , \\Theta ) ) \\Big )\n$$", + "text_format": "latex", + "bbox": [ + 179, + 213, + 813, + 260 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Feat2Vec has interesting theoretical properties. For example, it is well known that Factorization Machines can be used as a multi-label classifier: with at least two features, one can use one of the feature as the target label, and the other as the input feature to make a prediction. In such setting, the output indicates whether the label is associated with the input $( y = + 1 )$ ), or not $( y = 0 )$ ), and therefore the input can be associated with more than one label. With $n$ feature types, Feat2Vec is equivalent to optimizing a convex combination of the loss functions from $n$ individual Factorization Machines. In other words, it optimizes $n$ multi-label classifiers, where each classifier is optimized for a different target (i.e.,a specific feature group). We show the proof of this in the Appendix 1. ", + "bbox": [ + 173, + 279, + 825, + 390 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 EMPIRICAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 412, + 382, + 429 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 SUPERVISED EMBEDDINGS ", + "text_level": 1, + "bbox": [ + 176, + 444, + 401, + 459 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We now address our working hypotheses for evaluating supervised embeddings. For all our experiments we define a development set and a single test set which is $10 \\%$ of the dataset, and a part of the development set is used for early stopping or validating hyper-parameters. Since these datasets are large and require significant time to train on an Nvidia K80 GPU cluster, we report results on only a single training-test split. For the multi-label classification task in 4.1.1 we predict a probability for each document-label pair and use an evaluation metric called Area Under the Curve (AUC) of the Receiver Operating Characteristic (ROC). Since we only observe positive labels, for each positive label in the test set we sample negative labels according to the label frequency. This ensures that if a model merely predicts the labels according to their popularity, it would have an AUC of 0.5. A caveat of our evaluation strategy is that we could be underestimating the performance of our models—there is a small probability that the sampled negatives labels are false negatives. However, since we apply the same evaluation strategy consistently across our methods and baselines, the relative difference of the AUC is meaningful. We choose the AUC as a metric because it is popular for both classification and ranking problems. For the regression task in 4.1.2, we use mean squared error (MSE) as the evaluation metric. In preliminary experiments we noticed that regularization slows down convergence with no gains in prediction accuracy, so we avoid overfitting only by using early stopping. We share most of the code for the experiments online1 for reproducibility. ", + "bbox": [ + 174, + 472, + 825, + 707 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For our feature extraction function $\\phi$ for text, we use a Convolutional Neural Network (CNN) that has been shown to be effective for natural language tasks (Kalchbrenner et al., 2014; Weston et al., 2014). In Appendix B we describe this network and its hyper-parameters. Instead of tuning the hyper-parameters, we follow previously published guidelines (Zhang & Wallace, 2015). ", + "bbox": [ + 174, + 714, + 825, + 770 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1.1 IS Feat2Vec EFFECTIVE FOR COLD-START PREDICTIONS? ", + "text_level": 1, + "bbox": [ + 173, + 787, + 624, + 801 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We compare Feat2Vec with an extension of matrix factorization that can generalize to unseen items for text documents, Collaborative Topic Regression (CTR– Wang & Blei (2011)), a method with an open-source Python implementation2. We evaluate them on the CiteULike dataset which consists of pairs of scientific articles and the users who have added them to their personal libraries, and it contains 16,980 unique articles and 5,551 unique users. We use the models to predict users who may have added a given article to their library. We compare the performance of Feat2Vec with CTR using pre-defined cross-validation splits3. We use $1 \\%$ of the training set for early stopping. ", + "bbox": [ + 174, + 813, + 825, + 882 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/e15d23db6da66d622d6e2cb01403ebca6efe46c783462b3c56ffade7ab1be232.jpg", + "table_caption": [ + "Table 1: Yelp rating prediction " + ], + "table_footnote": [], + "table_body": "
MSEImprovement over Matrix Factorization
Matrix Factorization1.561=
Feat2Vec0.48069.2 %
DeepCoNN1.44119.6 %
", + "bbox": [ + 243, + 126, + 754, + 199 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 224, + 823, + 252 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For CTR we use the hyper-parameters reported by the authors as best, except for $r$ which we found had a significant impact on training time . We only consider $r \\in \\{ 5 , 1 0 , 1 5 \\}$ and choose the value which gives the best performance for CTR (details in Appendix A.2). On the warm-start condition, CTR has an AUC of 0.9356; however, it shows significant degradation in performance for unseen documents and it only performs slightly better than random chance with an AUC of 0.5047. On the other hand, Feat2Vec achieves AUC of 0.9401 on the warm-start condition, and it only degrades to 0.9124 on unseen documents. Feat2Vec can also be trained over ten times faster, since it can leverage GPUs.4 We also note that we have not tuned the architecture or hyper-parameters of the feature extraction function $\\phi$ and greater improvements are possible by optimizing them. ", + "bbox": [ + 174, + 260, + 825, + 386 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.1.2 COMPARISON WITH ALTERNATIVE CNN-BASED TEXT FACTORIZATION ", + "text_level": 1, + "bbox": [ + 176, + 401, + 714, + 415 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We now compare with a method called DeepCoNN, a deep network specifically designed for incorporating text into matrix factorization (Zheng et al., 2017)—which reportedly, is the state of the art for predicting customer ratings when textual reviews are available. For Feat2Vec we use the same feature extraction function (see Appendix B.1 for details) used by DeepCoNN. We evaluate on the Yelp dataset5, which consists of 4.7 million reviews of restaurants. For each user-item pair, DeepCoNN concatenates the text from all reviews for that item and all reviews by that user. The concatenated text is fed into a feature extraction function followed by a factorization machine. In contrast, for Feat2Vec, we build 3 feature groups: item identifiers (in this case, restaurants), users and review text. ", + "bbox": [ + 173, + 425, + 825, + 550 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 1 compares our methods to DeepCoNN’s published results because a public implementation is not available. We see that Feat2Vec provides a large performance increase when comparing the reported improvement, over Matrix Factorization, of the mean squared error. Our approach is more general, and we claim that it is also more efficient. Since DeepCoNN concatenates text, when the average reviews per user is $\\bar { n _ { u } }$ and reviews per item is $\\bar { n _ { i } }$ , each text is duplicated on average $\\bar { n _ { i } } \\times \\bar { n _ { u } }$ times per training epoch. In contrast, for Feat2Vec each review is seen only once per epoch. Thus it can be 1-2 orders of magnitude more efficient for datasets where $\\bar { n _ { i } } \\times \\bar { n _ { u } }$ is large. ", + "bbox": [ + 174, + 558, + 825, + 655 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 GENERAL-PURPOSE EMBEDDINGS ", + "text_level": 1, + "bbox": [ + 176, + 672, + 450, + 685 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2.1 DOES Feat2Vec ENABLE BETTER EMBEDDINGS? ", + "text_level": 1, + "bbox": [ + 174, + 698, + 562, + 712 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Ex ante, it is unclear to us how to evaluate the performance of an unsupervised embedding algorithm, but we felt that a reasonable task would be a ranking task one might practically attempt using our datasets. This task will assess the similarity of trained embeddings using unseen records in a left-out dataset. In order to test the relative performance of our learned embeddings, we train our unsupervised Feat2Vec algorithm and compare its performance in a targeted ranking task to Word2Vec’s CBOW algorithm for learning embeddings. In our evaluation approach, we compare the cosine similarity of the embeddings of two entities where these entities are known to be associated with each other since they appear in the same observation in a test dataset. In particular, in the movie dataset we compare the similarity of movie directors to those of actors who were cast in the same film for a left-out set of films. For our educational dataset, we compare rankings of textbooks by evaluating the similarity of textbook and user embeddings. We evaluate the rankings according to their mean percentile rank (MPR): ", + "bbox": [ + 173, + 722, + 825, + 834 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/1f42c7c8c0326e778ea84b89f9100615f0fae78a8842c700b9aebf3c8533ce4e.jpg", + "text": "$$\nM P R = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\frac { R _ { i } } { \\operatorname* { m a x } R }\n$$", + "text_format": "latex", + "bbox": [ + 415, + 157, + 581, + 202 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "where $R _ { i }$ is the rank of the entity under our evaluation procedure for observation $i$ . This measures on average how well we rank actual entities. A score of 0 would indicate perfect performance (i.e. top rank every test sample given), so a lower value is better under this metric. See the appendix $\\ S \\mathrm { A } . 1$ for further details on the experimental setup. ", + "bbox": [ + 174, + 204, + 825, + 260 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.2.2 DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 275, + 302, + 290 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Movies The Internet Movie Database (IMDB) is a publicly available dataset6 of information related to films, television programs and video games. Though in this paper, we focus only on data on its 465,136 movies. Table A.1 in the appendix $( \\ S \\mathrm { A } . 1 )$ summarizes the feature types we use. It contains information on writers, directors, and principal cast members attached to each film, along with metadata. ", + "bbox": [ + 173, + 299, + 825, + 369 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Education We use a dataset from an anonymized leading technology company that provides educational services. In this proprietary dataset, we have 57 million observations and 9 categorical feature types which include textbook identifier, user identifier, school identifier, and course the book is typically used with, along with other proprietary features. Here, each observation is an “interaction” a user had with a textbook. ", + "bbox": [ + 174, + 385, + 823, + 455 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.2.3 RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 470, + 292, + 484 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "After training, we use the cast members associated with the movies of the test set and attempt to predict the actual director the film was directed. We take the sum of the cast member embeddings, and rank the directors by cosine similarity of their embeddings to the summed cast member vector. If there is a cast member in the test dataset who did not appear in the training data, we exclude them from the summation. For the educational dataset, we simply use the user embedding directly to get the most similar textbooks. ", + "bbox": [ + 173, + 494, + 825, + 579 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 2 presents the results from our evaluation. Feat2Vec sizably outperforms CBOW in the MPR metric. In fact, Feat2Vec predicts the actual director $2 . 4 3 \\%$ of the times, while CBOW only does so $1 . 2 6 \\%$ of the time, making our approach almost 2 times better in terms of Top-1 Precision metric. We explore in greater detail the distribution of the rankings in the appendix in $\\ S \\mathrm { A } . 2$ . ", + "bbox": [ + 174, + 585, + 825, + 642 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/08b29bd4b1b2e13f28874831987363381016d9c171e0b44a7111feedd9419f20.jpg", + "table_caption": [ + "Table 2: Mean percentile rank " + ], + "table_footnote": [], + "table_body": "
DatasetFeat2VecCBOW
IMDB19.36%24.15%
Educational25.2%29.2%
", + "bbox": [ + 379, + 681, + 616, + 724 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.3 UNSUPERVISED Feat2Vec PERFORMANCE WITH CONTINUOUS INPUTS ", + "text_level": 1, + "bbox": [ + 173, + 747, + 699, + 762 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We now focus on how well Feat2Vec performs on a real-valued feature with a complex feature extraction function. We expect this task to highlight Feat2Vec’s advantage over token-based embedding learning algorithms, such as Word2Vec, since our rating embedding extraction function will require embeddings of numerically similar ratings to be close , while Word2Vec will treat two differing ratings tokens as completely different entities. We evaluate the prediction of the real-valued rating of movies in the test dataset by choosing the IMDB rating embedding most similar7 to the embedding of the movie’s director, and compute the Root Mean Squared Error (RMSE) of the predicted rating in the test dataset. We also vary $\\alpha _ { 1 }$ , the flattening hyperparameter for feature group sampling, to see what effect this hyperparameter has on our performance. Intuitively, a low $\\alpha _ { 1 }$ will greatly improve the quality of the ratings embeddings learned, since it has relatively few parameters and is otherwise sampled infrequently. At the same time, with low $\\alpha _ { 1 }$ the director feature will be sampled less since it is one of the most complex features to learn, so the learned director embeddings may be of poorer quality. Figure 3 displays the results of our experiment, benchmarked against the performance of Word2Vec’s CBOW algorithm in the prediction task. We also show as a baseline the RMSE of a random uniform variable over the range of possible ratings (0 to 10). As is evident from the plot, CBOW performs a bit better than a random prediction, but is also handily outperformed by Feat2Vec across all hyper-parameter settings. The algorithm’s performance does not seem very sensitive to the hyperparameter choice. ", + "bbox": [ + 173, + 773, + 825, + 886 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/617543a6a2a807e0a383fe6a61bcbf4ec4eb3fbaf9fd269a94493f8b5ea3637d.jpg", + "image_caption": [ + "Figure 3: RMSE in Ratings Task as a Function of $\\alpha _ { 1 }$ " + ], + "image_footnote": [], + "bbox": [ + 328, + 118, + 655, + 315 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 397, + 825, + 536 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 RELATION TO PRIOR WORK ", + "text_level": 1, + "bbox": [ + 176, + 583, + 437, + 599 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The original Factorization Machine formulation has been extended for multiple contexts. For example, Field-Aware Factrorization Machine (Juan et al., 2016) allows different weights for some feature interactions, but does not allow feature groups or feature extraction functions like Feat2Vec does. ", + "bbox": [ + 174, + 631, + 825, + 686 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Algorithms that calculate continuous representations of entities other than words have been proposed for biological sequences (Abrahamsson & Plotkin, 2009), of vertices in network graphs (Perozzi et al., 2014) or in machine translation for embeddings of complete sentences (Kiros et al., 2015). Generative Adversarial Networks (Goodfellow et al., 2014)(GANs) have been used to produce unsupervised embeddings of images effective for classification (Radford et al., 2015) and for generating natural language (Press et al., 2017). To our knowledge, GANs have not been used for jointly embedding multiple feature types. Adversarial training could be an alternative to NCE for unsupervised learning, but we leave this for future study. ", + "bbox": [ + 174, + 694, + 825, + 805 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We recently discovered a promising direction for an algorithm still in development called StarSpace (Wu et al., 2017) with similar goals from ours. Even though they intend to be able to embed all types of features, at the time of the writing of this paper, their pre-print method was limited to only work for bag of words. While Feat2Vec can jointly learn embeddings for all feature values in a dataset, StarSpace samples a single arbitrary feature. Our preliminary experiments suggest that sampling a single feature does not produce embeddings that generalize well. Nonetheless, a limitation of our work is that we do not compare with StarSpace, which future work may decide to do. ", + "bbox": [ + 174, + 813, + 825, + 924 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 102, + 318, + 117 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Embeddings have proven useful in a wide variety of contexts, but they are typically built from datasets with a single feature type as in the case of Word2Vec, or tuned for a single prediction task as in the case of Factorization Machine. We believe Feat2Vec is an important step towards generalpurpose methods, because it decouples feature extraction from prediction for datasets with multiple feature types, it is general-purpose, and its embeddings are easily interpretable. ", + "bbox": [ + 174, + 135, + 823, + 205 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In the supervised setting, Feat2Vec is able to calculate embeddings for whole passages of texts, and we show experimental results outperforming an algorithm specifically designed for text—even when using the same feature extraction CNN. This suggests that the need for ad-hoc networks should be situated in relationship to the improvements over a general-purpose method. ", + "bbox": [ + 174, + 212, + 825, + 267 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In the unsupervised setting, Feat2Vec’s embeddings are able to capture relationships across features that can be twice as better as Word2Vec’s CBOW algorithm on some evaluation metrics. Feat2Vec exploits the structure of a datasets to learn embeddings in a way that is structurally more sensible than existing methods. The sampling method, and loss function that we use have interesting theoretical properties. To the extent of our knowledge, Unsupervised Feat2Vec is the first method able to calculate continuous representations of data with arbitrary feature types. ", + "bbox": [ + 174, + 275, + 825, + 358 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Future work could study how to reduce the amount of human knowledge our approach requires; for example by automatically grouping features into entities, or by automatically choosing a feature extraction function. These ideas can extend to our codebase that we make available 8. Overall, we evaluate supervised and unsupervised Feat2Vec on 2 datasets each. Though further experimentation is necessary, we believe that our results are an encouraging step towards general-purpose embedding models. 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", + "bbox": [ + 171, + 496, + 826, + 882 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 80, + 828, + 912 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/bf34d4eebcb9f890edf4a4cbf4846f7ce55c933140f8d2fbbd7a590e3701c8c8.jpg", + "image_caption": [ + "Figure A.1: Feature Sampling Probabilities as a Function of $\\alpha _ { 1 }$ " + ], + "image_footnote": [], + "bbox": [ + 303, + 118, + 676, + 315 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/b759e4a92d6a5993a0d148cd18d1e17ac009a0c9ab41e3c7524175bd680517e8.jpg", + "table_caption": [ + "Table A.1: IMDB dataset features " + ], + "table_footnote": [], + "table_body": "
Feature Type NameType# of feats.Example for an instance
Runtime (minutes)Real-valued1116
IMDB rating (0-10)Real-valued17.8
# of IMDB rating votesReal-valued1435,682
Is adult film?Boolean2False
Movie releaes yearCategorical2712001
Movie titleText165,471“Ocean's”,“Eleven”
DirectorsBag of categories174,382‘Steven Soderbergh”
GenresBag of categories28“Crime”,“Thriller”
WritersBag of categories244,241“George Johnson”,“Jack Russell"
Principal cast members (actors)Bag of categories1,104,280“George Clooney”,“Brad Pitt”,“Julia Roberts”
", + "bbox": [ + 173, + 390, + 859, + 545 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A APPENDIXES ", + "text_level": 1, + "bbox": [ + 176, + 573, + 318, + 588 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 UNSUPERVISED RANKING EXPERIMENT DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 606, + 562, + 619 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For our evaluation, we define a testing set that was not used to tune the parameters of the model. For the IMDB dataset, we randomly select a $10 \\%$ sample of the observations that contain a director that appears at least twice in the database 9. We do this to guarantee that the set of directors in the left-out dataset appear during training at least once, so that each respective algorithm can learn something about the characteristics of these directors. For the educational dataset, our testing set only has observations of textbooks and users that appear at least 10 times in training. ", + "bbox": [ + 174, + 633, + 825, + 717 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For both Feat2Vec and CBOW, we perform cross-validation on the loss function, by splitting the $10 \\%$ of the training data randomly into a validation set, to determine the number of epochs to train, and then train the full training dataset with this number of epochs. 10 While regularization of the embeddings during training is possible, this did not dramatically change results, so we ignore this dimension of hyperparameters. ", + "bbox": [ + 174, + 724, + 825, + 794 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We rank left-out entity pairs in the test dataset using the ordinal ranking of the cosine similarity of target and input embeddings. For the IMDB dataset, the target is the director embedding, and the input embedding is the sum of the cast member embeddings. For the educational dataset, the target is the textbook embedding, and the input embedding is the user embedding. ", + "bbox": [ + 174, + 801, + 823, + 829 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For training Feat2Vec we set $\\alpha _ { 1 } = \\alpha _ { 2 } = 3 / 4$ in the IMDB dataset; and $\\alpha _ { 1 } = 0$ and $\\alpha _ { 2 } = 0 . 5$ for the educational. In each setting, $\\alpha _ { 2 }$ is set to the same flattening hyperparameter we use for CBOW to negatively sample words in a document. We learn $r = 5 0$ dimensional embeddings under both algorithms. ", + "bbox": [ + 174, + 138, + 825, + 194 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Below we describe how CBOW is implemented on our datasets for unsupervised experiments and what extraction functions are used to represent features in the IMDB dataset. ", + "bbox": [ + 173, + 202, + 823, + 229 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Word2Vec For every observation in each of the datasets, we create a document that tokenizes the same information that we feed into Feat2Vec. We prepend each feature value by its feature name, and we remove spaces from within features. In Figure A.2 we show an example document. Some features may allow multiple values (e.g., multiple writers, directors). To feed these features into the models, for convenience, we constraint the number of values, by truncating each feature to no more than 10 levels (and sometimes less if reasonable). This results in retaining the full set of information for well over $9 5 \\%$ of the values. We pad the sequences with a “null” category whenever necessary to maintain a fixed length. We do this consistently for both Word2Vec and Feat2Vec. We use the CBOW Word2Vec algorithm and set the context window to encompass all other tokens in a document during training, since the text in this application is unordered. ", + "bbox": [ + 174, + 246, + 825, + 386 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/330946bb7dcbf02917794a98c168e5aef01780c17fee4188796103aa1971d0ff.jpg", + "image_caption": [ + "Figure A.2: Sample document for Word2Vec for the Ocean’s Eleven movie " + ], + "image_footnote": [], + "bbox": [ + 321, + 406, + 678, + 428 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Feat2Vec Feature representation in Feat2Vec requires a feature extraction function for each feature type. Here, we explain how we build these functions: ", + "bbox": [ + 173, + 493, + 823, + 522 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• Bag of categories, categorical, and boolean: For all of the categorical variables, we learn a unique $r$ -dimensional embedding for each entity using a linear fully-connected layer (Equation 4). We do not require one-hot encodings, and thus we allow multiple categories to be active; resulting in a single embedding for the group that is the sum of the embeddings of the subfeatures. This is ordering-invariant: the embedding of “Brad Pitt” would be the same when he appears in a movie as a principal cast member, regardless whether he was 1st or 2nd star. Though, if he were listed as a director it may result in a different embedding. ", + "bbox": [ + 217, + 535, + 823, + 632 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• Text: We preprocess the text by removing non alpha-numeric characters, stopwords, and stemming the remaining words. We then follow the same approach that we did for categorical variables, summing learned word embeddings to a “title embedding” before interacting. It would be easy to use more sophisticated methods (e.g, convolutions), but we felt this would not extract further information. ", + "bbox": [ + 217, + 637, + 825, + 707 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• Real-valued: For all real-valued features, we pass these features through a 3-layer feedforward fully connected neural network that outputs a vector of dimension $r$ , which we treat as the feature’s embedding. Each intermediate layer has $r$ units with $\\mathtt { r e l u }$ activation functions. These real-valued features highlight one of the advantages of the Feat2Vec algorithm: using a numeric value as an input, Feat2Vec can learn a highly nonlinear relation mapping a real number to our high-dimensional embedding space. In contrast, Word2Vec would be unable to know ex ante that an IMDB rating of 5.5 is similar to 5.6. ", + "bbox": [ + 217, + 713, + 825, + 810 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.2 DISTRIBUTION OF IMDB DIRECTOR RANKINGS ", + "text_level": 1, + "bbox": [ + 176, + 827, + 547, + 842 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Figure A.3 shows the full distribution of rankings of the IMDB dataset, rather than summary statistics, in the form of a Cumulative Distribution Function (CDF) of all rankings calculated in the test dataset. The graphic makes it apparent for the vast majority of the ranking space, the rank CDF of Feat2Vec is to the left of CBOW, indicating a greater probability of a lower ranking under Feat2Vec. This is not, however, the case at the upper tail of ranking space, where it appears CBOW is superior. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "However, when we zoom-in on the absolute upper region of rankings (1 to 25), which might be a sensible length of ranks one might give as actual recommendatiosn, it is the case that up until rank 8 or so, Feat2Vec outperforms CBOW still. Intermediate rankings are still strong signals that our Feat2Vec algorithm is doing a better job of extracting information into embeddings, particularly those entities that appear sparsely in the training data and so are especially difficult to learn. ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/fe8ebcf2f1729c2c77fccf48acc7e972098c21fdd9cc58b0c0592dc3fc7497e5.jpg", + "image_caption": [ + "Figure A.3: Cumulative Distribution Function of Director Rankings (With Zoom-in to Top 25 Ranks) " + ], + "image_footnote": [], + "bbox": [ + 285, + 246, + 691, + 493 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.3 PROOF TO THEOREM 1 ", + "text_level": 1, + "bbox": [ + 176, + 643, + 375, + 659 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Theorem 1. The gradient for learning embeddings with Feat2Vec is a convex combination of the gradient from n targeted Factorization Machines for each feature in the data when each feature group is a singleton, where n is the total number of features in the dataset. ", + "bbox": [ + 174, + 684, + 825, + 728 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. Let $S _ { \\kappa _ { i } } ^ { + }$ denote the positively labeled records whose corresponding negative samples resample feature $\\kappa _ { i }$ . For convenience, suppress the inclusion of learned parameters θ in the notation in this section while understanding the feature extraction functions $\\vec { \\phi }$ implicitly include these parameters. We can express the loss function $L ( . )$ , the binary cross-entropy of the data given the Feat2Vec model, as follows: ", + "bbox": [ + 173, + 851, + 825, + 922 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/47fecd434e48b4e1db91a1fb4fa5fdd80a07cdb5dbddcce21637b1722fb3fe31.jpg", + "text": "$$\n\\begin{array} { r l } { L ( S ^ { + } | \\vec { \\phi } ) = \\displaystyle \\frac { 1 } { | S ^ { + } | } \\sum _ { \\tau ^ { \\prime } \\in S ^ { + } } \\Big ( \\log ( \\tilde { \\rho } ( y | \\vec { \\phi } - 1 | \\vec { \\phi } , \\vec { x } ^ { + } ) ) + \\underbrace { \\sum _ { \\tau ^ { \\prime } \\in S ^ { + } } ^ { \\vec { K } } \\log ( | \\vec { \\rho } ( y = 0 | | \\vec { \\phi } , \\vec { x } ^ { - } ) ) } _ { \\tau ^ { \\prime } - \\tau < 2 ( | \\vec { x } ^ { + } - \\vec { x } ^ { + } | ) ^ { \\tilde { \\phi } } ( \\vec { x } ^ { + } ) } \\Big . } & { } \\\\ { = \\frac { 1 } { | S ^ { + } | } \\sum _ { \\tau ^ { \\prime } \\in S ^ { + } } \\Big ( \\log ( \\tilde { \\rho } ( y | \\vec { \\phi } - 1 | \\vec { \\phi } , \\vec { x } ^ { + } ) , \\vec { x } ^ { + } + S _ { \\tau , y } ^ { + } ) \\rho ( \\vec { x } ^ { + } \\in S _ { \\tau , x } ^ { + } ) ) } \\\\ { + \\underbrace { \\sum _ { \\tau ^ { \\prime } \\in S ^ { + } } ^ { \\vec { K } } \\log ( \\tilde { \\rho } ( y = 0 | \\vec { \\phi } , \\vec { x } ^ { - } , \\vec { x } ^ { + } \\in S _ { \\tau , y } ^ { + } ) \\rho ( \\vec { x } ^ { + } \\in S _ { \\tau , x } ^ { + } ) ) } _ { \\tau ^ { \\prime } - \\tau ^ { \\prime } ( | \\vec { x } ^ { + } | ) ^ { \\tilde { \\phi } } ( \\vec { x } ^ { + } \\in S _ { \\tau , x } ^ { + } ) } \\Big ) } & { } \\\\ - \\displaystyle \\frac { 1 } { | S ^ { + } | } \\sum _ { \\tau ^ { \\prime } \\in S ^ { + } } ^ { \\vec { K } } \\log ( \\log ( \\frac { e ^ { - i ( \\tau ^ { + } \\cdot \\vec { \\phi } ) } p | \\vec { x } ^ { \\top } \\in S _ { \\tau , y } ^ { + } } { e ^ { i ( \\tau ^ { + } \\cdot \\vec { \\phi } ) } + P _ { 0 } ( | \\vec { x } ^ { + } | \\vec { x } ^ { + } , \\vec { x } ^ { + } \\in S _ { \\tau , x } ^ { + } ) } \\\\ + \\underbrace \\sum _ { \\tau ^ { \\prime } \\in S ^ { + } } ^ \\vec \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 169, + 122, + 717, + 354 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Note now that $P _ { \\mathcal { Q } } ( \\vec { x } | \\vec { x } ^ { + } , \\vec { x } ^ { + } \\in S _ { \\kappa _ { i } } ^ { + } )$ is simply the probability of the record’s feature value $\\vec { x } _ { f }$ under the second step noise distribution $\\mathcal { Q } _ { 2 } ( \\mathrm { X } _ { \\mathrm { f } } , \\alpha _ { 2 } )$ : $P _ { \\mathcal { Q } } ( \\vec { x } | \\vec { x } ^ { + } , \\vec { x } ^ { + } \\in S _ { \\kappa _ { i } } ^ { + } ) = P _ { \\mathcal { Q } _ { 2 } } ( \\vec { x } _ { f } )$ ", + "bbox": [ + 171, + 363, + 826, + 396 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/15dea8747a70f75436933770de2957cba27be9fd98b40a4c8856d0a502082cb8.jpg", + "text": "$$\n\\begin{array} { r l } & { = \\displaystyle \\frac { 1 } { | S ^ { + } | } \\displaystyle \\sum _ { i = 1 } ^ { n } \\sum _ { \\bar { x } ^ { + } \\in S _ { \\star _ { i } } ^ { + } } \\Big ( \\log ( \\frac { e ^ { s ( \\bar { x } ^ { + } , \\bar { \\phi } ) } p ( \\bar { x } ^ { + } \\in S _ { \\kappa _ { i } } ^ { + } ) } { e ^ { s ( \\bar { x } ^ { + } , \\bar { \\phi } ) } + P _ { Q _ { 2 } } ( \\vec { x } _ { \\kappa _ { i } } ^ { + } ) } ) + \\frac { k } { \\bar { x } ^ { - } \\sim Q ( \\cdot | \\vec { x } ^ { + } , i \\in S _ { \\star _ { i } } ^ { + } ) } \\log ( \\frac { P _ { Q _ { 2 } } ( \\vec { x } _ { f } ^ { - } ) p ( \\bar { x } ^ { + } \\in S _ { \\kappa _ { i } } ^ { + } ) } { e ^ { s ( \\bar { x } ^ { - } , \\bar { \\phi } ) } + P _ { Q _ { 2 } } ( \\vec { x } _ { f } ^ { - } ) ) } \\Big ) } \\\\ & { = \\displaystyle \\frac { 1 } { | S ^ { + } | } \\displaystyle \\sum _ { i = 1 } ^ { n } \\sum _ { \\bar { x } ^ { + } \\in S _ { \\star _ { i } } ^ { + } } \\Big ( \\log ( \\frac { e ^ { s ( \\bar { x } ^ { + } , \\bar { \\phi } ) } } { e ^ { s ( \\bar { x } ^ { + } , \\bar { \\phi } ) } + P _ { Q _ { 2 } } ( \\vec { x } _ { \\kappa _ { i } } ^ { + } ) } ) + \\log ( p ( \\bar { x } ^ { + } \\in S _ { \\kappa _ { i } } ^ { + } ) ^ { k + 1 } ) } \\\\ & { \\quad + \\left. \\frac { k } { \\bar { x } ^ { - } \\sim Q ( \\cdot | \\vec { x } ^ { + } , \\vec { x } ^ { + } \\in S _ { \\star _ { i } } ^ { + } ) } \\log ( \\frac { P _ { Q _ { 2 } } ( \\vec { x } _ { f } ^ { - } ) } { e ^ { s ( \\bar { x } ^ { - } , \\bar { \\phi } ) } + P _ { Q _ { 2 } } ( \\vec { x } _ { f } ^ { - } ) } ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 422, + 877, + 573 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We now drop the term containing the probability of assignment to a feature group $p ( \\vec { x } ^ { + } \\in S _ { \\kappa _ { i } } ^ { + }$ ) since it is outside of the learned model parameters $\\vec { \\phi }$ and fixed in advance: ", + "bbox": [ + 173, + 582, + 826, + 616 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/a61fd9c2b5b6d5d38837753040afe29c1eefebcf6fbc8ad957b46992b73584bf.jpg", + "text": "$$\n\\begin{array} { c } \\displaystyle \\propto \\displaystyle \\frac { 1 } { | S ^ { + } | } \\displaystyle \\sum _ { i = 1 } ^ { n } \\sum _ { \\vec { x } ^ { + } \\in S _ { \\mathbf { * } _ { i } } ^ { + } } \\left( \\log ( \\frac { e ^ { s ( \\vec { x } ^ { + } , \\vec { \\phi } ) } } { e ^ { s ( \\vec { x } ^ { + } , \\vec { \\phi } ) } + P _ { Q _ { 2 } } ( \\vec { x } _ { \\mathbf { * } _ { i } } ^ { + } ) } ) + \\displaystyle \\sum _ { \\vec { x } ^ { - } \\sim Q ( \\cdot , | \\vec { x } ^ { + } , \\vec { x } ^ { + } \\in S _ { \\mathbf { * } _ { i } } ^ { + } ) } ^ { k } \\log ( \\frac { P _ { Q _ { 2 } } ( \\vec { x } _ { f } ^ { - } ) } { e ^ { s ( \\vec { x } ^ { - } , \\vec { \\phi } ) } + P _ { Q _ { 2 } } ( \\vec { x } _ { f } ^ { - } ) } ) \\right) \\nonumber _ { \\vec { x } ^ { + } \\sim \\mathbb { S } _ { \\mathbf { \\ * } ^ { + } \\sim \\mathbb { S } _ { \\mathbf { \\ * } ^ { + } \\sim \\mathbb { S } _ { \\mathbf { \\ * } ^ { + } \\sim \\mathbb { S } _ { \\mathbf { \\ * } ^ { + } \\sim \\mathbb { S } _ { \\mathbf { \\ * } ^ { + } \\sim \\mathbb { S } _ { \\mathbf { \\ * } ^ { + } \\sim \\mathbb { S } _ { \\mathbf { \\ * } ^ { + } \\sim \\mathbb { S } _ { \\mathbf { \\ * } ^ { - } \\mathbb { S } _ { \\mathbf { \\ * } ^ { - } \\mathbb { S } _ { \\mathbf { \\ * } ^ { - } \\mathbb { S } _ { \\mathcal \\delta } } } } } } } } } } } } \\\\ \\displaystyle \\xrightarrow [ { \\vec { x } ^ { + } | \\to \\infty } ] { n } p ( \\vec { x } ^ { + } \\in S _ { \\mathbf { \\star } _ { i } } ^ { + } ) E \\Big [ \\log ( \\frac { e ^ { s ( \\vec { x } ^ { + } , \\vec { \\phi } ) } } { e ^ { s ( \\vec { x } ^ { + } , \\vec { \\phi } ) } + P _ { Q _ { 2 } } ( \\vec { x } _ { \\mathbf { \\star } _ { i } } ^ { + } ) } ) + \\sum _ { \\vec { x } ^ { - } \\sim Q ( \\cdot , | \\vec { x } ^ { + } , \\vec { x } ^ { + } \\in S _ { \\mathbf { \\star } _ { i } } ^ { + } ) } ^ { k } \\ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 202, + 643, + 861, + 786 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Thus, the loss function is just a convex combination of the loss functions of the targeted classifiers for each of the $p$ features, and by extension so is the gradient since: ", + "bbox": [ + 173, + 796, + 825, + 827 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/c4dafdfbf2ae2faefb9234cfc7a7859bbfa191742e09d5502e01e62ef87fb719.jpg", + "text": "$$\n\\frac { \\partial } { \\partial \\phi } \\sum _ { i = 1 } ^ { n } p ( \\vec { x } ^ { + } \\in S _ { \\mathrm { \\bf { k } } _ { i } } ^ { + } ) E \\Big [ L ( \\vec { x } | \\vec { \\phi } , \\mathrm { t a r g e t } = f ) \\Big ] = \\sum _ { i = 1 } ^ { n } p ( \\vec { x } ^ { + } \\in S _ { \\mathrm { \\bf { k } } _ { i } } ^ { + } ) \\frac { \\partial } { \\partial \\phi } E \\Big [ L ( \\vec { x } | \\vec { \\phi } , \\mathrm { t a r g e t } = f ) \\Big ]\n$$", + "text_format": "latex", + "bbox": [ + 202, + 849, + 794, + 891 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Thus the algorithm will, at each step, learn a convex combination of the gradient for a targeted classifier on feature $f$ , with weights proportional to the feature group sampling probabilities in step 1 of ", + "bbox": [ + 173, + 895, + 823, + 925 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "the sampling algorithm. Note that if feature groups are not singletons, the gradient from unsupervised Feat2Vec will analogously be a convex combination of $n$ gradients learned from supervised learning tasks on each of the $n$ feature groups. □ ", + "bbox": [ + 173, + 103, + 825, + 146 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B FEATURE EXTRACTION NETWORK FOR NATURAL LANGUAGE ", + "text_level": 1, + "bbox": [ + 174, + 167, + 709, + 183 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/71766fd860ac992bd9f8f7e92e427018a951fd40442b3983d9ec580cd9dd0aa9.jpg", + "image_caption": [ + "Figure A.4: Feature extraction network used for labelling tasks. We use $\\mathrm { f } { = } 1 0 0 0$ convolutional filters each of width 3 (words) " + ], + "image_footnote": [], + "bbox": [ + 245, + 202, + 712, + 354 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Here we describe the details of the feature extraction function $\\phi$ used in our experiments for supervised tasks in $\\ S 4 . 1$ . An overview of the network is given in Fig. A.4. We choose the most common words of each dataset to build a vocabulary of size $n$ , and convert the words of each document to a sequence of length $t$ of one-hot encodings of the input words. If the input text is shorter than $t$ , then we pad it with zeros; if the text is longer, we truncate it by discarding the trailing words. Therefore, for a vocabulary size $n$ , the input has dimensions $t \\times n$ . These $t \\times$ dimensional matrix is then passed through the following layers: ", + "bbox": [ + 174, + 412, + 825, + 511 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "1. We use an embedding layer to assign a $d$ -dimensional vector to each word in the input passage of text. This is done through a $d \\times n$ -dimensional lookup table, which results in an $t \\times d$ matrix. \n2. We extract features from the embeddings with functions called convolutional filters (LeCun et al., 1998) (also called feature maps). A convolutional filter is simply a matrix learned from an input. We learn $f$ filters that are applied on groups of $m$ adjacent word embeddings, thus each of our filters is a $d \\times m$ matrix of learned parameters. Filters are applied by computing the element-wise dot product of the filter along a sliding window of the entire input. The resulting output for each filter is a vector of length $t - m + 1$ . We also apply a ReLU activation to the output of each filter. \n3. Consider the case of inputs of different lengths. For very short texts, the output of the filters will be mostly zero since the input is zero-padded. To enforce learning from the features of the text, and not just its length we apply a function called 1-max pooling to the output of the filters: from the $t - m + 1$ output vector of each filter, we select the maximum value. This yields a vector of length $F$ , a representation of the passage which is independent of its length. \n4. We learn higher-level features from the convolutional filters. For this, we use a fully connected layer with $p$ units and a ReLU activation, \n5. During training (not in inference), we prevent the units from co-adapting too much with a dropout layer (Srivastava et al., 2014). Dropout is a form of regularization that for each mini-batch randomly drops a specified percentage of units. \n6. the final embedding for $x _ { j }$ (that is used in the factorization) is computed by a dense layer with $r$ output units and an activation function, where $r$ is the embedding size of our indexable items. ", + "bbox": [ + 210, + 521, + 825, + 882 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We set the maximum vocabulary size $n$ to 100,000 words, and input embedding size $d$ to 50 for all experiments. We initialize the input word embeddings and the label embeddings using ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Word2Vec(Mikolov et al., 2013) We have have not evaluated multiple architectures or hyperparameter settings and obtain good results on diverse datasets with the same architecture, which was designed followed recommendations from a large scale evaluation of CNN hyper parameters(Zhang & Wallace, 2015). We set the number of convolutional filters $f$ to 1,000, and the dropout rate to 0.1. The maximum sequence length $t$ was chosen according to the typical document length (350 words for CiteULike and 250 for Yelp). For the CTR dataset, because we use very small values of $r$ , due to the tendency of the ReLU units to‘die’ during training (output zero for all examples), which can have a significant impact, we used instead PReLU activations (He et al., 2015) for the final layer, since they do not suffer from this issue. ", + "bbox": [ + 174, + 103, + 825, + 228 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.1 FEATURE EXTRACTION FOR DEEPCONN COMPARISON ", + "text_level": 1, + "bbox": [ + 176, + 246, + 596, + 260 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The CNN architecture used for DeepCoNN (Zheng et al., 2017) is similar to the previous section. It consists of a word embedding lookup table, convolutional layer, 1-max pooling and a fully connected layer. We use the hyper-parameters that the authors report as best - 100 convolution filters and 50 units for the fully connected layer. We set the word embedding size to 100, the vocabulary size to 100,000 and the maximum document length to 250. ", + "bbox": [ + 174, + 271, + 825, + 342 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C HYPER-PARAMETERS FOR CTR ", + "text_level": 1, + "bbox": [ + 176, + 361, + 470, + 377 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "To compare Feat2Vec with Collaborative Topic Regression, we choose the embedding size $r \\in$ $\\{ 5 , 1 0 , { \\bar { 1 } } 5 \\}$ for which CTR performs best. The results are show in Table A.2. ", + "bbox": [ + 171, + 393, + 823, + 421 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/16cbc2fd80651930fba2741571ee3dbdb414bfd6819bbb2c4477846b456300a4.jpg", + "table_caption": [ + "Table A.2: Tuning embedding size for CTR " + ], + "table_footnote": [], + "table_body": "
r=5r=10r=15Time (mins.)
Matrix Fact.0.87230.89110.90461
Feat2Vec0.90810.93030.9401133
C.T.R0.87630.92340.93561425
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We believe that we are the first to propose a method for learning unsuper-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 341, + 425, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 341, + 425, + 356 + ], + "score": 1.0, + "content": "vised embeddings that leverage the structure of multiple feature types.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11, + "bbox_fs": [ + 141, + 210, + 470, + 356 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 373, + 206, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 208, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 208, + 389 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 442, + 411 + ], + "score": 1.0, + "content": "Informally, in machine learning a dense representation, or embedding of a vector", + "type": "text" + }, + { + "bbox": [ + 443, + 398, + 477, + 408 + ], + "score": 0.91, + "content": "\\vec { x } \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "is an-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 408, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 157, + 423 + ], + "score": 1.0, + "content": "other vector", + "type": "text" + }, + { + "bbox": [ + 158, + 410, + 189, + 421 + ], + "score": 0.91, + "content": "\\vec { y } \\in \\mathbb { R } ^ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 408, + 339, + 423 + ], + "score": 1.0, + "content": "that has much lower dimensionality", + "type": "text" + }, + { + "bbox": [ + 339, + 410, + 371, + 420 + ], + "score": 0.82, + "content": "( r \\ll n )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 408, + 505, + 423 + ], + "score": 1.0, + "content": "than the original representation,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 421, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 432 + ], + "score": 1.0, + "content": "and can be used to replace the original vector in downstream prediction tasks. Embeddings have", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "multiple advantages, as they enable more efficient training (Mikolov et al., 2013), and unsupervised", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "learning (Schnabel et al., 2015). For example, when applied to text, semantically similar words are", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 453, + 210, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 210, + 466 + ], + "score": 1.0, + "content": "mapped to nearby points.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 398, + 505, + 466 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 470, + 337, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 338, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 338, + 483 + ], + "score": 1.0, + "content": "We consider two kind of algorithms that use embeddings:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 468, + 338, + 483 + ] + }, + { + "type": "list", + "bbox": [ + 130, + 490, + 505, + 658 + ], + "lines": [ + { + "bbox": [ + 130, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 130, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "1. Unsupervised methods (sometimes referred as self-supervised methods) like", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 142, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "Word2Vec (Mikolov et al., 2013), are designed to provide embeddings that are use-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 141, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "ful for a wide-array of predictions tasks. For example, the loss function of the continuous", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 523, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 141, + 523, + 506, + 535 + ], + "score": 1.0, + "content": "bag of words (CBOW) algorithm of Word2Vec is tuned to predict the next word of a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 141, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "sequence; however, in practice, the embeddings produced are mostly used for other tasks,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 141, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "such as analogy solving (Mikolov et al., 2013), or sentiment analysis (Le & Mikolov,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 141, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "2014). In the context of this paper, we refer to the embeddings of an unsupervised method", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 567, + 455, + 580 + ], + "spans": [ + { + "bbox": [ + 141, + 567, + 455, + 580 + ], + "score": 1.0, + "content": "that can be used for a variety of auxiliary prediction tasks as general-purpose.", + "type": "text" + } + ], + "index": 33, + "is_list_end_line": true + }, + { + "bbox": [ + 130, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 130, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "2. Supervised methods, like matrix factorization, produce embeddings that are highly tuned", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 592, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 141, + 592, + 505, + 604 + ], + "score": 1.0, + "content": "to a prediction task. These embeddings may be interpretable but do not usually general-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 604, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 141, + 604, + 505, + 615 + ], + "score": 1.0, + "content": "ize to other tasks. We refer to these embeddings as task-specific. Matrix factorization and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 141, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "Word2Vec are unable to calculate embeddings for items that are not available during train-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 625, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 141, + 625, + 505, + 639 + ], + "score": 1.0, + "content": "ing (“cold-start” problem). While recent work using n-gram features (Bojanowski et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 141, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "2016) have addressed this limitation for supervised and unsupervised tasks, it can only be", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 142, + 648, + 297, + 660 + ], + "spans": [ + { + "bbox": [ + 142, + 648, + 297, + 660 + ], + "score": 1.0, + "content": "used for a single feature type—words.", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + } + ], + "index": 33, + "bbox_fs": [ + 130, + 491, + 506, + 660 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 667, + 504, + 689 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 681 + ], + "score": 1.0, + "content": "In this paper we propose Feat2Vec as a novel method that allows calculating embeddings of arbitrary", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 678, + 455, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 455, + 691 + ], + "score": 1.0, + "content": "feature types from both supervised and unsupervised data. Our main contributions are:", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 666, + 505, + 691 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 133, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 133, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "• Unsupervised Feat2Vec. Existing general-purpose dense representation methods are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 141, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 141, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "largely restricted to one or two feature types. 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To our knowledge, Feat2Vec is the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 141, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 141, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "first algorithm that is able to calculate general-purpose embeddings that are not tuned for a", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 104, + 370, + 118 + ], + "spans": [ + { + "bbox": [ + 141, + 104, + 370, + 118 + ], + "score": 1.0, + "content": "single specific prediction task for arbitrary feature types.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 44, + "bbox_fs": [ + 133, + 698, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 141, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 141, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 141, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "both words and documents (Le & Mikolov, 2014). To our knowledge, Feat2Vec is the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 141, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 141, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "first algorithm that is able to calculate general-purpose embeddings that are not tuned for a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 104, + 370, + 118 + ], + "spans": [ + { + "bbox": [ + 141, + 104, + 370, + 118 + ], + "score": 1.0, + "content": "single specific prediction task for arbitrary feature types.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 133, + 119, + 504, + 186 + ], + "lines": [ + { + "bbox": [ + 132, + 119, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 132, + 119, + 505, + 132 + ], + "score": 1.0, + "content": "• Supervised Feat2Vec. 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In Figure 2, we show how we address this problem by treating the words", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 366, + 689 + ], + "score": 1.0, + "content": "as indexed features, but placed within a structured feature group", + "type": "text" + }, + { + "bbox": [ + 366, + 678, + 379, + 688 + ], + "score": 0.87, + "content": "\\kappa _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 677, + 505, + 689 + ], + "score": 1.0, + "content": ", the group of word features. 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Consider a feature", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 132, + 177 + ], + "score": 1.0, + "content": "group", + "type": "text" + }, + { + "bbox": [ + 133, + 164, + 143, + 175 + ], + "score": 0.86, + "content": "\\vec { \\mathsf { K } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 164, + 505, + 177 + ], + "score": 1.0, + "content": ", that exists in very high dimensional space. For example, this could happen because we", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 188 + ], + "score": 1.0, + "content": "are modeling with one-hot encoding a categorical variable with large number of possible values. In", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 199 + ], + "score": 1.0, + "content": "such scenario, it is overwhelmingly costly to feed the model all negative labels, particularly if the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 197, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 197, + 210 + ], + "score": 1.0, + "content": "model is fairly sparse.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "A shortcut around this is a concept known as implicit sampling, where instead of using all of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 355, + 238 + ], + "score": 1.0, + "content": "possible negative labels, one simply samples a fixed number", + "type": "text" + }, + { + "bbox": [ + 355, + 225, + 368, + 236 + ], + "score": 0.71, + "content": "( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "from the set of possible negative", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 505, + 249 + ], + "score": 1.0, + "content": "labels for each positively labelled record. Word2Vec makes use of an algorithm called Negative", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "Sampling, that has little theoretical guarantees (Dyer, 2014). In short, their approach samples a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 291, + 271 + ], + "score": 1.0, + "content": "negative observation from a noise distribution", + "type": "text" + }, + { + "bbox": [ + 292, + 258, + 315, + 269 + ], + "score": 0.92, + "content": "\\mathcal { Q } _ { w 2 v }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 257, + 505, + 271 + ], + "score": 1.0, + "content": ", that is proportional to the empirical frequency", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 269, + 227, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 227, + 281 + ], + "score": 1.0, + "content": "of a word in the training data.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "We introduce a new implicit sampling method that enables learning unsupervised embeddings for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "structured feature sets. We can learn the correlation of features within a dataset by imputing negative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "labels, simply by generating unobserved records as our negative samples. Unlike Word2Vec, we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "do not constraint features types to be words. Features groups can be individual columns in a data", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 421, + 342 + ], + "score": 1.0, + "content": "matrix, but they need not to be. By grouping subfeatures using the parameter", + "type": "text" + }, + { + "bbox": [ + 422, + 331, + 429, + 340 + ], + "score": 0.44, + "content": "\\boldsymbol { \\mathsf { K } }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "in Equation 3, the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 354 + ], + "score": 1.0, + "content": "model can reason on more abstract entities in the data. By entity, we mean a particular feature group", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "value. For example, in our experiments on a movie dataset, we use a “genre” feature group, where", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "we group non-mutually exclusive indicators for movie genres including comedy, action, and drama", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 373, + 132, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 132, + 385 + ], + "score": 1.0, + "content": "films.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 108, + 390, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 199, + 403 + ], + "score": 1.0, + "content": "We start with a dataset", + "type": "text" + }, + { + "bbox": [ + 199, + 390, + 213, + 401 + ], + "score": 0.89, + "content": "S ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "of records with |~κ| feature groups. We then mark all observed records in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 399, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 410, + 415 + ], + "score": 1.0, + "content": "the training set as positive examples. For each positive record, we generate", + "type": "text" + }, + { + "bbox": [ + 411, + 402, + 418, + 411 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 399, + 505, + 415 + ], + "score": 1.0, + "content": "negative labels using", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 412, + 233, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 233, + 424 + ], + "score": 1.0, + "content": "the following 2-step algorithm:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 108, + 435, + 400, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 399, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 389, + 451 + ], + "score": 1.0, + "content": "Algorithm 1 Implicit sampling algorithm for unsupervised Feat2Vec:", + "type": "text" + }, + { + "bbox": [ + 390, + 437, + 399, + 447 + ], + "score": 0.29, + "content": "\\mathcal { Q }", + "type": "inline_equation" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 109, + 452, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 110, + 451, + 293, + 465 + ], + "spans": [ + { + "bbox": [ + 110, + 451, + 293, + 465 + ], + "score": 1.0, + "content": "1: function FEAT2VEC SAMPLE(S+, k, α1, α2)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 111, + 462, + 172, + 474 + ], + "spans": [ + { + "bbox": [ + 111, + 462, + 121, + 474 + ], + "score": 1.0, + "content": "2:", + "type": "text" + }, + { + "bbox": [ + 136, + 462, + 172, + 474 + ], + "score": 1.0, + "content": "S − ← ∅", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 110, + 470, + 201, + 484 + ], + "spans": [ + { + "bbox": [ + 110, + 470, + 122, + 484 + ], + "score": 1.0, + "content": "3:", + "type": "text" + }, + { + "bbox": [ + 136, + 472, + 201, + 484 + ], + "score": 1.0, + "content": "for ~x + ∈ S+ do", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 110, + 478, + 380, + 497 + ], + "spans": [ + { + "bbox": [ + 110, + 481, + 122, + 494 + ], + "score": 1.0, + "content": "4:", + "type": "text" + }, + { + "bbox": [ + 147, + 478, + 258, + 497 + ], + "score": 1.0, + "content": "Draw a random feature group", + "type": "text" + }, + { + "bbox": [ + 259, + 481, + 380, + 495 + ], + "score": 0.89, + "content": "\\kappa _ { i } \\sim \\mathcal { Q } _ { 1 } ( \\{ \\mathrm { p a r a m s } ( \\phi _ { i } ) \\} _ { i = 1 } ^ { | \\vec { \\kappa } | } , \\alpha _ { 1 } )", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 110, + 491, + 234, + 505 + ], + "spans": [ + { + "bbox": [ + 110, + 491, + 121, + 504 + ], + "score": 1.0, + "content": "5:", + "type": "text" + }, + { + "bbox": [ + 150, + 492, + 164, + 505 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 164, + 493, + 219, + 503 + ], + "score": 0.45, + "content": "j \\in \\{ 1 , \\ldots , k \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 492, + 234, + 505 + ], + "score": 1.0, + "content": "do", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 110, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 110, + 502, + 122, + 514 + ], + "score": 1.0, + "content": "6:", + "type": "text" + }, + { + "bbox": [ + 162, + 502, + 207, + 514 + ], + "score": 1.0, + "content": "~x − ← ~x +", + "type": "text" + }, + { + "bbox": [ + 333, + 505, + 340, + 512 + ], + "score": 0.42, + "content": "\\triangleright", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "set initially to be equal to the positive sample", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 110, + 511, + 360, + 525 + ], + "spans": [ + { + "bbox": [ + 110, + 511, + 122, + 524 + ], + "score": 1.0, + "content": "7:", + "type": "text" + }, + { + "bbox": [ + 162, + 511, + 293, + 525 + ], + "score": 1.0, + "content": "Draw a random feature group value", + "type": "text" + }, + { + "bbox": [ + 294, + 513, + 360, + 524 + ], + "score": 0.93, + "content": "\\tilde { x } \\sim \\mathcal { Q } _ { 2 } ( \\mathrm { X } _ { \\kappa _ { i } , \\alpha _ { 2 } } )", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 110, + 522, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 110, + 522, + 122, + 534 + ], + "score": 1.0, + "content": "8:", + "type": "text" + }, + { + "bbox": [ + 161, + 522, + 238, + 546 + ], + "score": 0.63, + "content": "\\begin{array} { l } { { { \\vec { x } _ { \\kappa _ { i } } ^ { - 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For each observation", + "type": "text" + }, + { + "bbox": [ + 343, + 610, + 357, + 620 + ], + "score": 0.88, + "content": "{ \\bar { x } } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 609, + 457, + 622 + ], + "score": 1.0, + "content": ", it randomly selects the", + "type": "text" + }, + { + "bbox": [ + 457, + 611, + 462, + 620 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "-th feature", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 232, + 633 + ], + "score": 1.0, + "content": "group from a noise distribution", + "type": "text" + }, + { + "bbox": [ + 232, + 621, + 255, + 633 + ], + "score": 0.91, + "content": "\\mathcal { Q } _ { 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 620, + 486, + 633 + ], + "score": 1.0, + "content": ". 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In", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "our application, we use the same class of noise distributions (flattened multinomial) for both levels", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 653, + 323, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 323, + 667 + ], + "score": 1.0, + "content": "of sampling, but this need not necessarily be the case.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 107, + 670, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 669, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 365, + 684 + ], + "score": 1.0, + "content": "We now describe the two noise distributions that we use. 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The function params calculates the complexity of a feature extraction", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 141, + 723 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 710, + 152, + 721 + ], + "score": 0.87, + "content": "\\phi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 709, + 367, + 723 + ], + "score": 1.0, + "content": ". 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By complexity, we mean the number of", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 300, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 301, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 301, + 95 + ], + "score": 1.0, + "content": "3.3 UNSUPERVISED LEARNING FROM DATA", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 116 + ], + "score": 1.0, + "content": "We now discuss how Feat2Vec can be used to learn embeddings in an unsupervised setting with no", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 114, + 223, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 223, + 127 + ], + "score": 1.0, + "content": "explicit target for prediction.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 106, + 102, + 505, + 127 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 131, + 505, + 208 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 505, + 145 + ], + "score": 1.0, + "content": "The training dataset for a Feat2Vec model consists of only the observed data. In natural language,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 155 + ], + "score": 1.0, + "content": "these would be documents written by humans. Since Feat2Vec (Equation 3) requires positive and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "score": 1.0, + "content": "negative examples, we also need to supply unobserved data as negative examples. Consider a feature", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 132, + 177 + ], + "score": 1.0, + "content": "group", + "type": "text" + }, + { + "bbox": [ + 133, + 164, + 143, + 175 + ], + "score": 0.86, + "content": "\\vec { \\mathsf { K } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 164, + 505, + 177 + ], + "score": 1.0, + "content": ", that exists in very high dimensional space. For example, this could happen because we", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 506, + 188 + ], + "score": 1.0, + "content": "are modeling with one-hot encoding a categorical variable with large number of possible values. In", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 199 + ], + "score": 1.0, + "content": "such scenario, it is overwhelmingly costly to feed the model all negative labels, particularly if the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 197, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 197, + 210 + ], + "score": 1.0, + "content": "model is fairly sparse.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 129, + 506, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 214, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "A shortcut around this is a concept known as implicit sampling, where instead of using all of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 355, + 238 + ], + "score": 1.0, + "content": "possible negative labels, one simply samples a fixed number", + "type": "text" + }, + { + "bbox": [ + 355, + 225, + 368, + 236 + ], + "score": 0.71, + "content": "( k )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "from the set of possible negative", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 505, + 249 + ], + "score": 1.0, + "content": "labels for each positively labelled record. Word2Vec makes use of an algorithm called Negative", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "Sampling, that has little theoretical guarantees (Dyer, 2014). In short, their approach samples a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 291, + 271 + ], + "score": 1.0, + "content": "negative observation from a noise distribution", + "type": "text" + }, + { + "bbox": [ + 292, + 258, + 315, + 269 + ], + "score": 0.92, + "content": "\\mathcal { Q } _ { w 2 v }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 257, + 505, + 271 + ], + "score": 1.0, + "content": ", that is proportional to the empirical frequency", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 269, + 227, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 227, + 281 + ], + "score": 1.0, + "content": "of a word in the training data.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 213, + 505, + 281 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "We introduce a new implicit sampling method that enables learning unsupervised embeddings for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "structured feature sets. We can learn the correlation of features within a dataset by imputing negative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "labels, simply by generating unobserved records as our negative samples. Unlike Word2Vec, we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "do not constraint features types to be words. Features groups can be individual columns in a data", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 421, + 342 + ], + "score": 1.0, + "content": "matrix, but they need not to be. By grouping subfeatures using the parameter", + "type": "text" + }, + { + "bbox": [ + 422, + 331, + 429, + 340 + ], + "score": 0.44, + "content": "\\boldsymbol { \\mathsf { K } }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "in Equation 3, the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 354 + ], + "score": 1.0, + "content": "model can reason on more abstract entities in the data. By entity, we mean a particular feature group", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "value. For example, in our experiments on a movie dataset, we use a “genre” feature group, where", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "we group non-mutually exclusive indicators for movie genres including comedy, action, and drama", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 373, + 132, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 132, + 385 + ], + "score": 1.0, + "content": "films.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 285, + 506, + 385 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 390, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 199, + 403 + ], + "score": 1.0, + "content": "We start with a dataset", + "type": "text" + }, + { + "bbox": [ + 199, + 390, + 213, + 401 + ], + "score": 0.89, + "content": "S ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "of records with |~κ| feature groups. We then mark all observed records in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 399, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 410, + 415 + ], + "score": 1.0, + "content": "the training set as positive examples. 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In", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "our application, we use the same class of noise distributions (flattened multinomial) for both levels", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 653, + 323, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 323, + 667 + ], + "score": 1.0, + "content": "of sampling, but this need not necessarily be the case.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 599, + 505, + 667 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 670, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 669, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 365, + 684 + ], + "score": 1.0, + "content": "We now describe the two noise distributions that we use. 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The function params calculates the complexity of a feature extraction", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 141, + 723 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 710, + 152, + 721 + ], + "score": 0.87, + "content": "\\phi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 709, + 367, + 723 + ], + "score": 1.0, + "content": ". 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This choice places", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 108 + ], + "score": 1.0, + "content": "more weight on features that have more parameters and thus are going to require more training", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 432, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 432, + 118 + ], + "score": 1.0, + "content": "iterations to properly learn. 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Figure A.1 in the Appendix provides a visualization of how the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 245 + ], + "score": 1.0, + "content": "feature sampling rate varies with the hyperparameter for features with differing levels of complexity.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 248, + 504, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 451, + 262 + ], + "score": 1.0, + "content": "Sampling Feature Group Values. 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The literature offers two possibilities: in the Negative Sampling that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 364, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 379 + ], + "score": 1.0, + "content": "Word2Vec follows, the duplicate negative samples are simply ignored (Dyer, 2014). 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In such setting,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 378, + 266 + ], + "score": 1.0, + "content": "the output indicates whether the label is associated with the input", + "type": "text" + }, + { + "bbox": [ + 378, + 254, + 415, + 265 + ], + "score": 0.87, + "content": "( y = + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 254, + 451, + 266 + ], + "score": 1.0, + "content": "), or not", + "type": "text" + }, + { + "bbox": [ + 451, + 254, + 482, + 265 + ], + "score": 0.86, + "content": "( y = 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "), and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 265, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 387, + 277 + ], + "score": 1.0, + "content": "therefore the input can be associated with more than one label. With", + "type": "text" + }, + { + "bbox": [ + 387, + 267, + 395, + 275 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 265, + 506, + 277 + ], + "score": 1.0, + "content": "feature types, Feat2Vec is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 399, + 288 + ], + "score": 1.0, + "content": "equivalent to optimizing a convex combination of the loss functions from", + "type": "text" + }, + { + "bbox": [ + 399, + 277, + 407, + 286 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "individual Factorization", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 288, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 266, + 298 + ], + "score": 1.0, + "content": "Machines. 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For the multi-label classification task in 4.1.1 we predict a probability", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "for each document-label pair and use an evaluation metric called Area Under the Curve (AUC) of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "the Receiver Operating Characteristic (ROC). Since we only observe positive labels, for each pos-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "itive label in the test set we sample negative labels according to the label frequency. This ensures", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "that if a model merely predicts the labels according to their popularity, it would have an AUC of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "score": 1.0, + "content": "0.5. A caveat of our evaluation strategy is that we could be underestimating the performance of our", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "models—there is a small probability that the sampled negatives labels are false negatives. However,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "since we apply the same evaluation strategy consistently across our methods and baselines, the rel-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "ative difference of the AUC is meaningful. We choose the AUC as a metric because it is popular", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "for both classification and ranking problems. For the regression task in 4.1.2, we use mean squared", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "error (MSE) as the evaluation metric. In preliminary experiments we noticed that regularization", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 537, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 552 + ], + "score": 1.0, + "content": "slows down convergence with no gains in prediction accuracy, so we avoid overfitting only by using", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 548, + 465, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 465, + 563 + ], + "score": 1.0, + "content": "early stopping. We share most of the code for the experiments online1 for reproducibility.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 375, + 506, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 248, + 578 + ], + "score": 1.0, + "content": "For our feature extraction function", + "type": "text" + }, + { + "bbox": [ + 249, + 567, + 256, + 578 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "for text, we use a Convolutional Neural Network (CNN) that", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "has been shown to be effective for natural language tasks (Kalchbrenner et al., 2014; Weston et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "score": 1.0, + "content": "2014). In Appendix B we describe this network and its hyper-parameters. Instead of tuning the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 599, + 459, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 459, + 612 + ], + "score": 1.0, + "content": "hyper-parameters, we follow previously published guidelines (Zhang & Wallace, 2015).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 567, + 506, + 612 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 624, + 382, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 384, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 384, + 637 + ], + "score": 1.0, + "content": "4.1.1 IS Feat2Vec EFFECTIVE FOR COLD-START PREDICTIONS?", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 644, + 505, + 699 + ], + "lines": [ + { + "bbox": [ + 107, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 107, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "We compare Feat2Vec with an extension of matrix factorization that can generalize to unseen items", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "for text documents, Collaborative Topic Regression (CTR– Wang & Blei (2011)), a method with an", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "open-source Python implementation2. We evaluate them on the CiteULike dataset which consists", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "of pairs of scientific articles and the users who have added them to their personal libraries, and it", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "contains 16,980 unique articles and 5,551 unique users. We use the models to predict users who", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 178, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 192 + ], + "score": 1.0, + "content": "may have added a given article to their library. We compare the performance of Feat2Vec with CTR", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 187, + 471, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 306, + 204 + ], + "score": 1.0, + "content": "using pre-defined cross-validation splits3. 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We use", + "type": "text" + }, + { + "bbox": [ + 306, + 190, + 321, + 200 + ], + "score": 0.85, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 187, + 471, + 204 + ], + "score": 1.0, + "content": "of the training set for early stopping.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 206, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 206, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 430, + 219 + ], + "score": 1.0, + "content": "For CTR we use the hyper-parameters reported by the authors as best, except for", + "type": "text" + }, + { + "bbox": [ + 431, + 209, + 437, + 216 + ], + "score": 0.68, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 206, + 505, + 219 + ], + "score": 1.0, + "content": "which we found", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 353, + 230 + ], + "score": 1.0, + "content": "had a significant impact on training time . We only consider", + "type": "text" + }, + { + "bbox": [ + 353, + 217, + 416, + 229 + ], + "score": 0.93, + "content": "r \\in \\{ 5 , 1 0 , 1 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 218, + 505, + 230 + ], + "score": 1.0, + "content": "and choose the value", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 228, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 506, + 241 + ], + "score": 1.0, + "content": "which gives the best performance for CTR (details in Appendix A.2). On the warm-start condition,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 239, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 253 + ], + "score": 1.0, + "content": "CTR has an AUC of 0.9356; however, it shows significant degradation in performance for unseen", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "documents and it only performs slightly better than random chance with an AUC of 0.5047. On the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "other hand, Feat2Vec achieves AUC of 0.9401 on the warm-start condition, and it only degrades", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "to 0.9124 on unseen documents. Feat2Vec can also be trained over ten times faster, since it can", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 282, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 296 + ], + "score": 1.0, + "content": "leverage GPUs.4 We also note that we have not tuned the architecture or hyper-parameters of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 295, + 461, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 214, + 307 + ], + "score": 1.0, + "content": "feature extraction function", + "type": "text" + }, + { + "bbox": [ + 214, + 295, + 222, + 306 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 295, + 461, + 307 + ], + "score": 1.0, + "content": "and greater improvements are possible by optimizing them.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 318, + 437, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 439, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 439, + 331 + ], + "score": 1.0, + "content": "4.1.2 COMPARISON WITH ALTERNATIVE CNN-BASED TEXT FACTORIZATION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 337, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 350 + ], + "score": 1.0, + "content": "We now compare with a method called DeepCoNN, a deep network specifically designed for in-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "corporating text into matrix factorization (Zheng et al., 2017)—which reportedly, is the state of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "score": 1.0, + "content": "art for predicting customer ratings when textual reviews are available. For Feat2Vec we use the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "same feature extraction function (see Appendix B.1 for details) used by DeepCoNN. We evaluate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 380, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 394 + ], + "score": 1.0, + "content": "on the Yelp dataset5, which consists of 4.7 million reviews of restaurants. For each user-item pair,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "DeepCoNN concatenates the text from all reviews for that item and all reviews by that user. The", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 404, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 505, + 414 + ], + "score": 1.0, + "content": "concatenated text is fed into a feature extraction function followed by a factorization machine. In", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "contrast, for Feat2Vec, we build 3 feature groups: item identifiers (in this case, restaurants), users", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 426, + 172, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 172, + 436 + ], + "score": 1.0, + "content": "and review text.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "Table 1 compares our methods to DeepCoNN’s published results because a public implementation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 104, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "is not available. We see that Feat2Vec provides a large performance increase when comparing the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "score": 1.0, + "content": "reported improvement, over Matrix Factorization, of the mean squared error. Our approach is more", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "general, and we claim that it is also more efficient. 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In contrast, for Feat2Vec each review is seen only once per epoch. Thus", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 507, + 443, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 374, + 521 + ], + "score": 1.0, + "content": "it can be 1-2 orders of magnitude more efficient for datasets where", + "type": "text" + }, + { + "bbox": [ + 374, + 509, + 408, + 519 + ], + "score": 0.91, + "content": "\\bar { n _ { i } } \\times \\bar { n _ { u } }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 507, + 443, + 521 + ], + "score": 1.0, + "content": "is large.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 533, + 276, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 278, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 278, + 545 + ], + "score": 1.0, + "content": "4.2 GENERAL-PURPOSE EMBEDDINGS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 107, + 553, + 344, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 553, + 345, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 345, + 565 + ], + "score": 1.0, + "content": "4.2.1 DOES Feat2Vec ENABLE BETTER EMBEDDINGS?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 504, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 504, + 585 + ], + "score": 1.0, + "content": "Ex ante, it is unclear to us how to evaluate the performance of an unsupervised embedding algorithm,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "but we felt that a reasonable task would be a ranking task one might practically attempt using our", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "datasets. This task will assess the similarity of trained embeddings using unseen records in a left-out", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "dataset. In order to test the relative performance of our learned embeddings, we train our unsuper-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 617, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 506, + 630 + ], + "score": 1.0, + "content": "vised Feat2Vec algorithm and compare its performance in a targeted ranking task to Word2Vec’s", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "CBOW algorithm for learning embeddings. 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MSEImprovement over Matrix Factorization
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We only consider", + "type": "text" + }, + { + "bbox": [ + 353, + 217, + 416, + 229 + ], + "score": 0.93, + "content": "r \\in \\{ 5 , 1 0 , 1 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 218, + 505, + 230 + ], + "score": 1.0, + "content": "and choose the value", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 228, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 506, + 241 + ], + "score": 1.0, + "content": "which gives the best performance for CTR (details in Appendix A.2). On the warm-start condition,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 239, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 253 + ], + "score": 1.0, + "content": "CTR has an AUC of 0.9356; however, it shows significant degradation in performance for unseen", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "documents and it only performs slightly better than random chance with an AUC of 0.5047. On the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "other hand, Feat2Vec achieves AUC of 0.9401 on the warm-start condition, and it only degrades", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "to 0.9124 on unseen documents. Feat2Vec can also be trained over ten times faster, since it can", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 282, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 296 + ], + "score": 1.0, + "content": "leverage GPUs.4 We also note that we have not tuned the architecture or hyper-parameters of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 295, + 461, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 214, + 307 + ], + "score": 1.0, + "content": "feature extraction function", + "type": "text" + }, + { + "bbox": [ + 214, + 295, + 222, + 306 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 295, + 461, + 307 + ], + "score": 1.0, + "content": "and greater improvements are possible by optimizing them.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 206, + 506, + 307 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 318, + 437, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 439, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 439, + 331 + ], + "score": 1.0, + "content": "4.1.2 COMPARISON WITH ALTERNATIVE CNN-BASED TEXT FACTORIZATION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 337, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 350 + ], + "score": 1.0, + "content": "We now compare with a method called DeepCoNN, a deep network specifically designed for in-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "corporating text into matrix factorization (Zheng et al., 2017)—which reportedly, is the state of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "score": 1.0, + "content": "art for predicting customer ratings when textual reviews are available. For Feat2Vec we use the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "same feature extraction function (see Appendix B.1 for details) used by DeepCoNN. We evaluate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 380, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 394 + ], + "score": 1.0, + "content": "on the Yelp dataset5, which consists of 4.7 million reviews of restaurants. For each user-item pair,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "DeepCoNN concatenates the text from all reviews for that item and all reviews by that user. The", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 404, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 505, + 414 + ], + "score": 1.0, + "content": "concatenated text is fed into a feature extraction function followed by a factorization machine. In", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "contrast, for Feat2Vec, we build 3 feature groups: item identifiers (in this case, restaurants), users", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 426, + 172, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 172, + 436 + ], + "score": 1.0, + "content": "and review text.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 336, + 506, + 436 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "Table 1 compares our methods to DeepCoNN’s published results because a public implementation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 104, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "is not available. We see that Feat2Vec provides a large performance increase when comparing the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "score": 1.0, + "content": "reported improvement, over Matrix Factorization, of the mean squared error. Our approach is more", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "general, and we claim that it is also more efficient. Since DeepCoNN concatenates text, when the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 484, + 504, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 214, + 500 + ], + "score": 1.0, + "content": "average reviews per user is", + "type": "text" + }, + { + "bbox": [ + 215, + 487, + 227, + 497 + ], + "score": 0.89, + "content": "\\bar { n _ { u } }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 484, + 321, + 500 + ], + "score": 1.0, + "content": "and reviews per item is", + "type": "text" + }, + { + "bbox": [ + 322, + 487, + 332, + 497 + ], + "score": 0.87, + "content": "\\bar { n _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 484, + 471, + 500 + ], + "score": 1.0, + "content": ", each text is duplicated on average", + "type": "text" + }, + { + "bbox": [ + 471, + 486, + 504, + 497 + ], + "score": 0.9, + "content": "\\bar { n _ { i } } \\times \\bar { n _ { u } }", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "times per training epoch. In contrast, for Feat2Vec each review is seen only once per epoch. Thus", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 507, + 443, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 374, + 521 + ], + "score": 1.0, + "content": "it can be 1-2 orders of magnitude more efficient for datasets where", + "type": "text" + }, + { + "bbox": [ + 374, + 509, + 408, + 519 + ], + "score": 0.91, + "content": "\\bar { n _ { i } } \\times \\bar { n _ { u } }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 507, + 443, + 521 + ], + "score": 1.0, + "content": "is large.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 442, + 506, + 521 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 533, + 276, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 278, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 278, + 545 + ], + "score": 1.0, + "content": "4.2 GENERAL-PURPOSE EMBEDDINGS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 107, + 553, + 344, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 553, + 345, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 345, + 565 + ], + "score": 1.0, + "content": "4.2.1 DOES Feat2Vec ENABLE BETTER EMBEDDINGS?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 504, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 504, + 585 + ], + "score": 1.0, + "content": "Ex ante, it is unclear to us how to evaluate the performance of an unsupervised embedding algorithm,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "but we felt that a reasonable task would be a ranking task one might practically attempt using our", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "datasets. 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It", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 269, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 104, + 269, + 506, + 285 + ], + "score": 1.0, + "content": "contains information on writers, directors, and principal cast members attached to each film, along", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 168, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 168, + 294 + ], + "score": 1.0, + "content": "with metadata.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 238, + 506, + 294 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 305, + 504, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "Education We use a dataset from an anonymized leading technology company that provides ed-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "ucational services. 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Even though they intend to be able to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "embed all types of features, at the time of the writing of this paper, their pre-print method was lim-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "ited to only work for bag of words. While Feat2Vec can jointly learn embeddings for all feature", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 702 + ], + "score": 1.0, + "content": "values in a dataset, StarSpace samples a single arbitrary feature. Our preliminary experiments sug-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "gest that sampling a single feature does not produce embeddings that generalize well. 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This suggests that the need for ad-hoc networks should be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 201, + 412, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 412, + 213 + ], + "score": 1.0, + "content": "situated in relationship to the improvements over a general-purpose method.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 218, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "In the unsupervised setting, Feat2Vec’s embeddings are able to capture relationships across features", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 229, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 240 + ], + "score": 1.0, + "content": "that can be twice as better as Word2Vec’s CBOW algorithm on some evaluation metrics. Feat2Vec", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "exploits the structure of a datasets to learn embeddings in a way that is structurally more sensible", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 251, + 504, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 504, + 263 + ], + "score": 1.0, + "content": "than existing methods. The sampling method, and loss function that we use have interesting theo-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 273 + ], + "score": 1.0, + "content": "retical properties. 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These ideas can extend to our codebase that we make available 8. Overall, we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "evaluate supervised and unsupervised Feat2Vec on 2 datasets each. Though further experimentation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 104, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "is necessary, we believe that our results are an encouraging step towards general-purpose embedding", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 344, + 141, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 141, + 356 + ], + "score": 1.0, + "content": "models.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 107, + 374, + 175, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 372, + 176, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 176, + 387 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 105, + 393, + 506, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. 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Runtime (minutes)Real-valued1116
IMDB rating (0-10)Real-valued17.8
# of IMDB rating votesReal-valued1435,682
Is adult film?Boolean2False
Movie releaes yearCategorical2712001
Movie titleText165,471“Ocean's”,“Eleven”
DirectorsBag of categories174,382‘Steven Soderbergh”
GenresBag of categories28“Crime”,“Thriller”
WritersBag of categories244,241“George Johnson”,“Jack Russell"
Principal cast members (actors)Bag of categories1,104,280“George Clooney”,“Brad Pitt”,“Julia Roberts”
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Feature Type NameType# of feats.Example for an instance
Runtime (minutes)Real-valued1116
IMDB rating (0-10)Real-valued17.8
# of IMDB rating votesReal-valued1435,682
Is adult film?Boolean2False
Movie releaes yearCategorical2712001
Movie titleText165,471“Ocean's”,“Eleven”
DirectorsBag of categories174,382‘Steven Soderbergh”
GenresBag of categories28“Crime”,“Thriller”
WritersBag of categories244,241“George Johnson”,“Jack Russell"
Principal cast members (actors)Bag of categories1,104,280“George Clooney”,“Brad Pitt”,“Julia Roberts”
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We learn", + "type": "text" + }, + { + "bbox": [ + 323, + 133, + 355, + 143 + ], + "score": 0.88, + "content": "r = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 133, + 505, + 144 + ], + "score": 1.0, + "content": "dimensional embeddings under both", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 155, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 155, + 156 + ], + "score": 1.0, + "content": "algorithms.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 504, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 173 + ], + "score": 1.0, + "content": "Below we describe how CBOW is implemented on our datasets for unsupervised experiments and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 415, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 415, + 183 + ], + "score": 1.0, + "content": "what extraction functions are used to represent features in the IMDB dataset.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 195, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 194, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 208 + ], + "score": 1.0, + "content": "Word2Vec For every observation in each of the datasets, we create a document that tokenizes the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 204, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 221 + ], + "score": 1.0, + "content": "same information that we feed into Feat2Vec. We prepend each feature value by its feature name,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "score": 1.0, + "content": "and we remove spaces from within features. In Figure A.2 we show an example document. Some", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "features may allow multiple values (e.g., multiple writers, directors). To feed these features into the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "models, for convenience, we constraint the number of values, by truncating each feature to no more", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 251, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 505, + 262 + ], + "score": 1.0, + "content": "than 10 levels (and sometimes less if reasonable). This results in retaining the full set of information", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 160, + 275 + ], + "score": 1.0, + "content": "for well over", + "type": "text" + }, + { + "bbox": [ + 161, + 261, + 181, + 272 + ], + "score": 0.88, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 260, + 505, + 275 + ], + "score": 1.0, + "content": "of the values. We pad the sequences with a “null” category whenever necessary", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "to maintain a fixed length. We do this consistently for both Word2Vec and Feat2Vec. We use", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "the CBOW Word2Vec algorithm and set the context window to encompass all other tokens in a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 294, + 396, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 396, + 307 + ], + "score": 1.0, + "content": "document during training, since the text in this application is unordered.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12.5 + }, + { + "type": "image", + "bbox": [ + 197, + 322, + 415, + 339 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 197, + 322, + 415, + 339 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 197, + 322, + 415, + 339 + ], + "spans": [ + { + "bbox": [ + 197, + 322, + 415, + 339 + ], + "score": 0.894, + "type": "image", + "image_path": "330946bb7dcbf02917794a98c168e5aef01780c17fee4188796103aa1971d0ff.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 197, + 322, + 415, + 339 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 153, + 355, + 457, + 367 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 152, + 354, + 459, + 368 + ], + "spans": [ + { + "bbox": [ + 152, + 354, + 459, + 368 + ], + "score": 1.0, + "content": "Figure A.2: Sample document for Word2Vec for the Ocean’s Eleven movie", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 504, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "Feat2Vec Feature representation in Feat2Vec requires a feature extraction function for each fea-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 403, + 338, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 338, + 415 + ], + "score": 1.0, + "content": "ture type. Here, we explain how we build these functions:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 133, + 424, + 504, + 501 + ], + "lines": [ + { + "bbox": [ + 135, + 424, + 504, + 437 + ], + "spans": [ + { + "bbox": [ + 135, + 424, + 504, + 437 + ], + "score": 1.0, + "content": "• Bag of categories, categorical, and boolean: For all of the categorical variables, we learn", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 140, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 140, + 435, + 181, + 448 + ], + "score": 1.0, + "content": "a unique", + "type": "text" + }, + { + "bbox": [ + 182, + 437, + 188, + 445 + ], + "score": 0.73, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "-dimensional embedding for each entity using a linear fully-connected layer", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 142, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "(Equation 4). We do not require one-hot encodings, and thus we allow multiple categories", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 456, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 141, + 456, + 505, + 470 + ], + "score": 1.0, + "content": "to be active; resulting in a single embedding for the group that is the sum of the embeddings", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 142, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "of the subfeatures. This is ordering-invariant: the embedding of “Brad Pitt” would be the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 141, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "same when he appears in a movie as a principal cast member, regardless whether he was 1st", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 489, + 499, + 503 + ], + "spans": [ + { + "bbox": [ + 141, + 489, + 499, + 503 + ], + "score": 1.0, + "content": "or 2nd star. Though, if he were listed as a director it may result in a different embedding.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 133, + 505, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 132, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 132, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "• Text: We preprocess the text by removing non alpha-numeric characters, stopwords, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 141, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "stemming the remaining words. We then follow the same approach that we did for cate-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 528, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 141, + 528, + 506, + 539 + ], + "score": 1.0, + "content": "gorical variables, summing learned word embeddings to a “title embedding” before inter-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 538, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 141, + 538, + 506, + 551 + ], + "score": 1.0, + "content": "acting. It would be easy to use more sophisticated methods (e.g, convolutions), but we felt", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 549, + 312, + 560 + ], + "spans": [ + { + "bbox": [ + 142, + 549, + 312, + 560 + ], + "score": 1.0, + "content": "this would not extract further information.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 133, + 565, + 505, + 642 + ], + "lines": [ + { + "bbox": [ + 133, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 133, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "• Real-valued: For all real-valued features, we pass these features through a 3-layer feed-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 141, + 575, + 452, + 588 + ], + "score": 1.0, + "content": "forward fully connected neural network that outputs a vector of dimension", + "type": "text" + }, + { + "bbox": [ + 452, + 578, + 458, + 586 + ], + "score": 0.67, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 575, + 505, + 588 + ], + "score": 1.0, + "content": ", which we", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 142, + 587, + 504, + 598 + ], + "spans": [ + { + "bbox": [ + 142, + 587, + 387, + 598 + ], + "score": 1.0, + "content": "treat as the feature’s embedding. Each intermediate layer has", + "type": "text" + }, + { + "bbox": [ + 388, + 588, + 394, + 596 + ], + "score": 0.69, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 587, + 437, + 598 + ], + "score": 1.0, + "content": "units with", + "type": "text" + }, + { + "bbox": [ + 438, + 587, + 462, + 597 + ], + "score": 0.32, + "content": "\\mathtt { r e l u }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 587, + 504, + 598 + ], + "score": 1.0, + "content": "activation", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 141, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "functions. These real-valued features highlight one of the advantages of the Feat2Vec al-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 141, + 608, + 506, + 621 + ], + "score": 1.0, + "content": "gorithm: using a numeric value as an input, Feat2Vec can learn a highly nonlinear relation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 619, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 141, + 619, + 506, + 631 + ], + "score": 1.0, + "content": "mapping a real number to our high-dimensional embedding space. In contrast, Word2Vec", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 142, + 631, + 451, + 642 + ], + "spans": [ + { + "bbox": [ + 142, + 631, + 451, + 642 + ], + "score": 1.0, + "content": "would be unable to know ex ante that an IMDB rating of 5.5 is similar to 5.6.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 108, + 655, + 335, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 336, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 336, + 668 + ], + "score": 1.0, + "content": "A.2 DISTRIBUTION OF IMDB DIRECTOR RANKINGS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Figure A.3 shows the full distribution of rankings of the IMDB dataset, rather than summary statis-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "tics, in the form of a Cumulative Distribution Function (CDF) of all rankings calculated in the test", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "dataset. The graphic makes it apparent for the vast majority of the ranking space, the rank CDF of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "Feat2Vec is to the left of CBOW, indicating a greater probability of a lower ranking under Feat2Vec.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "This is not, however, the case at the upper tail of ranking space, where it appears CBOW is superior.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 506, + 108 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 154 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 227, + 123 + ], + "score": 1.0, + "content": "For training Feat2Vec we set", + "type": "text" + }, + { + "bbox": [ + 227, + 110, + 294, + 123 + ], + "score": 0.91, + "content": "\\alpha _ { 1 } = \\alpha _ { 2 } = 3 / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 110, + 400, + 123 + ], + "score": 1.0, + "content": "in the IMDB dataset; and", + "type": "text" + }, + { + "bbox": [ + 400, + 111, + 432, + 122 + ], + "score": 0.91, + "content": "\\alpha _ { 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 110, + 450, + 123 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 450, + 111, + 489, + 122 + ], + "score": 0.91, + "content": "\\alpha _ { 2 } = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 110, + 506, + 123 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 237, + 133 + ], + "score": 1.0, + "content": "the educational. In each setting,", + "type": "text" + }, + { + "bbox": [ + 237, + 123, + 249, + 132 + ], + "score": 0.83, + "content": "\\alpha _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 120, + 505, + 133 + ], + "score": 1.0, + "content": "is set to the same flattening hyperparameter we use for CBOW", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 133, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 323, + 144 + ], + "score": 1.0, + "content": "to negatively sample words in a document. We learn", + "type": "text" + }, + { + "bbox": [ + 323, + 133, + 355, + 143 + ], + "score": 0.88, + "content": "r = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 133, + 505, + 144 + ], + "score": 1.0, + "content": "dimensional embeddings under both", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 155, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 155, + 156 + ], + "score": 1.0, + "content": "algorithms.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 110, + 506, + 156 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 504, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 173 + ], + "score": 1.0, + "content": "Below we describe how CBOW is implemented on our datasets for unsupervised experiments and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 415, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 415, + 183 + ], + "score": 1.0, + "content": "what extraction functions are used to represent features in the IMDB dataset.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 159, + 505, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 195, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 194, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 208 + ], + "score": 1.0, + "content": "Word2Vec For every observation in each of the datasets, we create a document that tokenizes the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 204, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 221 + ], + "score": 1.0, + "content": "same information that we feed into Feat2Vec. We prepend each feature value by its feature name,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "score": 1.0, + "content": "and we remove spaces from within features. In Figure A.2 we show an example document. Some", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "features may allow multiple values (e.g., multiple writers, directors). To feed these features into the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "models, for convenience, we constraint the number of values, by truncating each feature to no more", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 251, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 505, + 262 + ], + "score": 1.0, + "content": "than 10 levels (and sometimes less if reasonable). This results in retaining the full set of information", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 160, + 275 + ], + "score": 1.0, + "content": "for well over", + "type": "text" + }, + { + "bbox": [ + 161, + 261, + 181, + 272 + ], + "score": 0.88, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 260, + 505, + 275 + ], + "score": 1.0, + "content": "of the values. We pad the sequences with a “null” category whenever necessary", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "to maintain a fixed length. We do this consistently for both Word2Vec and Feat2Vec. We use", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "the CBOW Word2Vec algorithm and set the context window to encompass all other tokens in a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 294, + 396, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 396, + 307 + ], + "score": 1.0, + "content": "document during training, since the text in this application is unordered.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 194, + 506, + 307 + ] + }, + { + "type": "image", + "bbox": [ + 197, + 322, + 415, + 339 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 197, + 322, + 415, + 339 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 197, + 322, + 415, + 339 + ], + "spans": [ + { + "bbox": [ + 197, + 322, + 415, + 339 + ], + "score": 0.894, + "type": "image", + "image_path": "330946bb7dcbf02917794a98c168e5aef01780c17fee4188796103aa1971d0ff.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 197, + 322, + 415, + 339 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 153, + 355, + 457, + 367 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 152, + 354, + 459, + 368 + ], + "spans": [ + { + "bbox": [ + 152, + 354, + 459, + 368 + ], + "score": 1.0, + "content": "Figure A.2: Sample document for Word2Vec for the Ocean’s Eleven movie", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 504, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "Feat2Vec Feature representation in Feat2Vec requires a feature extraction function for each fea-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 403, + 338, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 338, + 415 + ], + "score": 1.0, + "content": "ture type. Here, we explain how we build these functions:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 391, + 505, + 415 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 424, + 504, + 501 + ], + "lines": [ + { + "bbox": [ + 135, + 424, + 504, + 437 + ], + "spans": [ + { + "bbox": [ + 135, + 424, + 504, + 437 + ], + "score": 1.0, + "content": "• Bag of categories, categorical, and boolean: For all of the categorical variables, we learn", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 140, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 140, + 435, + 181, + 448 + ], + "score": 1.0, + "content": "a unique", + "type": "text" + }, + { + "bbox": [ + 182, + 437, + 188, + 445 + ], + "score": 0.73, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "-dimensional embedding for each entity using a linear fully-connected layer", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 142, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "(Equation 4). We do not require one-hot encodings, and thus we allow multiple categories", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 456, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 141, + 456, + 505, + 470 + ], + "score": 1.0, + "content": "to be active; resulting in a single embedding for the group that is the sum of the embeddings", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 142, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "of the subfeatures. This is ordering-invariant: the embedding of “Brad Pitt” would be the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 141, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "same when he appears in a movie as a principal cast member, regardless whether he was 1st", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 489, + 499, + 503 + ], + "spans": [ + { + "bbox": [ + 141, + 489, + 499, + 503 + ], + "score": 1.0, + "content": "or 2nd star. Though, if he were listed as a director it may result in a different embedding.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 135, + 424, + 505, + 503 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 505, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 132, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 132, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "• Text: We preprocess the text by removing non alpha-numeric characters, stopwords, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 141, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "stemming the remaining words. We then follow the same approach that we did for cate-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 528, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 141, + 528, + 506, + 539 + ], + "score": 1.0, + "content": "gorical variables, summing learned word embeddings to a “title embedding” before inter-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 538, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 141, + 538, + 506, + 551 + ], + "score": 1.0, + "content": "acting. It would be easy to use more sophisticated methods (e.g, convolutions), but we felt", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 549, + 312, + 560 + ], + "spans": [ + { + "bbox": [ + 142, + 549, + 312, + 560 + ], + "score": 1.0, + "content": "this would not extract further information.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 132, + 504, + 506, + 560 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 565, + 505, + 642 + ], + "lines": [ + { + "bbox": [ + 133, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 133, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "• Real-valued: For all real-valued features, we pass these features through a 3-layer feed-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 141, + 575, + 452, + 588 + ], + "score": 1.0, + "content": "forward fully connected neural network that outputs a vector of dimension", + "type": "text" + }, + { + "bbox": [ + 452, + 578, + 458, + 586 + ], + "score": 0.67, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 575, + 505, + 588 + ], + "score": 1.0, + "content": ", which we", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 142, + 587, + 504, + 598 + ], + "spans": [ + { + "bbox": [ + 142, + 587, + 387, + 598 + ], + "score": 1.0, + "content": "treat as the feature’s embedding. Each intermediate layer has", + "type": "text" + }, + { + "bbox": [ + 388, + 588, + 394, + 596 + ], + "score": 0.69, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 587, + 437, + 598 + ], + "score": 1.0, + "content": "units with", + "type": "text" + }, + { + "bbox": [ + 438, + 587, + 462, + 597 + ], + "score": 0.32, + "content": "\\mathtt { r e l u }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 587, + 504, + 598 + ], + "score": 1.0, + "content": "activation", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 141, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "functions. These real-valued features highlight one of the advantages of the Feat2Vec al-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 141, + 608, + 506, + 621 + ], + "score": 1.0, + "content": "gorithm: using a numeric value as an input, Feat2Vec can learn a highly nonlinear relation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 619, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 141, + 619, + 506, + 631 + ], + "score": 1.0, + "content": "mapping a real number to our high-dimensional embedding space. In contrast, Word2Vec", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 142, + 631, + 451, + 642 + ], + "spans": [ + { + "bbox": [ + 142, + 631, + 451, + 642 + ], + "score": 1.0, + "content": "would be unable to know ex ante that an IMDB rating of 5.5 is similar to 5.6.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37, + "bbox_fs": [ + 133, + 564, + 506, + 642 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 655, + 335, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 336, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 336, + 668 + ], + "score": 1.0, + "content": "A.2 DISTRIBUTION OF IMDB DIRECTOR RANKINGS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Figure A.3 shows the full distribution of rankings of the IMDB dataset, rather than summary statis-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "tics, in the form of a Cumulative Distribution Function (CDF) of all rankings calculated in the test", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "dataset. The graphic makes it apparent for the vast majority of the ranking space, the rank CDF of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "Feat2Vec is to the left of CBOW, indicating a greater probability of a lower ranking under Feat2Vec.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "This is not, however, the case at the upper tail of ranking space, where it appears CBOW is superior.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 677, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "However, when we zoom-in on the absolute upper region of rankings (1 to 25), which might be a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "sensible length of ranks one might give as actual recommendatiosn, it is the case that up until rank", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "8 or so, Feat2Vec outperforms CBOW still. Intermediate rankings are still strong signals that our", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "Feat2Vec algorithm is doing a better job of extracting information into embeddings, particularly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 474, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 474, + 140 + ], + "score": 1.0, + "content": "those entities that appear sparsely in the training data and so are especially difficult to learn.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "image", + "bbox": [ + 175, + 195, + 423, + 391 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 175, + 195, + 423, + 391 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 175, + 195, + 423, + 391 + ], + "spans": [ + { + "bbox": [ + 175, + 195, + 423, + 391 + ], + "score": 0.971, + "type": "image", + "image_path": "fe8ebcf2f1729c2c77fccf48acc7e972098c21fdd9cc58b0c0592dc3fc7497e5.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 175, + 195, + 423, + 260.3333333333333 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 175, + 260.3333333333333, + 423, + 325.66666666666663 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 175, + 325.66666666666663, + 423, + 390.99999999999994 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 411, + 379, + 434 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 410, + 379, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 379, + 425 + ], + "score": 1.0, + "content": "Figure A.3: Cumulative Distribution Function of Director Rankings", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 422, + 241, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 241, + 436 + ], + "score": 1.0, + "content": "(With Zoom-in to Top 25 Ranks)", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 7.25 + }, + { + "type": "title", + "bbox": [ + 108, + 510, + 230, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 231, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 231, + 524 + ], + "score": 1.0, + "content": "A.3 PROOF TO THEOREM 1", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "Theorem 1. The gradient for learning embeddings with Feat2Vec is a convex combination of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "gradient from n targeted Factorization Machines for each feature in the data when each feature", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 565, + 406, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 406, + 578 + ], + "score": 1.0, + "content": "group is a singleton, where n is the total number of features in the dataset.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 674, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 104, + 672, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 672, + 152, + 691 + ], + "score": 1.0, + "content": "Proof. 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The gradient for learning embeddings with Feat2Vec is a convex combination of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "gradient from n targeted Factorization Machines for each feature in the data when each feature", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 565, + 406, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 406, + 578 + ], + "score": 1.0, + "content": "group is a singleton, where n is the total number of features in the dataset.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 542, + 505, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 674, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 104, + 672, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 672, + 152, + 691 + ], + "score": 1.0, + "content": "Proof. 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We also apply a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 517, + 318, + 530 + ], + "spans": [ + { + "bbox": [ + 141, + 517, + 318, + 530 + ], + "score": 1.0, + "content": "ReLU activation to the output of each filter.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 129, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 129, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "3. Consider the case of inputs of different lengths. For very short texts, the output of the filters", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 142, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "will be mostly zero since the input is zero-padded. To enforce learning from the features of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 141, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "the text, and not just its length we apply a function called 1-max pooling to the output of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 141, + 565, + 223, + 578 + ], + "score": 1.0, + "content": "the filters: from the", + "type": "text" + }, + { + "bbox": [ + 223, + 566, + 267, + 576 + ], + "score": 0.91, + "content": "t - m + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "output vector of each filter, we select the maximum value.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 141, + 576, + 264, + 590 + ], + "score": 1.0, + "content": "This yields a vector of length", + "type": "text" + }, + { + "bbox": [ + 264, + 577, + 273, + 586 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 576, + 506, + 590 + ], + "score": 1.0, + "content": ", a representation of the passage which is independent of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 588, + 184, + 600 + ], + "spans": [ + { + "bbox": [ + 141, + 588, + 184, + 600 + ], + "score": 1.0, + "content": "its length.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 128, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 128, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "4. We learn higher-level features from the convolutional filters. For this, we use a fully con-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 614, + 337, + 626 + ], + "spans": [ + { + "bbox": [ + 141, + 614, + 213, + 626 + ], + "score": 1.0, + "content": "nected layer with", + "type": "text" + }, + { + "bbox": [ + 213, + 616, + 220, + 626 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 614, + 337, + 626 + ], + "score": 1.0, + "content": "units and a ReLU activation,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 129, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 129, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "5. 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We use", + "type": "text" + }, + { + "bbox": [ + 392, + 290, + 423, + 300 + ], + "score": 0.74, + "content": "\\mathrm { f } { = } 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "convolutional filters", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 300, + 205, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 205, + 313 + ], + "score": 1.0, + "content": "each of width 3 (words)", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + } + ], + "index": 6.25 + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 356, + 340 + ], + "score": 1.0, + "content": "Here we describe the details of the feature extraction function", + "type": "text" + }, + { + "bbox": [ + 357, + 328, + 364, + 339 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "used in our experiments for super-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 163, + 351 + ], + "score": 1.0, + "content": "vised tasks in", + "type": "text" + }, + { + "bbox": [ + 163, + 339, + 181, + 350 + ], + "score": 0.84, + "content": "\\ S 4 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 338, + 506, + 351 + ], + "score": 1.0, + "content": ". An overview of the network is given in Fig. A.4. We choose the most common", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 311, + 362 + ], + "score": 1.0, + "content": "words of each dataset to build a vocabulary of size", + "type": "text" + }, + { + "bbox": [ + 312, + 351, + 319, + 360 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 349, + 506, + 362 + ], + "score": 1.0, + "content": ", and convert the words of each document to a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 183, + 372 + ], + "score": 1.0, + "content": "sequence of length", + "type": "text" + }, + { + "bbox": [ + 183, + 361, + 189, + 370 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 360, + 476, + 372 + ], + "score": 1.0, + "content": "of one-hot encodings of the input words. 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We also apply a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 517, + 318, + 530 + ], + "spans": [ + { + "bbox": [ + 141, + 517, + 318, + 530 + ], + "score": 1.0, + "content": "ReLU activation to the output of each filter.", + "type": "text" + } + ], + "index": 25, + "is_list_end_line": true + }, + { + "bbox": [ + 129, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 129, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "3. Consider the case of inputs of different lengths. For very short texts, the output of the filters", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 142, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "will be mostly zero since the input is zero-padded. To enforce learning from the features of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 141, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "the text, and not just its length we apply a function called 1-max pooling to the output of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 141, + 565, + 223, + 578 + ], + "score": 1.0, + "content": "the filters: from the", + "type": "text" + }, + { + "bbox": [ + 223, + 566, + 267, + 576 + ], + "score": 0.91, + "content": "t - m + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "output vector of each filter, we select the maximum value.", + "type": "text" + } + ], + "index": 29, + "is_list_end_line": true + }, + { + "bbox": [ + 141, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 141, + 576, + 264, + 590 + ], + "score": 1.0, + "content": "This yields a vector of length", + "type": "text" + }, + { + "bbox": [ + 264, + 577, + 273, + 586 + ], + "score": 0.82, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 576, + 506, + 590 + ], + "score": 1.0, + "content": ", a representation of the passage which is independent of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 588, + 184, + 600 + ], + "spans": [ + { + "bbox": [ + 141, + 588, + 184, + 600 + ], + "score": 1.0, + "content": "its length.", + "type": "text" + } + ], + "index": 31, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 128, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "4. We learn higher-level features from the convolutional filters. For this, we use a fully con-", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 614, + 337, + 626 + ], + "spans": [ + { + "bbox": [ + 141, + 614, + 213, + 626 + ], + "score": 1.0, + "content": "nected layer with", + "type": "text" + }, + { + "bbox": [ + 213, + 616, + 220, + 626 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 614, + 337, + 626 + ], + "score": 1.0, + "content": "units and a ReLU activation,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 129, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 129, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "5. During training (not in inference), we prevent the units from co-adapting too much with", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 141, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "a dropout layer (Srivastava et al., 2014). Dropout is a form of regularization that for each", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 651, + 379, + 664 + ], + "spans": [ + { + "bbox": [ + 141, + 651, + 379, + 664 + ], + "score": 1.0, + "content": "mini-batch randomly drops a specified percentage of units.", + "type": "text" + } + ], + "index": 36, + "is_list_end_line": true + }, + { + "bbox": [ + 129, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 129, + 666, + 239, + 679 + ], + "score": 1.0, + "content": "6. the final embedding for", + "type": "text" + }, + { + "bbox": [ + 239, + 668, + 250, + 679 + ], + "score": 0.86, + "content": "x _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "(that is used in the factorization) is computed by a dense layer", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 679, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 142, + 679, + 162, + 689 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 163, + 680, + 169, + 687 + ], + "score": 0.72, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 679, + 354, + 689 + ], + "score": 1.0, + "content": "output units and an activation function, where", + "type": "text" + }, + { + "bbox": [ + 355, + 680, + 361, + 687 + ], + "score": 0.76, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 679, + 505, + 689 + ], + "score": 1.0, + "content": "is the embedding size of our index-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 142, + 688, + 188, + 700 + ], + "spans": [ + { + "bbox": [ + 142, + 688, + 188, + 700 + ], + "score": 1.0, + "content": "able items.", + "type": "text" + } + ], + "index": 39, + "is_list_end_line": true + } + ], + "index": 27.5, + "bbox_fs": [ + 128, + 413, + 506, + 700 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 268, + 722 + ], + "score": 1.0, + "content": "We set the maximum vocabulary size", + "type": "text" + }, + { + "bbox": [ + 268, + 712, + 276, + 720 + ], + "score": 0.68, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 709, + 470, + 722 + ], + "score": 1.0, + "content": "to 100,000 words, and input embedding size", + "type": "text" + }, + { + "bbox": [ + 471, + 710, + 478, + 720 + ], + "score": 0.77, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "to 50", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "for all experiments. We initialize the input word embeddings and the label embeddings using", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 709, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 181 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "Word2Vec(Mikolov et al., 2013) We have have not evaluated multiple architectures or hyper-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "parameter settings and obtain good results on diverse datasets with the same architecture, which was", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "designed followed recommendations from a large scale evaluation of CNN hyper parameters(Zhang", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 350, + 127 + ], + "score": 1.0, + "content": "& Wallace, 2015). We set the number of convolutional filters", + "type": "text" + }, + { + "bbox": [ + 350, + 116, + 357, + 127 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "to 1,000, and the dropout rate to 0.1.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 234, + 139 + ], + "score": 1.0, + "content": "The maximum sequence length", + "type": "text" + }, + { + "bbox": [ + 235, + 127, + 240, + 136 + ], + "score": 0.5, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "was chosen according to the typical document length (350 words", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 478, + 150 + ], + "score": 1.0, + "content": "for CiteULike and 250 for Yelp). For the CTR dataset, because we use very small values of", + "type": "text" + }, + { + "bbox": [ + 478, + 139, + 484, + 147 + ], + "score": 0.68, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 137, + 505, + 150 + ], + "score": 1.0, + "content": ", due", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "to the tendency of the ReLU units to‘die’ during training (output zero for all examples), which can", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 157, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 157, + 506, + 173 + ], + "score": 1.0, + "content": "have a significant impact, we used instead PReLU activations (He et al., 2015) for the final layer,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 170, + 265, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 265, + 183 + ], + "score": 1.0, + "content": "since they do not suffer from this issue.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 108, + 195, + 365, + 206 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 366, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 366, + 208 + ], + "score": 1.0, + "content": "B.1 FEATURE EXTRACTION FOR DEEPCONN COMPARISON", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 215, + 505, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "The CNN architecture used for DeepCoNN (Zheng et al., 2017) is similar to the previous section. It", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 240 + ], + "score": 1.0, + "content": "consists of a word embedding lookup table, convolutional layer, 1-max pooling and a fully connected", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "layer. We use the hyper-parameters that the authors report as best - 100 convolution filters and 50", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 248, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 262 + ], + "score": 1.0, + "content": "units for the fully connected layer. We set the word embedding size to 100, the vocabulary size to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 259, + 315, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 315, + 272 + ], + "score": 1.0, + "content": "100,000 and the maximum document length to 250.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 286, + 288, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 289, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 289, + 301 + ], + "score": 1.0, + "content": "C HYPER-PARAMETERS FOR CTR", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 105, + 312, + 504, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 486, + 326 + ], + "score": 1.0, + "content": "To compare Feat2Vec with Collaborative Topic Regression, we choose the embedding size", + "type": "text" + }, + { + "bbox": [ + 486, + 313, + 505, + 324 + ], + "score": 0.78, + "content": "r \\in", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 322, + 414, + 335 + ], + "spans": [ + { + "bbox": [ + 107, + 323, + 151, + 335 + ], + "score": 0.82, + "content": "\\{ 5 , 1 0 , { \\bar { 1 } } 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 322, + 414, + 335 + ], + "score": 1.0, + "content": "for which CTR performs best. 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For the CTR dataset, because we use very small values of", + "type": "text" + }, + { + "bbox": [ + 478, + 139, + 484, + 147 + ], + "score": 0.68, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 137, + 505, + 150 + ], + "score": 1.0, + "content": ", due", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "to the tendency of the ReLU units to‘die’ during training (output zero for all examples), which can", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 157, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 157, + 506, + 173 + ], + "score": 1.0, + "content": "have a significant impact, we used instead PReLU activations (He et al., 2015) for the final layer,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 170, + 265, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 265, + 183 + ], + "score": 1.0, + "content": "since they do not suffer from this issue.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 81, + 506, + 183 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 195, + 365, + 206 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 366, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 366, + 208 + ], + "score": 1.0, + "content": "B.1 FEATURE EXTRACTION FOR DEEPCONN COMPARISON", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 215, + 505, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 215, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 506, + 228 + ], + "score": 1.0, + "content": "The CNN architecture used for DeepCoNN (Zheng et al., 2017) is similar to the previous section. It", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 240 + ], + "score": 1.0, + "content": "consists of a word embedding lookup table, convolutional layer, 1-max pooling and a fully connected", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "layer. We use the hyper-parameters that the authors report as best - 100 convolution filters and 50", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 248, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 262 + ], + "score": 1.0, + "content": "units for the fully connected layer. We set the word embedding size to 100, the vocabulary size to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 259, + 315, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 315, + 272 + ], + "score": 1.0, + "content": "100,000 and the maximum document length to 250.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 215, + 506, + 272 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 286, + 288, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 289, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 289, + 301 + ], + "score": 1.0, + "content": "C HYPER-PARAMETERS FOR CTR", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 105, + 312, + 504, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 486, + 326 + ], + "score": 1.0, + "content": "To compare Feat2Vec with Collaborative Topic Regression, we choose the embedding size", + "type": "text" + }, + { + "bbox": [ + 486, + 313, + 505, + 324 + ], + "score": 0.78, + "content": "r \\in", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 322, + 414, + 335 + ], + "spans": [ + { + "bbox": [ + 107, + 323, + 151, + 335 + ], + "score": 0.82, + "content": "\\{ 5 , 1 0 , { \\bar { 1 } } 5 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 322, + 414, + 335 + ], + "score": 1.0, + "content": "for which CTR performs best. The results are show in Table A.2.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 310, + 505, + 335 + ] + }, + { + "type": "table", + "bbox": [ + 189, + 366, + 422, + 423 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 218, + 345, + 394, + 357 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 217, + 345, + 393, + 358 + ], + "spans": [ + { + "bbox": [ + 217, + 345, + 393, + 358 + ], + "score": 1.0, + "content": "Table A.2: Tuning embedding size for CTR", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "table_body", + "bbox": [ + 189, + 366, + 422, + 423 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 189, + 366, + 422, + 423 + ], + "spans": [ + { + "bbox": [ + 189, + 366, + 422, + 423 + ], + "score": 0.979, + "html": "
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1: function FEAT2VEC_SAMPLE(S+,k,α1,α2)
2: si↑の
3:for x+∈S+do
4:
5:forj∈{1,...,k}do
6:x-←x+ > set initially to be equal to the positive sample
7:Draw a random feature group value 𝑥 ~ Q2(Xk , α2)
8:x←substitute the i-th feature type with the sampled one
9:S²←S+{χ-}
10:end for
11:end for
12:
return S-
13: end function
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Feature Type NameType# of feats.Example for an instance
Runtime (minutes)Real-valued1116
IMDB rating (0-10)Real-valued17.8
# of IMDB rating votesReal-valued1435,682
Is adult film?Boolean2False
Movie releaes yearCategorical2712001
Movie titleText165,471“Ocean's”,“Eleven”
DirectorsBag of categories174,382‘Steven Soderbergh”
GenresBag of categories28“Crime”,“Thriller”
WritersBag of categories244,241“George Johnson”,“Jack Russell"
Principal cast members (actors)Bag of categories1,104,280“George Clooney”,“Brad Pitt”,“Julia Roberts”
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", + "text_level": 1, + "bbox": [ + 196, + 122, + 802, + 148 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Joshua RobinsonMIT CSAIL & LIDSjoshrob@mit.edu", + "bbox": [ + 218, + 200, + 359, + 243 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Li Sun University of Pittsburgh lis118@pitt.edu ", + "bbox": [ + 410, + 200, + 570, + 243 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ke Yu University of Pittsburgh yu.ke@pitt.edu ", + "bbox": [ + 620, + 202, + 779, + 243 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Kayhan Batmanghelich University of Pittsburgh kayhan@pitt.edu ", + "bbox": [ + 220, + 263, + 383, + 306 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Stefanie Jegelka MIT CSAIL stefje@csail.mit.edu ", + "bbox": [ + 433, + 263, + 607, + 306 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Suvrit Sra \nMIT LIDS \nsuvrit@mit.edu", + "bbox": [ + 656, + 265, + 779, + 305 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 342, + 535, + 358 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The generalization of representations learned via contrastive learning depends crucially on what features of the data are extracted. However, we observe that the contrastive loss does not always sufficiently guide which features are extracted, a behavior that can negatively impact the performance on downstream tasks via “shortcuts”, i.e., by inadvertently suppressing important predictive features. We find that feature extraction is influenced by the difficulty of the so-called instance discrimination task (i.e., the task of discriminating pairs of similar points from pairs of dissimilar ones). Although harder pairs improve the representation of some features, the improvement comes at the cost of suppressing previously well represented features. In response, we propose implicit feature modification (IFM), a method for altering positive and negative samples in order to guide contrastive models towards capturing a wider variety of predictive features. Empirically, we observe that IFM reduces feature suppression, and as a result improves performance on vision and medical imaging tasks. The code is available at: https://github. com/joshr17/IFM. ", + "bbox": [ + 232, + 372, + 766, + 579 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 592, + 312, + 608 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Representations trained with contrastive learning are adept at solving various vision tasks including classification, object detection, instance segmentation, and more [5, 15, 44]. In contrastive learning, encoders are trained to discriminate pairs of positive (similar) inputs from a selection of negative (dissimilar) pairs. This task is called instance discrimination: It is often framed using the InfoNCE loss [14, 33], whose minimization forces encoders to extract input features that are sufficient to discriminate similar and dissimilar pairs. ", + "bbox": [ + 174, + 616, + 825, + 699 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, learning features that are discriminative during training does not guarantee a model will generalize. Many studies find inductive biases in supervised learning toward simple “shortcut” features and decision rules [16, 21, 32] which result in unpredictable model behavior under perturbations [22, 43] and failure outside the training distribution [2, 37]. Simplicity bias has various potential sources [11] including training methods [8, 29, 41] and architecture design [10, 17]. Bias towards shortcut decision rules also hampers transferability in contrastive learning [4], where it is in addition influenced by the instance discrimination task. These difficulties lead us to ask: can the contrastive instance discrimination task itself be modified to avoid learning shortcut solutions? ", + "bbox": [ + 174, + 705, + 825, + 816 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We approach this question by studying the relation between contrastive instance discrimination and feature learning. First, we theoretically explain why optimizing the InfoNCE loss alone does not guarantee avoidance of shortcut solutions that suppress (i.e., discard) certain input features [4, 11]. Second, despite this negative result, we show that it is still possible to trade off representation of one feature for another using simple methods for adjusting the difficulty of instance discrimination. However, these methods have an important drawback: improved learning of one feature often comes at the cost of harming another. That is, feature suppression is still prevalent. In response, we propose implicit feature modification, a technique that encourages encoders to discriminate instances using multiple input features. Our method introduces no computational overhead, reduces feature suppression (without trade-offs), and improves generalization on various downstream tasks. ", + "bbox": [ + 174, + 823, + 825, + 878 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/d6fbf1d29fd523c967273ded3a7ee69208a51fd99337cc1bb6d7349925dbcce9.jpg", + "image_caption": [ + "Figure 1: An ideal encoder would discriminate between instances using multiple distinguishing features instead of finding simple shortcuts that suppress features. We show that InfoNCE-trained encoders can suppress features (Sec. 2.2). However, making instance discrimination harder during training can trade off representation of different features (Sec. 2.3). To avoid the need for trade-offs we propose implicit feature modification (Sec. 3), which reduces suppression in general, and improves generalization (Sec. 4). " + ], + "image_footnote": [], + "bbox": [ + 181, + 84, + 818, + 208 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 306, + 825, + 391 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contributions. In summary, this paper makes the following main contributions: ", + "bbox": [ + 174, + 396, + 699, + 411 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. It analyzes feature suppression in contrastive learning, and explains why feature suppression can occur when optimizing the InfoNCE loss. 2. It studies the relation between instance discrimination tasks and feature learning; concretely, adjustments to instance discrimination difficulty leads to different features being learned. 3. It proposes implicit feature modification, a simple and efficient method that reduces the tendency to use feature suppressing shortcut solutions and improves generalization. ", + "bbox": [ + 212, + 416, + 825, + 508 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.1 Related work ", + "text_level": 1, + "bbox": [ + 174, + 518, + 307, + 534 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Unsupervised representation learning is enjoying a renaissance driven by steady advances in effective frameworks [3, 5, 15, 18, 33, 44, 45, 51]. As well as many effective contrastive methods, Siamese approaches that avoid representation collapse without explicitly use of negatives have also been proposed [6, 13, 51]. Pretext task design has been at the core of progress in self-supervised learning. Previously popular tasks include image colorization [54] and inpainting [35], and theoretical work shows pre-trained encoders can provably generalize if a pretext task necessitates the learning of features that solve downstream tasks [27, 39]. In contrastive learning, augmentation strategies are a key design component [5, 48, 50], as are negative mining techniques [9, 15, 25, 40]. While feature learning in contrastive learning has received less attention, recent work finds that low- and mid-level features are more important for transfer learning [55], and feature suppression can occur [4] just as with supervised learning [10, 16]. Combining contrastive learning with an auto-encoder has also been considered [28], but was found to harm representation of some features in order to avoid suppression of others. Our work is distinguished from prior work through our focus on how the design of the instance discrimination task itself affects which features are learned. ", + "bbox": [ + 174, + 540, + 826, + 732 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Feature suppression in contrastive learning ", + "text_level": 1, + "bbox": [ + 174, + 747, + 566, + 765 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Feature suppression refers to the phenomenon where, in the presence of multiple predictive input features, a model uses only a subset of them and ignores the others. The selected subset often corresponds to intuitively “simpler” features, e.g., color as opposed to shape. Such features lead to “shortcut” decision rules that might perform well on training data, but can harm generalization and lead to poor robustness to data shifts. Feature suppression has been identified as a common problem in deep learning [11], and both supervised and contrastive learning suffer from biases induced by the choice of optimizer and architecture. However, contrastive learning bears an additional potential source of bias: the choice of instance discrimination task. Which positive and negative pairs are presented critically affects which features are discriminative, and hence which features are learned. In this work we study the relation between feature suppression and instance discrimination. ", + "bbox": [ + 173, + 772, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "First, we explain why optimizing the InfoNCE loss is insufficient in general to avoid feature suppression, and show how it can lead to counter-intuitive generalization (Sec. 2.2). Given this negative result, we then ask if it is at least possible to control which features a contrastive encoder learns? We find that this is indeed the case, and that adjustments to the instance discrimination task lead to different features being learned (Sec. 2.3). However, the primary drawback of these adjustments is that improving one feature often comes at the cost of harming representation of another. That is, feature suppression is still prevalent. Addressing this drawback is the focus of Sec. 3. ", + "bbox": [ + 173, + 90, + 826, + 189 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 Setup and definition of feature suppression ", + "text_level": 1, + "bbox": [ + 173, + 198, + 511, + 213 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Formally, we assume that the data has underlying feature spaces ${ \\mathcal { Z } } ^ { 1 } , \\ldots , { \\mathcal { Z } } ^ { n }$ with a distribution $p _ { j }$ on each $\\mathcal { Z } ^ { j }$ . Each $j \\in [ n ]$ , corresponding to a latent space $\\mathcal { Z } ^ { j }$ , models a distinct feature. We write the product as $\\begin{array} { r } { \\mathcal { Z } ^ { S } = \\prod _ { j \\in S } \\mathcal { Z } ^ { j } } \\end{array}$ , and simply write $\\mathcal { Z }$ instead of $\\mathcal { Z } ^ { [ n ] }$ where $[ n ] = \\{ 1 , \\dots , n \\}$ . A set of features $z = ( z ^ { j } ) _ { j \\in [ n ] } \\in \\mathcal { Z }$ is generated by sampling each coordinate $z ^ { j } \\in \\mathcal { Z } ^ { j }$ independently, and we denote the measure on $\\mathcal { Z }$ induced by $z$ by $\\lambda$ . Further, let $\\lambda ( \\cdot | z ^ { S } )$ denote the conditional measure on $\\mathcal { Z }$ for fixed $z ^ { S }$ . For $S \\subseteq [ n ]$ we use $\\dot { z } ^ { S }$ to denote the projection of $z$ onto $\\mathcal { Z } ^ { S }$ . Finally, an injective map $g : { \\mathcal { Z } } { \\mathcal { X } }$ produces observations $x = g ( z )$ . ", + "bbox": [ + 173, + 217, + 825, + 324 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our aim is to train an encoder $f : \\mathcal { X } \\to \\mathbb { S } ^ { d - 1 }$ to map input data $x$ to the surface of the unit sphere $\\mathbb { S } ^ { d - 1 } = \\{ u \\in \\mathbb { R } ^ { d } : \\| u \\| _ { 2 } = 1 \\}$ in such a way that $f$ extracts useful information. To formally define feature suppression, we need the pushforward $h \\# \\nu ( V ) = \\nu ( h ^ { - 1 } ( V ) )$ of a measure $\\nu$ on a space $\\mathcal { U }$ for a measurable map $h : \\mathcal { U } \\to \\mathcal { V }$ and measurable $V \\subseteq \\nu$ , where $\\dot { h } ^ { - 1 } ( V )$ denotes the preimage. ", + "bbox": [ + 173, + 329, + 825, + 387 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Consider an encoder $f : \\mathcal { X } \\to \\mathbb { S } ^ { d - 1 }$ and features $S \\subseteq [ n ]$ . For each $z ^ { S } \\in \\mathcal { Z } ^ { S }$ , let $\\mu ( \\cdot | z ^ { S } ) =$ $( f \\circ g ) \\# \\lambda ( \\cdot | z ^ { S } )$ be the pushforward measure on $\\mathbb { S } ^ { d - 1 }$ by $f \\circ g$ of the conditional $\\lambda ( \\cdot | z ^ { S } )$ . ", + "bbox": [ + 173, + 392, + 823, + 424 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1. $f$ suppresses $S$ if for any pair $z ^ { S } , { \\bar { z } } ^ { S } \\in { \\mathcal { Z } } ^ { S }$ , we have $\\mu ( \\cdot | z ^ { S } ) = \\mu ( \\cdot | \\bar { z } ^ { S } )$ . ", + "bbox": [ + 209, + 424, + 707, + 440 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2. $f$ distinguishes $S$ if for any pair of distinct $z ^ { S } , { \\bar { z } } ^ { S } \\in { \\mathcal { Z } } ^ { S }$ , measures $\\mu ( \\cdot | z ^ { S } ) , \\mu ( \\cdot | \\bar { z } ^ { S } )$ have disjoint support. ", + "bbox": [ + 204, + 440, + 823, + 468 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Feature suppression is thus captured in a distributional manner, stating that $S$ is suppressed if the encoder distributes inputs in a way that is invariant to the value $z ^ { S }$ . Distinguishing features, meanwhile, asks that the encoder $f$ separates points with different features $z ^ { S }$ into disjoint regions. We consider training an encoder $f : \\mathcal { X } \\to \\mathbb { S } ^ { d - 1 }$ to optimize the InfoNCE loss [33, 14], ", + "bbox": [ + 173, + 469, + 826, + 525 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8372a061c0968a9519e8cfe8e8d5269415322e8facc55a3f6317562d5d24ff60.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { m } ( f ) = \\mathbb { E } _ { x , x ^ { + } , \\{ x _ { i } ^ { - } \\} _ { i = 1 } ^ { m } } \\bigg [ - \\log \\frac { e ^ { f ( x ) ^ { \\top } f ( x ^ { + } ) / \\tau } } { e ^ { f ( x ) ^ { \\top } f ( x ^ { + } ) / \\tau } + \\sum _ { i = 1 } ^ { m } e ^ { f ( x ) ^ { \\top } f ( x _ { i } ^ { - } ) / \\tau } } \\bigg ] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 258, + 527, + 738, + 569 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\tau$ is known as the temperature. Positive pairs $x , x ^ { + }$ are generated by first sampling $z \\sim \\lambda$ , then independently sampling two random augmentations $a , a ^ { + } \\sim A$ , $a : \\mathcal { X } \\to \\mathcal { X }$ from a distribution $\\mathcal { A }$ , and setting $x = a ( g ( z ) )$ and $x ^ { + } = a ^ { + } ( g ( z ) )$ . We assume $\\mathcal { A }$ samples the identity function $a ( x ) = x$ with non-zero probability (“ $x$ is similar to itself”), and that there are no collisions: $a ( x ) \\neq a ^ { \\prime } ( x ^ { \\prime } )$ for all $a , a ^ { \\prime }$ , and all $x \\neq x ^ { \\prime }$ . Each negative example $\\boldsymbol { x } _ { i } ^ { - }$ is generated as $x _ { i } ^ { - } = a _ { i } ( g ( z _ { i } ) )$ , by independently sampling features $z _ { i } \\sim \\lambda$ and an augmentation $a _ { i } \\sim { \\mathcal { A } }$ . ", + "bbox": [ + 173, + 571, + 826, + 659 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Why optimizing the InfoNCE loss can still lead to feature suppression ", + "text_level": 1, + "bbox": [ + 176, + 666, + 696, + 683 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Do optimal solutions to the InfoNCE loss automatically avoid shortcut solutions? Unfortunately, as we show in this section, this is not the case in general; there exist both optimal solutions of the InfoNCE loss that do and solutions that do not suppress a given feature. Following previous work [40, 49, 56], we analyze the loss as the number of negatives goes to infinity, ", + "bbox": [ + 173, + 686, + 826, + 742 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/0e366edf6fb558f7d4f2d56f7eee01b257f6e58842366efbc9f076ec2dd49679.jpg", + "text": "$$\n\\begin{array} { r l } { \\stackrel { \\cdot } { = } \\underset { m \\infty } { \\operatorname* { l i m } } \\{ \\mathcal { L } _ { m } ( f ) - \\log m - \\frac { 2 } { \\tau } \\} = \\frac { 1 } { 2 \\tau } \\mathbb { E } _ { x , x ^ { + } } \\Vert f ( x ) - f ( x ^ { + } ) \\Vert ^ { 2 } + \\mathbb { E } _ { x ^ { + } } \\log [ \\mathbb { E } _ { x ^ { - } } e ^ { f ( x ^ { + } ) ^ { \\top } f ( x ^ { - } ) / \\tau } ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 747, + 825, + 772 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We subtract $\\log m$ to ensure the limit is finite, and use $x ^ { - }$ to denote a random sample with the same distribution as $\\boldsymbol { x } _ { i } ^ { - }$ . Prop. 2.2 (proved in App. A) shows that, assuming the marginals $p _ { j }$ are uniform, the InfoNCE loss is optimized both by encoders that suppress feature $j$ , and by encoders that distinguish $j$ . ", + "bbox": [ + 173, + 776, + 825, + 834 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Suppose that $p _ { j }$ is uniform on $\\mathcal { Z } ^ { j } = \\mathbb { S } ^ { d - 1 }$ for all $j \\in [ n ]$ . Then for any feature $j \\in [ n ]$ there exists an encoder $f _ { \\mathrm { s u p p } }$ that suppresses feature $j$ and encoder $f _ { \\mathrm { d i s c } }$ that discriminates $j$ but both attain $\\operatorname* { m i n } { } _ { f }$ : measurable $\\mathcal { L } ( f )$ . ", + "bbox": [ + 174, + 838, + 825, + 882 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The condition that $p _ { j }$ is uniformly distributed on $\\mathcal { Z } ^ { j } = \\mathbb { S } ^ { d - 1 }$ is similar to conditions used in previous work [56]. Prop. 2.2 shows that empirical observations of feature suppression [4] (see also Fig. 3) ", + "bbox": [ + 174, + 882, + 823, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "are not simply due to a failure to sufficiently optimize the loss, but that the possibility of feature suppression is built into the loss. What does Prop. 2.2 imply for the generalization behavior of encoders? Besides explaining why feature suppression can occur, Prop. 2.2 also suggests another counter-intuitive possibility: lower InfoNCE loss may actually lead to worse performance on some tasks. ", + "bbox": [ + 174, + 90, + 825, + 160 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To empirically study whether this possibility manifests in practice, we use two datasets with known semantic features: (1) In the Trifeature data, [16] each image is $1 2 8 \\times 1 2 8$ and has three features: color, shape, and texture, each taking possible 10 values. See Fig. 10, App. C for sample images. (2) In the STL-digits data, samples combine MNIST digits and STL10 objects by placing copies of a randomly selected MNIST digit on top of an STL10 image. See Fig. 11 App. C for sample images. ", + "bbox": [ + 173, + 166, + 457, + 333 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We train encoders with ResNet-18 backbone using SimCLR [5]. To study correlations between the loss value and error on downstream tasks, we train 33 encoders on Trifea", + "bbox": [ + 174, + 339, + 457, + 395 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/58d9620e1c56307182ac968752d1752ee45a3c23d4152d77dc12bb6e84348c5e.jpg", + "image_caption": [ + "Figure 2: Linear readout error on different downstream tasks can be negatively correlated. Further, lower InfoNCE loss does not always yield not lower error: error rates on texture, shape and STL10 prediction are negatively correlated with InfoNCE loss. " + ], + "image_footnote": [], + "bbox": [ + 477, + 179, + 821, + 304 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "ture and 7 encoders on STL-digits with different hyperparameter settings (see App. C.2 for full details on training and hyperparameters). For Trifeature, we compute the Pearson correlation between InfoNCE loss and linear readout error when predicting {color, shape, texture}. Likewise, for STL-digits we compute correlations between the InfoNCE loss and MNIST and STL10 prediction error. ", + "bbox": [ + 174, + 395, + 825, + 449 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Fig. 2 shows that performance on different downstream tasks is not always positively correlated. For Trifeature, color error is negatively correlated with shape and texture, while for STL-digits there is a strong negative correlation between MNIST digit error and STL10 error. Importantly, lower InfoNCE loss is correlated with lower prediction error for color and MNIST-digit, but with larger error for shape, texture and STL10. Hence, lower InfoNCE loss can improve representation of some features (color, MNIST digit), but may actually hurt others. This conflict is likely due to the simpler color and MNIST digit features being used as shortcuts. Our observation is an important addition to the statement of Wang and Isola [49] that lower InfoNCE loss improves generalization: the situation is more subtle – whether lower InfoNCE helps generalization on a task depends on the use of shortcuts. ", + "bbox": [ + 174, + 457, + 825, + 582 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 Controlling feature learning via the difficulty of instance discrimination ", + "text_level": 1, + "bbox": [ + 174, + 598, + 710, + 613 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The previous section showed that the InfoNCE objective has solutions that suppress features. Next, we ask what factors determine which features are suppressed? Is there a way to target specific features and ensure they are encoded? One idea is to use harder positive and negative examples. Hard examples are precisely those that are not easily distinguishable using the currently extracted features. So, a focus on hard examples may change the scope of the captured features. To test this hypothesis, we consider two methods for adjusting the difficulty of positive and negative samples: ", + "bbox": [ + 174, + 617, + 826, + 700 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1. Temperature $\\tau$ in the InfoNCE loss (Eqn. 1). Smaller $\\tau$ places higher importance on positive an negative pairs with high similarity [47]. 2. Hard negative sampling method of Robinson et al. [40], which uses importance sampling to sample harder negatives. The method introduces a hardness concentration parameter $\\beta$ , with larger $\\beta$ corresponding to harder negatives (see [40] for full details). ", + "bbox": [ + 210, + 708, + 823, + 781 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Results reported in Fig. 3 (also Fig. 13 in App. C.2) show that varying instance discrimination difficulty—i.e., varying temperature $\\tau$ or hardness concentration $\\beta$ —enables trade-offs between which features are represented. On Trifeature, easier instance discrimination (large $\\tau$ , small $\\beta$ ) yields good performance on ‘color’—an “easy” feature for which a randomly initialized encoder already has high linear readout accuracy—while generalization on the harder texture and shape features is poor. The situation reverses for harder instance discrimination (small $\\tau$ , large $\\beta$ ). We hypothesize that the use of “easy” features with easy instance discrimination is analogous to simplicity biases in supervised deep networks [17, 21]. As with supervised learning [10, 17], we observe a bias for texture over shape in convolutional networks, with texture prediction always outperforming shape. ", + "bbox": [ + 174, + 786, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/ec74850ae8dbb37af5c9996ed86223161b43646369ceb4f7b0680fc8bb16fcb2.jpg", + "image_caption": [ + "Figure 3: Trifeature dataset [16]. The difficulty of instance discrimination affects which features are learned (Sec. 2.3). When instance discrimination is easy (big $\\tau$ , small $\\beta$ ), encoders represent color well and other features badly. When instance discrimination is hard (small $\\tau$ , big $\\beta$ ), encoders represent more challenging shape and texture features well, at the expense of color. " + ], + "image_footnote": [], + "bbox": [ + 176, + 88, + 821, + 265 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "That there are simple levers for controlling which features are learned already distinguishes contrastive learning from supervised learning, where attaining such control is less easy (though efforts have been made [23]). However, these results show that representation of one feature must be sacrificed in exchange for learning another one better. To understand how to develop methods for improving feature representation without suppressing others, the next result (proof in App. A) examines more closely why there is a relationship between (hard) instance discrimination tasks and feature learning. ", + "bbox": [ + 173, + 343, + 825, + 426 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "[Informal] Suppose that $p _ { j }$ is uniform on $\\mathcal { Z } ^ { j } = \\mathbb { S } ^ { d - 1 }$ for all $j \\in [ n ]$ . Further, for $S \\subseteq [ n ]$ suppose that $x , x ^ { + } , \\{ x _ { i } ^ { - } \\} _ { i }$ are conditioned on the event that they have the same features $S$ . Then any $f$ that minimizes the (limiting) InfoNCE loss suppresses features $S$ . ", + "bbox": [ + 174, + 431, + 825, + 477 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The positive and negative instances in Prop. 2.3 must be distinguished with features in $S ^ { c }$ . Relating this point to the above observations, assume that an encoder exclusively uses features $S$ . Any positives and negatives that do not (much) differ in features $S$ are difficult for the encoder. By Prop. 2.3, focusing the training on these difficult examples pushes the encoder to instead use features in $S ^ { c }$ , i.e., to learn new features. But at the same time, the proposition also says that a strong focus on such hard negative pairs leads to suppressing the originally used features $S$ , explaining the results in Fig. 3. While the two techniques for adjusting instance difficulty we studied were unable to avoid feature suppression, this insight forms the motivation for implicit feature modification, which we introduce next. ", + "bbox": [ + 173, + 482, + 826, + 608 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 Implicit feature modification for reducing feature suppression ", + "text_level": 1, + "bbox": [ + 174, + 626, + 720, + 643 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The previous section found that simple adjustments to instance discrimination difficulty could significantly alter which features a model learns. Prop. 2.3 suggests that this ability to modify which features are learned stems from holding features constant across positive and negative samples. However, these methods were unable to avoid trade-offs in feature representation (Fig. 3) since features that are held constant are themselves suppressed (Prop. 2.3). ", + "bbox": [ + 173, + 654, + 825, + 724 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To avoid this effect, we develop a technique that adaptively modifies samples to remove whichever features are used to discriminate a particular positive pair from negatives, then trains an encoder to discriminate instances using both the original features, and the features left over after modification. While a natural method for modifying features is to directly transform raw input data, it is very challenging to modify the semantics of an input in this way. So instead we propose modifying features by applying transformations to encoded samples $v = f ( x )$ . Since we modify the encoded samples, instead of raw inputs $x$ , we describe our method as implicit. ", + "bbox": [ + 173, + 729, + 825, + 828 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We set up our notation. Given batch $x , x ^ { + } , \\{ x _ { i } ^ { - } \\} _ { i = 1 } ^ { m }$ we write $ { \\boldsymbol { v } } \\ = \\ f ( { \\boldsymbol { { x } } } )$ , $v ^ { + } ~ = ~ f ( x ^ { + } )$ , and $v _ { i } ^ { - } = f ( x _ { i } ^ { - } )$ to denote the corresponding embeddings. As in Eqn. 1, the point-wise InfoNCE loss is, ", + "bbox": [ + 173, + 833, + 826, + 864 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/b3e23cbd76d022e8bdaccab60792aaebb4cd55c0c88c790899f63c021b9cd617.jpg", + "text": "$$\n\\ell ( v , v ^ { + } , \\{ v _ { i } ^ { - } \\} _ { i = 1 } ^ { m } ) = - \\log \\frac { e ^ { v ^ { \\top } v ^ { + } / \\tau } } { e ^ { v ^ { \\top } v ^ { + } / \\tau } + \\sum _ { i = 1 } ^ { m } e ^ { v ^ { \\top } v _ { i } ^ { - } / \\tau } } .\n$$", + "text_format": "latex", + "bbox": [ + 315, + 875, + 679, + 916 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "[Implicit feature modification] Given budget $\\varepsilon \\in \\mathbb { R } _ { + } ^ { m }$ , and encoder $f : \\mathcal { X } \\to \\mathbb { S } ^ { d }$ , an adversary removes features from $f$ that discriminates batch $x , \\stackrel { \\cdot } { x } ^ { + } , \\{ x _ { i } ^ { - } \\} _ { i = 1 } ^ { m }$ by maximizing the point-wise InfoNCE loss, \\`\"(v, v+, {v\u0000i }mi=1) = max\u0000+2B + ,{\u0000\u0000i 2B\" }mi=1 \\` $\\begin{array} { r } { \\ell _ { \\varepsilon } ( v , v ^ { + } , \\{ v _ { i } ^ { - } \\} _ { i = 1 } ^ { m } ) = \\operatorname* { m a x } _ { \\delta ^ { + } \\in \\mathcal { B } _ { \\varepsilon ^ { + } } , \\{ \\delta _ { i } ^ { - } \\in \\mathcal { B } _ { \\varepsilon _ { i } } \\} _ { i = 1 } ^ { m } } \\ell ( v , v ^ { + } + \\delta ^ { + } , \\{ v _ { i } ^ { - } + \\delta _ { i } ^ { - } \\} _ { i = 1 } ^ { m } ) } \\end{array}$ . ", + "bbox": [ + 174, + 90, + 825, + 141 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Here $B _ { \\varepsilon }$ denotes the $\\ell _ { 2 }$ -ball of radius $\\varepsilon$ . Implicit feature modification (IFM) removes components of the current representations that are used to discriminate positive and negative pairs. In other words, the embeddings of positive and negative samples are modified to remove well represented features. So, if the encoder is currently using a simple shortcut solution, IFM removes the features used, thereby encouraging the encoder to also discriminate instances using other features. By applying perturbations in the embedding space IFM can modify high level semantic features (see Fig. 4), which is extremely challenging when applying perturbations in input space. In order to learn new features using the perturbed loss while still learning potentially complementary information using the original InfoNCE objective, we propose optimizing the the multi-task objective $\\mathrm { m i n } _ { f } \\{ \\mathcal { L } ( f ) \\dot { + } \\alpha \\mathcal { L } _ { \\varepsilon } ( \\dot { f } ) \\} / 2$ where $\\mathcal { L } _ { \\varepsilon } = \\mathbb { E } \\ell _ { \\varepsilon }$ is the adversarial perturbed loss, and $\\mathcal { L }$ the standard InfoNCE loss. For simplicity, all experiments set the balancing parameter $\\alpha = 1$ unless explicitly noted, and all take $\\varepsilon ^ { + } , \\varepsilon _ { i } ^ { - }$ to be equal, and denote this single value by $\\varepsilon$ . Crucially, $\\ell _ { \\varepsilon }$ can be computed analytically and efficiently. For any $v , v ^ { + } , \\{ v _ { i } ^ { - } \\} _ { i = 1 } ^ { m } \\in \\bar { \\mathbb { R } } ^ { d }$ we have, ", + "bbox": [ + 173, + 142, + 826, + 321 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/6b64cbe5fbae51d1b2c970e3cd8ce2f5589d9210bccae48ab4ec129c9d4f66c9.jpg", + "text": "$$\n\\nabla _ { v _ { j } ^ { - } } \\ell = \\frac { e ^ { v ^ { \\top } v _ { j } ^ { - } / \\tau } } { e ^ { v ^ { \\top } v + / \\tau } + \\sum _ { i = 1 } ^ { m } e ^ { v ^ { \\top } v _ { i } ^ { - } / \\tau } } \\cdot \\frac { v } { \\tau } \\quad \\mathrm { a n d } \\quad \\nabla _ { v ^ { + } } \\ell = \\left( \\frac { e ^ { v ^ { \\top } v ^ { + } / \\tau } } { e ^ { v ^ { \\top } v ^ { + } / \\tau } + \\sum _ { i = 1 } ^ { m } e ^ { v ^ { \\top } v _ { i } ^ { - } / \\tau } } - 1 \\right) \\cdot \\frac { v } { \\tau } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 327, + 816, + 369 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In particular, $\\nabla _ { v _ { \\bot \\ldots } ^ { - } } \\ell \\propto v$ and $\\nabla _ { v ^ { + } } \\ell \\propto - v$ . This expression shows that the adversary perturbs $\\cdot ^ { v _ { j } ^ { - } }$ (resp. $v ^ { + }$ j) in the direction of the anchor $v$ (resp $- v ,$ ). Since the derivative directions are independent of $\\{ v _ { i } ^ { - } \\} _ { i = 1 } ^ { m }$ and $v ^ { + }$ , we can analytically compute optimal perturbations in $B _ { \\varepsilon }$ . Indeed, following the constant ascent direction shows the optimal updates are simply $v _ { i } ^ { - } v _ { i } ^ { - } + \\varepsilon _ { i } v$ and $v ^ { + } v ^ { + } - \\varepsilon ^ { + } v$ . The positive (resp. negative) perturbations increase (resp. decrease) cosine similarity to the anchor $\\sin ( v , v _ { i } ^ { - } + \\varepsilon _ { i } v ) 1$ as $\\varepsilon _ { i } \\to \\infty$ (resp. $\\sin ( v , v ^ { + } - \\bar { \\varepsilon } ^ { + } v ) - 1$ as $\\varepsilon ^ { + } \\to \\infty$ ). In Fig. 4 we visualize the newly synthesized $v _ { i } ^ { - } , v ^ { + }$ and find meaningful interpolation of semantics. Plugging the update rules for $v ^ { + }$ and ${ \\boldsymbol v } _ { i } ^ { - }$ into the point-wise InfoNCE loss yields, ", + "bbox": [ + 173, + 376, + 825, + 492 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/53e748cec07dfe622902369f99c6b5233004d1127704289f799c53d663e89050.jpg", + "text": "$$\n\\begin{array} { r } { \\ell _ { \\varepsilon } ( v , v ^ { + } , \\{ v _ { i } ^ { - } \\} _ { i = 1 } ^ { m } ) = - \\log \\frac { e ^ { ( v ^ { \\top } v ^ { + } - \\varepsilon ^ { + } ) / \\tau } } { e ^ { ( v ^ { \\top } v ^ { + } - \\varepsilon ^ { + } ) / \\tau } + \\sum _ { i = 1 } ^ { m } e ^ { ( v ^ { \\top } v _ { i } ^ { - } + \\varepsilon _ { i } ) / \\tau } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 497, + 718, + 540 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In other words, IFM amounts to simply perturbing the logits – reduce the positive logit by $\\varepsilon ^ { + } / \\tau$ and increase negative logits by $\\varepsilon _ { i } / \\tau$ . From this we see that $\\ell _ { \\varepsilon }$ is automatically symmetrized in the positive samples: perturbing $v$ instead of $v ^ { + }$ results in the exact same objective. Eqn. 2 shows that IFM re-weights each negative sample by a factor $e ^ { \\varepsilon _ { i } / \\tau }$ and positive samples by $e ^ { - \\varepsilon ^ { + } / \\tau }$ . ", + "bbox": [ + 173, + 545, + 825, + 604 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 Visualizing implicit feature modification ", + "text_level": 1, + "bbox": [ + 176, + 613, + 493, + 628 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "With implicit feature modification, newly synthesized data points do not directly correspond to any “true” input data point. However it is still possible to visualize the effects of implicit feature modification. To do this, assume access to a memory bank of input data $\\mathcal { M } = \\{ x _ { i } \\} _ { i }$ . A newly synthesized sample $s$ can be approximately visualized by retrieving the 1-nearest neighbour using cosine similarity arg $\\operatorname* { m i n } _ { x \\in { \\mathcal { M } } }$ $\\sin ( s , f ( x ) )$ and viewing the image $x$ as an approximation to $s$ . ", + "bbox": [ + 174, + 633, + 825, + 704 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Fig. 4 shows results using a ResNet-50 encoder trained using MoCo-v2 on ImageNet1K using the training set as the memory bank. For positive pair $v , v ^ { + }$ increasing $\\varepsilon$ causes the semantics of $v$ and $v ^ { + }$ to diverge. For $\\varepsilon = 0 . 1$ a different car with similar pose and color is generated, for $\\varepsilon = 0 . 2$ the pose and color then changes, and finally for $\\varepsilon = 1$ the pose, color and type of vehicle changes. For negative pair $v , v ^ { - }$ the reverse occurs. For $\\varepsilon = 0 . 1$ , $v ^ { - }$ is a vehicle with similar characteristics (number of windows, color etc.), and with $\\varepsilon = 0 . 2$ , the pose of the vehicle $v ^ { + }$ aligns with $v$ . Finally for $\\varepsilon = 1$ the pose and color of the perturbed negative sample become aligned to the anchor $v$ . In summary, implicit feature modification successfully modifies the feature content in positive and negative samples, thereby altering which features can be used to discriminate instances. ", + "bbox": [ + 173, + 708, + 825, + 834 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Related Work. Several works consider adversarial contrastive learning [19, 24, 26] using PGD (e.g. FGSM) attacks to alter samples in input space. Unlike our approach, PGD-based attacks require costly inner-loop optimization. Other work takes an adversarial viewpoint in input space for other self-supervised tasks e.g., rotations and jigsaws but uses an image-to-image network to simulate FGSM/PGD attacks [31], introducing comparable computation overheads. They note that low-level (i.e., pixel-level) shortcuts can be avoided using their method. All of these works differ from ours by applying attacks in input space, thereby focusing on lower-level features, whereas ours aims to modify high-level features. Fig. 5 compares IFM to this family of input-space adversarial methods by comparing to a top performing method ACL(DS) [24]. We find that ACL improves robust accuracy under $\\ell _ { \\infty }$ -attack on input space (see [24] for protocol details), whereas IFM improves standard accuracy (full details and discussion in Appdx. C.3). Synthesizing harder negatives in latent space using Mixup [53] has also been considered [25] but does not take an adversarial perspective. Other work, AdCo [20], also takes an adversarial viewpoint in latent space. There are several differences to our approach. AdCo perturbs all negatives using the same weighted combination of all the queries, whereas IFM perturbations are query specific. In other words, IFM makes instance discrimination harder point-wise, whereas AdCo perturbation makes the InfoNCE loss larger on average (see Fig. 4 for visualizations of instance dependent perturbation using IFM). AdCo also treats the negatives as learnable parameters, introducing $\\sim 1 M$ more parameters and $\\sim 7 \\%$ computational overhead, while IFM has no computational overhead and is implemented with only two lines of code (see Tab. 1 for empirical comparison). Finally, no previous work makes the connection between suppression of semantic features and adversarial methods in contrastive learning (see Fig. 6). ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/5afd711f6d1c8ba5155a029a38cb357006a49104919af8921fef36588c01ef36.jpg", + "image_caption": [ + "Figure 4: Visualizing implicit feature modification. Top row: progressively moving positive sample away from anchor. Bottom row: progressively moving negative sample away from anchor. In both cases, semantics such as color, orientation, and vehicle type are modified, showing the suitability of implicit feature modification for altering instance discrimination tasks. " + ], + "image_footnote": [], + "bbox": [ + 176, + 89, + 591, + 248 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/12e76d56fbd4c437b958a68c9ba5a8527e9b6f78defb2f0d01b3e203aa5892c9.jpg", + "image_caption": [ + "Figure 5: Comparison between IFM and ACL(DS). Under standard linear evaluation IFM performs best. ACL is suited to adversarial evaluation. " + ], + "image_footnote": [], + "bbox": [ + 614, + 93, + 808, + 248 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/13aac9c5a8c8da9fb88ce77335c13c805d3c88c58d0b67da7c450ce2eb3ce0da.jpg", + "image_caption": [ + "Figure 6: Trifeature dataset. Implicit feature modification reduces feature suppression, enhancing the representation of texture, shape and color features simultaneously. All results are average linear readout accuracy over three seeds and use a fixed value $\\varepsilon = 0 . 1$ to illustrate robustness to $\\varepsilon$ . " + ], + "image_footnote": [], + "bbox": [ + 183, + 348, + 810, + 438 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 501, + 825, + 722 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 Experimental results ", + "text_level": 1, + "bbox": [ + 174, + 732, + 380, + 750 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Implicit feature modification (IFM) can be used with any InfoNCE-based contrastive framework, and we write IFM-SimCLR, IFM-MoCo-v2 etc. to denote IFM applied within a specific framework. Code for IFM will be released publicly, and is also available in the supplementary material. ", + "bbox": [ + 176, + 752, + 825, + 795 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 Does implicit feature modification help avoid feature suppression? ", + "text_level": 1, + "bbox": [ + 174, + 808, + 671, + 821 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We study the effect IFM has on feature suppression by training ResNet-18 encoders for 200 epochs with $\\tau \\in \\{ 0 . 0 5 , 0 . 2 , 0 . 5 \\}$ on the Trifeature dataset [16]. Results are averaged over three seeds, with IFM using $\\varepsilon = 0 . 1$ for simplicity. Fig. 6 shows that IFM improves the linear readout accuracy across all three features for all temperature settings. The capability of IFM to enhance the representation of all features – i.e. reduce reliance on shortcut solutions – is an important contrast with tuning temperature $\\tau$ or using hard negatives, which Fig. 3 shows only trades-off which features are learned. ", + "bbox": [ + 174, + 828, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/26ba8877d15efa834c609010064f2443c1db3c4db9cb586a0504757a21b985bb.jpg", + "image_caption": [ + "Figure 7: IFM improves linear readout performance on all datasets for all $\\varepsilon \\in \\{ 0 . 0 5 , 0 . 1 , 0 . 2 \\}$ compared to baselines. Protocol uses 400 epochs of training with ResNet-50 backbone. " + ], + "image_footnote": [], + "bbox": [ + 176, + 89, + 821, + 202 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 Performance on downstream tasks ", + "text_level": 1, + "bbox": [ + 174, + 243, + 452, + 257 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Sec. 3.1 and Sec. 4.1 demonstrate that implicit feature modification is adept at altering high-level features of an input, and combats feature suppression. This section shows that these desirable traits translate into improved performance on object classification and medical imaging tasks. ", + "bbox": [ + 174, + 262, + 825, + 305 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Experimental setup for classification tasks. Having observed the positive effect IFM has on feature suppression, we next test if this feeds through to improved performance on real tasks of interest. We benchmark using both SimCLR and MoCo-v2 [5, 7] with standard data augmentation [5]. All encoders have ResNet-50 backbones and are trained for 400 epochs (with the exception of on ImageNet100, which is trained for 200 epochs). All encoders are evaluated using the test accuracy of a linear classifier trained on the full training dataset (see Appdx. C.4 for full setup details). ", + "bbox": [ + 173, + 314, + 825, + 397 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Classification tasks. Results given in Fig. 7 and Tab. 1 find that every value of $0 ~ < ~ \\varepsilon ~ \\le ~ 0 . 2$ improves performance across all datasets using both MoCo- $\\nu 2$ and SimCLR frameworks. We find ", + "bbox": [ + 173, + 402, + 346, + 512 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/c628452adce985bd88739d5ca35446a32ed4bf0bcd2313a9e29a9772dee3568d.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
MoCo-v2AdCo [20]IFM-MoCo-v2
mN/AN/A0.050.10.2
top-180.4±0.1178.9±0.2181.1±0.0280.9±0.2580.7±0.13
", + "bbox": [ + 356, + 406, + 833, + 469 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 1: Linear readout $( \\% )$ ) on ImageNet100, averaged over five seeds. \nIFM improves over MoCo-v2 for all settings of $\\varepsilon$ . ", + "bbox": [ + 356, + 474, + 825, + 503 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "that optimizing $\\mathcal { L } _ { \\varepsilon }$ $7 6 . 0 \\%$ average score across all eight runs in Fig. 7) performs similarly to the standard contrastive loss $( 7 5 . 9 \\%$ average score), and does worse than the IFM loss $( \\mathcal { L } + \\mathcal { L } _ { \\varepsilon } ) \\dot { / } 2$ . This suggests that $\\mathcal { L }$ and $\\mathcal { L } _ { \\varepsilon }$ learn complementary features. Tab. 1 benchmarks IFM on ImageNet100 [44] using MoCo-v2, observing improvements of $0 . 9 \\%$ . We also compare results on ImageNet100 to AdCo [20], another adversarial method for contrastive learning. We adopt the official code and use the exact same training and finetuning hyperparameters as for MoCo-v2 and IFM. For the AdCo-specific hyperparamters – negatives learning rate $l r _ { \\mathrm { n e g } }$ and negatives temperature $\\tau _ { \\mathrm { n e g } } - \\mathrm { w e }$ use a grid search over all combinations $l r _ { \\mathrm { n e g } } \\in \\{ 1 , \\bar { 2 } , 3 , 4 \\}$ and $\\tau _ { \\mathrm { n e g } } \\in \\mathsf { \\bar { \\{ 0 . 0 2 , 0 . 1 \\} } }$ , which includes the AdCo default ImageNet1K recommendations $l r _ { \\mathrm { n e g } } = 3$ and $\\tau _ { \\mathrm { n e g } } = 0 . 0 2$ [20]. The resulting AdCo performance of $7 8 . 9 \\%$ is slightly below MoCo-v2. However using their respective ImageNet1K default parameters AdCo and MoCo-v2 achieve $7 2 . 4 \\%$ and $7 1 . 8 \\%$ respectively, suggesting that the discrepancy between AdCo and MoCo-v2 may in part be due to the use of improved hyperparameters tuned on MoCo-v2. Note importantly, IFM is robust to the choice of $\\varepsilon$ : all values $\\varepsilon \\in \\{ 0 . 0 5 , 0 . 1 , 0 . 2 \\}$ were found to boost performance across all datasets and all frameworks. We emphasize that the MoCo-v2 baseline performance of $8 0 . 5 \\%$ on ImageNet100 is strong. Our hyperparameters, which we detail in Appdx. C.4.1, may be of interest to other works benchmarking MoCo-v2 on ImageNet100. ", + "bbox": [ + 173, + 513, + 826, + 733 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Medical images. To evaluate our method on a modality differing significantly from object-based images we consider the task of learning representations of medical images. We benchmark using the approach proposed by [42] which is a variant of MoCo-v2 that incorporates the anatomical context in the medical images. We evaluate our method on the COPDGene dataset [38], which is a multi-center observational study focused on the genetic epidemiology of Chronic obstructive pulmonary disease (COPD). See Appdx. C.5 for full background details on the COPDGene dataset, the five COPD related outcomes we use for evaluation, and our implementation. We perform regression analysis for continuous outcomes in terms of coefficient of determination (R-square), and logistic regression to predict ordinal outcomes and report the classification accuracy and the 1-off accuracy, i.e., the probability of the predicted category is within one class of true value. ", + "bbox": [ + 173, + 738, + 825, + 877 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Tab. 2 reports results. For fair comparison we use same experimental configuration for the baseline approach [42] and our method. We find that IFM yields improvements on all outcome predictions. ", + "bbox": [ + 173, + 883, + 823, + 911 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/d8d7ae890dc62850236c01565ee33ad7c66634f9cfa2409d122a8cb0ebb2185b.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
MethodlogFEV1pplogFEV1FVCCLECLE1-offPara-septalPara-septal l-offmMRCmMRC1-off
LossR-SquareAccuracy (%)
L(baseline)0.566±.0050.661±.00549.6±0.481.8±0.555.7±0.384.4±0.250.4±0.572.5±0.3
Lg,ε=0.10.591±.0080.681±.00849.4±0.481.9±0.355.6±0.385.1±0.250.3±0.872.7±0.4
IFM,ε=0.10.615±.0050.691±.00648.2±0.880.6±0.455.3±0.484.7±0.350.4±0.572.8±0.2
IFM,ε = 0.20.595±.0060.683±.00648.5±0.680.5±0.655.3±0.385.1±0.149.8±0.872.0±0.3
IFM,ε= 0.50.607±.0060.683±.00549.6±0.482.0±0.354.9±0.284.7±0.250.6±0.473.1±0.2
IFM,ε = 1.00.583±.0050.675±.00650.0±0.582.9±0.456.3±0.685.7±0.250.3±0.671.9±0.3
", + "bbox": [ + 176, + 88, + 820, + 188 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 2: Linear readout performance on COPDGene dataset. The values are the average of 5-fold cross validation with standard deviations. The bold face indicates the best average performance. IFM yields improvements on all phenotype predictions. ", + "bbox": [ + 173, + 194, + 825, + 237 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/b3a0c0f5b45705c48026a3b939aab0886ce1f3dbd5c0de0274447ad9ce01439a.jpg", + "image_caption": [ + "Figure 8: Label $\\left\\{ \\mathcal { D } , \\mathcal { D } _ { \\mathrm { R } } , \\mathcal { D } _ { \\mathrm { N R } } \\right\\}$ indicates which dataset was used to train the linear readout function. Improved performance of IFM on standard data $\\mathcal { D }$ can be attributed to improved representation of robust features $\\mathcal { D } _ { \\mathrm { R } }$ . See Sec. 4.3 for construction of robust $( \\mathcal { D } _ { \\mathtt { R } } )$ and non-robust $( \\mathcal { D } _ { \\mathrm { N R } } )$ datasets. " + ], + "image_footnote": [], + "bbox": [ + 178, + 246, + 820, + 324 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The gain is largest on spirometry outcome prediction, particularly logFEV1pp with improvement of $8 . { \\bar { 7 } } \\%$ with $\\varepsilon = 0 . 1$ . We found that at least $\\varepsilon = 0 . 5$ and 1.0 improve performance on all tasks. However, we note that not all features yield a statistically significant improvement with IFM. ", + "bbox": [ + 174, + 387, + 825, + 429 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.3 Further study on the impact of IFM on feature learning ", + "text_level": 1, + "bbox": [ + 173, + 446, + 601, + 462 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This section further studies the effect implicit feature modification has on what type of features are extracted. Specifically, we consider the impact on learning of robust (higher-level) vs. non-robust features (pixel-level features). Our methodology, which is similar to that of Ilyas et al. [22] for deep supervised learning, involves carefully perturbing inputs to obtain non-robust features. ", + "bbox": [ + 174, + 463, + 825, + 518 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Constructing non-robust features. Given encoder $f$ we finetune a linear probe (classifier) $h$ ontop of $f$ using training data (we do not use data augmentation). Once $h$ is trained, we consider each labeled example $( x , y )$ from training data $\\mathcal { D } _ { \\mathrm { t r a i n } } \\in \\{ \\mathrm { t i n y I m a g e N e t , S T L 1 0 , C I F A R 1 0 , C I F A R 1 0 0 } \\}$ . A hallucinated target label $t$ is sampled uniformly at random, and we perturb $x = x _ { 0 }$ until $h \\circ f$ predicts $t$ using repeated FGSM attacks [12] $x _ { k } \\gets x _ { k - 1 } - \\varepsilon \\mathrm { s i g n } ( \\nabla _ { x } \\ell ( h \\circ f ( x _ { k - 1 } ) , t ) )$ . At each step we check if arg maxi $h \\circ f ( x _ { k } ) _ { i } = t$ (we use the maximum of logits for inference) and stop iterating and set $x _ { \\mathrm { a d v } } = x _ { k }$ for the first $k$ for which the prediction is $t$ . This usually takes no more than a few FGSM steps with $\\varepsilon = 0 . 0 1$ . We form a dataset of “robust” features by adding $( x _ { \\mathrm { a d v } } , y )$ to $\\mathcal { D } _ { R }$ , and a dataset of “non-robust” features by adding $( x _ { \\mathrm { a d v } } , t )$ to $\\mathcal { D } _ { N R }$ . To a human the pair $( x _ { \\mathrm { a d v } } , t )$ will look mislabeled, but for the encoder $x _ { \\mathrm { a d v } }$ contains features predictive of $t$ . Finally, we re-finetune (i.e. re-train) linear classifier $g$ using $\\mathcal { D } _ { R }$ (resp. $\\mathcal { D } _ { N R }$ ). ", + "bbox": [ + 173, + 534, + 825, + 686 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Fig. 8 compares accuracy of the re-finetuned models on a test set of standard $\\mathcal { D } _ { \\mathrm { t e s t } }$ examples (no perturbations are applied to the test set). Note that $\\mathcal { D } _ { R }$ , $\\mathcal { D } _ { N R }$ depend on the original encoder $f$ . When re-finetuning $f$ we always use datasets $\\mathcal { D } _ { R }$ , $\\mathcal { D } _ { N R }$ formed via FGSM attacks on $f$ itself. So there is one set $\\mathcal { D } _ { R } , \\mathcal { D } _ { N R }$ for SimCLR, and another set for IFM. Fig. 8 shows that IFM achieves superior generalization $( \\mathcal { D } )$ compared to SimCLR by better representing robust features $( \\mathcal { D } _ { R } )$ . Representation of non-robust features $( \\mathcal { D } _ { N R } )$ is similar for IFM $( 5 \\bar { 5 } . 5 \\%$ average across all datasets) and SimCLR $( 5 6 . 7 \\%$ average). IFM is juxtaposed to the supervised adversarial training of Madry et al., which sacrifices standard supervised performance in exchange for not using non-robust features [30, 46]. ", + "bbox": [ + 173, + 693, + 825, + 803 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 Discussion ", + "text_level": 1, + "bbox": [ + 174, + 815, + 294, + 832 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This work studies the relation between contrastive instance discrimination and feature learning. While we focus specifically on contrastive learning, it would be of interest to also study feature learning for other empirically successful self-supervised methods [1, 6, 13, 51]. Understanding differences in feature learning biases between different methods may inform which methods are best suited for a given task, as well as point the way to further improved self-supervised techniques. ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Acknowledgments SJ was supported by NSF BIGDATA award IIS-1741341, NSF Convergence Accelerator Track D 2040636. SS acknowledges support from NSF-TRIPODS $^ +$ X:RES (1839258). JR was partially supported by a Two Sigma fellowship. KB acknowledges support from NIH (1R01HL141813-01), NSF (1839332 Tripod $+ \\mathrm { X }$ ), and a research grant from SAP SE Commonwealth Universal Research Enhancement (CURE) program awards research grants from the Pennsylvania Department of Health. Finally, we warmly thank Katherine Hermann and Andrew Lampinen for making the Trifeature dataset available for our use. ", + "bbox": [ + 174, + 92, + 826, + 188 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 208, + 266, + 223 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "[1] Adrien Bardes, Jean Ponce, and Yann LeCun. VICReg: Variance-invariance-covariance regularization for self-supervised learning. preprint arXiv:2105.04906, 2021. \n[2] Sara Beery, Grant Van Horn, and Pietro Perona. Recognition in terra incognita. 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