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+# RESERVOIR TRANSFORMERS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We demonstrate that transformers obtain impressive performance even when some of the layers are randomly initialized and never updated. Inspired by old and wellestablished ideas in machine learning, we explore a variety of non-linear “reservoir” layers interspersed with regular transformer layers, and show improvements in wall-clock compute time until convergence, as well as overall performance, on various machine translation and (masked) language modelling tasks.
+
+# 1 INTRODUCTION
+
+Transformers (Vaswani et al., 2017) have dominated natural language processing (NLP) in recent years, from large scale machine translation (Ott et al., 2018) to pre-trained (masked) language modeling (Devlin et al., 2018; Radford et al., 2018), and are becoming more popular in other fields as well, from reinforcement learning (Vinyals et al., 2019) to speech recognition (Baevski et al., 2019) and computer vision (Carion et al., 2020). Their success is enabled in part by ever increasing computational demands, which has naturally led to an increased interest in improving their efficiency. Scalability gains in transformers could facilitate bigger, deeper networks with longer contexts (Kitaev et al., 2020; Wang et al., 2020; Beltagy et al., 2020; Kaplan et al., 2020; Tay et al., 2020b). Conversely, improved efficiency could reduce environmental costs (Strubell et al., 2019) and hopefully help democratize the technology.
+
+In this work, we explore a simple question: if some layers of the transformer are kept frozen—i.e., never updated after random initialization—can we match the performance of fully learned transformers, while being more efficient? Surprisingly, the answer is resoundingly yes; and what is more, we find that freezing layers may actually improve performance.
+
+Beyond desirable efficiency gains, random layers are interesting for several additional reasons. Fixed randomly initialized networks (Gallicchio & Scardapane, 2020) converge to Gaussian processes in the limit of infinite width (Daniely et al., 2016), have intriguing interpretations in metric learning (Rosenfeld & Tsotsos, 2019; Giryes et al., 2016), and have been shown to provide excellent “priors” either for subsequent learning (Ulyanov et al., 2018) or pruning (Frankle & Carbin, 2018). Fixed layers allow for efficient low-cost hardware implementations (Schrauwen et al., 2007) and can be characterized using only a random number generator and its seed, which might have repercussions in distributed training and enables highly efficient deployment to edge devices. The strong performance of networks with fixed layers also sheds new light on the inner workings of BERT (Devlin et al., 2018), and layer-wise interpretations of such models (Rogers et al., 2020; Tenney et al., 2019). It appears that “not all layers are created equal” (Zhang et al., 2019) is true to such an extent that some layers can simply remain random and fixed.
+
+These ideas have a long history in machine learning. By Cover’s theorem (Cover, 1965), any highdimensional non-linear transformation is more likely to be linearly separable than its lower-or-equaldimensional input space. By Johnson-Lindenstrauss (Johnson & Lindenstrauss, 1984), random projections distort Euclidean distances very little under mild assumptions, which is useful e.g. for dimensionality reduction and random indexing (Sahlgren, 2005). Fixed random layers in neural networks pre-date deep learning by far (Gamba et al., 1961; Baum, 1988). Indeed, random kernel methods have been an impactful idea in machine learning (Rahimi & Recht, 2008; 2009).
+
+One way to think of such layers is as “reservoirs” (Lukosevi ˇ cius & Jaeger, 2009), where a highly ˇ non-linear high-dimensional black box representation is provided to a lightweight “readout” network, as in echo state networks (Jaeger, 2003) and liquid state machines (Maass et al., 2002). The benefit of such an approach is that the reservoir has fixed parameters and is computationally efficient, as it can be pre-computed and does not (necessarily) require backpropagation.
+
+In NLP, Wieting & Kiela (2019) showed that random sentence encoders present a strong baseline for text classification, with subsequent work showing applications in a variety of NLP tasks (Enguehard et al., 2019; Garg et al., 2020; Pilault et al., 2020). To our knowledge, this work is the first to examine this phenomenon in transformers, and the first to recursively alternate reservoirs with subsequent transformer layers acting as readout functions. We introduce “reservoir transformers”, wherein fixed random reservoir layers are interspersed with regular updateable transformer layers. The goal of this work is not necessarily to set a new state of the art, but to put our understanding of transformer models on a more solid footing by providing empirical evidence of their capabilities even when some of their parameters are fixed. Our contributions are as follows:
+
+• We introduce a new area under the convergence curve metric for measuring performanceefficiency trade-offs, and show that replacing regular transformer layers with reservoir layers leads to better results on that metric.   
+• We show that the addition of reservoir layers in fact leads to improved test set generalization on a variety of tasks in a variety of settings.   
+We show that pre-trained masked language modelling architectures like BERT and RoBERTa (Liu et al., 2019) can benefit from having some of their layers frozen, both during pre-training as well as when fine-tuning on downstream tasks.   
+• In addition, we experiment with different types of reservoir layers, including convolutional and recurrent neural network-based ones. We also show empirical evidence that the backward pass can be entirely skipped by approximating top-layer gradients using an approach we call backskipping, with a relatively small sacrifice in performance.
+
+# 2 APPROACH
+
+This paper is based on a very simple idea. Neural networks are trained via backpropagation, which involves consecutive steps of matrix addition and multiplication, i.e.,
+
+$$
+\theta _ { t + 1 }  \theta _ { t } - \eta \frac { \partial J } { \partial \theta _ { t } } ; \frac { \partial J } { \partial \theta _ { t } } = \frac { \partial J } { \partial L _ { n } } \frac { \partial L _ { n } } { \partial L _ { n - 1 } } \cdot \cdot \cdot \frac { \partial L _ { 1 } } { \partial L _ { 0 } } \frac { \partial L _ { 0 } } { \partial x }
+$$
+
+for some objective $J$ , parameterization $\theta$ and learning rate $\eta$ , with the gradient computed via the chain rule, where $L _ { i }$ is the $i$ -th layer of the neural network and $x$ is the input. Let $L \ =$ Transformer $( X )$ be a single layer in a Transformer network (Vaswani et al., 2017), i.e.,
+
+$$
+\begin{array} { r } { H = \mathbf { M } \mathbf { u } ] \mathrm { t i } \mathbf { H e a d } \mathbf { S e l f A t t n } ( \mathbf { L a y e r N o r m } ( X ) ) + X } \\ { L = \mathbf { F F N } ( \mathbf { L a y e r N o r m } ( H ) ) + H } \end{array}
+$$
+
+Now, during every “backward pass”, we compute the Jacobian for parameters $\theta ^ { L }$ at layer $L$ , which are used to update the parameters of $L$ , $\theta _ { t } ^ { L }$ , as well as to compute the next layer’s Jacobian, thus back-propagating the gradients. In this work however, for some of the layers, we still backpropagate through them to compute gradients for earlier layers, but we never update their parameters. As a result, these layers stay fixed at their random initialization, saving computational resources.
+
+# 2.1 BACKGROUND
+
+Naturally, never updating some of the parameters is computationally more efficient, as some matrix addition operations can be skipped in the backward pass, but why is this not detrimental to the performance of the network?
+
+In the early days of neural networks, the bottom layers were often kept fixed as “associators” (Block, 1962), or what Minsky & Papert (2017) called the Gamba perceptron (Gamba et al., 1961; Borsellino & Gamba, 1961). Fixed random networks (Baum, 1988; Schmidt et al., 1992; Pao et al., 1994) have been explored from many angles, including as “random kitchen sink” kernel machines (Rahimi & Recht, 2008; 2009), “extreme learning machines” (Huang et al., 2006) and reservoir computing (Jaeger, 2003; Maass et al., 2002; Lukosevi ˇ cius & Jaeger, 2009). In reservoir computing, in- ˇ put data are represented through fixed random high-dimensional non-linear representations, called “reservoirs”, which are followed by a regular (often but not necessarily linear) “readout” network to make the final classification decision.
+
+The theoretical justification for these approaches lies in two well-known results in machine learning: Cover’s theorem (Cover, 1965) on the separability of patterns states that high-dimensional non-linear transformations are more likely to be linearly separable; and the Johnson-Lindenstrauss lemma (Johnson & Lindenstrauss, 1984) shows that random projections distort Euclidean distances very little under mild assumptions.
+
+Practically, random layers can be seen as a cheap way to increase network depth. There are interesting advantages to this approach. Fixed layers are known to have particularly low-cost hardware requirements and can be easily implemented on high-bandwidth FPGAs with low power consumption (Hadaeghi et al., 2017; Tanaka et al., 2019), or on optical devices (Hicke et al., 2013). This might yield interesting possibilities for training in a distributed fashion across multiple devices, as well as for neurmorphic hardware (Neftci et al., 2017). This approach also facilitates lower-latency deployment of neural networks to edge devices, since weights can be shared simply by sending the seed number, assuming the random number generator is known on both ends.
+
+# 2.2 RESERVOIR TRANSFORMERS
+
+This work explores inserting random non-linear transformations, or what we call reservoir layers, into transformer networks. Specifically, we experiment with a variety of reservoir layers:
+
+• Transformer Reservoir: The standard transformer layer as described above, but with all parameters fixed after initialization, including the self-attention module.   
+• FFN Reservoir: A transformer-style fixed feed-forward layer without any self-attention, i.e., FFN(LayerNorm(Previous layer)) $^ +$ Previous layer.   
+• BiGRU Reservoir: A fixed bidirectional Gated Recurrent Unit (Cho et al., 2014) layer, which is closer in spirit to previous work on reservoir computing, most of which builds on recurrent neural network architectures. CNN Reservoir: A fixed Convolutional Neural Network (LeCun et al., 1998) layer, specifically light dynamical convolution layers (Wu et al., 2019), which are known to be competitive with transformers in sequence-to-sequence tasks.
+
+We find that all these approaches work well, to a certain extent. For clarity, we focus primarily on the first two reservoir layers, but include a broader comparison in Appendix A.
+
+In each case, contrary to traditional reservoir computing, our reservoir layers are interspersed throughout a regular transformer network, or what we call a reservoir transformer. A good justification for this approach is that while random projections are not learned and might introduce noise, subsequent normal transformer “readout” layers might allow us to recover from any adverse effects of randomness. For example, previous work has shown that ResNets, with all of their parameters fixed except for the scale and shift parameters of batch normalization, can still achieve high performance, simply by scaling and shifting random features (Frankle et al., 2020). Adding noise to the parameters of neural networks is also known to help convergence and generalization (Jim et al., 1995; 1996; Gulcehre et al., 2016; Noh et al., 2017).
+
+# 3 EVALUATION
+
+We evaluate the proposed approach on a variety of well-known tasks in natural language processing, namely: machine translation, language modelling and masked language model pre-training.
+
+In this work, we are not necessarily interested in obtaining the state of the art on any task or even in improving overall task performance via this method. The main objective is to examine efficiency, i.e. the relationship between compute time and task performance. This is closely related to efforts in Green AI, which are concerned with the trade-offs between compute, data, and performance (Schwartz et al., 2019). We propose a new metric for our purposes, the area under the convergence curve (AUCC): similarly to how the area under the receiver operating characteristic (Bradley, 1997, AUC-ROC) measures a classifier’s performance independent of the classification threshold, AUCC measures a model’s performance independent of the specific compute budget. Specifically, AUCC is computed as follows:
+
+![](images/f12829b70f6d779da8d13c4ab6c635efbc557e9a0d0480b52615b7795f025ea3.jpg)  
+Figure 1: Validation BLEU AUCC and test BLEU for IWSLT (high is good). Comparison of regular transformer and reservoir transformer with FFN or Transformer reservoir layers added.
+
+$$
+\int _ { t = 0 } ^ { \hat { T } } \sum _ { x , y \in \mathcal { D } } g _ { t } ( f ( x ) , y )
+$$
+
+where $f$ is the network and $g$ is the evaluation metric, measured until convergence time $\hat { T }$ , which is the maximum convergence time of all models included in the comparison. Note that time here is wall-clock time, not iterations. By convergence, we mean that validation performance has stopped improving, and hence the convergence curve whose area we measure plots the desired metric over time. Runs are averaged over multiple seeds and reported with standard deviation. We normalize raw AUCC scores by their maximum score to ensure a more easily interpretable $[ 0 - 1 ]$ range.
+
+One potential downside of this approach is that the AUCC metric could lead to higher scores for a model that converges quickly but to ultimately worse performance, if measured in a small window. We account for this by making sure that $\hat { T }$ is set sufficiently high. We include the raw validation curves in the appendix and also report test set generalization in each experiment.
+
+# 3.1 EXPERIMENTAL SETTINGS AND IMPLEMENTATION DETAILS
+
+We evaluate on IWSLT de-en (Cettolo et al., 2015) and WMT en-de (Bojar et al., 2014) for machine translation; enwiki8 (LLC, 2009) for language modelling; and experiment with RoBERTa (Liu et al., 2019) in our pretraining experiments. For IWSLT, we follow the pre-processing steps in Edunov et al. (2018). The train/val/test split is $\_$ sentences. For WMT, we follow the pre-processing steps in Ott et al. (2018). The train/val/test split is $4 . 5 \mathrm { M } / 1 6 . 5 \mathrm { k } / 3 \mathrm { k }$ sentences. For enwiki8, we follow the pre-processing steps in Dai et al. (2019). The train/val/test split is 1M/54k/56k sentences. For RoBERTa pretraining, we follow the pre-processing steps in Liu et al. (2019).
+
+We use 8 Volta V100 GPUs for WMT and enwik8, 32 V100 GPUs for RoBERTa and a single V100 for IWSLT. The hyperparameters for IWSLT14 and WMT16 were set to the best-performing values from Ott et al. (2018) and Kasai et al. (2020) respectively. The enwik8 experiment settings followed Bachlechner et al. (2020) and the RoBERTa experiments followed Liu et al. (2019). All experiments were conducted using fairseq (Ott et al., 2019). Our code and experimental settings will be made open source at [ANONYMIZED-GITHUB-URL].
+
+Table 1: Wall-clock time (averaged over multiple runs) saved for IWSLT for different model types and encoder depths. Max BLEU is for validation. Number of layers is for encoder, decoder depth is kept fixed at 2. Ratio is computed compared to comparable number of layers in the normal case.   
+
+<table><tr><td>Model</td><td>#Layers</td><td>Frozen</td><td>Max BLEU</td><td>Train time until max (in hours)</td><td>Ratio</td><td># Params Trainable (Total)</td><td>Train Time each epoch (in seconds)</td></tr><tr><td rowspan="4">Transformer</td><td>6</td><td>0</td><td>34.52 ± 0.07</td><td>2.548 ± 0.06</td><td>1</td><td>26.8M</td><td>122.73 ± 1.16</td></tr><tr><td>8</td><td>0</td><td>34.59 ± 0.11</td><td>2.557 ± 0.05</td><td>1</td><td>31.1M</td><td>142.28 ± 1.87</td></tr><tr><td>10</td><td>0</td><td>34.56 ± 0.05</td><td>3.173 ± 0.04</td><td>1</td><td>35.3M</td><td>161.66 ± 1.54</td></tr><tr><td>12</td><td>0</td><td>34.29 ± 0.12</td><td>3.521 ± 0.09</td><td>1</td><td>39.5M</td><td>172.45 ± 1.98</td></tr><tr><td rowspan="4">TReservoir</td><td>6</td><td>2</td><td>34.37 ± 0.12</td><td>2.422 ± 0.03</td><td>0.95</td><td>22.6M (26.8M)</td><td>120.59 ± 1.32</td></tr><tr><td>8</td><td>2</td><td>34.80 ± 0.07</td><td>2.450 ± 0.06</td><td>0.96</td><td>26.8M (31.1M)</td><td>134.49 ± 1.76</td></tr><tr><td>10</td><td>2</td><td>34.70 ± 0.03</td><td>2.831 ± 0.05</td><td>0.89</td><td>31.1M (35.3M)</td><td>144.42 ± 1.98</td></tr><tr><td>12</td><td>2</td><td>34.78 ± 0.04</td><td>3.476 ± 0.04</td><td>0.98</td><td>35.3M (39.5M)</td><td>159.43 ± 1.67</td></tr><tr><td rowspan="4">FFN Reservoir</td><td>6</td><td>2</td><td>34.43 ± 0.15</td><td>2.120 ± 0.04</td><td>0.83</td><td>22.6M (25.8M)</td><td>107.71 ± 1.73</td></tr><tr><td>8</td><td>2</td><td>34.56 ± 0.16</td><td>2.203 ± 0.06</td><td>0.86</td><td>26.8M (29.1M)</td><td>120.07 ± 1.65</td></tr><tr><td>10</td><td>2</td><td>34.66 ± 0.02</td><td>2.493 ± 0.05</td><td>0.79</td><td>31.1M (33.3M)</td><td>130.11 ± 1.43</td></tr><tr><td>12</td><td>2</td><td>34.76 ± 0.03</td><td>3.241 ± 0.04</td><td>0.92</td><td>35.3M (37.5M)</td><td>156.32 ± 1.87</td></tr><tr><td rowspan="4">LayerDrop</td><td>6</td><td>2</td><td>34.59 ± 0.15</td><td>2.364 ± 0.08</td><td>0.92</td><td>22.6M (26.8M)</td><td>119.30 ± 1.36</td></tr><tr><td>8</td><td>2</td><td>34.58 ± 0.16</td><td>2.554 ± 0.05</td><td>0.99</td><td>26.8M (31.1M)</td><td>138.62 ± 1.44</td></tr><tr><td>10</td><td>2</td><td>34.57 ± 0.07</td><td>3.404 ± 0.06</td><td>1.07</td><td>31.1M (35.3M)</td><td>140.88 ± 1.62</td></tr><tr><td>12</td><td>2</td><td>33.65 ± 0.24</td><td>3.251 ± 0.04</td><td>0.92</td><td>35.3M (39.5M)</td><td>160.85 ± 1.49</td></tr></table>
+
+All the experiments in this paper were run with 3 random seeds and the mean and standard deviation are reported. For the relatively small IWSLT, the $\hat { T }$ value in the AUCC metric was set to 4 hours. For WMT, which is larger, we set it to 20 hours. For enwiki8, it was 30 hours; and for the RoBERTa pre-training experiments, it was set to 60 hours.
+
+The projection weights in random layers were initialized using orthogonal initialization (Saxe et al., 2013), which makes sense since random orthogonal projections should be most informationpreserving, and which was found to work well empirically for initializing fixed random representations in previous work (Wieting & Kiela, 2019). Biases and layer norm parameters were initialized using their respective PyTorch defaults (based on Xavier init; Glorot & Bengio, 2010).
+
+We intersperse reservoir layers in alternating fashion starting from the middle. Specifically, we alternate one reservoir layer with one transformer layer, and place the alternating block in the middle. For example: a 7-layer encoder LLLLLLL in which we replace three layers with reservoirs becomes LRLRLRL, and with two becomes LLRLRLL. See Appendix C for a study comparing this strategy to alternative approaches (e.g., freezing in the bottom, middle or top).
+
+# 4 EXPERIMENTS
+
+In what follows, we first show our main result: reservoir transformers often have better AUCC metrics, less training time per epoch, less convergence time until the best validation performance is achieved, and even improved test set generalization metrics, on a variety of tasks. As a strong baseline method, we compare to LayerDrop (Fan et al., 2019). LayerDrop can also be seen as a method that dynamically bypasses parts of the computation during Transformer training in an attempt to improve efficiency, and is a suitable comparison to examine our methods.. We also examine whether we can minimize the expectation over the gradients of upper layers in the transformer network such that we do not have to pass the true gradients through the reservoir for further efficiency.
+
+# 4.1 MACHINE TRANSLATION
+
+Machine translation (MT) is one of the core tasks of NLP. We demonstrate on two well-known MT datasets, IWSLT’14 German-English and WMT’16 English-German, that reservoir transformers obtain a better AUCC. For the raw validation plots over time that were used to calculate the AUCC, please refer to Appendix F.
+
+Following Kasai et al. (2020), the architecture of the network is an N-layer reservoir transformer encoder, followed by a regular shallow one- or two-layer decoder. This design choice has been shown to lead to very good speed and efficiency trade-offs, and serves as a good baseline for our experiments. Moreover, shallow decoders make it easier to decide where to place reservoir layers (in the encoder) and makes it more straightforward to identify where performance gains come from.
+
+![](images/90e2f338200568e105b601ef0f6d470de812574f5e1411facb2387a68f64cf2e.jpg)  
+Figure 2: Validation BLEU AUCC and test BLEU for WMT (high is good). Comparison of regular transformer and reservoir transformer with FFN or Transformer reservoir layers added.
+
+![](images/fbf8f8b964d88c1ee463d7c528fdd11742523e63086217892f8a7d6bfafcc93f.jpg)  
+Figure 3: Validation BPC AUCC and test BPC on the enwik8 language modelling task (low is good). Comparison of regular and reservoir transformers for varying depths.
+
+Figure 1 shows the results for IWSLT. On the y-axis we show validation AUCC for the BLEU metric; on the $\mathbf { X } ^ { } -$ -axis we show the number of updatable layers in the encoder. The performance of a regular transformer encoder with 6 layers and a reservoir transformer encoder with 6 layers plus N additional reservoir layers are plotted for the same $\mathbf { X }$ -axis value to show the total number of updated layers. Plots for the total number of layers (updatable plus not-updatable, so essentially shifted versions) are shown in Appendix E. Table 1 shows the time it took to achieve the maximum validation BLEU score and how that relates to the regular transformer, demonstrating that reservoir transformers consistently converge faster in terms of wall-clock time, up to $22 \%$ as much with the same number of updateable layers. We save as much as $27 \%$ time until convergence a 24 layer model on WMT, as shown in Table 3. One other noticeable point is that we can see that the T Reservoir achieves similar performance to LayerDrop on IWSLT and WMT in terms of wall-clock per epoch and wall-clock time to the best performance. However, on both tasks, FFN Reservoir performs much better than LayerDrop in terms of efficiency per epoch and achieves better/similar performance in less time in each case. As a point of reference, a half hour gain on IWSLT translates to a gain of several days in the training of bigger transformer models like GPT-3 (Brown et al., 2020).
+
+We observe that reservoir transformers consistently perform better than, or are competitive to, regular transformers, both in terms of validation BLEU AUCC as well as test time BLEU, for all examined encoder depths.
+
+![](images/d4ae168b422fd5b5699ef8859159ebcc355c0e8a55b9feb4da847e6f8b55cace.jpg)  
+Figure 4: Downstream RoBERTa performance on SST-2 (left) and MultiNLI-matched (right).
+
+Figure 2 shows a similar trend for WMT. WMT is much larger and requires a much deeper encoder, as illustrated by the fact that a certain minimum depth is required for reservoir transformers to achieve a comparable validation AUCC. At test time, reservoir transformers outperform regular transformers for almost all encoder depths. The FFN reservoir transformer seems to work best in both cases, which is surprising because it does not have any self-attention component at all. This finding shows that self-attention, or the mechanism to summarize context information, should be learned if present. Once the context features have been gathered, a random projection via a fixed FFN module appears to be beneficial, at least for MT.
+
+# 4.2 LANGUAGE MODELLING
+
+To examine whether the same findings hold for other tasks, we evaluate on the enwiki8 (LLC, 2009) language modelling task. We examine the BPC (bits per character) rate for a variety of network depths (since the task is language modelling, these layers are in the decoder). The results show that we obtain consistently better BPC for lower depths, except for the 64-layer regular transformer, which appears to be particularly optimal for this task. We observe similar trends during test time.
+
+# 4.3 MASKED LANGUAGE MODEL PRETRAINING
+
+We train RoBERTa (Liu et al., 2019) models from scratch at a variety of depths, both in the normal and reservoir setting. We find that these networks show minor differences in their best perplexity and similar AUCC perplexity (see Appendix D). We then examine the performance of these models when fine-tuned on downstream tasks, specifically the well known SST-2 (Socher et al., 2013) and MultiNLI1 (Williams et al., 2017) tasks. When fine-tuning the reservoir models, we keep the reservoir layers fixed (including them in fine-tuning did not work very well, see Appendix D).
+
+Figure 4 shows the results of fine-tuning. We observe that the reservoir transformer outperforms normal RoBERTa at all depths in both tasks. At lower depth, the improvements are substantial. As a sanity check, we also experiment with freezing some of the layers in normal RoBERTa during fine-tuning (Transformer frozen finetuned) and show that this helps a little but is still outperformed by the reservoir transformer.
+
+These findings suggest that you can train a RoBERTa model without updating all of the layers, achieve similar perplexity at a similar computational cost, but with better downstream performance. The fact that some layers can be kept random and entirely fixed during training, without sacrificing any performance, raises intriguing questions for “BERTology” (Rogers et al., 2020) and for the study of what different layers in transformers learn.
+
+![](images/8dc191b8c7100f587adeeeb66439469b208ef40391084c7b050f44a7cb3a4cb5.jpg)  
+Figure 5: IWSLT comparison of normal v frozen v backskipped
+
+# 4.4 BACKSKIPPING
+
+With the reservoir transformers as described above, we obtain better efficiency by skipping the “gradient application” matrix addition step in some of the layers (i.e., updating the weights). One step further would be to investigate skipping the entire backward pass for reservoirs altogether, which would save us from having to do the much more expensive matrix multiplication for these layers that is required for the propagation of gradients. We report on preliminary experiments where in the backward pass we replace the gradients for the layer $L _ { i }$ going into the reservoir $L _ { i + 1 }$ with a noisy estimate (Jaderberg et al., 2017; Czarnecki et al., 2017). Promisingly, Oktay et al. (2020) recently asked “why spend resources on exact gradients when we’re going to use stochastic optimization?” and show that you can do randomized auto-differentiation quite successfully.
+
+Here, rather than minimizing the actual gradients $\frac { \partial L _ { i } } { \partial \theta ^ { L _ { i } } }$ , we minimize their expectation and train via continuous-action REINFORCE (Williams, 1992). That is, $L _ { i }$ becomes a policy $\pi _ { a }$ : $s  \mu$ whei.e., $\textstyle { \frac { 1 } { n } } \sum _ { i = 0 } ^ { n } ( { \dot { R } } ^ { i } - V ^ { i } ( a ) ) ^ { 2 }$ $a \sim \mathcal { N } ( \mu , 1 )$ . We train to miniREINFORCE loss $\mathbb { E } _ { a } \left[ \log ( { \bar { a } } ) \left( R - { \bar { V } } ( a ) \right) \right]$ ion loss via MSE,, where the value network $V$ acts as the baseline. $R$ is defined as the mean of the gradients of the top layer $L _ { i + 2 }$ , with the sign flipped. Thus, simply put, we train to minimize the expectation of the true gradients at the layer directly following the reservoir. We employ an annealing scheme where we first train the value network and propagate the true gradients during warmup. Afterwards, we anneal the probability of backskipping rather than performing a true backward pass (multiplying the probability by 0.99 every iteration until we only backskip). We experimented with setting $R$ to the negation of the total loss as well but found the current reward to work better. We call this approach backskipping.
+
+Figure 5 shows the results as validation BLEU over time. We observe that this approach helps especially during the earlier stages of training. Although it does not match the performance of the approach with true gradients quite yet, it actually performs competitively. Backskipping looks promising as an approach to further reduce computational costs, and would be even more efficient from a hardware perspective since the circuitry for such layers (which do not need to propagate gradients) can effectively be hardwired entirely.
+
+# 5 RELATED WORK
+
+Recent work has shown that modern NLP models are able to function with different numbers of layers for different examples (Elbayad et al., 2019; Fan et al., 2019); that different layers specialize for different purposes (Zhang et al., 2019); that layers can be compressed (Li et al., 2020); and, that layers can be reordered (Press et al., 2019). There is a growing body of work in efficient self-attention networks (Tay et al., 2020b), such as linear attention (Wang et al., 2020), on how to process long context information (Beltagy et al., 2020) and on approximations to make transformers more scalable (Kitaev et al., 2020; Katharopoulos et al., 2020). BigBIRD (Zaheer et al., 2020) provides random keys as additional inputs to its attention mechanism. Locality sensitive hashing (LSH) as employed e.g. in Reformer (Kitaev et al., 2020) utilizes a fixed random projection. Performer (Choromanski et al., 2020) computes the transformer’s multi-head attention weights as a fixed orthogonal random projection. Closely related to this work, Tay et al. (2020a) showed that randomized alignment matrices in their “Synthesizer” architecture are sufficient for many NLP tasks. While these works focus on random attention, we show that entire layers can be random and fixed. We also show that entire layers can be replaced by fixed random projections that do not have any attention whatsoever.
+
+Beyond transformers, random features have been extensively explored. Examples of this include FreezeOut (Brock et al., 2017), deep reservoir computing networks (Scardapane & Wang, 2017; Gallicchio & Micheli, 2017), as well as applications in domains as varied as text classification (Conneau et al., 2017; Zhang & Bowman, 2018; Wieting & Kiela, 2019) or music classification (Pons & Serra, 2019). It is well known that randomly initialized networks can display impressive performance on their own (Ulyanov et al., 2018; Rosenfeld & Tsotsos, 2019; Ramanujan et al., 2020), which underlies, for example, the recently popularized lottery ticket hypothesis (Frankle & Carbin, 2018; Zhou et al., 2019). We know that learning deep overparameterized networks appears to help in general (Li & Liang, 2018; Du et al., 2019). Our method represents an easy and cheap way to add both depth and parameters to transformer networks.
+
+# 6 CONCLUSION
+
+This work demonstrated that state-of-the-art transformer architectures can be trained without updating all of the layers. This complements a long history in machine learning of harnessing the power of random features. In most cases, “reservoir transformers” achieve better performance-efficiency trade-offs as measured by our newly introduced AUCC metric, and better test set generalization, on a variety of tasks and in a variety of settings. Future work includes further investigating hybrid networks and backskipping architectures, as well as utilizing pruning strategies at inference time, in order to try to obtain even better performance/efficiency trade-offs.
+
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+![](images/b02380450760b4c0b3afd0aed260a881b2948fde2eb7140d57dab6c452cac15a.jpg)  
+Figure 6: IWSLT comparison of different hybrid architectures with different reservoir layers.
+
+![](images/91e3cdb50285f6e0d0f5645a8f28b599dc264c710e34a17c857f20ebe2df2ca1.jpg)  
+Figure 7: IWSLT validation AUCC and test BLEU with 6-layer decoder.
+
+A HYBRID NETWORKS AND NON-TRANSFORMER RESERVOIRS
+
+We investigate whether reservoir layers need to be transformer-based (or transformers-withoutattention, i.e., FFN). We examine two different alternatives: bidirectional Gated Recurrent Units (Cho et al., 2014) and Convolutional Neural Networks (LeCun et al., 1998; Kim, 2014), specifically light dynamical convolutions (Wu et al., 2019). Figure 6 shows the results for these hybrids: depending on the setting, they may obtain a better AUCC than the regular transformer, but this is less consistent than with the other reservoir layers, most likely because these layers have different computational properties. It’s possible that these hybrids simply require further tuning, as we found e.g. up-projecting to help for BiGRUs, but studying this is outside of the scope of the current work.
+
+# B DEEP DECODERS
+
+We show that the same results hold for a 6-layer decoder on IWSLT (although less pronounced for AUCC, probably because the decoder is computationally heavier). See Figure 7 and Table 2.
+
+# C FREEZING STRATEGY
+
+We explored different strategies for the placement of reservoir layers and found the “alternating” strategy reported in the main body of the paper to work best. Generally, we found repetitive application of reservoirs to yield diminishing returns, as might be expected. See Figure 8.
+
+Table 2: Wall-clock time (averaged over multiple runs) saved for IWSLT for different model types and encoder depths. Max BLEU is for validation. Number of layers is for encoder, decoder depth is kept fixed at 6. Ratio is computed compared to comparable number of layers in the normal case.   
+
+<table><tr><td>Model</td><td>#Layers</td><td>Frozen</td><td>Max BLEU</td><td>Train time until max (in hours)</td><td>Ratio</td><td># Params Trainable (Total)</td><td>Train Time each epoch (in seconds)</td></tr><tr><td rowspan="4">Transformer</td><td>6</td><td>0</td><td>34.97 ± 0.05</td><td>1.984 ± 0.02</td><td>1</td><td>39.5M</td><td>177.84 ± 2.98</td></tr><tr><td>8</td><td>0</td><td>34.99 ± 0.08</td><td>2.161 ± 0.03</td><td>1</td><td>43.7M</td><td>206.59 ± 3.47</td></tr><tr><td>10</td><td>0</td><td>34.98 ± 0.04</td><td>2.345± 0.02</td><td>1</td><td>47.9M</td><td>236.72 ± 3.52</td></tr><tr><td>12</td><td>0</td><td>34.78 ± 0.11</td><td>2.535 ± 0.05</td><td>1</td><td>52.0M</td><td>265.90 ± 4.97</td></tr><tr><td rowspan="4">TReservoir</td><td>6</td><td>2</td><td>34.73 ± 0.11</td><td>1.838 ± 0.01</td><td>0.92</td><td>35.3M (39.5M)</td><td>166.11 ± 2.21</td></tr><tr><td>8</td><td>2</td><td>35.07 ± 0.05</td><td>1.912 ± 0.03</td><td>0.88</td><td>39.5M (43.7M)</td><td>190.08 ± 3.73</td></tr><tr><td>10</td><td>2</td><td>35.02 ± 0.01</td><td>1.970 ± 0.04</td><td>0.84</td><td>43.7M (47.9M)</td><td>204.42 ± 2.89</td></tr><tr><td>12</td><td>2</td><td>35.06 ± 0.02</td><td>2.429 ± 0.02</td><td>0.95</td><td>47.8M (52.0M)</td><td>236.41 ± 4.35</td></tr><tr><td rowspan="4">FFN Reservoir</td><td>6</td><td>2</td><td>34.85 ± 0.10</td><td>1.729 ± 0.03</td><td>0.87</td><td>35.3M (37.4M)</td><td>161.72 ± 2.32</td></tr><tr><td>8</td><td>2</td><td>34.99 ± 0.11</td><td>1.751 ± 0.02</td><td>0.81</td><td>39.5M (41.6M)</td><td>180.21 ± 2.68</td></tr><tr><td>10</td><td>2</td><td>34.92 ± 0.03</td><td>1.907 ± 0.02</td><td>0.81</td><td>43.7M (45.8M)</td><td>191.40 ± 2.49</td></tr><tr><td>12</td><td>2</td><td>35.16 ± 0.04</td><td>2.395 ± 0.01</td><td>0.94</td><td>47.8M (49.9M)</td><td>216.08 ± 2.57</td></tr><tr><td rowspan="4">LayerDrop</td><td>6</td><td>２2</td><td>34.51 ± 0.12</td><td>1.908 ± 0.04</td><td>0.96</td><td>35.3M (39.5M)</td><td>169.62 ± 3.16</td></tr><tr><td>8</td><td></td><td>34.77 ± 0.11</td><td>2.023 ± 0.02</td><td>0.94</td><td>39.5M (43.7M)</td><td>186.71 ± 2.17</td></tr><tr><td>10</td><td>2</td><td>34.06 ± 0.05</td><td>1.912 ± 0.02</td><td>0.97</td><td>43.7M (47.9M)</td><td>205.52 ± 3.31</td></tr><tr><td>12</td><td>2</td><td>34.08 ± 0.13</td><td>2.524 ± 0.01</td><td>0.99</td><td>47.8M (52.0M)</td><td>222.45 ± 2.21</td></tr></table>
+
+<table><tr><td>Model</td><td> # Layers</td><td>Frozen</td><td>Max BLEU</td><td>Train time until max (in hours)</td><td>Ratio</td><td># Params Trainable (Total)</td><td>Train Time each epoch (in hours)</td></tr><tr><td rowspan="4">Transformer</td><td>12</td><td>0</td><td>24.46 ± 0.04</td><td>15.15 ± 0.15</td><td>1</td><td>75.6M</td><td>0.505 ± 0.005</td></tr><tr><td>16</td><td>0</td><td>24.52 ± 0.03</td><td>16.05 ± 0.18</td><td></td><td>88.2M</td><td>0.643 ± 0.006</td></tr><tr><td>24</td><td>0</td><td>24.69 ± 0.05</td><td>17.61 ± 0.85</td><td>1</td><td>113.4M</td><td>0.877 ± 0.029</td></tr><tr><td>32</td><td>0</td><td>24.83 ± 0.04</td><td>18.42 ± 0.28</td><td>1</td><td>138.6M</td><td>1.036 ± 0.010</td></tr><tr><td rowspan="4">TReservoir</td><td>12</td><td>4</td><td>24.26 ± 0.08</td><td>14.11 ± 0.21</td><td>0.93</td><td>72.4M (75.6M)</td><td>0.472 ± 0.007</td></tr><tr><td>16</td><td>4</td><td>24.50 ± 0.05</td><td>15.25 ± 0.28</td><td>0.95</td><td>75.6M (88.2M)</td><td>0.596 ± 0.009</td></tr><tr><td>24</td><td>4</td><td>25.11 ± 0.07</td><td>15.89 ± 0.74</td><td>0.90</td><td>100.8M (113.4M)</td><td>0.776 ± 0.024</td></tr><tr><td>32</td><td>4</td><td>24.66 ± 0.04</td><td>16.38 ± 0.24</td><td>0.88</td><td>126.0M (138.6M)</td><td>0.998 ± 0.009</td></tr><tr><td rowspan="4">FFN Reservoir</td><td>12</td><td>4</td><td>24.42 ± 0.05</td><td>14.01 ± 0.09</td><td>0.92</td><td>72.4M (71.4M)</td><td>0.441 ± 0.003</td></tr><tr><td>16</td><td>4</td><td>24.65 ± 0.07</td><td>14.53 ± 0.17</td><td>0.91</td><td>75.6M (83.9M)</td><td>0.524 ± 0.006</td></tr><tr><td>24</td><td>4</td><td>24.93 ± 0.04</td><td>12.62 ± 1.53</td><td>0.71</td><td>100.8M (109.2M)</td><td>0.743 ± 0.018</td></tr><tr><td>32</td><td>4</td><td>24.98 ± 0.03</td><td>13.96 ± 0.19</td><td>0.73</td><td>126.0M (134.4M)</td><td>0.964 ± 0.007</td></tr><tr><td rowspan="4">LayerDrop</td><td>12</td><td>4</td><td>24.27 ± 0.03</td><td>14.61 ± 0.14</td><td>0.96</td><td>72.4M (75.6M)</td><td>0.489 ± 0.006</td></tr><tr><td>16</td><td>4</td><td>24.15 ± 0.06</td><td>15.55 ± 0.54</td><td>0.97</td><td>75.6M (88.2M)</td><td>0.597 ± 0.017</td></tr><tr><td>24</td><td>4</td><td>24.37 ± 0.05</td><td>16.25 ± 0.36</td><td>0.92</td><td>100.8M (113.4M)</td><td>0.823 ± 0.013</td></tr><tr><td>32</td><td>4</td><td>23.84 ± 0.03</td><td>15.27 ± 0.38</td><td>0.83</td><td>126.0M (138.6M)</td><td>1.028 ± 0.012</td></tr></table>
+
+Table 3: Wall-clock time (averaged over multiple runs) saved for WMT for different model types and encoder depths. Max BLEU is for validation. Number of layers is for encoder, decoder depth is kept fixed at 1. Ratio is computed compared to comparable number of layers in the normal case.
+
+# D ROBERTA RESULTS
+
+Here we present the additional RoBERTa results for convergence plot and AUCC in various decoder depth setting in Figure 10. As stated in the main paper, the difference of AUCC / Convergence Plot between RoBERTa model with or without Reservoir layers are limited. Moreover, we plot the downstream task performance for SST-2 and MNLI compared to the pretraining wall-clock time in Figure 9. It can be seen that the FFN Reservoir can achieve up to $\cdot$ and $10 \%$ pretraining time savings while matching the best performance of vanilla transformers for MNLI-m and SST2, respectively.
+
+# E RESERVOIR RESULTS FOR TOTAL LAYERS
+
+Here we present the shifted Reservoir Results for IWSLT14, WMT16, Enwik8 and RoBERTa finetuning in Figure 11, 12, 13, 14, respectively. We show the same results also hold when it comes to replace normal transformer blocks with Reservoir blocks at least for MT.
+
+Table 4: Wall-clock time (averaged over multiple runs) saved for IWSLT/WMT for different model types and encoder depths. $\cdot$ Max BLEU is for validation.   
+
+<table><tr><td>Mode1</td><td>#Layers</td><td>IWSLT-Dec2 Train time until 95% max (in hours)</td><td>Max BLEU (95%)</td><td>#Layers</td><td>IWSLT-Dec6 Train time until 95% max (in hours)</td><td>Max BLEU (95%)</td><td>#Layers</td><td>WMT-Dec1 Train time until 95% max (in hours)</td><td>Max BLEU (95%)</td></tr><tr><td rowspan="6">Transformer</td><td>6</td><td>0.647 ± 0.03</td><td>32.89 ± 0.04</td><td>6</td><td>0.642 ± 0.02</td><td>33.36 ± 0.03</td><td>16</td><td>3.788 ± 0.053</td><td>23.36 ± 0.06</td></tr><tr><td></td><td>0.711 ± 0.05</td><td>33.04 ± 0.03</td><td>8</td><td>0.765 ± 0.03</td><td>33.41 ± 0.08</td><td></td><td>3.820 ± 0.072</td><td>23.41 ± 0.05</td></tr><tr><td></td><td>0.808 ± 0.02</td><td>33.96 ± 0.08</td><td>10</td><td>0.898 ± 0.04</td><td>33.32 ± 0.07</td><td></td><td>5.262 ± 0.607</td><td>23.50 ± 0.03</td></tr><tr><td>12</td><td>1.037 ± 0.03</td><td>33.07 ± 0.09</td><td>12</td><td>1.037 ± 0.03</td><td>33.07 ± 0.11</td><td>五</td><td>6.212 ± 0.232</td><td>23.81 ±0.04</td></tr><tr><td></td><td>0.569 ± 0.02</td><td>32.78 ±0.03</td><td>6</td><td>0.599 ± 0.01</td><td>33.09 ± 0.05</td><td></td><td>3.563 ± 0.061</td><td>23.21 ± 0.04</td></tr><tr><td>8</td><td>0.619 ± 0.04</td><td>33.12 ± 0.05</td><td>8</td><td>0.726 ± 0.02</td><td>33.38 ± 0.09</td><td>1</td><td>3.603 ± 0.056</td><td>23.80 ± 0.06</td></tr><tr><td rowspan="5">T Reservoir</td><td></td><td>0.729 ± 0.04</td><td>33.13 ± 0.07</td><td>10</td><td>0.738 ± 0.03</td><td>33.37 ± 0.04</td><td>24</td><td>4.923 ± 0.771</td><td>23.75 ± 0.02</td></tr><tr><td>12</td><td>0.982 ± 0.02</td><td>33.03 ± 0.11</td><td>12</td><td>0.958 ± 0.01</td><td>33.46± 0.09</td><td>32</td><td>5.780 ± 0.214</td><td>23.71 ±0.03</td></tr><tr><td>6</td><td>0.521 ± 0.05</td><td>32.85 ± 0.02</td><td>6</td><td>0.594 ± 0.03</td><td>33.13 ± 0.04</td><td>12</td><td>3.417 ± 0.046</td><td>23.22 ± 0.07</td></tr><tr><td>8</td><td>0.533 ± 0.03</td><td>33.84 ± 0.04</td><td>8</td><td>0.651 ± 0.04</td><td>33.36 ± 0.06</td><td>16</td><td>3.527 ± 0.063</td><td>23.54 ± 0.05</td></tr><tr><td>10</td><td>0.614 ± 0.01</td><td>33.05 ± 0.08</td><td>10</td><td>0.627 ± 0.05</td><td>33.26 ± 0.03</td><td>24</td><td>4.197 ± 0.697</td><td>23.74 ± 0.06</td></tr><tr><td rowspan="5">LayerDrop</td><td>12</td><td>0.811 ± 0.02</td><td>33.26 ± 0.10</td><td>12</td><td>0.780 ± 0.02</td><td>33.46 ± 0.08</td><td>32</td><td>4.984 ± 0.321</td><td>23.82 ± 0.02</td></tr><tr><td></td><td>0.837 ±0.08</td><td>32.87 ± 0.05</td><td>6</td><td>0.706 ±0.01</td><td>33.08 ± 0.03</td><td>12</td><td>3.912 ± 0.068</td><td>23.33 ± 0.08</td></tr><tr><td>6</td><td>0.934 ± 0.07</td><td>33.12 ± 0.03</td><td>8</td><td>0.753 ± 0.04</td><td>33.14 ± 0.05</td><td>16</td><td>3.581 ± 0.076</td><td>23.17 ± 0.04</td></tr><tr><td>10</td><td>0.901 ± 0.06</td><td>33.18 ±0.02</td><td>10</td><td>0.691 ± 0.03</td><td>32.39 ± 0.05</td><td>3</td><td>4.875 ± 0.728</td><td>23.43 ± 0.07</td></tr><tr><td>12</td><td>0.914 ± 0.01</td><td>32.33 ± 0.06</td><td>12</td><td>0.803 ± 0.02</td><td>32.94 ± 0.10</td><td></td><td>5.980 ± 0.219</td><td>22.97 ± 0.08</td></tr></table>
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">#Layers</td><td colspan="3">IWSLT-Dec2 Train time</td><td colspan="3">IWSLT-Dec6</td><td rowspan="2"></td><td colspan="2">WMT-Dec1</td></tr><tr><td>until 99 % max (in hours)</td><td>Max BLEU (99%)</td><td>#Layers</td><td>Train time until 99 % max (in hours)</td><td>Max BLEU (99%)</td><td>#Layers</td><td>Train time until 99 % max (in hours)</td><td>Max BLEU (99%)</td></tr><tr><td rowspan="5">Transformer</td><td></td><td>1.454 ± 0.06</td><td>34.24 ± 0.05</td><td>6</td><td>1.297 ± 0.03</td><td>34.69 ± 0.05</td><td>1</td><td>9.961 ± 0.053</td><td></td><td>24.27 ± 0.04</td></tr><tr><td></td><td>1.475 ± 0.09</td><td>34.32 ± 0.09</td><td>8</td><td>1.390 ± 0.02</td><td></td><td></td><td></td><td>12.623 ± 0.072</td><td>24.35 ± 0.06</td></tr><tr><td>10</td><td>1.526 ± 0.04</td><td>34.25 ± 0.04</td><td></td><td></td><td>1.622 ± 0.05</td><td>34.75 ± 0.09 34.64 ± 0.03</td><td>3</td><td>13.412 ± 0.837</td><td>24.49 ± 0.07</td></tr><tr><td>12</td><td>2.259 ± 0.07</td><td>34.24 ± 0.11</td><td>10 12</td><td></td><td>1.748 ± 0.01</td><td>34.66 ± 0.08</td><td></td><td>15.117 ± 0.232</td><td>24.56 ± 0.02</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="5">TReservoir</td><td></td><td>1.257 ± 0.04</td><td>34.05 ± 0.09</td><td>6</td><td>1.291 ± 0.03</td><td>34.51 ± 0.10</td><td>1</td><td>8.314 ± 0.062</td><td></td><td>24.15 ± 0.06</td></tr><tr><td>10</td><td>1.472 ± 0.06</td><td>34.47 ± 0.05</td><td></td><td>1.339 ± 0.03</td><td>34.80 ± 0.04</td><td></td><td></td><td>9.221 ± 0.073</td><td>24.41 ± 0.05</td></tr><tr><td>12</td><td>1.530 ± 0.03</td><td>34.36 ± 0.02</td><td>10</td><td>1.419 ± 0.04</td><td></td><td>34.72 ± 0.03</td><td></td><td>10.413 ± 0.580</td><td>24.56 ± 0.03</td></tr><tr><td></td><td>2.043 ± 0.05</td><td>34.53 ± 0.07</td><td>12</td><td>1.642 ± 0.02</td><td></td><td>34.87 ± 0.02</td><td>3</td><td>11.465 ± 0.227</td><td>24.49 ±0.01</td></tr><tr><td>680</td><td>1.138 ± 0.03</td><td>34.10 ± 0.13</td><td>6</td><td>1.169 ± 0.02</td><td></td><td>34.71 ± 0.09</td><td></td><td>7.407 ± 0.087</td><td>24.33 ± 0.08</td></tr><tr><td rowspan="5">FFN Reservoir</td><td></td><td>1.101 ± 0.07</td><td>34.32 ± 0.11 34.36 ± 0.03</td><td>8 10</td><td>1.201 ± 0.03 1.276 ± 0.03</td><td>34.79 ±0.08 34.63 ± 0.03</td><td></td><td></td><td>9.336 ± 0.036</td><td>24.42 ± 0.05</td></tr><tr><td>12</td><td>1.281 ± 0.01</td><td></td><td></td><td></td><td></td><td></td><td></td><td>9.978 ± 0.546</td><td>24.91 ± 0.07</td></tr><tr><td></td><td>1.785 ± 0.03</td><td>34.42 ± 0.06</td><td>12</td><td>1.440 ± 0.01</td><td>34.87 ± 0.02</td><td></td><td></td><td>10.524 ± 0.341</td><td>24.96 ± 0.01</td></tr><tr><td>8</td><td>1.363 ± 0.05</td><td>34.58 ± 0.14</td><td>6</td><td>1.253 ± 0.01</td><td></td><td>34.42 ± 0.10</td><td></td><td>8.372 ± 0.059</td><td>24.17 ± 0.04</td></tr><tr><td></td><td>1.468 ± 0.03</td><td>34.50 ± 0.12</td><td>8</td><td>1.244 ± 0.04</td><td>34.44 ± 0.09</td><td></td><td></td><td>9.741 ± 0.043</td><td>23.93 ± 0.08</td></tr><tr><td rowspan="4">LayerDrop</td><td>10</td><td>1.678 ± 0.04</td><td>34.52 ± 0.07</td><td>10</td><td></td><td></td><td>33.83 ±0.06</td><td>16 3</td><td>10.145 ± 0.628</td><td>24.07 ± 0.09</td></tr><tr><td>12</td><td>2.071 ± 0.02</td><td>33.45 ± 0.23</td><td>12</td><td>1.343 ± 0.04 1.423 ± 0.02</td><td>33.97 ± 0.12</td><td></td><td></td><td>10.168 ± 0.329</td><td>23.81 ± 0.03</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 5: Wall-clock time (averaged over multiple runs) saved for IWSLT/WMT for different model types and encoder depths. $9 9 \%$ Max BLEU is for validation.
+
+# F VALIDATION PLOTS
+
+Here we present the validation plots for training a 8-layer encoder, 2-layer decoder model for IWSLT14, a 24-layer encoder, 1-layer decoder model for WMT14, a 48-layer decoder model for enwik8 and a 12-layer decoder model for RoBERTa for detailed steps to calculate the AUCC. It can be clearly observed that given the configurations from Section 3.1, all the models have converged. So when we compute the area under the convergence curve, this depicts the training efficiency of the model (basically time x performance) until convergence. Specifically, we set T sufficiently high for computing the AUCC, which is 4h for IWSLT, 20h for WMT, 30h for enwik8 and 60h for RoBERTa pretraning. From the training plot in the appendix, we can see that each model has converged at that point. The Reservoir model in Figure 15 has 2 layers frozen for IWSLT14, 8 layers frozen for enwik8, and 4 layers frozen for WMT14 and RoBERTa.
+
+# G ROBERTA PROBING
+
+We follow Jawahar et al. (2019) and investigate what the frozen layers in the Reservoir Transformer have actually “learned” (while being forzen) as measured by probing tasks, reported in Table 6. The results are gathered over 3 random seeds for reporting the mean and standard deviation. From the table, we can see that generally probing performance is quite similar between Transformer and the T Reservoir model. We also noticed that the representations collected after the frozen layer (3, 5, 7, 9) in the T Reservoir actually have significantly better performance over the regular Transformer representations across all the probing tasks. This has interesting repercussions for the study of “BERTology”, as it clearly shows, somewhat confusingly, that even completely random and frozen layers represent linguistic phenomena.
+
+![](images/bdd1e0f5437432a0076b1c35076cc1e6057826414f43599a327bf57236d29faf.jpg)  
+Figure 8: IWSLT with 2-layer decoder using different freezing strategy.
+
+![](images/8efb8273099ff74135796331759c0532b0ccf7448a7b83e8c437cb3b5dc2a6fb.jpg)  
+Figure 9: RoBERTa Reservoir Results, Pre-training versus downstream task plot for 12 layer RoBERTa. MNLI-m (left). SST-2 (right).
+
+![](images/16c67d74e5743eb685c5b29836cc09d4540ceae3fe880d614d534330288bb088.jpg)  
+Figure 10: RoBERTa Reservoir Results, Training plot for 12 layer RoBERTa (left). AUCC result (right).
+
+![](images/e38bef769826f1e91e25614fc341e46647bcb6ea7bc3f2a9aafc23ea160c51d5.jpg)  
+Figure 11: Validation BLEU AUCC and test BLEU for IWSLT (high is good). Comparison of regular transformer and reservoir transformer with FFN or Transformer reservoir layers added.
+
+![](images/a79c4059de5d7c1e244e50c3037c5c3ba9d32882a35b29d942cc4721a4bbab54.jpg)  
+Figure 12: Validation BLEU AUCC and test BLEU for WMT (high is good). Comparison of regular transformer and reservoir transformer with FFN or Transformer reservoir layers added.
+
+![](images/341111a25bde86caccbfa885e06e66acf9b016afdcaeecdf8630ea4dc0291102.jpg)  
+Figure 13: Validation BPC AUCC and test BPC on the enwik8 language modelling task (low is good). Comparison of regular and reservoir transformers for varying depths.
+
+![](images/dbab4e8809ef2a132617f00c1dc1acd380a0973768e0196c29742da904ec4002.jpg)  
+Figure 14: Downstream RoBERTa performance on SST-2 (left) and MultiNLI-matched (right).
+
+![](images/41ebdc6849103e9713e0791384d38890f3c671a95f174869f62c2c94346167cf.jpg)  
+Figure 15: IWSLT with 2-layer decoder validation plot (upper left). WMT with 24-layer decoder validation plot (upper right). Enwik8 with 48-layer decoder validation plot (lower left). RoBERTa with 12-layer decoder validation plot (lower right).
+
+<table><tr><td>Model</td><td>Layer</td><td>SentLen (Surface)</td><td>TreeDepth (Syntactic)</td><td>TopConst (Syntactic)</td><td>BShift (Syntactic)</td><td>Tense (Semantic)</td><td>SubjNum (Semantic)</td><td>ObjNum (Semantic)</td><td>SOMO (Semantic)</td><td>CoordInv (Semantic)</td></tr><tr><td rowspan="14">Transformer</td><td></td><td>84.56 ± 0.54</td><td>32.30 ± 0.41</td><td>54.40 ± 0.33</td><td>49.99 ± 0.01</td><td>80.98 ± 0.32</td><td>76.26 ± 0.09</td><td>50.01 ± 0.19</td><td>76.38 ± 0.61</td><td>54.33 ± 0.47</td></tr><tr><td></td><td>87.22 ± 0.07</td><td>33.63 ± 0.57</td><td>58.38 ± 0.20</td><td>50.12 ± 0.17</td><td>82.84 ± 0.68</td><td>78.65 ± 0.19</td><td>51.47 ± 0.53</td><td>78.00 ± 1.12</td><td>54.66 ± 0.55</td></tr><tr><td></td><td>84.25 ± 0.16</td><td>32.60 ± 0.17</td><td>54.41 ± 0.10</td><td>50.02 ± 0.01</td><td>81.72 ± 0.59</td><td>77.00 ± 0.13</td><td>51.32 ± 0.64</td><td>76.57 ± 1.13</td><td>54.13 ± 0.51</td></tr><tr><td></td><td>87.37 ± 0.20</td><td>32.59 ± 0.29</td><td>50.06 ± 0.21</td><td>69.76 ± 0.26</td><td>81.63 ± 1.17</td><td>76.47 ± 0.09</td><td>52.41 ± 1.49</td><td>76.15 ± 0.84</td><td>52.62 ± 1.34</td></tr><tr><td>5</td><td>84.61 ± 0.24</td><td>31.14 ± 0.48</td><td>44.76 ± 0.38</td><td>74.82 ± 0.11</td><td>80.16 ± 0.19</td><td>73.66 ± 0.16</td><td>52.95 ± 1.77</td><td>72.90 ± 0.21</td><td>51.26 ± 1.14</td></tr><tr><td>6</td><td>82.56 ± 0.25</td><td>30.31 ± 0.40</td><td>39.30 ± 0.40</td><td>78.80 ±0.38</td><td>81.88 ± 0.47</td><td>75.30 ± 0.07</td><td>56.21 ± 1.26</td><td>74.37 ± 0.16</td><td>51.44 ± 1.04</td></tr><tr><td></td><td>70.85 ± 0.13</td><td>26.65 ± 0.72</td><td>40.70 ± 0.13</td><td>78.98 ± 0.32</td><td>85.11 ± 0.31</td><td>72.03 ± 0.46</td><td>58.15 ± 0.46</td><td>68.71 ± 0.91</td><td>55.39 ± 0.27</td></tr><tr><td>8</td><td>66.23 ± 1.33</td><td>23.46 ± 0.44</td><td>25.19 ± 1.02</td><td>77.42 ± 0.27</td><td>80.35 ± 0.45</td><td>67.55 ± 0.99</td><td>54.94 ± 2.04</td><td>63.69 ± 2.32</td><td>50.58 ± 0.83</td></tr><tr><td>9</td><td>71.17 ± 0.29</td><td>31.21 ± 0.31</td><td>58.42 ± 0.29</td><td>85.55 ± 0.44</td><td>86.77 ± 0.19</td><td>80.30 ± 0.08</td><td>64.36 ± 1.20</td><td>81.68 ± 0.45</td><td>66.90 ± 0.49</td></tr><tr><td>10</td><td>73.19 ± 0.50</td><td>27.74 ± 0.53</td><td>41.01 ± 0.22</td><td>83.56 ± 0.96</td><td>86.13 ± 0.35</td><td>83.04 ± 0.04</td><td>62.01 ± 0.59</td><td>79.73 ± 0.21</td><td>62.60 ± 1.04</td></tr><tr><td>11</td><td>71.37 ± 0.42</td><td>30.22 ± 0.28</td><td>48.58 ± 0.35</td><td>84.40 ± 0.44</td><td>87.28 ± 0.59</td><td>82.34 ± 0.15</td><td>61.10 ± 0.14</td><td>80.00 ± 0.40</td><td>64.44 ± 0.38</td></tr><tr><td>12</td><td>71.66 ± 0.12</td><td>33.43 ± 0.18</td><td>64.38 ± 0.20</td><td>87.38 ± 0.02</td><td>88.41 ± 0.09</td><td>84.46 ± 0.25</td><td>63.01 ± 0.05</td><td>81.80 ± 0.27</td><td>65.72 ± 0.16</td></tr><tr><td></td><td>87.75 ± 0.10</td><td>31.60 ± 0.21</td><td>50.38 ± 0.23</td><td>50.00 ± 0.00</td><td>80.40 ± 0.18</td><td>76.47 ± 0.20</td><td>50.53 ± 0.14</td><td>73.48 ± 0.15</td><td></td></tr><tr><td rowspan="14">TReservoir</td><td>2</td><td>81.28 ± 0.23</td><td>34.20 ± 0.41</td><td></td><td>60.64 ± 0.65</td><td></td><td></td><td></td><td></td><td>53.55 ± 0.70</td></tr><tr><td></td><td></td><td></td><td>61.41 ± 0.42</td><td></td><td>81.50 ± 0.77</td><td>76.33 ± 0.08</td><td>50.73 ± 0.34</td><td>74.28 ± 0.67</td><td>56.82 ± 0.10</td></tr><tr><td>3</td><td>89.28 ± 0.09</td><td>36.42 ± 0.11</td><td>67.36 ± 0.45</td><td>75.64 ± 0.52</td><td>85.42 ± 0.18</td><td>80.53 ± 0.02</td><td>52.50 ± 1.80</td><td>78.47 ± 1.81</td><td>57.16 ± 0.27</td></tr><tr><td></td><td>74.31 ± 0.32</td><td>32.42 ± 0.83</td><td>55.19 ± 0.33</td><td>73.41 ± 0.00</td><td>79.56 ± 0.00</td><td>75.15 ± 0.08</td><td>53.68 ± 0.66</td><td>75.02 ± 0.19</td><td>56.89 ± 0.08</td></tr><tr><td></td><td>88.03 ± 0.22 74.55 ± 0.37</td><td>38.34 ± 0.64 33.13 ± 0.29</td><td>68.65 ± 0.29</td><td>82.25 ± 0.12</td><td>86.80 ± 0.02</td><td>82.27 ± 0.33</td><td>57.95 ± 0.24</td><td>80.82 ± 0.91</td><td>58.05 ± 0.10</td></tr><tr><td></td><td></td><td></td><td>52.70 ± 0.81</td><td>79.21 ± 0.13</td><td>85.70 ± 0.36</td><td>77.43 ± 0.03</td><td>57.26 ± 0.19</td><td>75.38 ± 0.66</td><td>51.95 ± 1.30</td></tr><tr><td></td><td>85.82 ± 0.37 71.69 ± 0.71</td><td>37.63 ± 0.13 30.32 ± 0.01</td><td>70.43 ± 0.05 48.44 ± 0.30</td><td>84.12 ± 0.35 79.12 ± 0.12</td><td>86.88 ± 0.07</td><td>82.86 ± 0.30</td><td>61.17 ± 0.21</td><td>80.79 ± 0.17</td><td>61.83 ± 0.95</td></tr><tr><td>8 9</td><td>85.86 ± 0.12</td><td>37.89 ± 0.03</td><td>69.53 ± 0.37</td><td>85.55 ± 0.12</td><td>84.75 ± 0.09 87.98 ± 0.22</td><td>79.23 ± 0.11 84.13 ± 0.01</td><td>59.53 ± 0.16</td><td>76.80 ± 0.41</td><td>57.34 ± 0.14</td></tr><tr><td></td><td>69.22 ± 0.23</td><td>25.58 ± 0.35</td><td>29.20 ± 0.58</td><td>78.57 ± 0.09</td><td></td><td></td><td>63.06± 0.01</td><td>82.55 ± 0.31</td><td>66.07 ± 0.05</td></tr><tr><td>10</td><td></td><td></td><td>47.56 ± 0.02</td><td></td><td>85.02 ± 0.03</td><td>75.68 ± 0.16</td><td>57.55 ± 1.57</td><td>74.70 ± 0.02</td><td>55.02 ± 0.64</td></tr><tr><td>11</td><td>65.70 ± 0.05</td><td>30.57 ± 0.03</td><td></td><td>81.20 ± 0.00</td><td>86.78 ± 0.02</td><td>83.73 ± 0.05</td><td>60.38 ± 0.17</td><td>80.59 ± 0.15</td><td>62.50 ± 0.11</td></tr><tr><td>12</td><td>70.61 ± 0.18</td><td>34.45± 0.20</td><td>64.19 ± 0.10</td><td>84.53 ± 0.03</td><td>87.48 ± 0.16</td><td>84.86 ± 0.14</td><td>62.75 ± 0.14</td><td>82.08 ± 0.03</td><td>64.73 ± 0.06</td></tr></table>
+
+Table 6: RoBERTa Probing Results. The line in bold text are the the frozen layers in the T Reservoir.

parse/train/B1X0mzZCW/B1X0mzZCW.md

@@ -0,0 +1,576 @@
+# FIDELITY-WEIGHTED LEARNING
+
+Mostafa Dehghani University of Amsterdam dehghani@uva.nl
+
+Arash Mehrjou MPI for Intelligent Systems amehrjou@tuebingen.mpg.de
+
+Stephan Gouws Google Brain sgouws@google.com
+
+Jaap Kamps University of Amsterdam kamps@uva.nl
+
+Bernhard Scholkopf ¨ MPI for Intelligent Systems bs@tuebingen.mpg.de
+
+# ABSTRACT
+
+Training deep neural networks requires many training samples, but in practice training labels are expensive to obtain and may be of varying quality, as some may be from trusted expert labelers while others might be from heuristics or other sources of weak supervision such as crowd-sourcing. This creates a fundamental qualityversus-quantity trade-off in the learning process. Do we learn from the small amount of high-quality data or the potentially large amount of weakly-labeled data? We argue that if the learner could somehow know and take the label-quality into account when learning the data representation, we could get the best of both worlds. To this end, we propose “fidelity-weighted learning” (FWL), a semi-supervised studentteacher approach for training deep neural networks using weakly-labeled data. FWL modulates the parameter updates to a student network (trained on the task we care about) on a per-sample basis according to the posterior confidence of its label-quality estimated by a teacher (who has access to the high-quality labels). Both student and teacher are learned from the data. We evaluate FWL on two tasks in information retrieval and natural language processing where we outperform state-of-the-art alternative semi-supervised methods, indicating that our approach makes better use of strong and weak labels, and leads to better task-dependent data representations.
+
+# 1 INTRODUCTION
+
+The success of deep neural networks to date depends strongly on the availability of labeled data which is costly and not always easy to obtain. Usually it is much easier to obtain small quantities of high-quality labeled data and large quantities of unlabeled data. The problem of how to best integrate these two different sources of information during training is an active pursuit in the field of semi-supervised learning (Chapelle et al., 2006). However, for a large class of tasks it is also easy to define one or more so-called “weak annotators”, additional (albeit noisy) sources of weak supervision based on heuristics or “weaker”, biased classifiers trained on e.g. non-expert crowd-sourced data or data from different domains that are related. While easy and cheap to generate, it is not immediately clear if and how these additional weakly-labeled data can be used to train a stronger classifier for the task we care about. More generally, in almost all practical applications machine learning systems have to deal with data samples of variable quality. For example, in a large dataset of images only a small fraction of samples may be labeled by experts and the rest may be crowd-sourced using e.g. Amazon Mechanical Turk (Veit et al., 2017). In addition, in some applications, labels are intentionally perturbed due to privacy issues (Wainwright et al., 2012; Papernot et al., 2017).
+
+Assuming we can obtain a large set of weakly-labeled data in addition to a much smaller training set of “strong” labels, the simplest approach is to expand the training set by including the weakly-supervised samples (all samples are equal). Alternatively, one may pretrain on the weak data and then fine-tune on observations from the true function or distribution (which we call strong data). Indeed, it has recently been shown that a small amount of expert-labeled data can be augmented in such a way by a large set of raw data, with labels coming from a heuristic function, to train a more accurate neural ranking model (Dehghani et al., 2017d). The downside is that such approaches are oblivious to the amount or source of noise in the labels.
+
+![](images/1b69b7a31813138afadabf804b83c9b5fe27e801d1f33369e0e1b3f602c9dd48.jpg)  
+Figure 1: Illustration of Fidelity-Weighted Learning: Step 1: Pre-train student on weak data, Step 2: Fit teacher to observations from the true function, and Step 3: Fine-tune student on labels generated by teacher, taking the confidence into account. Red dotted borders and blue solid borders depict components with trainable and non-trainable parameters, respectively.
+
+In this paper, we argue that treating weakly-labeled samples uniformly (i.e. each weak sample contributes equally to the final classifier) ignores potentially valuable information of the label quality. Instead, we propose Fidelity-Weighted Learning (FWL), a Bayesian semi-supervised approach that leverages a small amount of data with true labels to generate a larger training set with confidence-weighted weakly-labeled samples, which can then be used to modulate the fine-tuning process based on the fidelity (or quality) of each weak sample. By directly modeling the inaccuracies introduced by the weak annotator in this way, we can control the extent to which we make use of this additional source of weak supervision: more for confidently-labeled weak samples close to the true observed data, and less for uncertain samples further away from the observed data.
+
+We propose a setting consisting of two main modules. One is called the student and is in charge of learning a suitable data representation and performing the main prediction task, the other is the teacher which modulates the learning process by modeling the inaccuracies in the labels. We explain our approach in much more detail in Section 2, but at a high level it works as follows (see Figure 1): We pretrain the student network on weak data to learn an initial task-dependent data representation which we pass to the teacher along with the strong data. The teacher then learns to predict the strong data, but crucially, based on the student’s learned representation. This then allows the teacher to generate new labeled training data from unlabeled data, and in the process correct the student’s mistakes, leading to a better final data representation and better final predictor.
+
+We introduce the proposed FWL approach in more detail in Section 2. We then present our experimental setup in Section 3 where we evaluate FWL on a toy task and two real-world tasks, namely document ranking and sentence sentiment classification. In all cases, FWL outperforms competitive baselines and yields state-of-the-art results, indicating that FWL makes better use of the limited true labeled data and is thereby able to learn a better and more meaningful task-specific representation of the data. Section 4 provides analysis of the bias-variance trade-off and the learning rate, suggesting also to view FWL from the perspective of Vapnik’s learning with privileged information (LUPI) framework (Vapnik & Izmailov, 2015). Section 5 situates FWL relative to related work, and we end the paper by drawing the main conclusions in Section 6.
+
+# 2 FIDELITY-WEIGHTED LEARNING (FWL)
+
+In this section, we describe our proposed FWL approach for semi-supervised learning when we have access to weak supervision (e.g. heuristics or weak annotators). We assume we are given a large set of unlabeled data samples, a heuristic labeling function called the weak annotator, and a small set of highquality samples labeled by experts, called the strong dataset, consisting of tuples of training samples $x _ { i }$ and their true labels $y _ { i }$ , i.e. $\bar { \mathcal { D } _ { s } } \bar { = } \{ ( x _ { i } , y _ { i } ) \}$ . We consider the latter to be observations from the true target function that we are trying to learn. We use the weak annotator to generate labels for the unlabeled samples. Generated labels are noisy due to the limited accuracy of the weak annotator. This gives us the weak dataset consisting of tuples of training samples $x _ { i }$ and their weak labels $\tilde { y } _ { i }$ , i.e. $\mathcal { D } _ { w } = \bar { \{ ( x _ { i } , \tilde { y } _ { i } ) \} }$ . Note that we can generate a large amount of weak training data $\mathcal { D } _ { w }$ at almost no cost using the weak annotator. In contrast, we have only a limited amount of observations from the true function, i.e. $\left| \mathcal { D } _ { s } \right| \ll \left| \mathcal { D } _ { w } \right|$ .
+
+# Algorithm 1 Fidelity-Weighted Learning.
+
+1: Train the student on samples from the weakly-annotated data $D _ { w }$ .   
+2: Freeze the representation-learning component $\psi ( . )$ of the student and train teacher on the strong data $D _ { s } = ( \psi ( x _ { j } ) , y _ { j } )$ . Apply teacher to unlabeled samples $x _ { t }$ to obtain soft dataset $D _ { s w } = \{ ( x _ { t } , \bar { y } _ { t } ) \}$ where $\bar { y } _ { t } = T ( x _ { t } )$ is the soft label and for each instance $x _ { t }$ , the uncertainty of its label, $\Sigma ( x _ { t } )$ , is provided by the teacher.   
+3: Train the student on samples from $D _ { s w }$ with SGD and modulate the step-size $\eta _ { t }$ according to the per-sample quality estimated using the teacher (Equation 1).
+
+Our proposed setup comprises a neural network called the student and a Bayesian function approximator called the teacher. The training process consists of three phases which we summarize in Algorithm 1 and Figure 1.
+
+# Step 1 Pre-train the student on $\mathcal { D } _ { w }$ using weak labels generated by the weak annotator.
+
+The main goal of this step is to learn a task dependent representation of the data as well as pretraining the student. The student function is a neural network consisting of two parts. The first part $\bar { \psi } ( . )$ learns the data representation and the second part $\phi ( . )$ performs the prediction task (e.g. classification). Therefore the overall function is $\hat { y } = \phi ( \psi ( x _ { i } ) )$ . The student is trained on all samples of the weak dataset $\mathcal { D } _ { w } = \{ ( x _ { i } , \tilde { y } _ { i } ) \}$ . For brevity, in the following, we will refer to both data sample $x _ { i }$ and its representation $\psi ( x _ { i } )$ by $x _ { i }$ when it is obvious from the context. From the self-supervised feature learning point of view, we can say that representation learning in this step is solving a surrogate task of approximating the expert knowledge, for which a noisy supervision signal is provided by the weak annotator.
+
+Step 2 Train the teacher on the strong data $( \boldsymbol { \psi } ( x _ { j } ) , y _ { j } ) \in \mathcal { D } _ { s }$ represented in terms of the student representation $\psi ( . )$ and then use the teacher to generate a soft dataset $\mathcal { D } _ { s w }$ consisting of hsample,predicted label, confidencei for all data samples.
+
+We use a Gaussian process as the teacher to capture the label uncertainty in terms of the student representation, estimated w.r.t the strong data. We explain the finer details of the $\mathcal { G P }$ in Appendix C, and just present the overall description here. A prior mean and co-variance function is chosen for $\mathcal { G P }$ . The learned embedding function $\bar { \psi } ( \cdot )$ in Step 1 is then used to map the data samples to dense vectors as input to the $\mathcal { G P }$ . We use the learned representation by the student in the previous step to compensate lack of data in $\mathcal { D } _ { s }$ and the teacher can enjoy the learned knowledge from the large quantity of the weakly annotated data. This way, we also let the teacher see the data through the lens of the student.
+
+The $\mathcal { G P }$ is trained on the samples from $\mathcal { D } _ { s }$ to learn the posterior mean $m _ { \mathrm { p o s t } }$ (used to generate soft labels) and posterior co-variance $K _ { \mathrm { p o s t } } ( . , . )$ (which represents label uncertainty). We then create the soft dataset $\mathcal { D } _ { s w } = \{ ( x _ { t } , \bar { y } _ { t } ) \}$ using the posterior $\mathcal { G P }$ , input samples $x _ { t }$ from $\mathcal { D } _ { w } \cup \mathcal { D } _ { s }$ , and predicted labels $\bar { y } _ { t }$ with their associated uncertainties as computed by $T ( x _ { t } )$ and $\Sigma ( x _ { t } )$ :
+
+$$
+\begin{array} { r c l } { { T ( x _ { t } ) } } & { { = } } & { { g ( m _ { \mathrm { p o s t } } ( x _ { t } ) ) } } \\ { { \Sigma ( x _ { t } ) } } & { { = } } & { { h ( K _ { \mathrm { p o s t } } ( x _ { t } , x _ { t } ) ) } } \end{array}
+$$
+
+The generated labels are called soft labels. Therefore, we refer to $\mathcal { D } _ { s w }$ as a soft dataset. $g ( . )$ transforms the output of $\mathcal { G P }$ to the suitable output space. For example in classification tasks, $g ( . )$ would be the softmax function to produce probabilities that sum up to one. For multidimensional-output tasks where a vector of variances is provided by the $\mathcal { G P }$ , the vector $K _ { \mathrm { p o s t } } ( x _ { t } , x _ { t } )$ is passed through an aggregating function $h ( . )$ to generate a scalar value for the uncertainty of each sample. Note that we train $\mathcal { G P }$ only on the strong dataset $\mathcal { D } _ { s }$ but then use it to generate soft labels $\bar { y } _ { t } = T ( x _ { t } )$ and uncertainty $\Sigma ( x _ { t } )$ for samples belonging to $\mathcal { D } _ { s w } = \mathcal { D } _ { w } \cup \mathcal { D } _ { s }$ .
+
+In practice, we furthermore divide the space of data into several regions and assign each region a separate $\mathcal { G P }$ trained on samples from that region. This leads to a better exploration of the data space and makes use of the inherent structure of data. The algorithm called clustered $\mathcal { G P }$ gave better results compared to a single GP. See Appendix A for the detailed description and empirical observations which makes the use of multiple $\mathcal G \mathcal P \mathbf s$ reasonable.
+
+Step 3 Fine-tune the weights of the student network on the soft dataset, while modulating the magnitude of each parameter update by the corresponding teacher-confidence in its label.
+
+The student network of Step 1 is fine-tuned using samples from the soft dataset $\mathcal { D } _ { s w } = \{ ( x _ { t } , \bar { y } _ { t } ) \}$ where $\bar { y } _ { t } = T ( x _ { t } )$ . The corresponding uncertainty $\Sigma ( x _ { t } )$ of each sample is mapped to a confidence value according to Equation 1 below, and this is then used to determine the step size for each iteration of the stochastic gradient descent (SGD). So, intuitively, for data points where we have true labels, the uncertainty of the teacher is almost zero, which means we have high confidence and a large step-size for updating the parameters. However, for data points where the teacher is not confident, we down-weight the training steps of the student. This means that at these points, we keep the student function as it was trained on the weak data in Step 1.
+
+More specifically, we update the parameters of the student by training on $\mathcal { D } _ { s w }$ using SGD:
+
+$$
+\begin{array} { r c l } { { \pmb w } ^ { * } } & { = } & { \displaystyle \underset { { \pmb w } \in \mathcal { W } } { \mathrm { a r g m i n } } \frac { 1 } { N } \sum _ { ( \pmb { x } _ { t } , \bar { y } _ { t } ) \in \mathcal { D } _ { s w } } l ( \pmb { w } , \pmb { x } _ { t } , \bar { y } _ { t } ) + \mathcal { R } ( \pmb { w } ) , } \\ { { \pmb w } _ { t + 1 } } & { = } & { \pmb { w } _ { t } - \eta _ { t } \big ( \nabla l ( \pmb { w } , \pmb { x } _ { t } , \bar { y } _ { t } ) + \nabla \mathcal { R } ( \pmb { w } ) \big ) } \end{array}
+$$
+
+where $l ( \cdot )$ is the per-example loss, $\eta _ { t }$ is the total learning rate, $N$ is the size of the soft dataset $\mathcal { D } _ { s w }$ $\pmb { w }$ is the parameters of the student network, and $\mathcal { R } ( . )$ is the regularization term.
+
+We define the total learning rate as $\eta _ { t } = \eta _ { 1 } ( t ) \eta _ { 2 } ( x _ { t } )$ , where $\eta _ { 1 } ( t )$ is the usual learning rate of our chosen optimization algorithm that anneals over training iterations, and $\eta _ { 2 } ( x _ { t } )$ is a function of the label uncertainty $\Sigma ( x _ { t } )$ that is computed by the teacher for each data point. Multiplying these two terms gives us the total learning rate. In other words, $\eta _ { 2 }$ represents the fidelity (quality) of the current sample, and is used to multiplicatively modulate $\eta _ { 1 }$ . Note that the first term does not necessarily depend on each data point, whereas the second term does. We propose
+
+$$
+\eta _ { 2 } ( x _ { t } ) = \exp [ - \beta \Sigma ( x _ { t } ) ] ,
+$$
+
+to exponentially decrease the learning rate for data point $x _ { t }$ if its corresponding soft label $\bar { y } _ { t }$ is unreliable (far from a true sample). In Equation 1, $\beta$ is a positive scalar hyper-parameter. Intuitively, small $\beta$ results in a student which listens more carefully to the teacher and copies its knowledge, while a large $\beta$ makes the student pay less attention to the teacher, staying with its initial weak knowledge. More concretely speaking, as $\beta \to 0$ student places more trust in the labels $\bar { y } _ { t }$ estimated by the teacher and the student copies the knowledge of the teacher. On the other hand, as $\beta \to \infty$ , student puts less weight on the extrapolation ability of $\mathcal { G P }$ and the parameters of the student are not affected by the correcting information from the teacher.
+
+# 3 EXPERIMENTS
+
+In this section, we apply FWL first to a toy problem and then to two different real tasks: document ranking and sentiment classification. The neural networks are implemented in TensorFlow (Abadi et al., 2015; Tang, 2016). GPflow (Matthews et al., 2017) is employed for developing the $\mathcal { G P }$ modules. For both tasks, we evaluate the performance of our method compared to the following baselines:
+
+1. WA. The weak annotator, i.e. the unsupervised method used for annotating the unlabeled data.   
+2. NNW. The student trained only on weak data.   
+3. $\mathbf { N N _ { S } }$ . The student trained only on strong data.   
+4. $\mathbf { N N _ { S ^ { + } / W } }$ . The student trained on samples that are alternately drawn from $\mathcal { D } _ { w }$ without replacement, and $\mathcal { D } _ { s }$ with replacement. Since $\left| \mathcal { D } _ { s } \right| \ll \left| \mathcal { D } _ { w } \right|$ , it oversamples the strong data.   
+5. ${ \bf N N } _ { \bf W \to S }$ . The student trained on weak dataset $\mathcal { D } _ { w }$ and fine-tuned on strong dataset $\mathcal { D } _ { s }$ .   
+6. $\mathbf { N N } _ { \mathbf { W } ^ { \omega } \to \mathbf { S } }$ . The student trained on the weak data, but the step-size of each weak sample is weighted by a fixed value $0 \leq \omega \leq 1$ , and fine-tuned on strong data. As an approximation for the optimal value for $\omega$ , we have used the mean of $\eta _ { 2 }$ of our model (below).   
+7. FWL unsuprep. The representation in the first step is trained in an unsupervised way1 and the student is trained on examples labeled by the teacher using the confidence scores.   
+8. FWL $\backslash \Sigma$ . The student trained on the weakly labeled data and fine-tuned on examples labeled by the teacher without taking the confidence into account. This baseline is similar to (Veit et al., 2017).   
+9. FWL. Our FWL model, i.e. the student trained on the weakly labeled data and fine-tuned on examples labeled by the teacher using the confidence scores.
+
+In the following, we introduce each task and the results produced for it, more detail about the exact student network and teacher $\mathcal { G P }$ for each task are in the appendix.
+
+![](images/dfac384163e0e39d7d67efc7c1966a3e5bfdabd5bbf2f6a94c42fabb1021739e.jpg)  
+c) Fine-tuning the student based on observations from the true function. (d) Fine-tuning the student based on label/confidence from teacher.   
+Figure 2: Toy example: The true function we want to learn is $y = \sin ( x )$ and the weak function is $y = 2 s i n c ( x )$
+
+# 3.1 TOY PROBLEM
+
+We first apply FWL to a one-dimensional toy problem to illustrate the various steps. Let $f _ { t } ( x ) = \sin ( x )$ be the true function (red dotted line in Figure 2a) from which a small set of observations $\mathcal { D } _ { s } = \{ x _ { j } , y _ { j } \}$ is provided (red points in Figure 2b). These observation might be noisy, in the same way that labels obtained from a human labeler could be noisy. A weak annotator function $f _ { w } ( x ) = 2 s i n c ( \dot { x } )$ (magenta line in Figure 2a) is provided, as an approximation to $f _ { t } ( . )$ .
+
+The task is to obtain a good estimate of $f _ { t } ( . )$ given the set $\mathcal { D } _ { s }$ of strong observations and the weak annotator function $f _ { w } ( . )$ . We can easily obtain a large set of observations $\mathcal { D } _ { w } = \{ x _ { i } , \tilde { y } _ { i } \}$ from $f _ { w } ( . )$ with almost no cost (magenta points in Figure 2a).
+
+We consider two experiments:
+
+1. A neural network trained on weak data and then fine-tuned on strong data from the true function, which is the most common semi-supervised approach (Figure 2c).   
+2. A teacher-student framework working by the proposed FWL approach.
+
+As can be seen in Figure 2d, FWL by taking into account label confidence, gives a better approximation of the true hidden function. We repeated the above experiment 10 times. The average RMSE with respect to the true function on a set of test points over those 10 experiments for the student, were as follows:
+
+1. Student is trained on weak data (blue line in Figure 2a): 0.8406,   
+2. Student is trained on weak data then fine tuned on true observations (blue line in Figure 2c): 0.5451,   
+3. Student is trained on weak data, then fine tuned by soft labels and confidence information provided by the teacher (blue line in Figure 2d): 0.4143 (best).
+
+More details of the neural network and $\mathcal { G P }$ along with the specification of the data used in the above experiment are presented in Appendix C and E.1.
+
+# 3.2 DOCUMENT RANKING
+
+This task is the core information retrieval problem and is challenging as the ranking model needs to learn a representation for long documents and capture the notion of relevance between queries and documents. Furthermore, the size of publicly available datasets with query-document relevance judgments is unfortunately quite small $\mathord { \sim } 2 5 0$ queries). We employ a state-of-the-art pairwise neural ranker architecture as the student (Dehghani et al., 2017d). In this model, ranking is cast as a regression task. Given each training sample $x$ as a triple of query $q$ , and two documents $d ^ { + }$ and $d ^ { - }$ , the goal is to learn a function $\mathcal { F } : \{ \bar { < } q , d ^ { + } , d ^ { - } > \}  \mathbb { R }$ , which maps each data sample $x$ to a scalar output value $y$ indicating the probability of $d ^ { + }$ being ranked higher than $d ^ { - }$ with respect to $q$ .
+
+Table 1: Performance of FWL approach and baseline methods for ranking task. IJi indicates that the improvements with respect to the baseline $_ i$ are statistically significant at the 0.05 level using the paired two-tailed t-test with Bonferroni correction.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Robust04</td><td colspan="2">ClueWeb</td></tr><tr><td>MAP</td><td>nDCG@20</td><td>MAP</td><td>nDCG@20</td></tr><tr><td>1</td><td>WABM25</td><td>0.250337</td><td>0.4102-37</td><td>0.102137</td><td>0.2070-37</td></tr><tr><td>2</td><td>NNw (Dehghani etal.,2017d)</td><td>0.2702137</td><td>0.4290137</td><td>0.1297137</td><td>0.2201137</td></tr><tr><td>3</td><td>NNs</td><td>0.1790</td><td>0.3519</td><td>0.0782</td><td>0.1730</td></tr><tr><td>4</td><td>NNs+/W</td><td>0.27631237</td><td>0.43301237</td><td>0.13541237</td><td>0.23191237</td></tr><tr><td>5</td><td>NNw→S</td><td>0.28101237</td><td>0.4372*1237</td><td>0.1346*1237</td><td>0.23171237</td></tr><tr><td>6</td><td>NNwω→S</td><td>0.2899123457</td><td>0.4431*123457</td><td>0.132012347</td><td>0.2309*12347</td></tr><tr><td>7</td><td>FWLunsuprep</td><td>0.2211-37</td><td>0.3700-37</td><td>0.083137</td><td>0.1964-37</td></tr><tr><td>8</td><td>FWL\Ω</td><td>0.2980123457</td><td>0.4516123457</td><td>0.1386123457</td><td>0.2340123457</td></tr><tr><td>9</td><td>FWL</td><td>0.312412345678</td><td>0.4607-12345678</td><td>0.1472*12345678</td><td>0.2453412345678</td></tr></table>
+
+The student follows the architecture proposed in (Dehghani et al., 2017d). The first layer of the network, i.e. representation learning layer $\psi : \{ < q , d ^ { + } , d ^ { - } > \}  \mathbb { R } ^ { m }$ maps each input sample to an $m$ - dimensional real-valued vector. In general, besides learning embeddings for words, function $\psi$ learns to compose word embedding based on their global importance in order to generate query/document embeddings. The representation layer is followed by a simple fullyconnected feed-forward network with a sigmoidal output unit to predict the probability of ranking $d ^ { + }$ higher than $d ^ { - }$ . The general schema of the student is illustrated in Figure 3. More details are provided in Appendix B.1.
+
+The teacher is implemented by clustered $\mathcal { G P }$ algorithm. See Appendix C for more details.
+
+![](images/97ae3fab8de7809a7fd8b6bcc5d1bab3d66cde886c7306cbed33212009679576.jpg)  
+Figure 3: The student for the document ranking task.
+
+The weak annotator is BM25 (Robertson & Zaragoza, 2009), a well-known unsupervised method for scoring query-document pairs based on statistics of the matched terms. More details are provided in Appendix D.1.
+
+Description of the data with weak labels and data with true labels as well as the setup of the documentranking experiments is presented in Appendix E.2 in more details.
+
+Results and Discussions We conducted $\mathbf { k }$ -fold cross validation on $\mathcal { D } _ { s }$ (the strong data) and report two standard evaluation metrics for ranking: mean average precision (MAP) of the top-ranked 1,000 documents and normalized discounted cumulative gain calculated for the top 20 retrieved documents $( \mathrm { n D C G } @ 2 0 )$ . Table 1 shows the performance on both datasets. As can be seen, FWL provides a significant boost on the performance over all datasets. In the ranking task, the student is designed in particular to be trained on weak annotations (Dehghani et al., 2017d), hence training the network only on weak supervision, i.e. $\mathrm { N N } _ { \mathrm { W } }$ performs better than $\mathrm { N N } _ { \mathrm { S } }$ . This can be due to the fact that ranking is a complex task requiring many training samples, while relatively few data with true labels are available.
+
+Alternating between strong and weak data during training, i.e. $\mathrm { N N } _ { \mathrm { S } ^ { + } / \mathrm { W } }$ seems to bring little (but statistically significant) improvement. However, we can gain better results by the typical fine-tuning strategy, ${ \mathrm { N N } } _ { \mathrm { W } \to S }$ . Comparing the performance of $\mathrm { F W L } _ { u n s u p r e p }$ to FWL indicates that, first of all learning the representation of the input data downstream of the main task leads to better results compared to a task-independent unsupervised or self-supervised way. Also the dramatic drop in the performance compared to the FWL, emphasizes the importance of the preretraining the student on weakly labeled data. We can gain improvement by fine-tuning the $\mathrm { N N } _ { \mathrm { W } }$ using labels generated by the teacher without considering their confidence score, i.e. $\mathrm { F W L } \backslash \Sigma$ . This means we just augmented the fine-tuning process by generating a fine-tuning set using teacher which is better than $\mathcal { D } _ { s }$ in terms of quantity and $\mathcal { D } _ { w }$ in terms of quality. This baseline is equivalent to setting $\beta = 0$ in Equation 1. However, we see a big jump in performance when we use FWL to include the estimated label quality from the teacher, leading to the best overall results.
+
+# 3.3 SENTIMENT CLASSIFICATION
+
+In sentiment classification, the goal is to predict the sentiment (e.g., positive, negative, or neutral) of a sentence. Each training sample $x$ consists of a sentence $s$ and its sentiment label $\tilde { y }$ .
+
+Table 2: Performance of the proposed FWL approach and baseline methods for sentiment classification task. IJi indicates that the improvements with respect to the baseline#i are statistically significant, at the 0.05 level using the paired two-tailed t-test, with Bonferroni correction.   
+
+<table><tr><td></td><td>Method</td><td>SemEval-14</td><td>SemEval-15</td></tr><tr><td>1</td><td>WALexicon</td><td>0.5141</td><td>0.4471</td></tr><tr><td>2</td><td>NNw</td><td>0.6719*137</td><td>0.5606^1</td></tr><tr><td>3</td><td>NNs</td><td>0.630741</td><td>0.5811^12</td></tr><tr><td>4</td><td>NNs+/W</td><td>0.703241237</td><td>0.631941237</td></tr><tr><td>5</td><td>NNw→S</td><td>0.7080*1237</td><td>0.6441^1237</td></tr><tr><td>6</td><td>NNww →S</td><td>0.716612347</td><td>0.6603123457</td></tr><tr><td>7</td><td>FWLunsuprep</td><td>0.6588 13</td><td>0.6954*123</td></tr><tr><td>8</td><td>FWL \Ω</td><td>0.7202 123457</td><td>0.6590123457</td></tr><tr><td>9</td><td>FWL</td><td>0.7470 12345678</td><td>0.6830 12345678</td></tr><tr><td>10</td><td>SemEvalBest</td><td>0.7162</td><td>0.6618</td></tr></table>
+
+![](images/7785c02fa208908e0b9b79a800302cce469065d3f82bb134bb08a888b0d0fb0f.jpg)  
+Figure 4: The student for the sentiment classification task.
+
+The student for the sentiment classification task is a convolutional model which has been shown to perform best on the dataset we used (Deriu et al., 2017; Severyn & Moschitti, 2015a;b; Deriu et al., 2016). The first layer of the network learns the function $\psi ( . )$ which maps input sentence $s$ to a dense vector as its representation. The inputs are first passed through an embedding layer mapping the sentence to a matrix $S \in \mathbb { R } ^ { m \times | s | }$ , followed by a series of 1d convolutional layers with max-pooling. The representation layer is followed by feed-forward layers and a softmax output layer which returns the probability distribution over all three classes. Figure 4 presents the general schema of the architecture of the student. See Appendix B.2 for more details.
+
+The teacher for this task is modeled by a $\mathcal { G P }$ . See Appendix C for more details.
+
+The weak annotator is a simple unsupervised lexicon-based method (Hamdan et al., 2013; Kiritchenko et al., 2014), which estimate a distribution over sentiments for each sentence, based on sentiment labels of its terms. More details are provided in Appendix D.2.
+
+Specification of the data with weak labels and data with true labels along with the detailed experimental setup are given in Appendix E.3.
+
+Results and Discussion We report Macro-F1, the official SemEval metric, in Table 2. We see that the proposed FWL is the best performing approach.
+
+For this task, since the amount of data with true labels are larger compared to the ranking task, the performance of $\mathrm { N N } _ { \mathrm { S } }$ is acceptable. Alternately sampling from weak and strong data gives better results. Pretraining on weak labels then fine-tuning the network on true labels, further improves the performance. Weighting the gradient updates from weak labels during pretraining and fine-tuning the network with true labels, i.e. $\mathrm { N N } _ { \mathrm { W } ^ { \omega }  \mathrm { S } }$ seems to work quite well in this task. For this task, like ranking task, learning the representation in an unsupervised task independent fashion, i.e. $\mathrm { F W L } _ { u n s u p r e p }$ , does not lead to good results compared to the FWL. Similar to the ranking task, fine-tuning $\mathrm { N N } _ { \mathrm { S } }$ based on labels generated by $\mathcal { G P }$ instead of data with true labels, regardless of the confidence score, works better than standard fine-tuning.
+
+Besides the baselines, we also report the best performing systems which are also convolution-based models (Rouvier & Favre 2016 on SemEval-14; Deriu et al. 2016 on SemEval-15). Using FWL and taking the confidence into consideration outperforms the best systems and leads to the highest reported results on both datasets.
+
+# 4 ANALYSIS
+
+In this section, we provide further analysis of FWL by investigating the bias-variance trade-off and the learning rate.
+
+# 4.1 HANDLING THE BIAS-VARIANCE TRADE-OFF
+
+As mentioned in Section 2, $\beta$ is a hyperparameter that controls the contribution of weak and strong data to the training procedure. In order to investigate its influence, we fixed everything in the model and ran the fine-tuning step with different values of $\beta \in \{ 0 . 0 , 0 . 1 , 1 . 0 , 2 . 0 , 5 . 0 \}$ in all the experiments.
+
+![](images/b2a92c21cac9cdbf0b90efc933aa6b9744d79eb50578f010efdf620eca9d39f6.jpg)  
+Figure 6: Performance of FWL and the baseline model trained on different amount of data.
+
+![](images/694d21f7ed6442940097ac56ef876c4f8e7c4c860615229b53f8a03963d8e107.jpg)  
+Figure 5 illustrates the performance on the ranking (on Robust04 dataset) and sentiment classification tasks (on SemEval14 dataset). For both sentiment classification and ranking, $\beta = 1$ gives the best results (higher scores are better). We also experimented on the toy problem with different values of $\beta$ in three cases: 1) having 10 observations from the true function (same setup as Section 3.1), marked as “Toy Data” in the plot, 2) having only 5 observations from the true function, marked as “Toy Data \*” in the plot, and 3)   
+Figure 5: Effect of different values for $\beta$ .
+
+having $f ( x ) = x + 1$ as the weak function, which is an extremely bad approximator of the true function, marked as “Toy Data $\ast \ast \ast$ in the plot. For the “Toy Data” experiment, $\beta = 1$ turned out to be optimal (here, lower scores are better). However, for “Toy Data \*”, where we have an extremely small number of observations from the true function, setting $\beta$ to a higher value acts as a regularizer by relying more on weak signals, and eventually leads to better generalization. On the other hand, for “Toy Data \*\*”, where the quality of the weak annotator is extremely low, lower values of $\beta$ put more focus on the true observations. Therefore, $\beta$ lets us control the bias-variance trade-off in these extreme cases.
+
+# 4.2 A GOOD TEACHER IS BETTER THAN MANY OBSERVATIONS
+
+We now look at the rate of learning for the student as the amount of training data is varied. We performed two types of experiments for all tasks: In the first experiment, we use all the available strong data but consider different percentages of the entire weak dataset. In the second experiment, we fix the amount of weak data and provide the model with varying amounts of strong data. We use standard fine-tuning with similar setups as for the baseline models. Details on the experiments for the toy problem are provided in Appendix E.1.
+
+Figure 6 presents the results of these experiments. In general, for all tasks and both setups, the student learns faster when there is a teacher. One caveat is in the case where we have a very small amount of weak data. In this case the student cannot learn a suitable representation in the first step, and hence the performance of FWL is pretty low, as expected. It is highly unlikely that this situation occurs in reality as obtaining weakly labeled data is much easier than strong data.
+
+The empirical observation of Figure 6 that our model learns more with less data can also be seen as evidence in support of another perspective to FWL, called learning using privileged information (Vapnik & Izmailov, 2015). We elaborate more on this connection in Appendix F.
+
+# 4.3 SENSITIVITY OF THE FWL TO THE QUALITY OF THE WEAK ANNOTATOR
+
+Our proposed setup in FWL requires defining a so-called “weak annotator” to provide a source of weak supervision for unlabelled data. In Section 4.1 we discussed the role of parameter $\beta$ for controlling the bias-variance trade-off by trying two weak annotators for the toy problem. Now, in this section, we study how the quality of the weak annotator may affect the performance of the FWL, for the task of document ranking as a real-world problem.
+
+To do so, besides BM25 (Robertson & Zaragoza, 2009), we use three other weak annotators:
+
+vector space model (Salton & Yang, 1973) with binary term occurrence (BTO) weighting schema and vector space model with TF-IDF weighting schema, which are both weaker than BM25, and $\mathbf { B M } 2 5 \mathbf { + R M } 3$ (Abdul-jaleel et al., 2004) that uses RM3 as the pseudo-relevance feedback method on top of BM25, leading to better labels.
+
+Figure 7 illustrates the performance of these four weak annotators in terms of their mean average precision (MAP) on the test data, versus the performance of FWL given the corresponding weak annotator. As it is expected, the performance of FWL depends on the quality of the employed weak annotator. The percentage of improvement of FWL over its corresponding weak annotator on the test data is also presented in Figure 7. As can be seen, the better the performance of the weak annotator is, the less the improvement of the FWL would be.
+
+![](images/efcf7abf8c6c1adf9e3dd3725ef5384214c99d32f8cd9afca5c5fd38c0b73ed5.jpg)  
+Figure 7: Performance of FWL versus performance of the corespondence weak annotator in the document ranking task, on Robust04 dataset.
+
+# 4.4 FROM MODIFYING THE LEARNING RATE TO WEIGHTED SAMPLING
+
+FWL provides confidence score based on the certainty associated with each generated label $\bar { y } _ { t }$ , given sample $\boldsymbol { x } _ { t } \in \mathcal { D } _ { s w }$ . We can translate the confidence score as how likely including $\left( x _ { t } , \bar { y } _ { t } \right)$ in the training set for the student model improves the performance, and rather than using this score as the multiplicative factor in the learning rate, we can use it to bias sampling procedure of mini-batches so that the frequency of training samples are proportional to the confidence score of their labels.
+
+We design an experiment to try FWL with this setup $( \mathrm { F W L } _ { s } )$ , in which we keep the architectures of the student and the teacher and the procedure of the first two steps of the FWL fixed, but we changed the step 3 as follows: Given the soft dataset $\mathcal { D } _ { s w }$ , consisting of $x _ { t }$ , its label $\bar { y } _ { t }$ and the associated confidence score generated by the teacher, we normalize the confidence scores over all training samples and set the normalized score of each sample as its probability to be sampled. Afterward, we train the student model by mini-batches sampled from this set with respect to the probabilities associated with each sample, but without considering the original confidence scores in parameter updating. This means the more confident the teacher is about the generated label for each sample, the more chance that sample has to be seen by the student model.
+
+![](images/77e8b6c15d488e1f6f725a3f8df084f3de33f92eb3344822e11e5e31d448b275.jpg)  
+Figure 8: Performance of FWL and $\mathrm { F W L } _ { s }$ with respect to different batch of data for the task of document ranking (Robust04 dataset) and sentiment classification (SemEval14 dataset).
+
+Figure 8 illustrates the performance of both FWL and $\mathrm { F W L } _ { s }$ trained on different amount of data sampled from $\mathcal { D } _ { s w }$ , in the document ranking and sentiment classification tasks. As can be seen, compared to FWL, the performance of $\mathrm { F W L } _ { s }$ increases rapidly in the beginning but it slows down afterward. We have looked into the sampling procedure and noticed that the confidence scores provided by the teacher form a rather skewed distribution and there is a strong bias in $\mathrm { F W L } _ { s }$ toward sampling from data points that are either in or closed to the points in $\mathcal { D } _ { s }$ , as $\mathcal { G P }$ has less uncertainty around these points and the confidence scores are high. We observed that the performance of $\mathrm { F W L } _ { s }$ gets closer to the performance of FWL after many epochs, while FWL had already a log convergence. The skewness of the confidence distribution makes $\mathrm { F W L } _ { s }$ to have a tendency for more exploitation than exploration, however, FWL has more chance to explore the input space, while it controls the effect of updates on the parameters for samples based on their merit.
+
+# 5 RELATED WORK
+
+In this section, we position our FWL approach relative to related work.
+
+Learning from imperfect labels has been thoroughly studied in the literature (Frenay & Verleysen, ´ 2014). The imperfect (weak) signal can come from non-expert crowd workers, be the output of other models that are weaker (for instance with low accuracy or coverage), biased, or models trained on data from different related domains. Among these forms, in the distant supervision setup, a heuristic labeling rule (Deriu et al., 2016; Severyn & Moschitti, 2015b) or function (Dehghani et al., 2017d)
+
+which can be relying on a knowledge base (Mintz et al., 2009; Min et al., 2013; Han & Sun, 2016) is employed to devise noisy labels.
+
+Learning from weak data sometimes aims at encoding various forms of domain expertise or cheaper supervision from lay annotators. For instance, in the structured learning, the label space is pretty complex and obtaining a training set with strong labels is extremely expensive, hence this class of problems leads to a wide range of works on learning from weak labels (Roth, 2017). Indirect supervision is considered as a form of learning from weak labels that is employed in particular in the structured learning, in which a companion binary task is defined for which obtaining training data is easier (Chang et al., 2010; Raghunathan et al., 2016). In the response-based supervision, the model receives feedback from interacting with an environment in a task, and converts this feedback into a supervision signal to update its parameters (Roth, 2017; Clarke et al., 2010; Riezler et al., 2014). Constraint-based supervision is another form of weak supervision in which constraints that are represented as weak label distributions are taken as signals for updating the model parameters. For instance, physics-based constraints on the output (Stewart & Ermon, 2017) or output constraints on execution of logical forms (Clarke et al., 2010).
+
+In the proposed FWL model, we can employ these approaches as the weak annotator to provide imperfect labels for the unlabeled data, however, a small amount of data with strong labels is also needed, which put our model in the class of semi-supervised models. In the semi-supervised setup, some ideas were developed to utilize weakly or even unlabeled data. For instance, the idea of self(incremental)-training (Rosenberg et al., 2005), pseudo-labeling (Lee, 2013; Hinton et al., 2014), and Co-training (Blum & Mitchell, 1998) are introduced for augmenting the training set by unlabeled data with predicted labels. Some research used the idea of self-supervised (or unsupervised) feature learning (Noroozi & Favaro, 2016; Dosovitskiy et al., 2016; Donahue et al., 2017) to exploit different labelings that are freely available besides or within the data, and to use them as intrinsic signals to learn general-purpose features. These features, that are learned using a proxy task, are then used in a supervised task like object classification/detection or description matching.
+
+As a common approach in semi-supervised learning, the unlabeled set can be used for learning the distribution of the data. In particular for neural networks, greedy layer-wise pre-training of weights using unlabeled data is followed by supervised fine-tuning (Hinton et al., 2006; Deriu et al., 2017; Severyn & Moschitti, 2015b;a; Go et al., 2009). Other methods learn unsupervised encoding at multiple levels of the architecture jointly with a supervised signal (Ororbia II et al., 2015; Weston et al., 2012).
+
+Alternatively, some noise cleansing methods have been proposed to remove or correct mislabeled samples (Brodley & Friedl, 1999). There are some studies showing that weak or noisy labels can be leveraged by modifying the loss function (Reed et al., 2015; Patrini et al., 2017; 2016; Vahdat, 2017) or changing the update rule to avoid imperfections of the noisy data (Malach & Shalev-Shwartz, 2017; Dehghani et al., 2017b;c).
+
+One direction of research focuses on modeling the pattern of the noise or weakness in the labels. For instance, methods that use a generative model to correct weak labels such that a discriminative model can be trained more effectively (Ratner et al., 2016; Rekatsinas et al., 2017; Varma et al., 2017). Furthermore, methods that aim at capturing the pattern of the noise by inserting an extra layer (Goldberger & Ben-Reuven, 2017) or a separate module tries to infer better labels from noisy ones and use them to supervise the training of the network (Sukhbaatar et al., 2015; Veit et al., 2017; Dehghani et al., 2017b). Our proposed FWL can be categorized in this class as the teacher tries to infer better labels and provide certainty information which is incorporated as the update rule for the student model.
+
+# 6 CONCLUSION
+
+Training neural networks using large amounts of weakly annotated data is an attractive approach in scenarios where an adequate amount of data with true labels is not available, a situation which often arises in practice. In this paper, we introduced fidelity-weighted learning (FWL), a new student-teacher framework for semi-supervised learning in the presence of weakly labeled data. We applied FWL to document ranking and sentiment classification, and empirically verified that FWL speeds up the training process and improves over state-of-the-art semi-supervised alternatives.
+
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+# APPENDICES
+
+We moved additional details to the appendices in order to keep the main text focused on the overall idea of the Fidelity-Weighted Learning approach. Specifically, we include further details on the clustered Gaussian process approach (Appendix A); on the student network architectures (Appendix B); on the teacher Gaussian process model (Appendix C); on the weak annotators (Appendix D); on the experimental data and setup (Appendix E); and on the connection to “learning with privileged information” (Appendix F).
+
+# A DETAILED DESCRIPTION OF CLUSTERED GP
+
+We suggest using several ${ \mathcal { G P } } { = } \{ G P _ { c _ { i } } \}$ to explore the entire data space more effectively. Even though inducing points and stochastic methods make $\mathcal { G P s }$ more scalable we still observed poor performance when the entire dataset was modeled by a single ${ \mathcal { G P } }$ . Therefore, the reason for using multiple $\mathcal { G P s }$ is mainly empirical inspired by (Shen et al., 2006) which is explained in the following:
+
+We used Sparse Gaussian Process implemented in GPflow. The algorithm is scalable in the sense that it is not $O ( N ^ { 3 } )$ as original $\mathcal { G P }$ is. It introduces inducing points in the data space and defines a variational lower bound for the marginal likelihood. The variational bound can now be optimized by stochastic methods which make the algorithm applicable in large datasets. However, the tightness of the bound depends on the location of inducing points which are found through the optimization process. We empirically observed that a single $\mathcal { G P }$ does not give a satisfactory accuracy on left-out test dataset. We hypothesized that this can be due to the inability of the algorithm to find good inducing points when the number of inducing points is restricted to just a few. Then we increased the number of inducing points $M$ which trades off the scalability of the algorithm because it scales with $O ( N M ^ { 2 } )$ . Moreover, apart from scalability which is partly solved by stochastic methods, we argue that the structure of the entire space may not be explored well by a single ${ \mathcal { G P } }$ and its inducing points. We guess this can be due to the observation that our datasets are distributed in a highly sparse way within the high dimensional embedding space. We also tried to cure the problem by means of PCA to reduce input dimensions and give a denser representation, but it did not result in a considerable improvement. The results are presented in Tabel 3.
+
+Table 3: Performance of FWL using a single $\mathcal { G P }$ , a single $\mathcal { G P }$ after applying PCA on the input data, and the clustered $\mathcal { G P }$ as the teacher.   
+
+<table><tr><td rowspan="3">Method</td><td colspan="4">Document Ranking</td><td colspan="2">Sentiment Classification</td></tr><tr><td colspan="2">Robust04</td><td colspan="2">ClueWeb</td><td>Robust04</td><td>ClueWeb</td></tr><tr><td>MAP</td><td>nDCG@20</td><td>MAP</td><td>nDCG@20</td><td>F1</td><td>F1</td></tr><tr><td>FWLgp</td><td>0.2614</td><td>0.4192</td><td>0.1205</td><td>0.2121</td><td>0.6904</td><td>0.6173</td></tr><tr><td>FWLPCA→9P</td><td>0.2864</td><td>0.4411</td><td>0.1331</td><td>0.2388</td><td>0.7022</td><td>0.6340</td></tr><tr><td>FWLClustered 9P</td><td>0.3124</td><td>0.4607</td><td>0.1472</td><td>0.2453</td><td>0.7470</td><td>0.6830</td></tr></table>
+
+We may be able to argue that clustered $\mathcal { G P }$ makes better use of the data structure roughly close to the idea of KISS-GP (Wilson & Nickisch, 2015). In inducing point methods, it is normally assumed that $M \ll N$ $M$ is the number of inducing points and $N$ is the number of training samples) for computational and storage saving. However, we have this intuition that few number of inducing points make the model unable to explore the inherent structure of data. By employing several GPs, we were able to use a large number of inducing points even when $M > N$ $M$ is the total number of inducing points) which seemingly better exploits the structure of datasets. Because our work was not aimed to be a close investigation of GP, we considered clustered $\mathcal { G P }$ as the engineering side of the work which is a tool to give us a measure of confidence. Other tools such as a single $\mathcal { G P }$ with inducing points that form a Kronecker or Toeplitz covariance matrix are also conceivable. Therefore, we do not of course claim that we have proposed a new method of inference for GPs. Here is practical description of clustered $\mathcal { G P }$ algorithm:
+
+Clustered $\mathcal { G P }$ : Let $N$ be the size of the dataset on which we train the teacher. Assume we allocate $K$ teachers to the entire data space. Therefore, each ${ \mathcal { G P } }$ sees a dataset of size $n { = } N / K$ . Then we use a simple clustering method (e.g. k-means) to find centroids of $K$ clusters $C _ { 1 } , C _ { 2 } , . . . , C _ { K }$ where $C _ { i }$ consists of samples $\{ x _ { i , 1 } , x _ { i , 2 } , . . . , x _ { i , n } \}$ . We take the centroid $c _ { i }$ of cluster $C _ { i }$ as the representative sample for all its content. Note that $c _ { i }$ does not necessarily belong to $\{ x _ { i , 1 } , x _ { i , 2 } , . . . , x _ { i , n } \}$ . We assign each cluster a $\mathcal { G P }$ trained by samples belonging to that cluster. More precisely, cluster $C _ { i }$ is assigned a $\mathcal { G P }$ whose data points are $\{ x _ { i , 1 } , x _ { i , 2 } , . . . , x _ { i , n } \}$ . Because there is no dependency among different clusters, we train them in parallel to speed-up the procedure more.
+
+The pseudo-code of the clustered $\mathcal { G P }$ is presented in Algorithm 2. When the main issue is computational resources (when the number of inducing points for each $\mathcal { G P }$ is large), we can first choose the number $_ n$ which is the maximum size of the dataset on which our resources allow to train a $\mathcal { G P }$ , then find the number of clusters $K = N / n$ accordingly. The rest of the algorithm remains unchanged.
+
+# Algorithm 2 Clustered Gaussian processes.
+
+1: Let $N$ be the sample size, $n$ the sample size of each cluster, $K$ the number of clusters, and $c _ { i }$ the center of cluster $_ { i }$ .   
+2: Run K-means with $K$ clusters over all samples with true labels $\mathcal { D } _ { s } = \{ x _ { i } , y _ { i } \}$ .
+
+$$
+\mathrm { K - m e a n s } ( x _ { i } )  c _ { 1 } , c _ { 2 } , . . . , c _ { K }
+$$
+
+where $c _ { i }$ represents the center of cluster $C _ { i }$ containing samples $D _ { s } ^ { c _ { i } } = \{ x _ { i , 1 } , x _ { i , 2 } , . . . x _ { i , n } \}$ .
+
+3: Assign each of $K$ clusters a Gaussian process and train them in parallel to approximate the label of each sample.
+
+$$
+\begin{array} { r c l } { \mathcal { G P } _ { c _ { i } } ( { m } _ { \mathrm { p o s t } } ^ { c _ { i } } , K _ { \mathrm { p o s t } } ^ { c _ { i } } ) } & { = } & { \mathcal { G P } ( { m } _ { \mathrm { p r i o r } } , K _ { \mathrm { p r i o r } } ) | D _ { s } ^ { c _ { i } } = \{ ( \psi ( x _ { s , c _ { i } } ) , y _ { s , c _ { i } } ) \} } \\ { { T } _ { c _ { i } } ( x _ { t } ) } & { = } & { g ( { m } _ { \mathrm { p o s t } } ^ { c _ { i } } ( x _ { t } ) ) } \\ { \Sigma _ { c _ { i } } ( x _ { t } ) } & { = } & { h ( K _ { \mathrm { p o s t } } ^ { c _ { i } } ( x _ { t } , x _ { t } ) ) } \end{array}
+$$
+
+where ${ \mathcal G P } _ { c _ { i } }$ is trained on $\mathcal { D } _ { s } ^ { c _ { i } }$ containing samples belonging to the cluster $c _ { i }$ . Other elements are defined in Section 2
+
+4: Use trained teacher $T _ { c _ { i } } ( . )$ to evaluate the soft label and uncertainty for samples from $\mathcal { D } _ { s w }$ to compute $\eta _ { 2 } ( x _ { t } )$ required for step 3 of Algorithm 1. We use $T ( . )$ as a wrapper for all teachers $\{ T _ { c _ { i } } \}$ .
+
+# B DETAILED ARCHITECTURE OF THE STUDENTS
+
+# B.1 RANKING TASK
+
+For the ranking task, the employed student is proposed in (Dehghani et al., 2017d). The first layer of the network models function $\psi$ that learns the representation of the input data samples, i.e. $( q , d ^ { + } , d ^ { - } )$ , and consists of three components: (1) an embedding function $\varepsilon : \mathcal { V } \to \mathbb { R } ^ { m }$ (where $\nu$ denotes the vocabulary set and $m$ is the number of embedding dimensions), (2) a weighting function $\omega : \mathcal { V }  \mathbb { R }$ , and (3) a compositionality function $\Theta : ( \mathbb { R } ^ { m } , \mathbb { R } ) ^ { n } \to \mathbb { R } ^ { m }$ . More formally, the function $\psi$ is defined as:
+
+$$
+\begin{array} { r l } { \psi ( q , d ^ { + } , d ^ { - } ) = [ \odot _ { i = 1 } ^ { | q | } ( \varepsilon ( t _ { i } ^ { q } ) , \omega ( t _ { i } ^ { q } ) ) \mid \mid } & { } \\ { \odot _ { i = 1 } ^ { | d ^ { + } | } ( \varepsilon ( t _ { i } ^ { d ^ { + } } ) , \omega ( t _ { i } ^ { d ^ { + } } ) ) \mid \mid } & { } \\ { \odot _ { i = 1 } ^ { | d ^ { - } | } ( \varepsilon ( t _ { i } ^ { d ^ { - } } ) , \omega ( t _ { i } ^ { d ^ { - } } ) ) \mid , } & { } \end{array}
+$$
+
+where $t _ { i } ^ { q }$ and $t _ { i } ^ { d }$ denote the $i ^ { t h }$ term in query $q$ respectively document $d$ . The embedding function $\varepsilon$ maps each term to a dense $m$ - dimensional real value vector, which is learned during the training phase. The weighting function $\omega$ assigns a weight to each term in the vocabulary. It has been shown that $\omega$ simulates the effect of inverse document frequency (IDF), which is an important feature in information retrieval (Dehghani et al., 2017d).
+
+The compositionality function $\odot$ projects a set of $_ n$ embedding-weighting pairs to an $m$ - dimensional representation, independent from the value of $n$ :
+
+$$
+\bigcirc _ { i = 1 } ^ { n } ( \varepsilon ( t _ { i } ) , \omega ( t _ { i } ) ) = \frac { \sum _ { i = 1 } ^ { n } \exp ( \omega ( t _ { i } ) ) \cdot \varepsilon ( t _ { i } ) } { \sum _ { j = 1 } ^ { n } \exp ( \omega ( t _ { j } ) ) } ,
+$$
+
+which is in fact the normalized weighted element-wise summation of the terms’ embedding vectors. Again, it has been shown that having global term weighting function along with embedding function improves the performance of ranking as it simulates the effect of inverse document frequency (IDF). In our experiments, we initialize the embedding function $\varepsilon$ with word2vec embeddings (Mikolov et al., 2013) pre-trained on Google News and the weighting function $\omega$ with IDF.
+
+The representation layer is followed by a simple fully connected feed-forward network with $l$ hidden layers followed by a softmax which receives the vector representation of the inputs processed by the representation learning layer and outputs a prediction $\tilde { y }$ . Each hidden layer $z _ { k }$ in this network computes $z _ { k } = \alpha ( \bar { W _ { k } } z _ { k - 1 } + b _ { k } )$ , where $W _ { k }$ and $b _ { k }$ denote the weight matrix and the bias term corresponding to the $k ^ { t h }$ hidden layer and $\alpha ( . )$ is the non-linearity. These layers follow a sigmoid output. We employ the cross entropy loss:
+
+$$
+\mathcal { L } _ { t } = \sum _ { i \in B } [ - y _ { i } \log ( \hat { y } _ { i } ) - ( 1 - y _ { i } ) \log ( 1 - \hat { y } _ { i } ) ] ,
+$$
+
+where $B$ is a batch of data samples.
+
+# B.2 SENTIMENT CLASSIFICATION TASK
+
+The student for the sentiment classification task is a convolutional model which has been shown to perform best in the dataset we used (Deriu et al., 2017; Severyn & Moschitti, 2015a;b; Deriu et al., 2016). The first layer of the network learns the function $\psi$ which maps input sentence $s$ to a vector as its representation consists of an embedding function $\varepsilon : \mathcal { V } \to \mathbb { R } ^ { m }$ , where $\nu$ denotes the vocabulary set and $m$ is the number of embedding dimensions.
+
+This function maps the sentence to a matrix $S \in \mathbb { R } ^ { m \times | s | }$ , where each column represents the embedding of a word at the corresponding position in the sentence. Matrix $S$ is passed through a convolution layer. In this layer, a set of $f$ filters is applied to a sliding window of length $h$ over $S$ to generate a feature map matrix $C$ . Each feature map $c _ { i }$ for a given filter $F$ is generated by $\begin{array} { r } { c _ { i } = \sum _ { k , j } S [ i : i + h ] _ { k , j } F _ { k , j } } \end{array}$ , where $S [ i : i + h ]$ denotes the concatenation of word vectors from position $_ { i }$ to $i + h$ . The concatenation of all $c _ { i }$ produces a feature vector $c \in \mathbb { R } ^ { | s | - h + 1 }$ . The vectors $c$ are then aggregated over all $f$ filters into a feature map matrix $C \in \mathbb { R } ^ { f \times ( | s | - h + 1 ) }$ .
+
+We also add a bias vector $b \in R ^ { f }$ to the result of a convolution. Each convolutional layer is followed by a non-linear activation function (we use ReLU(Nair & Hinton, 2010)) which is applied element-wise. Afterward, the output is passed to the max pooling layer which operates on columns of the feature map matrix $C$ returning the largest value: $p o o l ( c _ { i } ) : \mathbb { R } ^ { 1 \times ( | s | - h + 1 ) } \to \mathbb { R }$ (see Figure 4). This architecture is similar to the state-of-the-art model for Twitter sentiment classification from Semeval 2015 and 2016 (Severyn & Moschitti, 2015b; Deriu et al., 2016).
+
+We initialize the embedding matrix with word2vec embeddings (Mikolov et al., 2013) pretrained on a collection of 50M tweets.
+
+The representation layer then is followed by a feed-forward layer similar to the ranking task (with different width and depth) but with softmax instead of sigmoid as the output layer which returns $\hat { y } _ { i }$ , the probability distribution over all three classes. We employ the cross entropy loss:
+
+$$
+\mathcal { L } _ { t } = \sum _ { i \in B } \sum _ { k \in K } - y _ { i } ^ { k } \log ( \hat { y } _ { i } ^ { k } ) ,
+$$
+
+where $B$ is a batch of data samples, and $K$ is a set of classes.
+
+# C DETAILED ARCHITECTURE OF THE TEACHERS
+
+We use Gaussian Process as the teacher in all the experiments. For each task, either regression or (multi-class) classification, in order to generate soft labels, we pass the mean of ${ \mathcal { G P } }$ through the same function $g ( . )$ that is applied on the output of the student network for that task, e.g. softmax, or sigmoid. For binary classification or one dimensional regression, $\Sigma ( x _ { t } )$ is scalar and $h ( . )$ is identity. For multi-class classification or multi-dimensional regression tasks, $h ( . )$ is an aggregation function that takes variance over several dimensions and outputs a single measure of variance. As a reasonable choice, the aggregating function $h ( . )$ in our sentiment classification task (three classes) is mean of variances over dimensions.
+
+In the teacher, linear combinations of different kernels are used for different tasks in our experiments.
+
+Toy Problem: We use standard Gaussian process regression2 with this kernel:
+
+$$
+k ( x _ { i } , x _ { j } ) = k _ { \mathrm { R B F } } ( x _ { i } , x _ { j } ) + k _ { \mathrm { W h i t e } } ( x _ { i } , x _ { j } )
+$$
+
+Document Ranking: We use sparse variational GP regression3 (Titsias, 2009) with this kernel:
+
+$$
+k ( x _ { i } , x _ { j } ) = k _ { \mathrm { M a t e r n 3 / 2 } } ( x _ { i } , x _ { j } ) + k _ { \mathrm { L i n e a r } } ( x _ { i } , x _ { j } ) + k _ { \mathrm { W h i t e } } ( x _ { i } , x _ { j } )
+$$
+
+Sentiment Classification: We use sparse variational GP for multiclass classification4 (Hensman et al., 2015) with the following kernel:
+
+$$
+k ( x _ { i } , x _ { j } ) = k _ { \mathrm { R B F } } \left( x _ { i } , x _ { j } \right) + k _ { \mathrm { L i n e a r } } \left( x _ { i } , x _ { j } \right) + k _ { \mathrm { W h i t e } } \left( x _ { i } , x _ { j } \right)
+$$
+
+where,
+
+$$
+{ \begin{array} { r l } { k _ { \mathrm { R B F } } ( x _ { i } , x _ { j } ) = \exp \left( { \frac { \left\| x _ { i } - x _ { j } \right\| ^ { 2 } } { 2 l ^ { 2 } } } \right) } & { } \\ { k _ { \mathrm { M a t e r n 3 / 2 } } ( x _ { i } , x _ { j } ) = \left( 1 + { \frac { { \sqrt { 3 } } \left\| x _ { i } - x _ { j } \right\| } { l } } \right) \exp \left( - { \frac { { \sqrt { 3 } } \left\| x _ { i } - x _ { j } \right\| } { l } } \right) } & { } \\ { k _ { \mathrm { L i n e a r } } ( x _ { i } , x _ { j } ) = \sigma _ { 0 } ^ { 2 } + x _ { i } . x _ { j } } & { } \\ { k _ { \mathrm { W h i t e } } ( x _ { i } , x _ { j } ) = c o n s t a n t . v a l u e , } & { { \forall x _ { 1 } = x _ { 2 } \mathrm { a n d } } 0 { \mathrm { ~ o t h e r w i s e } } } \end{array} }
+$$
+
+We empirically found $l = 1$ satisfying value for the length scale of RBF and Matern3/2 kernels. We also set $\sigma _ { 0 } = 0$ to obtain a homogeneous linear kernel. The constant value of $K _ { W h i t e } ( . , . )$ determines the level of noise in the labels. This is different from the noise in weak labels. This term explains the fact that even in true labels there might be a trace of noise due to the inaccuracy of human labelers.
+
+We set the number of clusters in the clustered $\mathcal { G P }$ algorithm for the ranking task to 50 and for the sentiment classification task to 30.
+
+# D WEAK ANNOTATORS
+
+# D.1 DOCUMENT RANKING
+
+The weak annotator in the document ranking task is BM25 (Robertson & Zaragoza, 2009), a well-known unsupervised retrieval method. This method heuristically scores a given pair of query-document based on the statistics of their matched terms. In the pairwise document ranking setup, $\tilde { y } _ { i }$ for a given sample $x _ { j } = ( q , d ^ { + } , d ^ { - } )$ is the probability of document $d ^ { + }$ being ranked higher than $d ^ { - }$ $: \tilde { y } _ { i } = P _ { q , d ^ { + } , d ^ { - } } = s _ { q , d ^ { + } } / s _ { q , d ^ { + } } + s _ { q , d ^ { - } }$ , where $s _ { q , d }$ is the score obtained from the weak annotator.
+
+# D.2 SENTIMENT CLASSIFICATION
+
+The weak annotator for the sentiment classification task is a simple lexicon-based method (Hamdan et al., 2013; Kiritchenko et al., 2014). We use SentiWordNet03 (Baccianella et al., 2010) to assign probabilities (positive, negative and neutral) for each token in set $\mathcal { D } _ { w }$ . We use a bag-of-words model for the sentence-level probabilities (i.e. just averaging the distributions of the terms), yielding a noisy label $\tilde { y } _ { i } \in \mathbb { R } ^ { | K | }$ , where $| K | = 3$ is the number of classes. We found empirically that using soft labels from the weak annotator works better than assigning a single hard label.
+
+# E DATA COLLECTION, PARAMETERS AND SETUP
+
+# E.1 TOY PROBLEM
+
+Weak/True Data In all the experiments with the toy problem, we have randomly sampled 100 data points from the weak function and 10 data points from the true function. We introduce a small amount of noise to the observation of the true function to model the noise in the human labeled data.
+
+Setup The neural network employed in the toy problem experiments is a simple feed-forward network with the depth of 3 layers and width of 128 neurons per layer. We have used tanh as the nonlinearity for the intermediate layers and a linear output layer. As the optimizer, we used Adam (Kingma & Ba, 2015) and the initial learning rate has been set to 0.001. For the teacher in the toy problem, we fit only one $\mathcal { G P }$ on all the data points (i.e. no clustering). Also during fine-tuning, we set $\beta = 1$ .
+
+Setup of experiments in Section 4.2 We fixed everything in the model and tried running the fine-tuning step with different values for $\beta \in \{ 0 . 0 , 0 . 1 , 1 . 0 , 2 . 0 , 5 . 0 \}$ in all the experiments. For the experiments on toy problem in Section 4.2, the reported numbers are averaged over 10 trials. In the first experiment (i.e. Figure 6a), the size of sampled data data is: $| \mathcal { D } _ { s } | = 5 0$ and $\left| \mathcal { D } _ { w } \right| = \mathrm { { 1 0 0 } }$ (Fixed) and for the second one (i.e. Figure 6a): $\left| \mathcal { D } _ { w } \right| = 1 0 0$ and $| \mathcal { D } _ { s } | = 1 0$ (fixed).
+
+# E.2 RANKING TASK
+
+Collections We use two standard TREC collections for the task of ad-hoc retrieval: The first collection (Robust04) consists of $5 0 0 \mathrm { k }$ news articles from different news agencies as a homogeneous collection. The second collection (ClueWeb) is ClueWeb09 Category B, a large-scale web collection with over 50 million English documents, which is considered as a heterogeneous collection. Spam documents were filtered out using the Waterloo spam scorer 5 (Cormack et al., 2011) with the default threshold $7 0 \%$ .
+
+Data with true labels We take query sets that contain human-labeled judgments: a set of 250 queries (TREC topics 301–450 and 601–700) for the Robust04 collection and a set of 200 queries (topics 1-200) for the experiments on the ClueWeb collection. For each query, we take all documents judged as relevant plus the same number of documents judged as non-relevant and form pairwise combinations among them.
+
+Data with weak labels We create a query set $Q$ using the unique queries appearing in the AOL query logs (Pass et al., 2006). This query set contains web queries initiated by real users in the AOL search engine that were sampled from a three-month period from March 2006 to May 2006. We applied standard pre-processing Dehghani et al. (2017d;a) on the queries: We filtered out a large volume of navigational queries containing URL substrings (“http”, “www.”, “.com”, “.net”, “.org”, “.edu”). We also removed all non-alphanumeric characters from the queries. For each dataset, we took queries that have at least ten hits in the target corpus using our weak annotator method. Applying all these steps, We collect 6.15 million queries to train on in Robust04 and 6.87 million queries for ClueWeb. To prepare the weakly labeled training set $\mathcal { D } _ { w }$ , we take the top 1,000 retrieved documents using BM25 for each query from training query set $Q$ , which in total leads to $\sim | Q | \dot { \times } 1 0 ^ { 6 }$ training samples.
+
+Setup For the evaluation of the whole model, we conducted a 3-fold cross-validation. However, for each dataset, we first tuned all the hyper-parameters of the student in the first step on the set with true labels using batched GP bandits with an expected improvement acquisition function (Desautels et al., 2014) and kept the optimal parameters of the student fixed for all the other experiments. The size and number of hidden layers for the student is selected from $\{ 6 4 , 1 2 8 , 2 5 6 , 5 1 2 \}$ . The initial learning rate and the dropout parameter were selected from $\lbrace 1 0 ^ { - 3 } , 1 0 ^ { - 5 } \rbrace$ and $\{ 0 . 0 , 0 . 2 , 0 . 5 \}$ , respectively. We considered embedding sizes of $\{ 3 0 0 , 5 0 0 \}$ . The batch size in our experiments was set to 128. We use ReLU (Nair & Hinton, 2010) as a non-linear activation function $\alpha$ in student. We use the Adam optimizer (Kingma & Ba, 2015) for training, and dropout (Srivastava et al., 2014) as a regularization technique.
+
+At inference time, for each query, we take the top 2,000 retrieved documents using BM25 as candidate documents and re-rank them using the trained models. We use the Indri6 implementation of BM25 with default parameters (i.e., $k _ { 1 } = 1 . 2$ , $b { = } 0 . 7 5$ , and $k _ { 3 } = 1 , 0 0 0$ ).
+
+# E.3 SENTIMENT CLASSIFICATION TASK
+
+Collections We test our model on the twitter message-level sentiment classification of SemEval-15 Task 10B (Rosenthal et al., 2015). Datasets of SemEval-15 subsume the test sets from previous editions of SemEval, i.e. SemEval-13 and SemEval-14. Each tweet was preprocessed so that URLs and usernames are masked.
+
+Data with true labels We use train (9,728 tweets) and development (1,654 tweets) data from SemEval-13 for training and SemEval-13-test (3,813 tweets) for validation. To make your results comparable to the official runs on SemEval we us SemEval-14 (1,853 tweets) and SemEval-15 (2,390 tweets) as test sets (Rosenthal et al., 2015; Nakov et al., 2016).
+
+Data with weak labels We use a large corpus containing 50M tweets collected during two months for both, training the word embeddings and creating the weakly annotated set $\mathcal { D } _ { w }$ using the lexicon-based method explained in Section 3.3.
+
+Setup Similar to the document ranking task, we tuned hyper-parameters for the student in the first step with respect to the true labels of the validation set using batched GP bandits with an expected improvement acquisition function (Desautels et al., 2014) and kept the optimal parameters fixed for all the other experiments. The size and number of hidden layers for the classifier and is selected from $\{ 3 2 , 6 4 , 1 2 8 \}$ . We tested the model with both, 1 and 2 convolutional layers. The number of convolutional feature maps and the filter width is selected from $\{ 2 0 0 , 3 0 0 \}$ and $\{ 3 , 4 , 5 \}$ , respectively. The initial learning rate and the dropout parameter were selected from $\{ 1 E - 3 , 1 E - 5 \}$ and $\{ \mathrm { \bar { 0 } . 0 , 0 . \bar { 2 } , 0 . 5 } \}$ , respectively. We considered embedding sizes of $\lbrace 1 0 0 , 2 0 0 \rbrace$ and the batch size in these experiments was set to 64. ReLU (Nair & Hinton, 2010) is used as a non-linear activation function in student. Adam optimizer (Kingma & Ba, 2015) is used for training, and dropout (Srivastava et al., 2014) as a regularizer.
+
+# F CONNECTION WITH VAPNIK’S LEARNING USING PRIVILEGED INFORMATION
+
+In this section, we highlight the connections of our work with Vapnik’s learning using privileged information (LUPI) (Vapnik & Vashist, 2009; Vapnik & Izmailov, 2015). FWL makes use of information from a small set of correctly labeled data to improve the performance of a semi-supervised learning algorithm. The main idea behind LUPI comes from the fact that humans learn much faster than machines. This can be due to the role that an
+
+Intelligent Teacher plays in human learning. In this framework, the training data is a collection of triplets
+
+$$
+\{ ( x _ { 1 } , y _ { 1 } , x _ { 1 } ^ { * } ) , . . . , ( x _ { n } , y _ { n } , x _ { n } ^ { * } ) \} { \sim } P ^ { n } ( x , y , x ^ { * } )
+$$
+
+where each $( x _ { i } , y _ { i } )$ is a pair of feature-label and $\boldsymbol { x } _ { i } ^ { * }$ is the additional information provided by an intelligent teacher to ease the learning process for the student. Additional information for each $( x _ { i } , y _ { i } )$ is available only during training time and the learning machine must only rely on $x _ { i }$ at test time. The theory of LUPI studies how to leverage such a teaching signal $\boldsymbol { x } _ { i } ^ { * }$ to outperform learning algorithms utilizing only the normal features $x _ { i }$ . For example, MRI brain images can be augmented with high-level medical or even psychological descriptions of Alzheimer’s disease to build a classifier that predicts the probability of Alzheimer’s disease from an MRI image at test time. It is known from statistical learning theory (Vapnik, 1998) that the following bound for test error is satisfied with probability $1 - \delta$ :
+
+$$
+R ( f ) \leq R _ { n } ( f ) + O { \left( { \left( \frac { | \mathcal { F } | _ { V C } - \log \sigma } { n } \right) } ^ { \alpha } \right) } ,
+$$
+
+where $R _ { n } ( f )$ denotes the training error over $n$ samples, $| \mathcal { F } | _ { V C }$ is the VC dimension of the space of functions from which $f$ is chosen, and $\alpha \in [ 0 . 5 , 1 ]$ . When the classes are not separable, $\alpha { = } 0 . 5$ i.e. the machine learns at a slow rate of $O ( n ^ { - 1 / 2 } )$ . For easier problems where classes are separable, $\alpha = 1$ resulting in a learning rate of $O ( n ^ { - 1 } )$ . The difference between these two cases is severe. The same error bound achieved for a separable problem with 10 thousand data points is only obtainable for a non-separable problem when 100 million data points are provided. This is prohibitive even when obtaining large datasets is not so costly. The theory of LUPI shows that an intelligent teacher can reduce $\alpha$ resulting in a faster learning process for the student. In this paper, we proposed a teacher-student framework for semi-supervised learning. Similar to LUPI, in FWL a student is supposed to solve the main prediction task while an intelligent teacher provides additional information to improve its learning. In addition, we first train the student network so that it obtains initial knowledge of weakly labeled data and learns a good data representation. Then the teacher is trained on truly labeled data enjoying the representation learned by the student. This extends LUPI in a way that the teacher provides privileged information that is most useful for the current state of student’s knowledge. FWL also extends LUPI by introducing several teachers each of which is specialized to correct student’s knowledge related to a specific region of the data space.
+
+Figure 6(a) provides evidence for the assumption that privileged information in our task can accelerate the learning process of the student. It shows how the privileged information from an intelligent teacher affects the exponent $\alpha$ of the error bound in Equation 10. Figure 6(b) shows the test error for various number of samples $| \mathcal { D } _ { s } |$ with true label. As expected, In both extremes where $| \mathcal { D } _ { s } |$ is too small or too large, the performance of our model becomes close to the models without a teacher. The reason is that student has enough strong samples to learn a good model of true function. In more realistic cases where $\left| \mathcal { D } _ { s } \right| \ll \left| \mathcal { D } _ { w } \right|$ but $| \mathcal { D } _ { s } |$ is still large enough to be informative about $| \mathcal { D } _ { w } |$ , our model gives a lower test error than models without the intelligent teacher.
+
+The theory of LUPI was first developed and proved for support vector machines by Vapnik as a method for knowledge transfer. Hinton introduced Dark knowledge as a spiritually close idea in the context of neural networks (Hinton et al., 2006). He proposed to use a large network or an ensemble of networks for training and a smaller network at test time. It turned out that compressing knowledge of a large system into a smaller system can improve the generalization ability. It was shown in (Lopez-Paz et al., 2016) that dark knowledge and LUPI can be unified under a single umbrella, called generalized distillation. The core idea of these models is machinesteaching-machines. As the name suggests, a machine is learning the knowledge embedded in another machine. In our case, student is correcting his knowledge by receiving privileged information about label uncertainty from teacher.
+
+Our framework extends the core idea of LUPI in the following directions:
+
+• Trainable teacher: It is often assumed that the teacher in LUPI framework has some additional true information. We show that when this extra information is not available, one can still use the LUPI setup and define an implicit teacher whose knowledge is learned from the true data. In this approach, the performance of the final student-teacher system depends on a clever answer to the following question: which information should be considered as the privileged knowledge of teacher.   
+• Bayesian teacher: The proposed teacher is Bayesian. It provides posterior uncertainty of the label of each sample.   
+• Mutual representation: We introduced module $\psi ( . )$ which learns a mutual embedding (representation) for both student and teacher. This is in particular interesting because it defines a two-way channel between teacher and student.   
+• Multiple teachers: We proposed a scalable method to introduce several teachers such that each teacher is specialized in a particular region of the data space.

parse/train/Q_64PF6XNut/Q_64PF6XNut.md

@@ -0,0 +1,349 @@
+# Stochastic Optimization of Areas Under Precision-Recall Curves with Provable Convergence
+
+Qi Qi †∗, Youzhi Luo‡∗, Zhao $\mathbf { X } \mathbf { u } ^ { \ddag * }$ , Shuiwang $\mathbf { J } \mathbf { i } ^ { \ddag }$ , Tianbao Yang† †Department of Computer Science, The University of Iowa ‡Department of Computer Science & Engineering, Texas A&M University {qi-qi,tianbao-yang} $@$ uiowa.edu, {yzluo,zhaoxu,sji} $@$ tamu.edu
+
+# Abstract
+
+Areas under ROC (AUROC) and precision-recall curves (AUPRC) are common metrics for evaluating classification performance for imbalanced problems. Compared with AUROC, AUPRC is a more appropriate metric for highly imbalanced datasets. While stochastic optimization of AUROC has been studied extensively, principled stochastic optimization of AUPRC has been rarely explored. In this work, we propose a principled technical method to optimize AUPRC for deep learning. Our approach is based on maximizing the averaged precision (AP), which is an unbiased point estimator of AUPRC. We cast the objective into a sum of coupled compositional functions with inner functions dependent on random variables of the outer level. We propose efficient adaptive and non-adaptive stochastic algorithms named SOAP with provable convergence guarantee under mild conditions by leveraging recent advances in stochastic compositional optimization. Extensive experimental results on image and graph datasets demonstrate that our proposed method outperforms prior methods on imbalanced problems in terms of AUPRC. To the best of our knowledge, our work represents the first attempt to optimize AUPRC with provable convergence. The SOAP has been implemented in the libAUC library at https://libauc.org/.
+
+# 1 Introduction
+
+Although deep learning (DL) has achieved tremendous success in various domains, the standard DL methods have reached a plateau as the traditional objective functions in DL are no longer sufficient to model all requirements in new applications, which slows down the democratization of AI. For instance, in healthcare applications, data is often highly imbalanced, e.g., patients suffering from rare diseases are much less than those suffering from common diseases. In these applications, accuracy (the proportion of correctly predicted examples) is deemed as an inappropriate metric for evaluating the performance of a classifier. Instead, area under the curve (AUC), including area under ROC curve (AUROC) and area under the Precision-Recall curve (AUPRC), is widely used for assessing the performance of a model. However, optimizing accuracy on training data does not necessarily lead to a satisfactory solution to maximizing AUC [12].
+
+To break the bottleneck for further advancement, DL must be empowered with the capability of efficiently handling novel objectives such as AUC. Recent studies have demonstrated great success along this direction by maximizing AUROC [60]. For example, Yuan et al. [60] proposed a robust deep AUROC maximization method with provable convergence and achieved great success for classification of medical image data. However, to the best of our knowledge, novel DL by maximizing AUPRC has not yet been studied thoroughly. Previous studies [14, 20] have found that when dealing with highly skewed datasets, Precision-Recall (PR) curves could give a more informative picture of an algorithm’s performance, which entails the development of efficient stochastic optimization algorithms for DL by maximizing AUPRC.
+
+Compared with maximizing AUROC, maximizing AUPRC is more challenging. The challenges for optimization of AUPRC are two-fold. First, the analytical form of AUPRC by definition involves a complicated integral that is not readily estimated from model predictions of training examples. In practice, AUPRC is usually computed based on some point estimators, e.g., trapezoidal estimators and interpolation estimators of empirical curves, non-parametric average precision estimator, and parametric binomial estimator [3]. Among these estimators, non-parametric average precision (AP) is an unbiased estimate in the limit and can be directly computed based on the prediction scores of samples, which lends itself well to the task of model parameters optimization. Second, a surrogate function for AP is highly complicated and non-convex. In particular, an unbiased stochastic gradient is not readily computed, which makes existing stochastic algorithms such as SGD provide no convergence guarantee. Most existing works for maximizing AP-like function focus on how to compute an (approximate) gradient of the objective function [4, 6, 8, 11, 24, 38, 40, 43, 47, 48], which leave stochastic optimization of AP with provable convergence as an open question.
+
+Can we design direct stochastic optimization algorithms both in SGD-style and Adam-style for maximizing AP with provable convergence guarantee?
+
+In this paper, we propose a systematic and principled solution for addressing this question towards maximizing AUPRC for DL. By using a surrogate loss in lieu of the indicator function in the definition of AP, we cast the objective into a sum of non-convex compositional functions, which resembles a two-level stochastic compositional optimization problem studied in the literature [52, 53]. However, different from existing two-level stochastic compositional functions, the inner functions in our problem are dependent on the random variable of the outer level, which requires us developing a tailored stochastic update for computing an error-controlled stochastic gradient estimator. Specifically, a key feature of the proposed method is to maintain and update two scalar quantities associated with each positive example for estimating the stochastic gradient of the individual precision score at the threshold specified by its prediction score. By leveraging recent advances in stochastic compositional optimization, we propose both adaptive (Adam-style) and non-adaptive (SGD-style) algorithms, and establish their convergence under mild conditions. We conduct comprehensive empirical studies on class imbalanced graph and image datasets for learning graph neural networks and deep convolutional neural networks, respectively. We demonstrate that the proposed method can consistently outperform prior approaches in terms of AUPRC. In addition, we show that our method achieves better results when the sample distribution is highly imbalanced between classes and is insensitive to mini-batch size.
+
+# 2 Related Work
+
+AUROC Optimization. AUROC optimization 2 has attracted significant attention in the literature. Recent success of DL by optimizing AUROC on large-scale medical image data has demonstrated the importance of large-scale stochastic optimization algorithms and the necessity of accurate surrogate function [60]. Earlier papers [25, 28] focus on learning a linear model based on the pairwise surrogate loss and could suffer from a high computational cost, which could be as high as quadratic of the size of training data. To address the computational challenge, online and stochastic optimization algorithms have been proposed [18, 35, 42, 58, 63]. Recently, [21, 22, 36, 57] proposed stochastic deep AUC maximization algorithms by formulating the problem as non-convex strongly-concave minmax optimization problem, and derived fast convergence rate under PL condition, and in federated learning setting as well [21]. More recently, Yuan et al. [60] demonstrated the success of their methods on medical image classification tasks, e.g., X-ray image classification, melanoma classification based on skin images. However, an algorithm that maximizes the AUROC might not necessarily maximize AUPRC, which entails the development of efficient algorithms for DL by maximizing AUPRC.
+
+AUPRC Optimization. AUPRC optimization is much more challenging than AUROC optimization since the objective is even not decomposable over pairs of examples. Although AUPRC optimization has been considered in the literature (cf. [15, 47, 41] and references therein), efficient scalable algorithms for DL with provable convergence guarantee is still lacking. Some earlier works tackled this problem by using traditional optimization techniques, e.g., hill climbing search [37], cuttingplane method [61], dynamic programming [50], and by developing acceleration techniques in the framework of SVM [39]. These approaches are not scalable to big data for DL. There is a long list of studies in information retrieval [5, 11, 38, 47] and computer vision [4, 6, 8, 9, 24, 40, 48, 43], which have made efforts towards maximizing the AP score. However, most of them focus on how to compute an approximate gradient of the AP function or its smooth approximation, and provide no convergence guarantee for stochastic optimization based on mini-batch averaging. Due to lack of principled design, these previous methods when applied to deep learning are sensitive to the mini-batch size [6, 47, 48] and usually require a large mini-batch size in order to achieve good performance. In contrast, our stochastic algorithms are designed in a principled way to guarantee convergence without requiring a large mini-batch size as confirmed by our studies as well. Recently, [15] formulates the objective function as a constrained optimization problem using a surrogate function, and then casts it into a min-max saddle-point problem, which facilitates the use of stochastic min-max algorithms. However, they do not provide any convergence analysis for AUPRC maximization. In contrast, this is the first work that directly optimizes a surrogate function of AP (an unbaised estimator of AUPRC in the limit) and provides theoretical convergence guarantee for the proposed stochastic algorithms.
+
+Stochastic Compositional Optimization. Optimization of a two-level compositional function in the form of $\mathbb { E } _ { \xi } [ \bar { f } ( \mathbb { E } _ { \zeta } [ g ( \mathbf { w } ; \zeta ) ] ; \xi ) ]$ where $\xi$ and $\zeta$ are independent random variables, or its finite-sum variant has been studied extensively in the literature [1, 10, 52, 27, 30, 31, 33, 34, 46, 53, 59, 62, 45]. In this paper, we formulate the surrogate function of AP into a similar but more complicated two-level compositional function of the form $\mathbb { E } _ { \xi } [ f ( \mathbb { E } _ { \zeta } g ( { \mathbf w } ; \zeta , \xi ) ) ]$ , where $\xi$ and $\zeta$ are independent and $\xi$ has a finite support. The key difference between our formulated compositional function and the ones considered in previous work is that the inner function $g ( \mathbf { w } ; \zeta , \xi )$ also depends on the random variable $\xi$ of the outer level. Such subtle difference will complicate the algorithm design and the convergence analysis as well. Nevertheless, the proposed algorithm and its convergence analysis are built on previous studies of stochastic two-level compositional optimization.
+
+# 3 The Proposed Method
+
+Notations. We consider binary classification problem. Denote by $\left( \mathbf { x } , y \right)$ a data pair, where $\mathbf { x } \in \mathbb { R } ^ { d }$ denotes the input data and $y \in \{ 1 , - 1 \}$ denotes its class label. Let $h ( \mathbf { x } ) = h _ { \mathbf { w } } ( \mathbf { x } )$ denote the predictive function parameterized by a parameter vector $\mathbf { w } \in \mathbb { R } ^ { D }$ (e.g., a deep neural network). Denote by $\mathbf { I } ( \cdot )$ an indicator function that outputs 1 if the argument is true and zero otherwise. To facilitate the presentation, denote by $X$ a random data, by $Y$ its label and by $F = h ( X )$ its prediction score. Let $\mathcal { D } = \{ ( \mathbf { x } _ { 1 } , y _ { 1 } ) , : . . , ( \mathbf { x } _ { n } , y _ { n } ) \}$ denote the set of all training examples and $\mathbf { \bar { \mathcal { D } } } _ { + } = \{ \mathbf { x } _ { i } : y _ { i } = 1 \}$ denote the set of all positive examples. Let $n _ { + } = | \mathcal { D } _ { + } |$ denote the number of positive examples. $\mathbf { x } _ { i } \sim \mathcal { D }$ means that $\mathbf { x } _ { i }$ is randomly sampled from $\mathcal { D }$ .
+
+# 3.1 Background on AUPRC and its estimator AP
+
+Following the work of Bamber [2], AUPRC is an average of the precision weighted by the probability of a given threshold, which can be expressed as
+
+$$
+A = \int _ { - \infty } ^ { \infty } \dot { \operatorname* { P r } } ( Y = 1 | F \geq c ) d \operatorname* { P r } ( F \leq c | Y = 1 ) ,
+$$
+
+where $\operatorname* { P r } ( Y = 1 | F \geq c )$ is the precision at the threshold value of $c$ . The above integral is an importance-sampled Monte Carlo integral, by which we may interpret AUPRC as the fraction of positive examples among those examples whose output values exceed a randomly selected threshold $c \sim F ( X ) | Y = 1$ .
+
+For a finite set of examples $\mathcal { D } = \{ ( \mathbf { x } _ { i } , y _ { i } ) , i = 1 , \dots , n \}$ with the prediction score for each example $\mathbf { x } _ { i }$ given by $h _ { \mathbf { w } } ( \mathbf { x } _ { i } )$ , we consider to use AP to approximate AUPRC, which is given by
+
+$$
+\mathrm { A P } = \frac { 1 } { n _ { + } } \sum _ { i = 1 } ^ { n } \mathbf { I } ( y _ { i } = 1 ) \frac { \displaystyle \sum _ { s = 1 } ^ { n } \mathbf { \hat { I } } ( y _ { s } = 1 ) \mathbf { I } ( h _ { \mathbf { w } } ( \mathbf { x } _ { s } ) \geq h _ { \mathbf { w } } ( \mathbf { x } _ { i } ) ) } { \displaystyle \sum _ { s = 1 } ^ { n } \mathbf { I } ( h _ { \mathbf { w } } ( \mathbf { x } _ { s } ) \geq h _ { \mathbf { w } } ( \mathbf { x } _ { i } ) ) } ,
+$$
+
+where $n _ { + }$ denotes the number of positive examples. It can be shown that AP is an unbiased estimator in the limit $n \to \infty$ [3].
+
+However, the non-continuous indicator function $\mathbf { I } ( h _ { \mathbf { w } } ( \mathbf { x } _ { s } ) \geq h _ { \mathbf { w } } ( \mathbf { x } _ { i } ) )$ in both numerator and denominator in (1) makes the optimization non-tractable. To tackle this, we use a loss function $\ell ( \mathbf { w } ; \mathbf { x } _ { s } , \mathbf { x } _ { i } )$ as a surrogate function of $\mathbf { I } ( h _ { \mathbf { w } } ( \mathbf { x } _ { s } ) \geq h _ { \mathbf { w } } ( \mathbf { x } _ { i } ) )$ . One can consider different surrogate losses, e.g., hinge loss, squared hinge loss, and smoothed hinge loss, and exponential loss. In this paper, we will consider a smooth surrogate loss function to facilitate the development of an optimization algorithm, e.g., a squared hinge loss $\ell ( \mathbf { w } ; \mathbf { x } _ { s } ; \mathbf { x } _ { i } ) = ( \mathrm { m a x } \{ m - ( h _ { \mathbf { w } } ( \mathbf { x } _ { i } ) - \dot { h } _ { \mathbf { w } } ( \mathbf { x } _ { s } ) ) , 0 \} ) ^ { \dot { 2 } }$ , where $m$ is a margin parameter. Note that we do not require $\ell$ to be a convex function, hence one can also consider non-convex surrogate loss such as ramp loss. As a result, our problem becomes
+
+$$
+\operatorname* { m i n } _ { \mathbf { w } } P ( \mathbf { w } ) = \frac { 1 } { n _ { + } } \sum _ { \mathbf { x } _ { i } \in \mathcal { D } _ { + } } \frac { - \displaystyle \sum _ { s = 1 } ^ { n } \mathbf { I } ( y _ { s } = \mathrm { 1 } ) \ell ( \mathbf { w } ; \mathbf { x } _ { s } ; \mathbf { x } _ { i } ) } { \displaystyle \sum _ { s = 1 } ^ { n } \ell ( \mathbf { w } ; \mathbf { x } _ { s } ; \mathbf { x } _ { i } ) } .
+$$
+
+# 3.2 Stochastic Optimization of AP (SOAP)
+
+We cast the problem into a finite-sum of compositional functions. To this end, let us define a few notations:
+
+$\begin{array} { r l } & { \operatornamewithlimits { m a x } _ { j } \operatorname { m a x } _ { j } , } \\ & { g ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } ) = [ g _ { 1 } ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } ) , g _ { 2 } ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } ) ] ^ { \top } = [ \ell ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } ) \mathbf { I } ( y _ { j } = 1 ) , \ell ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } ) ] ^ { \top } } \end{array}$ $g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) = \mathbb { E } _ { \mathbf { x } _ { j } \sim \mathcal { D } } [ g ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } ) ]$ ,
+
+where $g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) : \mathbb { R } ^ { d }  \mathbb { R } ^ { 2 }$ . Let $\begin{array} { r } { f ( \mathbf { s } ) = - \frac { s _ { 1 } } { s _ { 2 } } : \mathbb { R } ^ { 2 }  \mathbb { R } } \end{array}$ . Then, we can write the objective function for maximizing AP as a sum of compositional functions:
+
+$$
+P ( \mathbf { w } ) = \frac { 1 } { n _ { + } } \sum _ { \mathbf { x } _ { i } \in \mathcal { D } _ { + } } ^ { } f ( g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) ) = \mathbb { E } _ { \mathbf { x } _ { i } \sim \mathcal { D } _ { + } } [ f ( g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) ) ] .
+$$
+
+We refer to the above problem as an instance of two-level stochastic coupled compositional functions. It is similar to the two-level stochastic compositional functions considered in literature [52, 53] but with a subtle difference. The difference is that in our formulation the inner function $g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) = \mathbb { E } _ { \mathbf { x } _ { j } \sim \mathcal { D } } [ g ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } ) ]$ depends on the random variable $\mathbf { x } _ { i }$ of the outer level. This difference makes the proposed algorithm slightly complicated by estimating $g _ { \mathbf { x } _ { i } } ( \mathbf { w } )$ separately for each positive example. It also complicates the analysis of the proposed algorithms. Nevertheless, we can still employ the techniques developed for optimizing stochastic compositional functions to design the algorithms and develop the analysis for optimizing the objective (4).
+
+In order to motivate the proposed method, let us consider how to compute the gradient of $P ( \mathbf { w } )$ . Let the gradient of $g _ { \mathbf { x } _ { i } } ( \mathbf { w } )$ be denoted by $\nabla _ { \mathbf { w } } g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) ^ { \top } = ( \nabla _ { \mathbf { w } } [ g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) ] _ { 1 } , \nabla _ { \mathbf { w } } [ g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) ] _ { 2 } )$ . Then we have
+
+$$
+\begin{array} { l }  { \displaystyle { \boldsymbol { \mathbf { \mathit { v } } } } ) \mathrm { ~ b e ~ d e n o t e d ~ t o y ~ V _ w } g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) ^ { \prime } = ( \nabla _ { \mathbf { w } } | g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) | _ { 1 } , \nabla _ { \mathbf { w } } | g _ { \mathbf { x } _ { i } } ( \mathbf { w } } \\ { { \nabla _ { \mathbf { w } } } P ( \mathbf { w } ) = \displaystyle \frac { 1 } { n _ { + } } \sum _ { \mathbf { x } _ { i } \in \mathcal { D } _ { + } } \nabla _ { \mathbf { w } } g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) ^ { \top } \nabla f ( g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) ) } \\ { { \displaystyle ~ = \frac { 1 } { n _ { + } } \sum _ { \mathbf { x } _ { i } \in \mathcal { D } _ { + } } \nabla _ { \mathbf { w } } g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) ^ { \top } \left( \frac { - 1 } { \left[ g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) \right] _ { 2 } } , \frac { \left[ g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) \right] _ { 1 } } { \left( \left[ g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) \right] _ { 2 } \right) ^ { 2 } } \right) ^ { \top } } . } \end{array}
+$$
+
+The major cost for computing $\nabla _ { \mathbf { w } } P ( \mathbf { w } )$ lies at evaluating $g _ { \mathbf { x } _ { i } } ( \mathbf { w } )$ and its gradient $\nabla _ { \mathbf { w } } g _ { \mathbf { x } _ { i } } ( \mathbf { w } )$ , which involves passing through all examples in $\mathcal { D }$ .
+
+To this end, we will approximate these quantities by stochastic samples. The gradient $\nabla _ { \mathbf { w } } g _ { \mathbf { x } _ { i } } ( \mathbf { w } )$ can be simply approximated by the stochastic gradient, i.e.,
+
+$$
+\begin{array} { r } { \hat { \nabla } _ { \mathbf { w } } g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) = \left( \begin{array} { c } { \frac { 1 } { B } \breve { \sum } _ { \mathbf { x } _ { j } \in B } \mathbf { I } ( y _ { j } = 1 ) \nabla \ell ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } ) } \\ { \frac { 1 } { B } \sum _ { \mathbf { x } _ { j } \in B } \nabla \ell ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } ) } \end{array} \right) , } \end{array}
+$$
+
+where $\boldsymbol { B }$ denote a set of $B$ random samples from $\mathcal { D }$ . For estimating $g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) = \mathbb { E } _ { \mathbf { x } _ { j } \sim \mathcal { D } } g ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } )$ , however, we need to ensure its approximation error is controllable due to the compositional structure such that the convergence can be guaranteed. We borrow a technique from the literature of stochastic compositional optimization [52] by using moving average estimator for estimating $g _ { \mathbf { x } _ { i } } ( \mathbf { w } )$ for all positive examples. To this end, we will maintain a matrix $\mathbf { u } = [ \mathbf { u } ^ { 1 } , \mathbf { u } ^ { 2 } ]$ with each column indexable by any positive example, i.e., $\mathbf { u } _ { \mathbf { x } _ { i } } ^ { 1 } , \mathbf { u } _ { \mathbf { x } _ { i } } ^ { 2 }$ correspond to the moving average estimator of $[ g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) ] _ { 1 }$ and $[ g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) ] _ { 2 }$ , respectively. The matrix $\mathbf { u }$ is updated by the subroutine UG in Algorithm 2, where $\gamma \in ( 0 , 1 )$ is a parameter. It is notable that in Step 3 of Algorithm 2, we clip the moving average update of $\mathbf { u } _ { \mathbf { x } _ { i } } ^ { 2 }$ by a lower bound $u _ { 0 }$ , which is a given parameter. This step can ensure the division in computing the stochastic gradient estimator in (7) always valid and is also important for convergence
+
+# Algorithm 1: SOAP
+
+1: Input: $\gamma , \alpha , u _ { 0 }$ , and other parameters for SGD-stype update or Adam-stype update.   
+2: Initialize w1 ∈ Rd, u ∈ R|n+|×2   
+3: for $t = 1 , \dots , T$ do   
+4: Draw a batch of $B _ { + }$ positive samples denoted by $B _ { + }$ .   
+5: Draw a batch of $B$ samples denoted by $\boldsymbol { B }$ .   
+6: ${ \bf u } = { \bf U } { \bf G } ( B , B _ { + } , { \bf u } , { \bf w } _ { t } , \gamma , u _ { 0 } )$   
+7: Compute (biased) Stochastic Gradient Estimator
+
+$$
+G ( \mathbf { w } _ { t } ) = \frac { 1 } { B _ { + } } \sum _ { \mathbf { x } _ { i } \in \mathcal { B } _ { + } } \sum _ { \mathbf { x } _ { j } \in B } \frac { ( \mathbf { u } _ { \mathbf { x } _ { i } } ^ { 1 } - \mathbf { u } _ { \mathbf { x } _ { i } } ^ { 2 } \mathbf { I } ( \mathbf { y } _ { j } = 1 ) ) \nabla \ell ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } ) } { B ( \mathbf { u } _ { \mathbf { x } _ { i } } ^ { 2 } ) ^ { 2 } }
+$$
+
+8: Update $\mathbf { w } _ { t + 1 }$ by a SGD-style method or by a Adam-style method
+
+$$
+\mathbf { w } _ { t + 1 } = \mathbf { U } \mathbf { W } ( \mathbf { w } _ { t } , G ( \mathbf { w } _ { t } ) )
+$$
+
+9: end for   
+10: Return: last solution.
+
+analysis. With these stochastic estimators, we can compute an estimate of $\nabla P ( \mathbf { w } )$ by equation (7), where $B _ { + }$ includes a batch of sampled positive data. With this stochastic gradient estimator, we can employ SGD-style method and Adam-style shown in Algorithm 3 to update the model parameter w. The final algorithm named as SOAP is presented in Algorithm 1.
+
+$$
+\overline { { \mathrm { A l g o r i t h m ~ 2 : ~ U G } ( \mathcal { B } , \mathcal { B } _ { + } , \mathbf { u } , \mathbf { w } _ { t } , \gamma , u _ { 0 } ) } }
+$$
+
+$$
+\overline { { \mathbf { A l g o r i t h m 3 : U W } ( \mathbf { w } _ { t } , G ( \mathbf { w } _ { t } ) ) } }
+$$
+
+1: for each positive $\mathbf { x } _ { i } \in B _ { + }$ do
+
+$$
+[ \overset { \mathbf { \hat { g } } } { \mathbf { x } } _ { i } ( \mathbf { w } _ { t } ) ] _ { 1 } = \frac { 1 } { | \mathcal { B } | } \sum _ { \underset { y _ { j } = 1 } { x _ { j } \in \mathcal { B } } } \ell ( \mathbf { w } _ { t } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } )
+$$
+
+$$
+\mathbf { w } _ { t + 1 } = \mathbf { w } _ { t } - \alpha G ( \mathbf { w } _ { t } )
+$$
+
+$$
+[ \tilde { g } _ { { \bf x } _ { i } } ( { \bf w } _ { t } ) ] _ { 2 } = \frac { 1 } { | B | } \sum _ { { \bf x } _ { j } \in B } \ell ( { \bf w } _ { t } ; { \bf x } _ { j } , { \bf x } _ { i } )
+$$
+
+$$
+\begin{array} { r l } & { \mathbf { \tau } , \epsilon , \eta _ { 1 } , \eta _ { 2 } ) } \\ & { \quad h _ { t + 1 } = \eta _ { 1 } h _ { t } + ( 1 - \eta _ { 1 } ) G ( \mathbf { w } _ { t } ) } \\ & { \quad v _ { t + 1 } = \eta _ { 2 } \hat { v } _ { t } + ( 1 - \eta _ { 2 } ) ( G ( \mathbf { w } _ { t } ) ) ^ { 2 } } \\ & { \quad \mathbf { w } _ { t + 1 } = \mathbf { w } _ { t } - \alpha \frac { h _ { t + 1 } } { \sqrt { \epsilon + \hat { v } _ { t + 1 } } } } \end{array}
+$$
+
+# 3: Compute
+
+$$
+\begin{array} { r l } & { \mathbf { u } _ { \mathbf { x } _ { i } } ^ { 1 } = ( 1 - \gamma ) \mathbf { u } _ { \mathbf { x } _ { i } } ^ { 1 } + \gamma [ \tilde { g } _ { \mathbf { x } _ { i } } ( \mathbf { w } _ { t } ) ] _ { 1 } } \\ & { \mathbf { u } _ { \mathbf { x } _ { i } } ^ { 2 } = \operatorname* { m a x } ( ( 1 - \gamma ) \mathbf { u } _ { \mathbf { x } _ { i } } ^ { 2 } + \gamma [ \tilde { g } _ { \mathbf { x } _ { i } } ( \mathbf { w } _ { t } ) ] _ { 2 } , u _ { 0 } ) } \end{array}
+$$
+
+4: end for   
+5: Return u
+
+# 3.3 Convergence Analysis
+
+In this subsection, we present the convergence results of SOAP and also highlight its convergence analysis. To this end, we first present the following assumption.
+
+Assumption 1. Assume that (a) there exists $\Delta _ { 1 }$ such that $P ( \mathbf { w } _ { 1 } ) - \operatorname* { m i n } _ { \mathbf { w } } P ( \mathbf { w } ) \leq \Delta _ { 1 }$ ; $( b )$ there exist $C , M > 0$ such that $\ell ( \mathbf { w } ; \mathbf { x } _ { i } , \mathbf { x } _ { i } ) \geq C$ for any $\mathbf { x } _ { i } \in \mathcal { D } _ { + }$ , $\ell ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } ) \le M$ , and $\ell ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } )$ is Lipscthiz continuous and smooth with respect to w for any $\mathbf { x } _ { i } \in \mathcal { D } _ { + } , \mathbf { x } _ { j } \in \mathcal { D } ,$ ; (c) there exists $V > 0$ such that $\begin{array} { r } { \mathbb { E } _ { \mathbf { x } _ { j } \sim \mathcal { D } } [ \| g ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } ) - g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) \| ^ { 2 } ] \leq V } \end{array}$ , and $\begin{array} { r } { \mathbb { E } _ { \mathbf { x } _ { j } \sim \mathcal { D } } [ \| \nabla g ( \mathbf { w } ; \mathbf { x } _ { j } , \mathbf { x } _ { i } ) - \nabla g _ { \mathbf { x } _ { i } } ( \mathbf { w } ) \| ^ { 2 } ] \leq V } \end{array}$ for any $\mathbf { x } _ { i }$ .
+
+With a bounded score function $h _ { \mathbf { w } } ( \mathbf { x } )$ the above assumption can be easily satisfied. Based on the above assumption, we can prove that the objective function $P ( \mathbf { w } )$ is smooth.
+
+Lemma 1. Suppose Assumption $^ { l }$ holds, then there exists $L > 0$ such that $P ( \cdot )$ is $L$ -smooth. In addition, there exists $u _ { 0 } \geq C / n$ such that gxi(w $\begin{array} { r } { v ) \in \Omega = \{ \mathbf { u } \in \mathbb { R } ^ { 2 } , 0 \leq [ \mathbf { u } ] _ { 1 } \leq M , u _ { 0 } \leq [ \mathbf { u } ] _ { 2 } \leq } \end{array}$ $M \}$ , $\forall \mathbf { x } _ { i } \in \mathcal { D } _ { + }$ .
+
+Next, we highlight the convergence analysis of SOAP employing the SGD-stype update and include that for employing Adam-style update in the supplement. Without loss of generality, we assume $| \boldsymbol { B } _ { + } | = 1$ and the positive sample in $B _ { + }$ is randomly selected from $\mathcal { D } _ { + }$ with replacement. When the context is clear, we abuse the notations $g _ { i } ( \mathbf { w } )$ and $\mathbf { u } _ { i }$ to denote $g _ { \mathbf { x } _ { i } } ( \mathbf { w } )$ and $\mathbf { u } _ { \mathbf { x } _ { i } }$ below, respectively. We first establish the following lemma following the analysis of non-convex optimization.
+
+Lemma 2. With $\alpha \leq 1 / 2$ , running $T$ iterations of $S O A P$ (SGD-style) updates, we have $\frac { \alpha } { 2 } \mathbb { E } [ \sum _ { t = 1 } ^ { T } \| \nabla P ( \mathbf { w } _ { t } ) \| ^ { 2 } ] \leq \mathbb { E } [ \sum _ { t } ( P ( \mathbf { w } _ { t } ) - P ( \mathbf { w } _ { t + 1 } ) ) ] + \frac { \alpha C _ { 1 } } { 2 } \mathbb { E } [ \sum _ { t = 1 } ^ { T } \| g _ { i _ { t } } ( \mathbf { w } _ { t } ) - \mathbf { u } _ { i _ { t } } \| ^ { 2 } ] + \alpha ^ { 2 } T C _ { 2 } ,$ where $i _ { t }$ denotes the index of the sampled positive data at iteration $t$ , $C _ { 1 }$ and $C _ { 2 }$ are proper constants.
+
+Our key contribution is the following lemma that bounds the second term in the above upper bound.
+
+Lemma 3. Suppose Assumption 1 holds, with u initialized by (6) for every $\mathbf { x } _ { i } \in \mathcal { D } _ { + }$ we have
+
+$$
+\mathbb { E } [ \sum _ { t = 1 } ^ { T } \| g _ { i _ { t } } ( \mathbf { w } _ { t } ) - \mathbf { u } _ { i _ { t } } \| ^ { 2 } ] \leq \frac { n _ { + } V } { \gamma } + \gamma V T + 2 \frac { n _ { + } ^ { 2 } \alpha ^ { 2 } T C _ { 3 } } { \gamma ^ { 2 } } ,
+$$
+
+where $C _ { 3 }$ is a proper constant.
+
+Remark: The innovation of proving the above lemma is by grouping $\mathbf { u } _ { i _ { t } } , t = 1 , \dots , T$ into $n _ { + }$ groups corresponding to the $n _ { + }$ positive examples, and then establishing the recursion of the error $\| g _ { i _ { t } } ( \mathbf { w } _ { t } ) - \mathbf { u } _ { i _ { t } } \| ^ { 2 }$ within each group, and then summing up these recursions together.
+
+Based on the two lemmas above, we establish the following convergence of SOAP with a SGD-style update.
+
+Theorem 1. Suppose Assumption 1 holds, let the parameters be $\begin{array} { r } { \alpha = \frac { 1 } { n _ { + } ^ { 2 / 5 } T ^ { 3 / 5 } } , \gamma = \frac { n _ { + } ^ { 2 / 5 } } { T ^ { 2 / 5 } } } \end{array}$ , $\forall t \in$ $1 , \cdots , T _ { \mathrm { { \scriptsize ~ 2 } } }$ , and $T > n _ { + }$ . Then after running $T$ iterations, SOAP with a $S G D$ -style update satisfies $\begin{array} { r } { \mathbb { E } \left[ \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \| \nabla P ( \mathbf { w } _ { t } ) \| ^ { 2 } \right] \leq O ( \frac { n _ { + } ^ { 2 / 5 } } { T ^ { 2 / 5 } } ) } \end{array}$ $O$
+
+Remark: To the best of our knowledge, this is the first time a stochastic algorithm was proved to converge for AP maximization.
+
+Similarly, we can establish the following convergence of SOAP by employing an Adam-style update, specifically the AMSGrad update.
+
+Theorem 2. Suppose Assumption $^ { l }$ holds, let the parameters $\eta _ { 1 } \leq \sqrt { \eta _ { 2 } } \leq 1$ , $\begin{array} { r } { \alpha = \frac { 1 } { n _ { + } ^ { 2 / 5 } T ^ { 3 / 5 } } , \gamma = } \end{array}$ n2/5+ $\frac { n _ { + } ^ { 2 / 5 } } { T ^ { 2 / 5 } }$ , $\forall t \in 1 , \cdots , T$ , and $T > n _ { + }$ . Then after running $T$ iterations, SOAP with an AMSGRAD update satisfies $\widehat { \sf z } \left[ \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \| \nabla P ( \mathbf { w } _ { t } ) \| ^ { 2 } \right] \leq O ( \frac { n _ { + } ^ { 2 / 5 } } { T ^ { 2 / 5 } } )$ , where $O$ suppresses constant numbers.
+
+# 4 Experiments
+
+In this section, we evaluate the proposed method through comprehensive experiments on imbalanced datasets. We show that the proposed method can outperform prior state-of-the-art methods for imbalanced classification problems. In addition, we conduct experiments on (i) the effects of imbalance ratio; (ii) the insensitivity to batch size and (iii) the convergence speed on testing data; and observe that our method (i) is more advantageous when data is more imbalanced, (ii) is not sensitive to batch size, and (iii) converges faster than baseline methods.
+
+Our proposed optimization algorithm is independent of specific datasets and tasks. Therefore, we perform experiments on both graph and image prediction tasks. In particular, the graph prediction tasks in the contexts of molecular property prediction and drug discovery suffer from very severe imbalance problems as positive labels are very rare while negative samples are abundantly available. Thus, we choose to use graph data intensively in our experiments. Additionally, the graph data we use allow us to vary the imbalance ratio to observe the performance change of different methods.
+
+In all experiments, we compare our method with the following baseline methods. CB-CE refers to a method using a class-balanced weighed cross entropy loss function, in which the weights for positive and negative samples are adjusted with the strategy proposed by Cui et al. [13]. Focal is to up-weight the penalty on hard examples using focal loss [32]. LDAM refers to training with labeldistribution-aware margin loss [7]. AUC-M is an AUROC maximization method using a surrogate loss [60]. In addition, we compare with three methods for optimizing AUPRC or AP, namely, the MinMax method [15] - a method for optimizing a discrete approximation of AUPRC, SmoothAP [4] - a method that optimizes a smoothed approximation of AP, and FastAP - a method that uses soft histogram binning to approximate the gradient of AP [6]. For all of these methods, we use the
+
+Table 1: The test AUPRC on the image datasets with two ResNet models. We report the average AUPRC and standard deviation (within brackets) over 5 runs.   
+
+<table><tr><td>Datasets</td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td></tr><tr><td>Networks</td><td>ResNet18</td><td>ResNet34</td><td>ResNet18</td><td>ResNet34</td></tr><tr><td>CE</td><td>0.7155 (± 0.0058)</td><td>0.6844(± 0.0031)</td><td>0.5946 (± 0.0031)</td><td>0.5792 (± 0.0028)</td></tr><tr><td>CB-CE</td><td>0.7325 (± 0.0039)</td><td>0.6936(±0.0021)</td><td>0.6165 (± 0.0096)</td><td>0.5632(± 0.0129)</td></tr><tr><td>Focal</td><td>0.7183(± 0.0082)</td><td>0.6943(± 0.0007)</td><td>0.6107(± 0.0093)</td><td>0.5585(± 0.0285)</td></tr><tr><td>LDAM</td><td>0.7346 (± 0.0125)</td><td>0.6745(± 0.0043)</td><td>0.6153 (± 0.0100)</td><td>0.5662(± 0.0212)</td></tr><tr><td>AUC-M</td><td>0.7399(± 0.0013)</td><td>0.6825(± 0.0089)</td><td>0.6103 (± 0.0075)</td><td>0.5306(± 0.0230)</td></tr><tr><td>SmoothAP</td><td>0.7365 (± 0.0088)</td><td>0.6909 (± 0.0049)</td><td>0.6071(± 0.0143)</td><td>0.5208 (± 0.0505)</td></tr><tr><td>FastAP</td><td>0.7028 (± 0.0341)</td><td>0.6798 (± 0.0032)</td><td>0.5618(± 0.0351)</td><td>0.5151(± 0.0450)</td></tr><tr><td>MinMax</td><td>0.7228 (± 0.0118)</td><td>0.6806(± 0.0027)</td><td>0.6071(± 0.0064)</td><td>0.5518(± 0.0030)</td></tr><tr><td>SOAP</td><td>0.7629(± 0.0014)</td><td>0.7012(± 0.0056)</td><td>0.6251 (± 0.0053)</td><td>0.6001(± 0.0060)</td></tr></table>
+
+SGD-style with momentum optimization for image prediction tasks and the Adam-style optimization algorithms for graph prediction tasks and unless specified otherwise. We refer to imbalance ratio as the number of positive samples over the total number of examples of a considered set. The hyper-parameters of all methods are fine tuned using cross-validation with training/validation splits mentioned below. For AP maximization methods, we use a sigmoid function to produce the prediction score. For simplicity, we set $u _ { 0 } = 0$ for SOAP and encounter no numerical problems in experiments. As SOAP requires positive samples for updating u to approximate the gradient of surrogate objective, we use a data sampler which samples a few positive examples (e.g., 2) and some negative examples per iteration. The same sampler applies to all methods for fair comparison. The code for reproducing the results is released here [44].
+
+# 4.1 Image Classification
+
+Data. We first conduct experiments on three image datasets: CIFAR10, CIFAR100 and Melanoma dataset [49]. We construct imbalanced version of CIFAR10 and CIFAR100 for binary classification. In particular, for each dataset we manually take the last half of classes as positive class and first half of classes as negative class. To construct highly imbalanced data, we remove $98 \%$ of the positive images from the training data and keep the test data unchanged (i.e., the testing data is still balanced). And we split the training dataset into train/validation set at $80 \% / 2 0 \%$ ratio. The Melanoma dataset is from a medical image Kaggle competition, which serves as a natural real imbalanced image dataset. It contains 33,126 labeled medical images, among which 584 images are related to malignant melanoma and labelled as positive samples. Since the test set used by Kaggle organization is not available, we manually split the training data into train/validation/test set at $8 0 \% / 1 0 \% / 1 0 \%$ ratio and report the achieved AUPRC on the test set by our method and baselines. The images of Melanoma dataset are always resized to have a resolution of $3 8 4 \times 3 8 4$ in our experiments.
+
+Setup. We use two ResNet [23] models, i.e., ResNet18 and ResNet34, as the backbone networks for image classification. For all methods except for CE, the ResNet models are initialized with a model pre-trained by CE with a SGD optimizer. We tune the learning rate in a range $\{ 1 \mathrm { e } \mathrm { - } 5 $ , 1e-4, 1e-3, 1e-2} and the weight decay parameter in a range $\{ 1 \mathrm { e } { - } 6 , 1 \mathrm { e } { - } 5 , 1 \mathrm { e } { - } 4 \}$ . Then the last fully connected layer is randomly re-initialized and the network is trained by different methods with the same weight decay parameter but other hyper-parameters individually tuned for fair comparison, e.g., we tune $\gamma$ of SOAP in a range $\{ 0 . 9 , 0 . 9 9 , 0 . 9 9 9 \}$ , and tune $m$ in $\{ 0 . 5 , 1 , 2 , 5 , 1 0 \}$ . We refer to this scheme as two-stage training, which is widely used for imbalanced data [60]. We consistently observe that this strategy can bring the model to a good initialization state and improve the final performance of our method and baselines.
+
+Results. Table 1 shows the AUPRC on testing sets of CIFAR-10 and CIFAR-100. We report the results on Melanoma in Table 3. We can observe that the proposed method SOAP outperforms all baselines. It is also striking to see that on Melanoma dataset, our proposed SOAP can outperform all baselines by a large margin, and all other methods have very poor performance. The reason is that the testing set of Melanoma is also imbalanced (imbalanced ratio $\mathrm { \Omega } = 1 . 7 2 \%$ ), while the testing sets of CIFAR-10 and CIFAR-100 are balanced. We also observe that the AUROC maximization (AUC-M) does not necessarily optimize AUPRC. We also plot the final PR curves in Figure 3 in the supplement.
+
+Table 2: The test AUPRC values on the HIV and MUV datasets with three graph neural network models. We report the average AUPRC and standard deviation (within brackets) over 3 runs.   
+
+<table><tr><td>Dataset</td><td>Method</td><td>GINE</td><td>MPNN</td><td>ML-MPNN</td></tr><tr><td rowspan="6">HIV</td><td>CE CB-CE</td><td>0.2774 (± 0.0101) 0.3082 (± 0.0101)</td><td>0.3197 (± 0.0050) 0.3056 (± 0.0018)</td><td>0.2988 (± 0.0076) 0.3291 (± 0.0189)</td></tr><tr><td>Focal</td><td></td><td>0.3136 (± 0.0197)</td><td>0.3279 (± 0.0173)</td></tr><tr><td>LDAM</td><td>0.3179 (± 0.0068)</td><td></td><td></td></tr><tr><td>AUC-M</td><td>0.2904 (± 0.0008)</td><td>0.2994 (± 0.0128)</td><td>0.3044 (± 0.0116)</td></tr><tr><td>SmothAP</td><td>0.2998 (± 0.0010)</td><td>0.2786 (± 0.0456)</td><td>0.3305 (± 0.0165)</td></tr><tr><td>FastAP</td><td>0.2686 (± 0.0007)</td><td>0.3276 (± 0.0063)</td><td>0.3235 (± 0.0092)</td></tr><tr><td rowspan="6"></td><td>MinMax</td><td>0.0169 (± 0.0031) 0.2874(± 0.0073)</td><td>0.0826 (± 0.0112) 0.3119 (± 0.0075)</td><td>0.0202 (± 0.0002) 0.3098 (± 0.0167)</td></tr><tr><td>SOAP</td><td>0.3385 (± 0.0024)</td><td>0.3401 (± 0.0045)</td><td>0.3547 (± 0.0077)</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>CE</td><td>0.0017 (±0.0001)</td><td>0.0021 (±0.0002)</td><td>0.0025 (±0.0004)</td></tr><tr><td>CB-CE</td><td>0.0055 (±0.0011)</td><td>0.0483 (±0.0083)</td><td>0.0121 (±0.0016)</td></tr><tr><td>Focal</td><td>0.0041 (±0.0007)</td><td>0.0281 (±0.0141)</td><td>0.0122 (±0.0001)</td></tr><tr><td rowspan="6">MUV</td><td>LDAM</td><td>0.0044(±0.0022)</td><td>0.0118 (±0.0098)</td><td>0.0059 (±0.0021)</td></tr><tr><td>AUC-M</td><td>0.0026 (±0.0001)</td><td>0.0040 (±0.0012)</td><td>0.0028 (±0.0012)</td></tr><tr><td>SmoothAP</td><td>0.0073 (±0.0012)</td><td>0.0068(±0.0038)</td><td>0.0029 (±0.0005)</td></tr><tr><td>FastAP</td><td>0.0016 (±0.0000)</td><td>0.0023 (±0.0021)</td><td>0.0022 (±0.0012)</td></tr><tr><td>MinMax</td><td>0.0028 (±0.0008)</td><td>0.0027 (±0.0005)</td><td>0.0043 (±0.0015)</td></tr><tr><td>SOAP</td><td>0.0254 (±0.0261)</td><td>0.3352 (±0.0008)</td><td>0.0236 (±0.0038)</td></tr></table>
+
+# 4.2 Graph Classification for Molecular Property Prediction
+
+Data. To further demonstrate the advantages of our method, we conduct experiments on two graph classification datasets. We use the datasets HIV and MUV from the MoleculeNet [55], which is a benchmark for molecular property prediction. The HIV dataset has 41,913 molecules from the Drug Therapeutics Program (DTP), and the positive samples are molecules tested to have inhibition ability to HIV. The MUV dataset has 93,127 molecules from the PubChem library, and molecules are labelled by whether a bioassay property exists or not. Note that the MUV dataset provides labels of 17 properties in total and we only conduct experiments to predict the third property as this property is more imbalanced. The percentage of positive samples in HIV and MUV datasets are $3 . 5 1 \%$ and $0 . 2 0 \%$ , respectively. We use the split of train/validation/test set provided by MoleculeNet. Molecules are treated as 2D graphs in our experiments, and we use the feature extraction procedure of MoleculeKit [54] to obtain node features of graphs. The same data preprocessing is used for all of our experiments on graph data.
+
+Setup. Many recent studies have shown that graph neural networks (GNNs) are powerful models for graph data analysis [29, 17, 16]. Hence, we use three different GNNs as the backbone network for graph classification, including the message passing neural network (MPNN) [19], an invariant of graph isomorphism network [56] named by GINE [26], and the multi-level message passing neural network (ML-MPNN) proposed by Wang et al. [54]. We use the same two-stage training scheme with a similar hyper-parameter tuning. We pre-train the networks by Adam with 100 epochs and a tuned initial learning rate 0.0005, which is decayed by half after 50 epochs.
+
+Results. The achieved AUPRC on the test set by all methods are presented in Table 2. Results show that our method can outperform all baselines by a large margin in terms of AUPRC, regardless of which model structure is used. These results clearly demonstrate that our method is effective for classification problems in which the sample distribution is highly imbalanced between classes.
+
+# 4.3 Graph Classification for Drug Discovery
+
+Data. In addition to molecular property prediction, we explore applying our method to drug discovery. Recent studies have shown that GNNs are effective in drug discovery through predicting the antibacterial property of chemical compounds [51]. Such application scenarios involves training a GNN model on labeled datasets and making predictions on a large library of chemical compounds so as to discover new antibiotic. However, because the positive samples in the training data, i.e., compounds known to have antibacterial property, are very rare, there exists very severe class imbalance.
+
+We show that our method can serve as a useful solution to the above problem. We conduct experiments on the MIT AICURES dataset from an open challenge (https://www.aicures.mit.edu/tasks)
+
+Table 3: The test AUPRC values on the MIT AICURES dataset with two graph neural networks, and on the Kaggle Melanoma dataset with two CNN models. We report the average AUPRC and standard deviation (within brackets) from 3 independent runs over 3 different train/validation/test splits.   
+
+<table><tr><td>Data</td><td colspan="2">MITAICURES</td><td colspan="2">Kaggle Melanoma</td></tr><tr><td>Networks</td><td>GINE</td><td>MPNN</td><td>ResNet18</td><td>ResNet34</td></tr><tr><td>CE</td><td>0.5037 (± 0.0718)</td><td>0.6282 (± 0.0634)</td><td>0.0701 (± 0.0031)</td><td>0.0582 (± 0.0016)</td></tr><tr><td>CB-CE</td><td>0.5655 (± 0.0453)</td><td>0.6308 (± 0.0263)</td><td>0.0631 (± 0.0065)</td><td>0.0721 (± 0.0054)</td></tr><tr><td>Focal</td><td>0.5143 (± 0.1062)</td><td>0.5875 (± 0.0774)</td><td>0.0549 (± 0.0083)</td><td>0.0663 (± 0.0034)</td></tr><tr><td>LDAM</td><td>0.5236 (± 0.0551)</td><td>0.6489 (± 0.0556)</td><td>0.0547 (± 0.0046)</td><td>0.0539 (± 0.0069)</td></tr><tr><td>AUC-M</td><td>0.5149 (± 0.0748)</td><td>0.5542 (± 0.0474)</td><td>0.1013 (± 0.0071)</td><td>0.0972 (± 0.0035)</td></tr><tr><td>SmothAP</td><td>0.2899 (± 0.0220)</td><td>0.4081 (± 0.0352)</td><td>0.1981 (± 0.0527)</td><td>0.2787 (± 0.0232)</td></tr><tr><td>FastAP</td><td>0.4777 (± 0.0896)</td><td>0.4518 (± 0.1495)</td><td>0.0324 (± 0.0087)</td><td>0.0359 (± 0.0062)</td></tr><tr><td>MinMax</td><td>0.5292 (± 0.0330)</td><td>0.5774 (± 0.0468)</td><td>0.0593 (± 0.0037)</td><td>0.0663 (± 0.0084)</td></tr><tr><td>SOAP</td><td>0.6639 (± 0.0515)</td><td>0.6547 (± 0.0616)</td><td>0.2624 (± 0.0410)</td><td>0.3152 (± 0.0337)</td></tr></table>
+
+![](images/90d2dec53afb889421a1515746678fc07644b82ce755a7edcef57f94abeeb611.jpg)  
+Figure 1: Comparison of convergence of different methods in terms of test AUPRC scores on CIFAR-10, CIFAR100 and MIT AICURES data.
+
+in drug discovery. The dataset consists of 2097 molecules. There are 48 positive samples that have antibacterial activity to Pseudomonas aeruginosa, which is the pathogen leading to secondary lungs infections of COVID-19 patients. We conduct experiments on three random train/validation/test splits at $8 0 \% / 1 0 \% / 1 0 \%$ ratio, and report the average AUPRC on the test set over three splits.
+
+Setup. Following the setup in Sec. 4.2, we use three GNNs: MPNN, GINE and ML-MPNN. We use the same two-stage training scheme with a similar hyper-parameter tuning. We pre-train GNNs by the Adam method for 100 epochs with a batch size of 64 and a tuned learning rate of 0.0005, which is decayed by half at the 50th epoch. Due to the limit of space, Table 3 only reports GINE and MPNN results. Please refer to Table 6 in the supplement for the full results of all three GNNs.
+
+Results. The average test AUPRC from three independent runs over three splits are summarized in Table 3, Table 6. We can see that our SOAP can consistently outperform all baselines on all three GNN models. Our proposed optimization method can significantly improve the achieved AUPRC of GNN models, indicating that models tend to assign higher confidence scores to molecules with antibacterial activity. This can help identify a larger number of candidate drugs.
+
+We have employed the proposed AUPRC maximization method for improving the testing performance on MIT AICures Challenge and achieved the 1st place. For details, please refer to [54].
+
+# 4.4 Ablation Studies
+
+Effects of Imbalance Ratio. We now study the effects of imbalance ratio on the performance improvements of our method. We use two datasets Tox21 and ToxCast from the MoleculeNet [55]. The Tox21 and ToxCast contain 8014 and 8589 molecules, respectively. There are 12 property prediction tasks in Tox21, and we conduct experiments on Task 0 and Task 2. Similarly, we select Task 12 and Task 8 of ToxCast for experiments. We use the split of train/validation/test set provided by MoleculeNet. The imbalanced ratios on the training sets are $4 . 1 4 \%$ for Task 0 of Tox21, $1 2 . 0 0 \%$ for Task 2 of Tox21, $2 . 9 7 \%$ for Task 12 of ToxCast, $8 . 6 7 \%$ for Task 8 of ToxCast.
+
+Following Sec. 4.2, we test three neural network models MPNN, GINE and ML-MPNN. The hyperparameters for training models are also the same as those in Sec. 4.2. We present the results of Tox21 and ToxCast in Table 5 in the supplement. Our SOAP can consistently achieve improved performance when the data is extremely imbalanced. However, it sometimes fails to do so if the imbalance ratio is not too low. Clearly, the improvements from our method are higher when the imbalance ratio of labels is lower. In other words, our method is more advantageous for data with extreme class imbalance.
+
+![](images/29ef093eea13be28581165e8fe9a97c3ac058f0edf548c4af3653833214d1db9.jpg)  
+Figure 2: Left most: insensitivity to batch size of SOAP. Right two: consistency between AP and Surrogate Objective $\mathbf { \nabla } _ { - } P ( \mathbf { w } )$ vs Iterations on CIFAR10 and CIFAR100.
+
+Table 4: The test AUPRC over 3 independent runs by SOAP with different surrogate functions.   
+
+<table><tr><td>Data</td><td colspan="2">CIFAR10</td><td colspan="2">CIFAR100</td></tr><tr><td>Networks</td><td>ResNet18</td><td>ResNet34</td><td>ResNet18</td><td>ResNet34</td></tr><tr><td>Squared Hinge</td><td>0.7629 (±0.0014)</td><td>0.7012 (±0.0056)</td><td>0.6251 (±0.0053)</td><td>0.6001(±0.0060)</td></tr><tr><td>Logistic</td><td>0.7542 (±0.0024)</td><td>0.6968 (±0.0121)</td><td>0.6378 (±0.0031)</td><td>0.5923 (±0.0101)</td></tr><tr><td>Sigmoid</td><td>0.7652 (±0.0035)</td><td>0.6983 (±0.0084)</td><td>0.6271 (±0.0043)</td><td>0.5832 (±0.0054)</td></tr><tr><td>Data</td><td>HIV</td><td></td><td>MUV</td><td></td></tr><tr><td>Networks</td><td>GINE</td><td>MPNN</td><td>GINE</td><td>MPNN</td></tr><tr><td>Squared Hinge</td><td>0.3485(±0.0083)</td><td>0.3401(±0.0045)</td><td>0.0354(±0.0025)</td><td>0.3365 (±0.0008)</td></tr><tr><td>Logistic</td><td>0.3436 (±0.0043)</td><td>0.3617 (±0.0031)</td><td>0.0493 (±0.0261)</td><td>0.3352 (±0.0008)</td></tr><tr><td>Sigmoid</td><td>0.3387 (±0.0051)</td><td>0.3629 (±0.0063)</td><td>0.0298 (±0.0043)</td><td>0.3362 (±0.0009)</td></tr></table>
+
+Insensitivity to Batch Size. We conduct experiments on CIFAR-10 and CIFAR-100 data by varying the mini-batch size for the SOAP algorithm and report results in Figure 2 (Left most). We can see that SOAP is not sensitive to the mini-batch size. This is consistent with our theory. In contrast, many previous methods for AP maximization are sensitive to the mini-batch size [47, 48, 6].
+
+Convergence Speed. We report the convergence curves of different methods for maximizing AUPRC or AP in Figure 1 on different datasets. We can see that the proposed SOAP algorithms converge much faster than other baseline methods.
+
+More Surrogate Losses. To verify the generality of SOAP, we evaluate the performance of SOAP with two more different surrogate loss functions $\ell ( \mathbf { w } ; \mathbf { x } _ { s } , \mathbf { x } _ { i } )$ as a surrogate function of the indicator I(hw(xs) ≥ hw(xi)), namely, the logistic loss, \`(w; xs, xi) = − log 11+exp(−c(\`(hw(xi)−hw(xs))) , and the sigmoid loss, \`(w; xs, xi) = 11+exp(c(\`(hw(xi)−hw(xs))) where $c$ is a hyperparameter. We tune $c \in \{ 1 , 2 \}$ in our experiments. We conduct experiments on CIFAR10, CIFAR100 following the experimental setting in Section 4.1 for the image data. For the graph data, we conduct experiments on HIV, MUV data following the experimental setting in Section 4.2. We report the results in Table 4. We can observe that SOAP has similar results with different surrogate loss functions.
+
+Consistency. Finally, we show the consistency between the Surrogate Objective - $\mathbf { \nabla } \cdot P ( \mathbf { w } )$ and AP by plotting the convergence curves on different datasets in Figure 2 (Right two). It is obvious two see the consistency between our surrogate objective and the true AP.
+
+# 5 Conclusions and Outlook
+
+In this work, we have proposed a stochastic method to optimize AUPRC that can be used in deep learning for tackling highly imbalanced data. Our approach is based on maximizing the averaged precision, and we cast the objective into a sum of coupled compositional functions. We proposed efficient adaptive and non-adaptive stochastic algorithms with provable convergence guarantee to compute the solutions. Extensive experimental results on graph and image datasets demonstrate that our proposed method can achieve promising results, especially when the class distribution is highly imbalanced. One limitation of SOAP is its convergence rate is still slow. In the future, we will consider to improve the convergence rate to address the limitation of the present work.
+
+# Acknowledgments
+
+We thank Bokun Wang for discussing the proofs, and thank anonymous reviewers for constructive comments. Q.Q contributed to the algorithm design, analysis, and experiments under supervision of T.Y. Y.L and Z.X contributed to the experiments under supervision of S.J. Q.Q and T.Y were partially supported by NSF Career Award #1844403, NSF Award #2110545 and NSF Award #1933212. Y.L, Z.X and S.J were partially supported by NSF IIS-1955189.
+
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parse/train/Q_64PF6XNut/Q_64PF6XNut_content_list.json

@@ -0,0 +1,1532 @@
+[
+    {
+        "type": "text",
+        "text": "Stochastic Optimization of Areas Under Precision-Recall Curves with Provable Convergence ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Qi Qi †∗, Youzhi Luo‡∗, Zhao $\\mathbf { X } \\mathbf { u } ^ { \\ddag * }$ , Shuiwang $\\mathbf { J } \\mathbf { i } ^ { \\ddag }$ , Tianbao Yang† †Department of Computer Science, The University of Iowa ‡Department of Computer Science & Engineering, Texas A&M University {qi-qi,tianbao-yang} $@$ uiowa.edu, {yzluo,zhaoxu,sji} $@$ tamu.edu ",
+        "bbox": [
+            254,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            319,
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+            335
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Areas under ROC (AUROC) and precision-recall curves (AUPRC) are common metrics for evaluating classification performance for imbalanced problems. Compared with AUROC, AUPRC is a more appropriate metric for highly imbalanced datasets. While stochastic optimization of AUROC has been studied extensively, principled stochastic optimization of AUPRC has been rarely explored. In this work, we propose a principled technical method to optimize AUPRC for deep learning. Our approach is based on maximizing the averaged precision (AP), which is an unbiased point estimator of AUPRC. We cast the objective into a sum of coupled compositional functions with inner functions dependent on random variables of the outer level. We propose efficient adaptive and non-adaptive stochastic algorithms named SOAP with provable convergence guarantee under mild conditions by leveraging recent advances in stochastic compositional optimization. Extensive experimental results on image and graph datasets demonstrate that our proposed method outperforms prior methods on imbalanced problems in terms of AUPRC. To the best of our knowledge, our work represents the first attempt to optimize AUPRC with provable convergence. The SOAP has been implemented in the libAUC library at https://libauc.org/. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Although deep learning (DL) has achieved tremendous success in various domains, the standard DL methods have reached a plateau as the traditional objective functions in DL are no longer sufficient to model all requirements in new applications, which slows down the democratization of AI. For instance, in healthcare applications, data is often highly imbalanced, e.g., patients suffering from rare diseases are much less than those suffering from common diseases. In these applications, accuracy (the proportion of correctly predicted examples) is deemed as an inappropriate metric for evaluating the performance of a classifier. Instead, area under the curve (AUC), including area under ROC curve (AUROC) and area under the Precision-Recall curve (AUPRC), is widely used for assessing the performance of a model. However, optimizing accuracy on training data does not necessarily lead to a satisfactory solution to maximizing AUC [12]. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To break the bottleneck for further advancement, DL must be empowered with the capability of efficiently handling novel objectives such as AUC. Recent studies have demonstrated great success along this direction by maximizing AUROC [60]. For example, Yuan et al. [60] proposed a robust deep AUROC maximization method with provable convergence and achieved great success for classification of medical image data. However, to the best of our knowledge, novel DL by maximizing AUPRC has not yet been studied thoroughly. Previous studies [14, 20] have found that when dealing with highly skewed datasets, Precision-Recall (PR) curves could give a more informative picture of an algorithm’s performance, which entails the development of efficient stochastic optimization algorithms for DL by maximizing AUPRC. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Compared with maximizing AUROC, maximizing AUPRC is more challenging. The challenges for optimization of AUPRC are two-fold. First, the analytical form of AUPRC by definition involves a complicated integral that is not readily estimated from model predictions of training examples. In practice, AUPRC is usually computed based on some point estimators, e.g., trapezoidal estimators and interpolation estimators of empirical curves, non-parametric average precision estimator, and parametric binomial estimator [3]. Among these estimators, non-parametric average precision (AP) is an unbiased estimate in the limit and can be directly computed based on the prediction scores of samples, which lends itself well to the task of model parameters optimization. Second, a surrogate function for AP is highly complicated and non-convex. In particular, an unbiased stochastic gradient is not readily computed, which makes existing stochastic algorithms such as SGD provide no convergence guarantee. Most existing works for maximizing AP-like function focus on how to compute an (approximate) gradient of the objective function [4, 6, 8, 11, 24, 38, 40, 43, 47, 48], which leave stochastic optimization of AP with provable convergence as an open question. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Can we design direct stochastic optimization algorithms both in SGD-style and Adam-style for maximizing AP with provable convergence guarantee? ",
+        "bbox": [
+            178,
+            328,
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+            357
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a systematic and principled solution for addressing this question towards maximizing AUPRC for DL. By using a surrogate loss in lieu of the indicator function in the definition of AP, we cast the objective into a sum of non-convex compositional functions, which resembles a two-level stochastic compositional optimization problem studied in the literature [52, 53]. However, different from existing two-level stochastic compositional functions, the inner functions in our problem are dependent on the random variable of the outer level, which requires us developing a tailored stochastic update for computing an error-controlled stochastic gradient estimator. Specifically, a key feature of the proposed method is to maintain and update two scalar quantities associated with each positive example for estimating the stochastic gradient of the individual precision score at the threshold specified by its prediction score. By leveraging recent advances in stochastic compositional optimization, we propose both adaptive (Adam-style) and non-adaptive (SGD-style) algorithms, and establish their convergence under mild conditions. We conduct comprehensive empirical studies on class imbalanced graph and image datasets for learning graph neural networks and deep convolutional neural networks, respectively. We demonstrate that the proposed method can consistently outperform prior approaches in terms of AUPRC. In addition, we show that our method achieves better results when the sample distribution is highly imbalanced between classes and is insensitive to mini-batch size. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            621,
+            321,
+            637
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "AUROC Optimization. AUROC optimization 2 has attracted significant attention in the literature. Recent success of DL by optimizing AUROC on large-scale medical image data has demonstrated the importance of large-scale stochastic optimization algorithms and the necessity of accurate surrogate function [60]. Earlier papers [25, 28] focus on learning a linear model based on the pairwise surrogate loss and could suffer from a high computational cost, which could be as high as quadratic of the size of training data. To address the computational challenge, online and stochastic optimization algorithms have been proposed [18, 35, 42, 58, 63]. Recently, [21, 22, 36, 57] proposed stochastic deep AUC maximization algorithms by formulating the problem as non-convex strongly-concave minmax optimization problem, and derived fast convergence rate under PL condition, and in federated learning setting as well [21]. More recently, Yuan et al. [60] demonstrated the success of their methods on medical image classification tasks, e.g., X-ray image classification, melanoma classification based on skin images. However, an algorithm that maximizes the AUROC might not necessarily maximize AUPRC, which entails the development of efficient algorithms for DL by maximizing AUPRC. ",
+        "bbox": [
+            174,
+            643,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "AUPRC Optimization. AUPRC optimization is much more challenging than AUROC optimization since the objective is even not decomposable over pairs of examples. Although AUPRC optimization has been considered in the literature (cf. [15, 47, 41] and references therein), efficient scalable algorithms for DL with provable convergence guarantee is still lacking. Some earlier works tackled this problem by using traditional optimization techniques, e.g., hill climbing search [37], cuttingplane method [61], dynamic programming [50], and by developing acceleration techniques in the framework of SVM [39]. These approaches are not scalable to big data for DL. There is a long list of studies in information retrieval [5, 11, 38, 47] and computer vision [4, 6, 8, 9, 24, 40, 48, 43], which have made efforts towards maximizing the AP score. However, most of them focus on how to compute an approximate gradient of the AP function or its smooth approximation, and provide no convergence guarantee for stochastic optimization based on mini-batch averaging. Due to lack of principled design, these previous methods when applied to deep learning are sensitive to the mini-batch size [6, 47, 48] and usually require a large mini-batch size in order to achieve good performance. In contrast, our stochastic algorithms are designed in a principled way to guarantee convergence without requiring a large mini-batch size as confirmed by our studies as well. Recently, [15] formulates the objective function as a constrained optimization problem using a surrogate function, and then casts it into a min-max saddle-point problem, which facilitates the use of stochastic min-max algorithms. However, they do not provide any convergence analysis for AUPRC maximization. In contrast, this is the first work that directly optimizes a surrogate function of AP (an unbaised estimator of AUPRC in the limit) and provides theoretical convergence guarantee for the proposed stochastic algorithms. ",
+        "bbox": [
+            174,
+            829,
+            823,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Stochastic Compositional Optimization. Optimization of a two-level compositional function in the form of $\\mathbb { E } _ { \\xi } [ \\bar { f } ( \\mathbb { E } _ { \\zeta } [ g ( \\mathbf { w } ; \\zeta ) ] ; \\xi ) ]$ where $\\xi$ and $\\zeta$ are independent random variables, or its finite-sum variant has been studied extensively in the literature [1, 10, 52, 27, 30, 31, 33, 34, 46, 53, 59, 62, 45]. In this paper, we formulate the surrogate function of AP into a similar but more complicated two-level compositional function of the form $\\mathbb { E } _ { \\xi } [ f ( \\mathbb { E } _ { \\zeta } g ( { \\mathbf w } ; \\zeta , \\xi ) ) ]$ , where $\\xi$ and $\\zeta$ are independent and $\\xi$ has a finite support. The key difference between our formulated compositional function and the ones considered in previous work is that the inner function $g ( \\mathbf { w } ; \\zeta , \\xi )$ also depends on the random variable $\\xi$ of the outer level. Such subtle difference will complicate the algorithm design and the convergence analysis as well. Nevertheless, the proposed algorithm and its convergence analysis are built on previous studies of stochastic two-level compositional optimization. ",
+        "bbox": [
+            173,
+            318,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 The Proposed Method ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            477,
+            392,
+            494
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Notations. We consider binary classification problem. Denote by $\\left( \\mathbf { x } , y \\right)$ a data pair, where $\\mathbf { x } \\in \\mathbb { R } ^ { d }$ denotes the input data and $y \\in \\{ 1 , - 1 \\}$ denotes its class label. Let $h ( \\mathbf { x } ) = h _ { \\mathbf { w } } ( \\mathbf { x } )$ denote the predictive function parameterized by a parameter vector $\\mathbf { w } \\in \\mathbb { R } ^ { D }$ (e.g., a deep neural network). Denote by $\\mathbf { I } ( \\cdot )$ an indicator function that outputs 1 if the argument is true and zero otherwise. To facilitate the presentation, denote by $X$ a random data, by $Y$ its label and by $F = h ( X )$ its prediction score. Let $\\mathcal { D } = \\{ ( \\mathbf { x } _ { 1 } , y _ { 1 } ) , : . . , ( \\mathbf { x } _ { n } , y _ { n } ) \\}$ denote the set of all training examples and $\\mathbf { \\bar { \\mathcal { D } } } _ { + } = \\{ \\mathbf { x } _ { i } : y _ { i } = 1 \\}$ denote the set of all positive examples. Let $n _ { + } = | \\mathcal { D } _ { + } |$ denote the number of positive examples. $\\mathbf { x } _ { i } \\sim \\mathcal { D }$ means that $\\mathbf { x } _ { i }$ is randomly sampled from $\\mathcal { D }$ . ",
+        "bbox": [
+            173,
+            505,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Background on AUPRC and its estimator AP ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            642,
+            527,
+            659
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Following the work of Bamber [2], AUPRC is an average of the precision weighted by the probability of a given threshold, which can be expressed as ",
+        "bbox": [
+            173,
+            671,
+            825,
+            700
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a976050dbb5078e6682f351b86ecb539572212ef3b3d3edb2d4eab2b65b035d8.jpg",
+        "text": "$$\nA = \\int _ { - \\infty } ^ { \\infty } \\dot { \\operatorname* { P r } } ( Y = 1 | F \\geq c ) d \\operatorname* { P r } ( F \\leq c | Y = 1 ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            699,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\operatorname* { P r } ( Y = 1 | F \\geq c )$ is the precision at the threshold value of $c$ . The above integral is an importance-sampled Monte Carlo integral, by which we may interpret AUPRC as the fraction of positive examples among those examples whose output values exceed a randomly selected threshold $c \\sim F ( X ) | Y = 1$ . ",
+        "bbox": [
+            174,
+            732,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For a finite set of examples $\\mathcal { D } = \\{ ( \\mathbf { x } _ { i } , y _ { i } ) , i = 1 , \\dots , n \\}$ with the prediction score for each example $\\mathbf { x } _ { i }$ given by $h _ { \\mathbf { w } } ( \\mathbf { x } _ { i } )$ , we consider to use AP to approximate AUPRC, which is given by ",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/70afd67ee4e0ff0ca30538c3d62ef4bb1589388e4e1b37c71e78eb0a47593cc9.jpg",
+        "text": "$$\n\\mathrm { A P } = \\frac { 1 } { n _ { + } } \\sum _ { i = 1 } ^ { n } \\mathbf { I } ( y _ { i } = 1 ) \\frac { \\displaystyle \\sum _ { s = 1 } ^ { n } \\mathbf { \\hat { I } } ( y _ { s } = 1 ) \\mathbf { I } ( h _ { \\mathbf { w } } ( \\mathbf { x } _ { s } ) \\geq h _ { \\mathbf { w } } ( \\mathbf { x } _ { i } ) ) } { \\displaystyle \\sum _ { s = 1 } ^ { n } \\mathbf { I } ( h _ { \\mathbf { w } } ( \\mathbf { x } _ { s } ) \\geq h _ { \\mathbf { w } } ( \\mathbf { x } _ { i } ) ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            820,
+            697,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $n _ { + }$ denotes the number of positive examples. It can be shown that AP is an unbiased estimator in the limit $n \\to \\infty$ [3]. ",
+        "bbox": [
+            171,
+            883,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "However, the non-continuous indicator function $\\mathbf { I } ( h _ { \\mathbf { w } } ( \\mathbf { x } _ { s } ) \\geq h _ { \\mathbf { w } } ( \\mathbf { x } _ { i } ) )$ in both numerator and denominator in (1) makes the optimization non-tractable. To tackle this, we use a loss function $\\ell ( \\mathbf { w } ; \\mathbf { x } _ { s } , \\mathbf { x } _ { i } )$ as a surrogate function of $\\mathbf { I } ( h _ { \\mathbf { w } } ( \\mathbf { x } _ { s } ) \\geq h _ { \\mathbf { w } } ( \\mathbf { x } _ { i } ) )$ . One can consider different surrogate losses, e.g., hinge loss, squared hinge loss, and smoothed hinge loss, and exponential loss. In this paper, we will consider a smooth surrogate loss function to facilitate the development of an optimization algorithm, e.g., a squared hinge loss $\\ell ( \\mathbf { w } ; \\mathbf { x } _ { s } ; \\mathbf { x } _ { i } ) = ( \\mathrm { m a x } \\{ m - ( h _ { \\mathbf { w } } ( \\mathbf { x } _ { i } ) - \\dot { h } _ { \\mathbf { w } } ( \\mathbf { x } _ { s } ) ) , 0 \\} ) ^ { \\dot { 2 } }$ , where $m$ is a margin parameter. Note that we do not require $\\ell$ to be a convex function, hence one can also consider non-convex surrogate loss such as ramp loss. As a result, our problem becomes ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1019437517de1d71d0ba2c33dd363f8415b60c012e40501ea57b72cc8b210626.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\mathbf { w } } P ( \\mathbf { w } ) = \\frac { 1 } { n _ { + } } \\sum _ { \\mathbf { x } _ { i } \\in \\mathcal { D } _ { + } } \\frac { - \\displaystyle \\sum _ { s = 1 } ^ { n } \\mathbf { I } ( y _ { s } = \\mathrm { 1 } ) \\ell ( \\mathbf { w } ; \\mathbf { x } _ { s } ; \\mathbf { x } _ { i } ) } { \\displaystyle \\sum _ { s = 1 } ^ { n } \\ell ( \\mathbf { w } ; \\mathbf { x } _ { s } ; \\mathbf { x } _ { i } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            200,
+            679,
+            267
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Stochastic Optimization of AP (SOAP) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            284,
+            480,
+            299
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We cast the problem into a finite-sum of compositional functions. To this end, let us define a few notations: ",
+        "bbox": [
+            173,
+            309,
+            823,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\begin{array} { r l } & { \\operatornamewithlimits { m a x } _ { j } \\operatorname { m a x } _ { j } , } \\\\ & { g ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } ) = [ g _ { 1 } ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } ) , g _ { 2 } ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } ) ] ^ { \\top } = [ \\ell ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } ) \\mathbf { I } ( y _ { j } = 1 ) , \\ell ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } ) ] ^ { \\top } } \\end{array}$ $g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) = \\mathbb { E } _ { \\mathbf { x } _ { j } \\sim \\mathcal { D } } [ g ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } ) ]$ , ",
+        "bbox": [
+            189,
+            333,
+            782,
+            371
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { 2 }$ . Let $\\begin{array} { r } { f ( \\mathbf { s } ) = - \\frac { s _ { 1 } } { s _ { 2 } } : \\mathbb { R } ^ { 2 }  \\mathbb { R } } \\end{array}$ . Then, we can write the objective function for maximizing AP as a sum of compositional functions: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3d10090ec4b5ba392f3b1f8763ced43b6900ba6f16cf42be27ca96017d13c61b.jpg",
+        "text": "$$\nP ( \\mathbf { w } ) = \\frac { 1 } { n _ { + } } \\sum _ { \\mathbf { x } _ { i } \\in \\mathcal { D } _ { + } } ^ { } f ( g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) ) = \\mathbb { E } _ { \\mathbf { x } _ { i } \\sim \\mathcal { D } _ { + } } [ f ( g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We refer to the above problem as an instance of two-level stochastic coupled compositional functions. It is similar to the two-level stochastic compositional functions considered in literature [52, 53] but with a subtle difference. The difference is that in our formulation the inner function $g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) = \\mathbb { E } _ { \\mathbf { x } _ { j } \\sim \\mathcal { D } } [ g ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } ) ]$ depends on the random variable $\\mathbf { x } _ { i }$ of the outer level. This difference makes the proposed algorithm slightly complicated by estimating $g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } )$ separately for each positive example. It also complicates the analysis of the proposed algorithms. Nevertheless, we can still employ the techniques developed for optimizing stochastic compositional functions to design the algorithms and develop the analysis for optimizing the objective (4). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In order to motivate the proposed method, let us consider how to compute the gradient of $P ( \\mathbf { w } )$ . Let the gradient of $g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } )$ be denoted by $\\nabla _ { \\mathbf { w } } g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) ^ { \\top } = ( \\nabla _ { \\mathbf { w } } [ g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) ] _ { 1 } , \\nabla _ { \\mathbf { w } } [ g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) ] _ { 2 } )$ . Then we have ",
+        "bbox": [
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+            553,
+            823,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/50579e9e2355f07a82429246a9b29596362532b4d96a791ddcce606574a73b5b.jpg",
+        "text": "$$\n\\begin{array} { l }  { \\displaystyle { \\boldsymbol { \\mathbf { \\mathit { v } } } } ) \\mathrm { ~ b e ~ d e n o t e d ~ t o y ~ V _ w } g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) ^ { \\prime } = ( \\nabla _ { \\mathbf { w } } | g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) | _ { 1 } , \\nabla _ { \\mathbf { w } } | g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } } \\\\ { { \\nabla _ { \\mathbf { w } } } P ( \\mathbf { w } ) = \\displaystyle \\frac { 1 } { n _ { + } } \\sum _ { \\mathbf { x } _ { i } \\in \\mathcal { D } _ { + } } \\nabla _ { \\mathbf { w } } g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) ^ { \\top } \\nabla f ( g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) ) } \\\\ { { \\displaystyle ~ = \\frac { 1 } { n _ { + } } \\sum _ { \\mathbf { x } _ { i } \\in \\mathcal { D } _ { + } } \\nabla _ { \\mathbf { w } } g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) ^ { \\top } \\left( \\frac { - 1 } { \\left[ g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) \\right] _ { 2 } } , \\frac { \\left[ g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) \\right] _ { 1 } } { \\left( \\left[ g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) \\right] _ { 2 } \\right) ^ { 2 } } \\right) ^ { \\top } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The major cost for computing $\\nabla _ { \\mathbf { w } } P ( \\mathbf { w } )$ lies at evaluating $g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } )$ and its gradient $\\nabla _ { \\mathbf { w } } g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } )$ , which involves passing through all examples in $\\mathcal { D }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To this end, we will approximate these quantities by stochastic samples. The gradient $\\nabla _ { \\mathbf { w } } g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } )$ can be simply approximated by the stochastic gradient, i.e., ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/203dafb705bbf0ad8ceb45b7618e1e27a7945680dac51d88d5deb1b04a9bb913.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { \\nabla } _ { \\mathbf { w } } g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) = \\left( \\begin{array} { c } { \\frac { 1 } { B } \\breve { \\sum } _ { \\mathbf { x } _ { j } \\in B } \\mathbf { I } ( y _ { j } = 1 ) \\nabla \\ell ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } ) } \\\\ { \\frac { 1 } { B } \\sum _ { \\mathbf { x } _ { j } \\in B } \\nabla \\ell ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } ) } \\end{array} \\right) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            732,
+            684,
+            773
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { B }$ denote a set of $B$ random samples from $\\mathcal { D }$ . For estimating $g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) = \\mathbb { E } _ { \\mathbf { x } _ { j } \\sim \\mathcal { D } } g ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } )$ , however, we need to ensure its approximation error is controllable due to the compositional structure such that the convergence can be guaranteed. We borrow a technique from the literature of stochastic compositional optimization [52] by using moving average estimator for estimating $g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } )$ for all positive examples. To this end, we will maintain a matrix $\\mathbf { u } = [ \\mathbf { u } ^ { 1 } , \\mathbf { u } ^ { 2 } ]$ with each column indexable by any positive example, i.e., $\\mathbf { u } _ { \\mathbf { x } _ { i } } ^ { 1 } , \\mathbf { u } _ { \\mathbf { x } _ { i } } ^ { 2 }$ correspond to the moving average estimator of $[ g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) ] _ { 1 }$ and $[ g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) ] _ { 2 }$ , respectively. The matrix $\\mathbf { u }$ is updated by the subroutine UG in Algorithm 2, where $\\gamma \\in ( 0 , 1 )$ is a parameter. It is notable that in Step 3 of Algorithm 2, we clip the moving average update of $\\mathbf { u } _ { \\mathbf { x } _ { i } } ^ { 2 }$ by a lower bound $u _ { 0 }$ , which is a given parameter. This step can ensure the division in computing the stochastic gradient estimator in (7) always valid and is also important for convergence ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: SOAP ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1: Input: $\\gamma , \\alpha , u _ { 0 }$ , and other parameters for SGD-stype update or Adam-stype update.   \n2: Initialize w1 ∈ Rd, u ∈ R|n+|×2   \n3: for $t = 1 , \\dots , T$ do   \n4: Draw a batch of $B _ { + }$ positive samples denoted by $B _ { + }$ .   \n5: Draw a batch of $B$ samples denoted by $\\boldsymbol { B }$ .   \n6: ${ \\bf u } = { \\bf U } { \\bf G } ( B , B _ { + } , { \\bf u } , { \\bf w } _ { t } , \\gamma , u _ { 0 } )$   \n7: Compute (biased) Stochastic Gradient Estimator ",
+        "bbox": [
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+            112,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e075d00e99ccbd0c615bc2e89b85df223c7274770050911cb71d05fcf6f1e768.jpg",
+        "text": "$$\nG ( \\mathbf { w } _ { t } ) = \\frac { 1 } { B _ { + } } \\sum _ { \\mathbf { x } _ { i } \\in \\mathcal { B } _ { + } } \\sum _ { \\mathbf { x } _ { j } \\in B } \\frac { ( \\mathbf { u } _ { \\mathbf { x } _ { i } } ^ { 1 } - \\mathbf { u } _ { \\mathbf { x } _ { i } } ^ { 2 } \\mathbf { I } ( \\mathbf { y } _ { j } = 1 ) ) \\nabla \\ell ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } ) } { B ( \\mathbf { u } _ { \\mathbf { x } _ { i } } ^ { 2 } ) ^ { 2 } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            217,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "8: Update $\\mathbf { w } _ { t + 1 }$ by a SGD-style method or by a Adam-style method ",
+        "bbox": [
+            178,
+            265,
+            648,
+            280
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d40fd248156cf4318eff39d85fdac3a70bf1dba09e0148dd1aeac127de88b39b.jpg",
+        "text": "$$\n\\mathbf { w } _ { t + 1 } = \\mathbf { U } \\mathbf { W } ( \\mathbf { w } _ { t } , G ( \\mathbf { w } _ { t } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            434,
+            286,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "9: end for   \n10: Return: last solution. ",
+        "bbox": [
+            176,
+            311,
+            346,
+            339
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "analysis. With these stochastic estimators, we can compute an estimate of $\\nabla P ( \\mathbf { w } )$ by equation (7), where $B _ { + }$ includes a batch of sampled positive data. With this stochastic gradient estimator, we can employ SGD-style method and Adam-style shown in Algorithm 3 to update the model parameter w. The final algorithm named as SOAP is presented in Algorithm 1. ",
+        "bbox": [
+            173,
+            352,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a69e5de9f3b2e8fb0c51c21b16a328e23a532b57a95321546471b77bdc2194f8.jpg",
+        "text": "$$\n\\overline { { \\mathrm { A l g o r i t h m ~ 2 : ~ U G } ( \\mathcal { B } , \\mathcal { B } _ { + } , \\mathbf { u } , \\mathbf { w } _ { t } , \\gamma , u _ { 0 } ) } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            171,
+            419,
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+            436
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f39745a7167e7a2bfe75240fcebb43dbf9b4015ff7309c696ab23a5163997fe4.jpg",
+        "text": "$$\n\\overline { { \\mathbf { A l g o r i t h m 3 : U W } ( \\mathbf { w } _ { t } , G ( \\mathbf { w } _ { t } ) ) } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1: for each positive $\\mathbf { x } _ { i } \\in B _ { + }$ do ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a816cb96a7f67ebdb3b4785249af292f24f02110df01eb782e8b7bbddab6f5c4.jpg",
+        "text": "$$\n[ \\overset { \\mathbf { \\hat { g } } } { \\mathbf { x } } _ { i } ( \\mathbf { w } _ { t } ) ] _ { 1 } = \\frac { 1 } { | \\mathcal { B } | } \\sum _ { \\underset { y _ { j } = 1 } { x _ { j } \\in \\mathcal { B } } } \\ell ( \\mathbf { w } _ { t } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            464,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/030def322c3a682dcc66daa7b4e5085ae7ad2a052d9935c38971c57e430a9c21.jpg",
+        "text": "$$\n\\mathbf { w } _ { t + 1 } = \\mathbf { w } _ { t } - \\alpha G ( \\mathbf { w } _ { t } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            612,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d21354ea4d884ac4cece7597747e36317bf54dc11979f0957c246a2c4fc6e1b9.jpg",
+        "text": "$$\n[ \\tilde { g } _ { { \\bf x } _ { i } } ( { \\bf w } _ { t } ) ] _ { 2 } = \\frac { 1 } { | B | } \\sum _ { { \\bf x } _ { j } \\in B } \\ell ( { \\bf w } _ { t } ; { \\bf x } _ { j } , { \\bf x } _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            235,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/582ea618f734814cfdac8ff8d937dd64f7ee4bff2898b2f049f614372d1ee857.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbf { \\tau } , \\epsilon , \\eta _ { 1 } , \\eta _ { 2 } ) } \\\\ & { \\quad h _ { t + 1 } = \\eta _ { 1 } h _ { t } + ( 1 - \\eta _ { 1 } ) G ( \\mathbf { w } _ { t } ) } \\\\ & { \\quad v _ { t + 1 } = \\eta _ { 2 } \\hat { v } _ { t } + ( 1 - \\eta _ { 2 } ) ( G ( \\mathbf { w } _ { t } ) ) ^ { 2 } } \\\\ & { \\quad \\mathbf { w } _ { t + 1 } = \\mathbf { w } _ { t } - \\alpha \\frac { h _ { t + 1 } } { \\sqrt { \\epsilon + \\hat { v } _ { t + 1 } } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3: Compute ",
+        "text_level": 1,
+        "bbox": [
+            181,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/556b45f87b0d3fa26f561f14c36db7c17347b97d379d56d3ca04e8b456c551d7.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbf { u } _ { \\mathbf { x } _ { i } } ^ { 1 } = ( 1 - \\gamma ) \\mathbf { u } _ { \\mathbf { x } _ { i } } ^ { 1 } + \\gamma [ \\tilde { g } _ { \\mathbf { x } _ { i } } ( \\mathbf { w } _ { t } ) ] _ { 1 } } \\\\ & { \\mathbf { u } _ { \\mathbf { x } _ { i } } ^ { 2 } = \\operatorname* { m a x } ( ( 1 - \\gamma ) \\mathbf { u } _ { \\mathbf { x } _ { i } } ^ { 2 } + \\gamma [ \\tilde { g } _ { \\mathbf { x } _ { i } } ( \\mathbf { w } _ { t } ) ] _ { 2 } , u _ { 0 } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4: end for   \n5: Return u ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 Convergence Analysis ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we present the convergence results of SOAP and also highlight its convergence analysis. To this end, we first present the following assumption. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1. Assume that (a) there exists $\\Delta _ { 1 }$ such that $P ( \\mathbf { w } _ { 1 } ) - \\operatorname* { m i n } _ { \\mathbf { w } } P ( \\mathbf { w } ) \\leq \\Delta _ { 1 }$ ; $( b )$ there exist $C , M > 0$ such that $\\ell ( \\mathbf { w } ; \\mathbf { x } _ { i } , \\mathbf { x } _ { i } ) \\geq C$ for any $\\mathbf { x } _ { i } \\in \\mathcal { D } _ { + }$ , $\\ell ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } ) \\le M$ , and $\\ell ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } )$ is Lipscthiz continuous and smooth with respect to w for any $\\mathbf { x } _ { i } \\in \\mathcal { D } _ { + } , \\mathbf { x } _ { j } \\in \\mathcal { D } ,$ ; (c) there exists $V > 0$ such that $\\begin{array} { r } { \\mathbb { E } _ { \\mathbf { x } _ { j } \\sim \\mathcal { D } } [ \\| g ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } ) - g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) \\| ^ { 2 } ] \\leq V } \\end{array}$ , and $\\begin{array} { r } { \\mathbb { E } _ { \\mathbf { x } _ { j } \\sim \\mathcal { D } } [ \\| \\nabla g ( \\mathbf { w } ; \\mathbf { x } _ { j } , \\mathbf { x } _ { i } ) - \\nabla g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } ) \\| ^ { 2 } ] \\leq V } \\end{array}$ for any $\\mathbf { x } _ { i }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "With a bounded score function $h _ { \\mathbf { w } } ( \\mathbf { x } )$ the above assumption can be easily satisfied. Based on the above assumption, we can prove that the objective function $P ( \\mathbf { w } )$ is smooth. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. Suppose Assumption $^ { l }$ holds, then there exists $L > 0$ such that $P ( \\cdot )$ is $L$ -smooth. In addition, there exists $u _ { 0 } \\geq C / n$ such that gxi(w $\\begin{array} { r } { v ) \\in \\Omega = \\{ \\mathbf { u } \\in \\mathbb { R } ^ { 2 } , 0 \\leq [ \\mathbf { u } ] _ { 1 } \\leq M , u _ { 0 } \\leq [ \\mathbf { u } ] _ { 2 } \\leq } \\end{array}$ $M \\}$ , $\\forall \\mathbf { x } _ { i } \\in \\mathcal { D } _ { + }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Next, we highlight the convergence analysis of SOAP employing the SGD-stype update and include that for employing Adam-style update in the supplement. Without loss of generality, we assume $| \\boldsymbol { B } _ { + } | = 1$ and the positive sample in $B _ { + }$ is randomly selected from $\\mathcal { D } _ { + }$ with replacement. When the context is clear, we abuse the notations $g _ { i } ( \\mathbf { w } )$ and $\\mathbf { u } _ { i }$ to denote $g _ { \\mathbf { x } _ { i } } ( \\mathbf { w } )$ and $\\mathbf { u } _ { \\mathbf { x } _ { i } }$ below, respectively. We first establish the following lemma following the analysis of non-convex optimization. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2. With $\\alpha \\leq 1 / 2$ , running $T$ iterations of $S O A P$ (SGD-style) updates, we have $\\frac { \\alpha } { 2 } \\mathbb { E } [ \\sum _ { t = 1 } ^ { T } \\| \\nabla P ( \\mathbf { w } _ { t } ) \\| ^ { 2 } ] \\leq \\mathbb { E } [ \\sum _ { t } ( P ( \\mathbf { w } _ { t } ) - P ( \\mathbf { w } _ { t + 1 } ) ) ] + \\frac { \\alpha C _ { 1 } } { 2 } \\mathbb { E } [ \\sum _ { t = 1 } ^ { T } \\| g _ { i _ { t } } ( \\mathbf { w } _ { t } ) - \\mathbf { u } _ { i _ { t } } \\| ^ { 2 } ] + \\alpha ^ { 2 } T C _ { 2 } ,$ where $i _ { t }$ denotes the index of the sampled positive data at iteration $t$ , $C _ { 1 }$ and $C _ { 2 }$ are proper constants. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our key contribution is the following lemma that bounds the second term in the above upper bound. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3. Suppose Assumption 1 holds, with u initialized by (6) for every $\\mathbf { x } _ { i } \\in \\mathcal { D } _ { + }$ we have ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/518c270fd4a5f3c4c64b817a63899c960926c08c348d47f076ce654a0dd1f73f.jpg",
+        "text": "$$\n\\mathbb { E } [ \\sum _ { t = 1 } ^ { T } \\| g _ { i _ { t } } ( \\mathbf { w } _ { t } ) - \\mathbf { u } _ { i _ { t } } \\| ^ { 2 } ] \\leq \\frac { n _ { + } V } { \\gamma } + \\gamma V T + 2 \\frac { n _ { + } ^ { 2 } \\alpha ^ { 2 } T C _ { 3 } } { \\gamma ^ { 2 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $C _ { 3 }$ is a proper constant. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark: The innovation of proving the above lemma is by grouping $\\mathbf { u } _ { i _ { t } } , t = 1 , \\dots , T$ into $n _ { + }$ groups corresponding to the $n _ { + }$ positive examples, and then establishing the recursion of the error $\\| g _ { i _ { t } } ( \\mathbf { w } _ { t } ) - \\mathbf { u } _ { i _ { t } } \\| ^ { 2 }$ within each group, and then summing up these recursions together. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Based on the two lemmas above, we establish the following convergence of SOAP with a SGD-style update. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Suppose Assumption 1 holds, let the parameters be $\\begin{array} { r } { \\alpha = \\frac { 1 } { n _ { + } ^ { 2 / 5 } T ^ { 3 / 5 } } , \\gamma = \\frac { n _ { + } ^ { 2 / 5 } } { T ^ { 2 / 5 } } } \\end{array}$ , $\\forall t \\in$ $1 , \\cdots , T _ { \\mathrm { { \\scriptsize ~ 2 } } }$ , and $T > n _ { + }$ . Then after running $T$ iterations, SOAP with a $S G D$ -style update satisfies $\\begin{array} { r } { \\mathbb { E } \\left[ \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\| \\nabla P ( \\mathbf { w } _ { t } ) \\| ^ { 2 } \\right] \\leq O ( \\frac { n _ { + } ^ { 2 / 5 } } { T ^ { 2 / 5 } } ) } \\end{array}$ $O$ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark: To the best of our knowledge, this is the first time a stochastic algorithm was proved to converge for AP maximization. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Similarly, we can establish the following convergence of SOAP by employing an Adam-style update, specifically the AMSGrad update. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. Suppose Assumption $^ { l }$ holds, let the parameters $\\eta _ { 1 } \\leq \\sqrt { \\eta _ { 2 } } \\leq 1$ , $\\begin{array} { r } { \\alpha = \\frac { 1 } { n _ { + } ^ { 2 / 5 } T ^ { 3 / 5 } } , \\gamma = } \\end{array}$ n2/5+ $\\frac { n _ { + } ^ { 2 / 5 } } { T ^ { 2 / 5 } }$ , $\\forall t \\in 1 , \\cdots , T$ , and $T > n _ { + }$ . Then after running $T$ iterations, SOAP with an AMSGRAD update satisfies $\\widehat { \\sf z } \\left[ \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\| \\nabla P ( \\mathbf { w } _ { t } ) \\| ^ { 2 } \\right] \\leq O ( \\frac { n _ { + } ^ { 2 / 5 } } { T ^ { 2 / 5 } } )$ , where $O$ suppresses constant numbers. ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we evaluate the proposed method through comprehensive experiments on imbalanced datasets. We show that the proposed method can outperform prior state-of-the-art methods for imbalanced classification problems. In addition, we conduct experiments on (i) the effects of imbalance ratio; (ii) the insensitivity to batch size and (iii) the convergence speed on testing data; and observe that our method (i) is more advantageous when data is more imbalanced, (ii) is not sensitive to batch size, and (iii) converges faster than baseline methods. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our proposed optimization algorithm is independent of specific datasets and tasks. Therefore, we perform experiments on both graph and image prediction tasks. In particular, the graph prediction tasks in the contexts of molecular property prediction and drug discovery suffer from very severe imbalance problems as positive labels are very rare while negative samples are abundantly available. Thus, we choose to use graph data intensively in our experiments. Additionally, the graph data we use allow us to vary the imbalance ratio to observe the performance change of different methods. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In all experiments, we compare our method with the following baseline methods. CB-CE refers to a method using a class-balanced weighed cross entropy loss function, in which the weights for positive and negative samples are adjusted with the strategy proposed by Cui et al. [13]. Focal is to up-weight the penalty on hard examples using focal loss [32]. LDAM refers to training with labeldistribution-aware margin loss [7]. AUC-M is an AUROC maximization method using a surrogate loss [60]. In addition, we compare with three methods for optimizing AUPRC or AP, namely, the MinMax method [15] - a method for optimizing a discrete approximation of AUPRC, SmoothAP [4] - a method that optimizes a smoothed approximation of AP, and FastAP - a method that uses soft histogram binning to approximate the gradient of AP [6]. For all of these methods, we use the ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/be4e7a45a863b1007af8005d676d1a2826855d0efbfb30979ed64ad95da63a16.jpg",
+        "table_caption": [
+            "Table 1: The test AUPRC on the image datasets with two ResNet models. We report the average AUPRC and standard deviation (within brackets) over 5 runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Datasets</td><td colspan=\"2\">CIFAR-10</td><td colspan=\"2\">CIFAR-100</td></tr><tr><td>Networks</td><td>ResNet18</td><td>ResNet34</td><td>ResNet18</td><td>ResNet34</td></tr><tr><td>CE</td><td>0.7155 (± 0.0058)</td><td>0.6844(± 0.0031)</td><td>0.5946 (± 0.0031)</td><td>0.5792 (± 0.0028)</td></tr><tr><td>CB-CE</td><td>0.7325 (± 0.0039)</td><td>0.6936(±0.0021)</td><td>0.6165 (± 0.0096)</td><td>0.5632(± 0.0129)</td></tr><tr><td>Focal</td><td>0.7183(± 0.0082)</td><td>0.6943(± 0.0007)</td><td>0.6107(± 0.0093)</td><td>0.5585(± 0.0285)</td></tr><tr><td>LDAM</td><td>0.7346 (± 0.0125)</td><td>0.6745(± 0.0043)</td><td>0.6153 (± 0.0100)</td><td>0.5662(± 0.0212)</td></tr><tr><td>AUC-M</td><td>0.7399(± 0.0013)</td><td>0.6825(± 0.0089)</td><td>0.6103 (± 0.0075)</td><td>0.5306(± 0.0230)</td></tr><tr><td>SmoothAP</td><td>0.7365 (± 0.0088)</td><td>0.6909 (± 0.0049)</td><td>0.6071(± 0.0143)</td><td>0.5208 (± 0.0505)</td></tr><tr><td>FastAP</td><td>0.7028 (± 0.0341)</td><td>0.6798 (± 0.0032)</td><td>0.5618(± 0.0351)</td><td>0.5151(± 0.0450)</td></tr><tr><td>MinMax</td><td>0.7228 (± 0.0118)</td><td>0.6806(± 0.0027)</td><td>0.6071(± 0.0064)</td><td>0.5518(± 0.0030)</td></tr><tr><td>SOAP</td><td>0.7629(± 0.0014)</td><td>0.7012(± 0.0056)</td><td>0.6251 (± 0.0053)</td><td>0.6001(± 0.0060)</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "SGD-style with momentum optimization for image prediction tasks and the Adam-style optimization algorithms for graph prediction tasks and unless specified otherwise. We refer to imbalance ratio as the number of positive samples over the total number of examples of a considered set. The hyper-parameters of all methods are fine tuned using cross-validation with training/validation splits mentioned below. For AP maximization methods, we use a sigmoid function to produce the prediction score. For simplicity, we set $u _ { 0 } = 0$ for SOAP and encounter no numerical problems in experiments. As SOAP requires positive samples for updating u to approximate the gradient of surrogate objective, we use a data sampler which samples a few positive examples (e.g., 2) and some negative examples per iteration. The same sampler applies to all methods for fair comparison. The code for reproducing the results is released here [44]. ",
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+    {
+        "type": "text",
+        "text": "4.1 Image Classification ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Data. We first conduct experiments on three image datasets: CIFAR10, CIFAR100 and Melanoma dataset [49]. We construct imbalanced version of CIFAR10 and CIFAR100 for binary classification. In particular, for each dataset we manually take the last half of classes as positive class and first half of classes as negative class. To construct highly imbalanced data, we remove $98 \\%$ of the positive images from the training data and keep the test data unchanged (i.e., the testing data is still balanced). And we split the training dataset into train/validation set at $80 \\% / 2 0 \\%$ ratio. The Melanoma dataset is from a medical image Kaggle competition, which serves as a natural real imbalanced image dataset. It contains 33,126 labeled medical images, among which 584 images are related to malignant melanoma and labelled as positive samples. Since the test set used by Kaggle organization is not available, we manually split the training data into train/validation/test set at $8 0 \\% / 1 0 \\% / 1 0 \\%$ ratio and report the achieved AUPRC on the test set by our method and baselines. The images of Melanoma dataset are always resized to have a resolution of $3 8 4 \\times 3 8 4$ in our experiments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Setup. We use two ResNet [23] models, i.e., ResNet18 and ResNet34, as the backbone networks for image classification. For all methods except for CE, the ResNet models are initialized with a model pre-trained by CE with a SGD optimizer. We tune the learning rate in a range $\\{ 1 \\mathrm { e } \\mathrm { - } 5 $ , 1e-4, 1e-3, 1e-2} and the weight decay parameter in a range $\\{ 1 \\mathrm { e } { - } 6 , 1 \\mathrm { e } { - } 5 , 1 \\mathrm { e } { - } 4 \\}$ . Then the last fully connected layer is randomly re-initialized and the network is trained by different methods with the same weight decay parameter but other hyper-parameters individually tuned for fair comparison, e.g., we tune $\\gamma$ of SOAP in a range $\\{ 0 . 9 , 0 . 9 9 , 0 . 9 9 9 \\}$ , and tune $m$ in $\\{ 0 . 5 , 1 , 2 , 5 , 1 0 \\}$ . We refer to this scheme as two-stage training, which is widely used for imbalanced data [60]. We consistently observe that this strategy can bring the model to a good initialization state and improve the final performance of our method and baselines. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Results. Table 1 shows the AUPRC on testing sets of CIFAR-10 and CIFAR-100. We report the results on Melanoma in Table 3. We can observe that the proposed method SOAP outperforms all baselines. It is also striking to see that on Melanoma dataset, our proposed SOAP can outperform all baselines by a large margin, and all other methods have very poor performance. The reason is that the testing set of Melanoma is also imbalanced (imbalanced ratio $\\mathrm { \\Omega } = 1 . 7 2 \\%$ ), while the testing sets of CIFAR-10 and CIFAR-100 are balanced. We also observe that the AUROC maximization (AUC-M) does not necessarily optimize AUPRC. We also plot the final PR curves in Figure 3 in the supplement. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/2d9aa651e413074ea0c3561b89862957896a900970264699b9aea85465d26d3f.jpg",
+        "table_caption": [
+            "Table 2: The test AUPRC values on the HIV and MUV datasets with three graph neural network models. We report the average AUPRC and standard deviation (within brackets) over 3 runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Method</td><td>GINE</td><td>MPNN</td><td>ML-MPNN</td></tr><tr><td rowspan=\"6\">HIV</td><td>CE CB-CE</td><td>0.2774 (± 0.0101) 0.3082 (± 0.0101)</td><td>0.3197 (± 0.0050) 0.3056 (± 0.0018)</td><td>0.2988 (± 0.0076) 0.3291 (± 0.0189)</td></tr><tr><td>Focal</td><td></td><td>0.3136 (± 0.0197)</td><td>0.3279 (± 0.0173)</td></tr><tr><td>LDAM</td><td>0.3179 (± 0.0068)</td><td></td><td></td></tr><tr><td>AUC-M</td><td>0.2904 (± 0.0008)</td><td>0.2994 (± 0.0128)</td><td>0.3044 (± 0.0116)</td></tr><tr><td>SmothAP</td><td>0.2998 (± 0.0010)</td><td>0.2786 (± 0.0456)</td><td>0.3305 (± 0.0165)</td></tr><tr><td>FastAP</td><td>0.2686 (± 0.0007)</td><td>0.3276 (± 0.0063)</td><td>0.3235 (± 0.0092)</td></tr><tr><td rowspan=\"6\"></td><td>MinMax</td><td>0.0169 (± 0.0031) 0.2874(± 0.0073)</td><td>0.0826 (± 0.0112) 0.3119 (± 0.0075)</td><td>0.0202 (± 0.0002) 0.3098 (± 0.0167)</td></tr><tr><td>SOAP</td><td>0.3385 (± 0.0024)</td><td>0.3401 (± 0.0045)</td><td>0.3547 (± 0.0077)</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>CE</td><td>0.0017 (±0.0001)</td><td>0.0021 (±0.0002)</td><td>0.0025 (±0.0004)</td></tr><tr><td>CB-CE</td><td>0.0055 (±0.0011)</td><td>0.0483 (±0.0083)</td><td>0.0121 (±0.0016)</td></tr><tr><td>Focal</td><td>0.0041 (±0.0007)</td><td>0.0281 (±0.0141)</td><td>0.0122 (±0.0001)</td></tr><tr><td rowspan=\"6\">MUV</td><td>LDAM</td><td>0.0044(±0.0022)</td><td>0.0118 (±0.0098)</td><td>0.0059 (±0.0021)</td></tr><tr><td>AUC-M</td><td>0.0026 (±0.0001)</td><td>0.0040 (±0.0012)</td><td>0.0028 (±0.0012)</td></tr><tr><td>SmoothAP</td><td>0.0073 (±0.0012)</td><td>0.0068(±0.0038)</td><td>0.0029 (±0.0005)</td></tr><tr><td>FastAP</td><td>0.0016 (±0.0000)</td><td>0.0023 (±0.0021)</td><td>0.0022 (±0.0012)</td></tr><tr><td>MinMax</td><td>0.0028 (±0.0008)</td><td>0.0027 (±0.0005)</td><td>0.0043 (±0.0015)</td></tr><tr><td>SOAP</td><td>0.0254 (±0.0261)</td><td>0.3352 (±0.0008)</td><td>0.0236 (±0.0038)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.2 Graph Classification for Molecular Property Prediction ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Data. To further demonstrate the advantages of our method, we conduct experiments on two graph classification datasets. We use the datasets HIV and MUV from the MoleculeNet [55], which is a benchmark for molecular property prediction. The HIV dataset has 41,913 molecules from the Drug Therapeutics Program (DTP), and the positive samples are molecules tested to have inhibition ability to HIV. The MUV dataset has 93,127 molecules from the PubChem library, and molecules are labelled by whether a bioassay property exists or not. Note that the MUV dataset provides labels of 17 properties in total and we only conduct experiments to predict the third property as this property is more imbalanced. The percentage of positive samples in HIV and MUV datasets are $3 . 5 1 \\%$ and $0 . 2 0 \\%$ , respectively. We use the split of train/validation/test set provided by MoleculeNet. Molecules are treated as 2D graphs in our experiments, and we use the feature extraction procedure of MoleculeKit [54] to obtain node features of graphs. The same data preprocessing is used for all of our experiments on graph data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Setup. Many recent studies have shown that graph neural networks (GNNs) are powerful models for graph data analysis [29, 17, 16]. Hence, we use three different GNNs as the backbone network for graph classification, including the message passing neural network (MPNN) [19], an invariant of graph isomorphism network [56] named by GINE [26], and the multi-level message passing neural network (ML-MPNN) proposed by Wang et al. [54]. We use the same two-stage training scheme with a similar hyper-parameter tuning. We pre-train the networks by Adam with 100 epochs and a tuned initial learning rate 0.0005, which is decayed by half after 50 epochs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Results. The achieved AUPRC on the test set by all methods are presented in Table 2. Results show that our method can outperform all baselines by a large margin in terms of AUPRC, regardless of which model structure is used. These results clearly demonstrate that our method is effective for classification problems in which the sample distribution is highly imbalanced between classes. ",
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+    {
+        "type": "text",
+        "text": "4.3 Graph Classification for Drug Discovery ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Data. In addition to molecular property prediction, we explore applying our method to drug discovery. Recent studies have shown that GNNs are effective in drug discovery through predicting the antibacterial property of chemical compounds [51]. Such application scenarios involves training a GNN model on labeled datasets and making predictions on a large library of chemical compounds so as to discover new antibiotic. However, because the positive samples in the training data, i.e., compounds known to have antibacterial property, are very rare, there exists very severe class imbalance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We show that our method can serve as a useful solution to the above problem. We conduct experiments on the MIT AICURES dataset from an open challenge (https://www.aicures.mit.edu/tasks) ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/0ac738d203c35d2c043bcf37ec4d960939969014efa2f17c8349057bdb0dad78.jpg",
+        "table_caption": [
+            "Table 3: The test AUPRC values on the MIT AICURES dataset with two graph neural networks, and on the Kaggle Melanoma dataset with two CNN models. We report the average AUPRC and standard deviation (within brackets) from 3 independent runs over 3 different train/validation/test splits. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Data</td><td colspan=\"2\">MITAICURES</td><td colspan=\"2\">Kaggle Melanoma</td></tr><tr><td>Networks</td><td>GINE</td><td>MPNN</td><td>ResNet18</td><td>ResNet34</td></tr><tr><td>CE</td><td>0.5037 (± 0.0718)</td><td>0.6282 (± 0.0634)</td><td>0.0701 (± 0.0031)</td><td>0.0582 (± 0.0016)</td></tr><tr><td>CB-CE</td><td>0.5655 (± 0.0453)</td><td>0.6308 (± 0.0263)</td><td>0.0631 (± 0.0065)</td><td>0.0721 (± 0.0054)</td></tr><tr><td>Focal</td><td>0.5143 (± 0.1062)</td><td>0.5875 (± 0.0774)</td><td>0.0549 (± 0.0083)</td><td>0.0663 (± 0.0034)</td></tr><tr><td>LDAM</td><td>0.5236 (± 0.0551)</td><td>0.6489 (± 0.0556)</td><td>0.0547 (± 0.0046)</td><td>0.0539 (± 0.0069)</td></tr><tr><td>AUC-M</td><td>0.5149 (± 0.0748)</td><td>0.5542 (± 0.0474)</td><td>0.1013 (± 0.0071)</td><td>0.0972 (± 0.0035)</td></tr><tr><td>SmothAP</td><td>0.2899 (± 0.0220)</td><td>0.4081 (± 0.0352)</td><td>0.1981 (± 0.0527)</td><td>0.2787 (± 0.0232)</td></tr><tr><td>FastAP</td><td>0.4777 (± 0.0896)</td><td>0.4518 (± 0.1495)</td><td>0.0324 (± 0.0087)</td><td>0.0359 (± 0.0062)</td></tr><tr><td>MinMax</td><td>0.5292 (± 0.0330)</td><td>0.5774 (± 0.0468)</td><td>0.0593 (± 0.0037)</td><td>0.0663 (± 0.0084)</td></tr><tr><td>SOAP</td><td>0.6639 (± 0.0515)</td><td>0.6547 (± 0.0616)</td><td>0.2624 (± 0.0410)</td><td>0.3152 (± 0.0337)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/90d2dec53afb889421a1515746678fc07644b82ce755a7edcef57f94abeeb611.jpg",
+        "image_caption": [
+            "Figure 1: Comparison of convergence of different methods in terms of test AUPRC scores on CIFAR-10, CIFAR100 and MIT AICURES data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "in drug discovery. The dataset consists of 2097 molecules. There are 48 positive samples that have antibacterial activity to Pseudomonas aeruginosa, which is the pathogen leading to secondary lungs infections of COVID-19 patients. We conduct experiments on three random train/validation/test splits at $8 0 \\% / 1 0 \\% / 1 0 \\%$ ratio, and report the average AUPRC on the test set over three splits. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Setup. Following the setup in Sec. 4.2, we use three GNNs: MPNN, GINE and ML-MPNN. We use the same two-stage training scheme with a similar hyper-parameter tuning. We pre-train GNNs by the Adam method for 100 epochs with a batch size of 64 and a tuned learning rate of 0.0005, which is decayed by half at the 50th epoch. Due to the limit of space, Table 3 only reports GINE and MPNN results. Please refer to Table 6 in the supplement for the full results of all three GNNs. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Results. The average test AUPRC from three independent runs over three splits are summarized in Table 3, Table 6. We can see that our SOAP can consistently outperform all baselines on all three GNN models. Our proposed optimization method can significantly improve the achieved AUPRC of GNN models, indicating that models tend to assign higher confidence scores to molecules with antibacterial activity. This can help identify a larger number of candidate drugs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We have employed the proposed AUPRC maximization method for improving the testing performance on MIT AICures Challenge and achieved the 1st place. For details, please refer to [54]. ",
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+        "type": "text",
+        "text": "4.4 Ablation Studies ",
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+        "text": "Effects of Imbalance Ratio. We now study the effects of imbalance ratio on the performance improvements of our method. We use two datasets Tox21 and ToxCast from the MoleculeNet [55]. The Tox21 and ToxCast contain 8014 and 8589 molecules, respectively. There are 12 property prediction tasks in Tox21, and we conduct experiments on Task 0 and Task 2. Similarly, we select Task 12 and Task 8 of ToxCast for experiments. We use the split of train/validation/test set provided by MoleculeNet. The imbalanced ratios on the training sets are $4 . 1 4 \\%$ for Task 0 of Tox21, $1 2 . 0 0 \\%$ for Task 2 of Tox21, $2 . 9 7 \\%$ for Task 12 of ToxCast, $8 . 6 7 \\%$ for Task 8 of ToxCast. ",
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+        "type": "text",
+        "text": "Following Sec. 4.2, we test three neural network models MPNN, GINE and ML-MPNN. The hyperparameters for training models are also the same as those in Sec. 4.2. We present the results of Tox21 and ToxCast in Table 5 in the supplement. Our SOAP can consistently achieve improved performance when the data is extremely imbalanced. However, it sometimes fails to do so if the imbalance ratio is not too low. Clearly, the improvements from our method are higher when the imbalance ratio of labels is lower. In other words, our method is more advantageous for data with extreme class imbalance. ",
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+        "img_path": "images/29ef093eea13be28581165e8fe9a97c3ac058f0edf548c4af3653833214d1db9.jpg",
+        "image_caption": [
+            "Figure 2: Left most: insensitivity to batch size of SOAP. Right two: consistency between AP and Surrogate Objective $\\mathbf { \\nabla } _ { - } P ( \\mathbf { w } )$ vs Iterations on CIFAR10 and CIFAR100. "
+        ],
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+        "table_caption": [
+            "Table 4: The test AUPRC over 3 independent runs by SOAP with different surrogate functions. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Data</td><td colspan=\"2\">CIFAR10</td><td colspan=\"2\">CIFAR100</td></tr><tr><td>Networks</td><td>ResNet18</td><td>ResNet34</td><td>ResNet18</td><td>ResNet34</td></tr><tr><td>Squared Hinge</td><td>0.7629 (±0.0014)</td><td>0.7012 (±0.0056)</td><td>0.6251 (±0.0053)</td><td>0.6001(±0.0060)</td></tr><tr><td>Logistic</td><td>0.7542 (±0.0024)</td><td>0.6968 (±0.0121)</td><td>0.6378 (±0.0031)</td><td>0.5923 (±0.0101)</td></tr><tr><td>Sigmoid</td><td>0.7652 (±0.0035)</td><td>0.6983 (±0.0084)</td><td>0.6271 (±0.0043)</td><td>0.5832 (±0.0054)</td></tr><tr><td>Data</td><td>HIV</td><td></td><td>MUV</td><td></td></tr><tr><td>Networks</td><td>GINE</td><td>MPNN</td><td>GINE</td><td>MPNN</td></tr><tr><td>Squared Hinge</td><td>0.3485(±0.0083)</td><td>0.3401(±0.0045)</td><td>0.0354(±0.0025)</td><td>0.3365 (±0.0008)</td></tr><tr><td>Logistic</td><td>0.3436 (±0.0043)</td><td>0.3617 (±0.0031)</td><td>0.0493 (±0.0261)</td><td>0.3352 (±0.0008)</td></tr><tr><td>Sigmoid</td><td>0.3387 (±0.0051)</td><td>0.3629 (±0.0063)</td><td>0.0298 (±0.0043)</td><td>0.3362 (±0.0009)</td></tr></table>",
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+        "type": "text",
+        "text": "Insensitivity to Batch Size. We conduct experiments on CIFAR-10 and CIFAR-100 data by varying the mini-batch size for the SOAP algorithm and report results in Figure 2 (Left most). We can see that SOAP is not sensitive to the mini-batch size. This is consistent with our theory. In contrast, many previous methods for AP maximization are sensitive to the mini-batch size [47, 48, 6]. ",
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+    {
+        "type": "text",
+        "text": "Convergence Speed. We report the convergence curves of different methods for maximizing AUPRC or AP in Figure 1 on different datasets. We can see that the proposed SOAP algorithms converge much faster than other baseline methods. ",
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+    {
+        "type": "text",
+        "text": "More Surrogate Losses. To verify the generality of SOAP, we evaluate the performance of SOAP with two more different surrogate loss functions $\\ell ( \\mathbf { w } ; \\mathbf { x } _ { s } , \\mathbf { x } _ { i } )$ as a surrogate function of the indicator I(hw(xs) ≥ hw(xi)), namely, the logistic loss, \\`(w; xs, xi) = − log 11+exp(−c(\\`(hw(xi)−hw(xs))) , and the sigmoid loss, \\`(w; xs, xi) = 11+exp(c(\\`(hw(xi)−hw(xs))) where $c$ is a hyperparameter. We tune $c \\in \\{ 1 , 2 \\}$ in our experiments. We conduct experiments on CIFAR10, CIFAR100 following the experimental setting in Section 4.1 for the image data. For the graph data, we conduct experiments on HIV, MUV data following the experimental setting in Section 4.2. We report the results in Table 4. We can observe that SOAP has similar results with different surrogate loss functions. ",
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+        "type": "text",
+        "text": "Consistency. Finally, we show the consistency between the Surrogate Objective - $\\mathbf { \\nabla } \\cdot P ( \\mathbf { w } )$ and AP by plotting the convergence curves on different datasets in Figure 2 (Right two). It is obvious two see the consistency between our surrogate objective and the true AP. ",
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+        "type": "text",
+        "text": "5 Conclusions and Outlook ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this work, we have proposed a stochastic method to optimize AUPRC that can be used in deep learning for tackling highly imbalanced data. Our approach is based on maximizing the averaged precision, and we cast the objective into a sum of coupled compositional functions. We proposed efficient adaptive and non-adaptive stochastic algorithms with provable convergence guarantee to compute the solutions. Extensive experimental results on graph and image datasets demonstrate that our proposed method can achieve promising results, especially when the class distribution is highly imbalanced. One limitation of SOAP is its convergence rate is still slow. In the future, we will consider to improve the convergence rate to address the limitation of the present work. ",
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+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We thank Bokun Wang for discussing the proofs, and thank anonymous reviewers for constructive comments. Q.Q contributed to the algorithm design, analysis, and experiments under supervision of T.Y. Y.L and Z.X contributed to the experiments under supervision of S.J. Q.Q and T.Y were partially supported by NSF Career Award #1844403, NSF Award #2110545 and NSF Award #1933212. Y.L, Z.X and S.J were partially supported by NSF IIS-1955189. ",
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+    {
+        "type": "text",
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In 7th International Conference on Learning Representations, 2019.   \n[57] Yan, Y., Xu, Y., Lin, Q., Liu, W., and Yang, T. Optimal epoch stochastic gradient descent ascent methods for min-max optimization. In Advances in Neural Information Processing Systems 33 (NeurIPS), 2020.   \n[58] Ying, Y., Wen, L., and Lyu, S. Stochastic online auc maximization. In Advances in Neural Information Processing Systems, pp. 451–459, 2016.   \n[59] Yu, Y. and Huang, L. Fast stochastic variance reduced ADMM for stochastic composition optimization. In Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI), pp. 3364–3370, 2017.   \n[60] Yuan, Z., Yan, Y., Sonka, M., and Yang, T. Robust deep auc maximization: A new surrogate loss and empirical studies on medical image classification. arXiv preprint arXiv:2012.03173, 2020.   \n[61] Yue, Y., Finley, T., Radlinski, F., and Joachims, T. A support vector method for optimizing average precision. In Proceedings of the 30th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, SIGIR ’07, pp. 271–278, New York, NY, USA, 2007. Association for Computing Machinery.   \n[62] Zhang, J. and Xiao, L. A composite randomized incremental gradient method. In Chaudhuri, K. and Salakhutdinov, R. (eds.), Proceedings of the 36th International Conference on Machine Learning (ICML), volume 97, pp. 7454–7462, 2019.   \n[63] Zhao, P., Hoi, S. C. H., Jin, R., and Yang, T. Online auc maximization. In ICML, pp. 233–240, 2011. ",
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parse/train/X3TdREzbZN/X3TdREzbZN.md

@@ -0,0 +1,391 @@
+# From Canonical Correlation Analysis to Self-supervised Graph Neural Networks
+
+Hengrui Zhang1⇤ , Qitian $\mathbf { W } \mathbf { u } ^ { 2 }$ , Junchi $\mathbf { Y a n } ^ { 2 }$ , David Wipf3, Philip S. $\mathbf { V } \mathbf { u } ^ { 1 }$
+
+1 Department of Computer Science, University of Illinois at Chicago 2 Department of Computer Science and Engineering, Shanghai Jiao Tong University 3AWS Shanghai AI Lab hzhan55@uic.edu, {echo740, yanjunchi}@sjtu.edu.cn daviwipf@amazon.com, psyu@uic.edu
+
+# Abstract
+
+We introduce a conceptually simple yet effective model for self-supervised representation learning with graph data. It follows the previous methods that generate two views of an input graph through data augmentation. However, unlike contrastive methods that focus on instance-level discrimination, we optimize an innovative feature-level objective inspired by classical Canonical Correlation Analysis. Compared with other works, our approach requires none of the parameterized mutual information estimator, additional projector, asymmetric structures, and most importantly, negative samples which can be costly. We show that the new objective essentially 1) aims at discarding augmentation-variant information by learning invariant representations, and 2) can prevent degenerated solutions by decorrelating features in different dimensions. Our theoretical analysis further provides an understanding for the new objective which can be equivalently seen as an instantiation of the Information Bottleneck Principle under the self-supervised setting. Despite its simplicity, our method performs competitively on seven public graph datasets.
+
+The code is available at: https://github.com/hengruizhang98/CCA-SSG.
+
+# 1 Introduction
+
+Self-supervised learning (SSL) has been a promising paradigm for learning useful representations without costly labels $\textcircled { 7 } , \textcircled { 4 6 } , \textcircled { 5 }$ . In general, it learns representations via a proxy objective between inputs and self-defined signals, among which contrastive methods [46, 40, 16, 5, 12] have achieved impressive performance on learning image representations by maximizing the mutual information of two views (or augmentations) of the same input. Such methods can be interpreted as a discrimination of a joint distribution (positive pairs) from the product of two marginal ones (negative pairs) $ { \mathbb { B } } 5 0  { \mathbb { I } }$ .
+
+Inspired by the success of contrastive learning in vision [17, 46, 40, 5, 16, 12, 6], similar methods have been adapted to learning graph neural networks [48, 15, 33, 57, 58]. Although these models have achieved impressive performance, they require complex designs and architectures. For example, DGI [48] and MVGRL $\dot { \mathbb { I } } \dot { \Sigma } \mathbb { I }$ rely on a parameterized mutual information estimator to discriminate positive node-graph pairs from negative ones; GRACE $\mathbb { \lVert \Xi \rVert }$ and GCA $\left[ \left[ 5 8 \right] \right]$ harness an additional MLP-projector to guarantee sufficient capacity. Moreover, negative pairs sampled or constructed from data often play an indispensable role in providing effective contrastive signals and have a large impact on performance. Selecting proper negative samples is often nontrivial for graph-structured data, not to mention the extra storage cost for prohibitively large graphs. BGRL $\textcircled { 1 3 9 } \textcircled { 1 }$ is a recent endeavor on targeting a negative-sample-free approach for GNN learning through asymmetric architectures [12, 6]. However, it requires additional components, e.g., an exponential moving average (EMA) and StopGradient, to empirically avoid degenerated solutions, leading to a more intricate architecture.
+
+Table 1: Technical comparison of self-supervised node representation learning methods. We provide a conceptual comparison with more self-supervised methods in Appendix $\boxed { \mathbf { G } }$ Target denotes the comparison pair, N/G/F denotes node/graph/feature respectively. MI-Estimator: parameterized mutual information estimator. Proj/Pred: additional (MLP) projector or predictor. Asymmetric: asymmetric architectures such as EMA and Stop-Gradient, or two separate encoders for two branches. Neg examples: requiring negative examples to prevent trivial solutions. Space denotes space requirement for storing all the pairs. Our method is simple without any listed component and memory-efficient.   
+
+<table><tr><td></td><td>Methods</td><td>Target</td><td>MI-Estimator</td><td>Proj/Pred</td><td>Asymmetric</td><td>Neg examples</td><td>Space</td></tr><tr><td></td><td>DGI 4</td><td>N-G</td><td></td><td></td><td></td><td>?</td><td>O(N)</td></tr><tr><td>[rrrereeiter</td><td>MVGRL [15]</td><td>N-G</td><td>?</td><td></td><td>√</td><td></td><td>O(N)</td></tr><tr><td></td><td>GRACE[ 国</td><td>N-N</td><td></td><td>√</td><td></td><td>√</td><td>O(N2)</td></tr><tr><td></td><td>GCA 8</td><td>N-N</td><td></td><td></td><td></td><td></td><td>O(N2)</td></tr><tr><td></td><td>BGRL ③9</td><td>N-N</td><td></td><td></td><td>√</td><td></td><td>O(N)</td></tr><tr><td></td><td>CCA-SSG (Ours)</td><td>F-F</td><td></td><td>-</td><td>-</td><td>=</td><td>0(D²)</td></tr></table>
+
+Deviating from the large body of previous works on contrastive learning, in this paper we take a new perspective to address SSL on graphs. We introduce Canonical Correlation Analysis inspired Self-Supervised Learning on Graphs (CCA-SSG), a simple yet effective approach that opens the way to a new SSL objective and frees the model from intricate designs. It follows the common practice of prior arts, generating two views of an input graph through random augmentation and acquiring node representations through a shared GNN encoder. Differently, we propose to harness a non-contrastive and non-discriminative feature-level objective, which is inspired by the well-studied Canonical Correlation Analysis (CCA) methods [18, 10, 11, 14, 2, 4]. More specifically, the new objective aims at maximizing the correlation between two augmented views of the same input and meanwhile decorrelating different (feature) dimensions of a single view’s representation. We show that the objective 1) essentially pursuits discarding augmentation-variant information and preserving augmentation-invariant information, and 2) can prevent dimensional collapse $\mathbb { \ m }$ (i.e., different dimensions capture the same information) in nature. Furthermore, our theoretical analysis sheds more lights that under mild assumptions, our model is an instantiation of Information Bottleneck Principle [43, 44, 37] under SSL settings [53, 9, 45].
+
+To sum up, as shown in Table $^ { 1 , }$ our new objective induces a simple and light model without reliance on negative pairs [48, 15, 57, 58], a parameterized mutual information estimator [48, 15], an additional projector or predictor [57, 58, 39] or asymmetric architectures $\mathbb { \lVert 3 9 , \rVert 5 \rVert }$ . We provide a thorough evaluation for the model on seven node classification benchmarks. The empirical results demonstrate that despite its simplicity, CCA-SSG can achieve very competitive performance in general and even superior test accuracy in five datasets. It is worth noting that our approach is agnostic to the input data format, which means that it can potentially be applied to other scenarios beyond graph-structured data (such as vision, language, etc.). We leave such a technical extension for future works.
+
+# Our contributions are as follows:
+
+1) We introduce a non-contrastive and non-discriminative objective for self-supervised learning, which is inspired by Canonical Correlation Analysis methods. It does not rely on negative samples, and can naturally remove the complicated components. Based on it we propose CCA-SSG, a simple yet effective framework for learning node representations without supervision (see Section 3).
+
+2) We theoretically prove that the proposed objective aims at keeping augmentation-invariant information while discarding augmentation-variant one, and possesses an inherent relationship to an embodiment of Information Bottleneck Principle under self-supervised settings (see Section 4).
+
+3) Experimental results show that without complex designs, our method outperforms state-of-the-art self-supervised methods MVGRL $\mathbb { \lVert 1 5 \rVert }$ and GCA $\left[ \left[ 5 8 \right] \right]$ on 5 out of 7 benchmarks. We also provide thorough ablation studies on the effectiveness of the key components of CCA-SSG (see Section $\bigtriangledown$
+
+# 2 Related Works and Background
+
+Contrastive Learning on Graphs. Contrastive methods [46, 40, 17, 16, 5, 12] have been shown to be effective for unsupervised learning in vision, which have also been adapted to graphs. Inspired by the local-global mutual information maximization viewpoints [17], DGI $\boxed { \boxed { 4 8 } } \vert$ and InfoGraph $\mathbf { \widehat { \left[ \left. 3 8 \right. \right] } }$ put forward unsupervised schemes for node and graph representation learning, respectively. MVGRL [15] generalizes CMC $\textcircled { | 4 0 | }$ to graph-structured data by introducing graph diffusion $\mathbb { \left[ \left[ 2 3 \right] \right] }$ to create another view for a graph. GCC $\pmb { \mathbb { B 3 } } ] |$ adopts InfoNCE loss $\lVert \overline { { 4 6 } } \rVert$ and MoCo-based negative pool $\mathbb { \left[ \left[ 1 6 \right] \right] }$ for largescale GNN pretraining. GRACE [57], GCA $\pmb { \Vert 5 8 \Vert }$ and GraphCL [52] follow the spirit of SimCLR [5] and learn node/graph representations by directly treating other nodes/graphs as negative samples. BGRL $\textcircled { \ 3 9 } \textcircled { }$ targets a negative-sample-free model, inspired by BYOL [12], on node representation learning. But it still requires complex asymmetric architectures.
+
+Feature-level Self-supervised Objectives. The above-mentioned methods all focus on instancelevel contrastive learning. To address their drawbacks, some recent works have been turning to feature-level objectives. For example, Contrastive Clustering $\mathbb { \left. 2 5 \right. }$ regards different feature dimensions as different clusters, thus combining the cluster-level discrimination with instance-level discrimination. W-MSE $\pmb { \mathbb { B } } ] \mathbf { l }$ performs a differentiable whitening operation on learned embeddings, which implicitly scatters data points in embedding space. Barlow Twins $\mathbb { \left[ \left. 5 3 \right] \right. }$ borrows the idea of redundancy reduction and adopts a soft decorrelation term that makes the cross-correlation matrix of two views’ representations close to an identity matrix. By contrast, our method is based on the classical Canonical Correlation Analysis, working by correlating the representations of two views from data augmentation and meanwhile decorrelating different feature dimensions of each view’s representation.
+
+Canonical Correlation Analysis. CCA is a classical multivariate analysis method, which is first introduced in $\mathbb { \lVert \rVert }$ . For two random variables $X \in \mathbb { R } ^ { m }$ and $Y \in \mathbb { R } ^ { n }$ , their covariance matrix is $\Sigma _ { X Y } = C o v ( X , Y )$ . CCA aims at seeking two vectors $a \in \mathbb { R } ^ { m }$ and $b \in \mathbb { R } ^ { n }$ such that the correlation $\begin{array} { r } { \rho = \operatorname { c o r r } ( a ^ { \top } X , b ^ { \top } Y ) = \frac { a ^ { \top } \Sigma _ { X Y } b } { \sqrt { a ^ { \top } \Sigma _ { X X } a } \sqrt { b ^ { \top } \Sigma _ { Y Y } b } } } \end{array}$ is maximized. Formally, the objective is
+
+$$
+\operatorname* { m a x } _ { a , b } a ^ { \top } \Sigma _ { X Y } b , \ \mathrm { s . t . } \ a ^ { \top } \Sigma _ { X X } a = b ^ { \top } \Sigma _ { Y Y } b = 1 .
+$$
+
+For multi-dimensional cases, CCA seeks two sets of vectors maximizing their correlation and subjected to the constraint that they are uncorrelated with each other $\mathbb { \ m }$ . Later studies apply CCA to multi-view learning with deep models $\scriptstyle \left\| 2 \right\| , \scriptstyle \left\| 1 \right\| , \scriptstyle \left\| 4 \right\|$ , by replacing the linear transformation with neural networks. Concretely, assuming $X _ { 1 } , X _ { 2 }$ as two views of an input data, it optimizes
+
+$$
+\operatorname* { m a x } _ { \theta _ { 1 } , \theta _ { 2 } } { \mathrm { T r } } \left( P _ { \theta _ { 1 } } ^ { \top } ( X _ { 1 } ) P _ { \theta _ { 2 } } ( X _ { 2 } ) \right) { \mathrm { ~ s . t . ~ } } P _ { \theta _ { 1 } } ^ { \top } ( X _ { 1 } ) P _ { \theta _ { 1 } } ( X _ { 1 } ) = P _ { \theta _ { 2 } } ^ { \top } ( X _ { 2 } ) P _ { \theta _ { 2 } } ( X _ { 2 } ) = I .
+$$
+
+where $P _ { \theta _ { 1 } }$ and $P _ { \theta _ { 2 } }$ are two feedforward neural networks and $I$ is an identity matrix. Despite its preciseness, such computation is really expensive $\mathbb { H }$ . Fortunately, soft CCA $\boxed { \boxed { 4 } }$ removes the hard decorrelation constraint by adopting the following Lagrangian relaxation:
+
+$$
+\operatorname* { m i n } _ { \theta _ { 1 } , \theta _ { 2 } } \mathcal { L } _ { d i s t } \left( P _ { \theta _ { 1 } } ( X _ { 1 } ) , P _ { \theta _ { 2 } } ( X _ { 2 } ) \right) + \lambda \left( \mathcal { L } _ { S D L } ( P _ { \theta _ { 1 } } ( X _ { 1 } ) ) + \mathcal { L } _ { S D L } ( P _ { \theta _ { 2 } } ( X _ { 2 } ) ) \right) ,
+$$
+
+where $\mathcal { L } _ { d i s t }$ measures correlation between two views’ representations and $\mathcal { L } _ { S D L }$ (called stochastic decorrelation loss) computes an $L _ { 1 }$ distance between $P _ { \theta _ { i } } ( X _ { i } )$ and an identity matrix, for $i = 1 , 2$ .
+
+# 3 Approach
+
+# 3.1 Model Framework
+
+In this paper we focus on self-supervised node representation learning, where we consider a single graph $\mathbf { G } = ( \mathbf { X } , \mathbf { A } )$ . $\mathbf { X } \in \mathbb { R } ^ { N \times F }$ and $\mathbf { A } \in \mathbb { R } ^ { \hat { N } \times N }$ denote node features and adjacency matrix respectively. Here $N$ is the number of nodes within the graph and $F$ denotes feature dimension.
+
+Our model simply consists of three parts: 1) a random graph augmentation generator $\tau$ . 2) a GNNbased graph encoder $f _ { \theta }$ where $\theta$ denotes its parameters. 3) a novel feature-level objective function based on Canonical Correlation Analysis. Fig. $\bigstar$ is an illustration of the proposed model.
+
+![](images/c87f1f3bc8e8fb64f5fd3a7e3267721f58c1d3093627cc197f3fc2996a4dd263.jpg)  
+Figure 1: Illustration of the proposed model: given an input graph, we first generate two views through random augmentations: edge dropping and node feature masking. The two views are subsequently put into a shared GNN encoder to generate representations. The loss function is applied on the column-normalized embedding matrix of the two views. Note that this simple yet effective pipeline can also be conceptually applied for other data like vision and texts, which we leave for future works.
+
+# Algorithm 1: PyTorch-style code for CCA-SSG
+
+# f: encoder network # lambda: trade-off # D: embedding dimension # g: input graph # feat: node features
+
+# generate two views through random augmentation   
+g1, feat1 $=$ augment(g, feat)   
+$\mathtt { g 2 }$ , feat2 $=$ augment(g, feat)   
+$\mathsf { \bar { z } 1 } \ = \ \pounds ( \mathsf { g 1 }$ , feat1) # embedding of the 1st view   
+$\mathsf { z } 2 \ = \ \pounds ( \mathsf { g } 2$ , feat2) $\#$ embedding of the 2st view
+
+# batch normalization z1_norm $=$ ((z1 - z1.mean(0)) / z1.std(0))/ sqrt(N) z2_norm $=$ ((z2 - z2.mean(0)) / z2.std(0))/ sqrt(N)
+
+# covariance matrix of each view $\mathtt { c 1 \_ = }$ torch.mm(z1_norm.T(), z1_norm) $c 2 \ =$ torch.mm(z2_norm.T(), z2_norm)
+
+iden $=$ torch.eye(D) loss_inv $=$ (z1_norm - z2_norm).pow(2).sum() loss_dec_1 $=$ (c1 - iden).pow(2).sum() loss_dec_2 $=$ (c2 - iden).pow(2).sum() loss_dec $=$ loss_dec_1 $^ +$ loss_dec_2 loss $=$ loss_inv $^ +$ lambda \* loss_dec
+
+Graph augmentations. We consider the standard pipeline for random graph augmentation that has been commonly used in previous works $1 5 7 , 1 3 9 \parallel$ . To be specific, we harness two ways for augmentation: edge dropping and node feature masking. Edge dropping randomly drops a fraction of edges from the original graph, while node feature masking randomly masks a fraction of features for all the nodes. In this way, $\tau$ is composed of all the possible graph transformation operations and each $t \sim \tau$ denotes a specific graph transformation for graph $G$ . Note that we use commonly adopted augmentation methods to stay our focus on the design of objective function and conduct fair comparison with existing approaches. More complicated random augmentations $ { \mathbb { B } } 2  { \mathbb { I } } 5 8  { \mathbb { I } }$ can also be readily plugged into our model. Details for the used augmentation functions are in Appendix E.
+
+Training. In each training iteration, we first randomly sample two graph transformations $t _ { A }$ and $t _ { B }$ from $\tau$ , and then generate two views $\tilde { \mathbf { G } } _ { A } = ( \tilde { \mathbf { X } } _ { A } , \tilde { \mathbf { A } } _ { A } )$ and $\tilde { \mathbf { G } } _ { B } = ( \bar { \mathbf { X } } _ { B } , \tilde { \mathbf { A } } _ { B } )$ according to the transformations. The two views are subsequently fed into a shared GNN encoder to generate the node embeddings of the two views: $\mathbf { Z } _ { A } = f _ { \theta } ( \mathbf { \bar { X } } _ { A } , \mathbf { \bar { A } } _ { A } )$ , $\mathbf { Z } _ { B } = f _ { \theta } ( \tilde { \mathbf { X } } _ { B } , \tilde { \mathbf { A } } _ { B } )$ , where $\mathbf { Z } _ { A } , \mathbf { Z } _ { B } \in \mathbb { R } ^ { N \times D }$ and $D$ denotes embedding dimension. We further normalize the node embeddings along instance dimension so that each feature dimension has a 0-mean and $1 / \sqrt { N }$ -standard deviation distribution:
+
+$$
+\tilde { \mathbf { Z } } = \frac { \mathbf { Z } - \mu ( \mathbf { Z } ) } { \sigma ( \mathbf { Z } ) * \sqrt { N } }
+$$
+
+The normalized $\tilde { \mathbf { Z } } _ { A }$ , $\tilde { \mathbf { Z } } _ { B }$ will be used to compute a feature-level objective in Section $3 . 2 .$ To help better understand the proposed framework, we provide the PyTorch-style pseudocode for training CCA-SSG in Algorithm 1.
+
+Inference. To generate node embeddings for downstream tasks, we put the original graph ${ \bf G } =$ $( \mathbf { X } , \mathbf { A } )$ into the trained graph neural network $f _ { \theta }$ and obtain node embeddings $\mathbf { Z } = f _ { \theta } ( \mathbf { X } , \mathbf { A } )$ .
+
+# 3.2 Learning Objective
+
+Canonical Correlation Analysis has shown its great power in multi-view learning like instance recognition $\pmb { \Vert 4 \Vert }$ . However, it still remains unexplored to leverage CCA for self-supervised learning. Note that in SSL, one generates two sets of data from the same input through transformation or random data augmentation, which could be regraded as two views of the input data. This inspires us to introduce the following objective for self-supervised representation learning:
+
+$$
+\mathcal { L } = \underbrace { \left\| \tilde { \mathbf { Z } } _ { A } - \tilde { \mathbf { Z } } _ { B } \right\| _ { F } ^ { 2 } } _ { \mathrm { i n v a r i a n c e ~ t e r m } } + \lambda \underbrace { \left( \left\| \tilde { \mathbf { Z } } _ { A } ^ { \top } \tilde { \mathbf { Z } } _ { A } - \mathbf { I } \right\| _ { F } ^ { 2 } + \left\| \tilde { \mathbf { Z } } _ { B } ^ { \top } \tilde { \mathbf { Z } } _ { B } - \mathbf { I } \right\| _ { F } ^ { 2 } \right) } _ { \mathrm { d e c o r r e l a t i o n ~ t e r m } }
+$$
+
+where $\lambda$ is a non-negative hyperparameter trading off two terms. Note that minimizing the invariance term is essentially maximizing the correlation between two views as their representations are already normalized. In SSL, as the two augmented views come randomly from the same distribution, we can adopt one encoder $f _ { \theta }$ that is shared across two branches and seek for a regularization that encourages different feature dimensions to capture distinct semantics via the decorrelation term.
+
+We next provide a variance-covariance perspective to the new objective, following similar lines of reasoning in [41, 42]. Assume that input data come from a distribution $\begin{array} { r } { \pmb { x } \sim p ( \pmb { x } ) } \end{array}$ and $\pmb { s }$ is a view of $_ { \textbf { \em x } }$ through random augmentation $\pmb { s } \sim p _ { a u g } ( \cdot | \pmb { x } )$ . Denote $z _ { s }$ as the representation of $\pmb { s }$ , then minimizing the invariance term, by expectation, is to minimize the variance of the normalized representation $\tilde { z } _ { s }$ , conditioned on $_ { \textbf { \em x } }$ . Also, minimizing the decorrelation term is to push the off-diagonal elements of the covariance matrix (given by two $\tilde { z } _ { s }$ ’s) close to 0. Formally, we have
+
+$$
+\mathcal { L } _ { i n v } = \Big | \Big | \tilde { \mathbf { Z } } _ { A } - \tilde { \mathbf { Z } } _ { B } \Big | \Big | _ { F } ^ { 2 } = \sum _ { i = 1 } ^ { N } \sum _ { k = 1 } ^ { D } ( \tilde { z } _ { i , j } ^ { A } - \tilde { z } _ { i , j } ^ { B } ) ^ { 2 } \cong \mathbb { E } _ { x } \left[ \sum _ { k = 1 } ^ { D } \mathbb { V } _ { s | x } [ \tilde { z } _ { s , k } ] \right] * 2 N ,
+$$
+
+$$
+\mathcal { L } _ { d e c } = \Big \| \tilde { \mathbf { Z } } _ { S } ^ { \top } \tilde { \mathbf { Z } } _ { S } - \mathbf { I } \Big \| _ { F } ^ { 2 } = \| \mathbf { C o v } _ { s } [ \tilde { z } ] - I \| _ { F } ^ { 2 } \cong \sum _ { i \neq j } \big ( \rho _ { i , j } ^ { z _ { s } } \big ) ^ { 2 } , \mathrm { ~ f o r ~ } \tilde { \mathbf { Z } } _ { S } \in \{ \tilde { \mathbf { Z } } _ { A } , \tilde { \mathbf { Z } } _ { B } \} ,
+$$
+
+where $\rho$ is the Pearson correlation coefficient.
+
+# 3.3 Advantages over Contrastive Methods
+
+In this subsection we provide a systematic comparison with previous self-supervised methods for node representation learning, including DGI $\lVert \overline { { 4 8 } } \rVert$ , MVGRL $\mathbb { \left. \overline { { \Omega } } \right. }$ , GRACE [57], GCA [58] and BGRL $\mathbb { B } 9 ]$ , and highlight the merits of CCA-SSG. A quick overview is presented in Table 1.
+
+No reliance on negative samples. Most of previous works highly rely on negative pairs to avoid collapse or interchangeable, trivial/degenerated solutions [48, 15, 57, 58]. E.g., DGI and MVGRL generate negative examples by corrupting the graph structure severely, and GRACE/GCA treats all the other nodes within a graph as negative examples. However, for self-supervised learning on graphs, it is non-trivial to construct informative negative examples since nodes are structurally connected, and selecting negative examples in an arbitrary manner may lead to large variance for stochastic gradients and slow training convergence [51]. The recently proposed BGRL model adopts asymmetric encoder architectures for SSL on graphs without the use of negative samples. However, though BGRL could avoid collapse empirically, it still remains as an open problem concerning its theoretical guarantee for preventing trivial solutions [41]. Compared with these methods, our model does not rely on negative pairs and asymmetric encoders. The feature decorrelation term can naturally prevent trivial solutions caused by the invariance term. We discuss the collapse issue detailedly in Appendix B.
+
+No MI estimator, projector network nor asymmetric architectures. Most previous works rely on additional components besides the GNN encoder to estimate some score functions in final objectives. DGI and MVGRL require a parameterized estimator to approximate mutual information between two views, and GRACE leverages a MLP projector followed by an InfoNCE estimator. BGRL harnesses asymmetric encoder architecture which consists of EMA (Exponential Moving Average), Stop-Gradient and an additional projector. MVGRL also induces asymmetric architectures as it adopts two different GNNs for the input graph and the diffusion graph respectively. In contrast, our approach requires no additional components except a single GNN encoder.
+
+Better efficiency and scalability to large graphs. Consider a graph with $N$ nodes. DGI and MVGRL contrast node embeddings with graph embedding, which would require $O ( N )$ space cost.
+
+GRACE treats two views of the same node as positive pairs and treat views of different nodes as negative pairs, which would take $O ( N ^ { 2 } )$ space. BGRL focuses only on positive pairs, which will also take $O ( N )$ space. By contrast, our method works on feature dimension. If we embed each node into a $D$ -dimensional vector, the computation of the loss function would require $O ( D ^ { 2 } )$ space. This indicates that the memory cost does not grow consistently as the size of graph increases. As a result, our method is promising for handling large-scale graphs without prohibitively large space costs.
+
+# 4 Theoretical Insights with Connection to Information Theory
+
+In this section we provide some analysis of the proposed objective function: 1) Interpretation of the loss function with entropy and mutual information. 2) The connection between the proposed objective and the Information Bottleneck principle. 3) Why the learned representations would be informative to downstream tasks. The proofs of propositions, theorems and corollaries are in Appendix D.
+
+Notations. Denote the random variable of input data as $X$ and the downstream task as $T$ (it could be the label $Y$ if the downstream task is classification). Note that in SSL, we have no access to $T$ in training and here we introduce the notation for our analysis. Define $S$ as the self-supervised signal (i.e., an augmented view of $X$ ), and $S$ shares the same space as $X$ . Our model learns a representation for the input, denoted by $Z _ { X }$ and its views, denoted by $Z _ { S }$ . $Z _ { X } = f _ { \theta } ( X ) , Z _ { S } = f _ { \theta } ( S ) , f _ { \theta } ( . )$ is a encoder shared by the original data and its views, which is parameterized by $\theta$ . The target of representation learning is to learn a optimal encoder parameter $\theta$ . Furthermore, for random variable $A , B , C$ , we use $I ( A , B )$ to denote the mutual information between $A$ and $B$ , $I ( A , B | C )$ to denote conditional mutual information of $A$ and $B$ on a given $C$ , $H ( A )$ for the entropy, and ${ \dot { H } } ( A | B )$ for conditional entropy. The proofs of propositions, theorems and corollaries are in Appendix D.
+
+# 4.1 An Entropy and Mutual Information Interpretation of the Objective
+
+We first introduce an assumption about the distributions of $P ( Z _ { S } )$ and $P ( Z _ { S } | X )$ .
+
+Assumption 1. (Gaussian assumption of $P ( Z _ { S } | X )$ and $P ( Z _ { S } ) _ { . }$ ):
+
+$$
+P ( Z _ { S } | X ) = \mathcal { N } ( \mu _ { X } , \Sigma _ { X } ) , P ( Z _ { S } ) = \mathcal { N } ( \mu , \Sigma ) .
+$$
+
+With Assumption $^ { 1 , }$ we can arrive at the following propositions:
+
+Proposition 1. In expectation, minimizing Eq. $\textcircled{6}$ is equivalent to minimizing the entropy of $Z _ { S }$ conditioned on input $X$ , i.e.,
+
+$$
+\operatorname* { m i n } _ { \theta } \mathcal { L } _ { i n v } \cong \operatorname* { m i n } _ { \theta } H ( Z _ { S } | X ) .
+$$
+
+Proposition 2. Minimizing Eq. $( 7 )$ is equivalent to maximizing the entropy of $Z _ { S }$ , i.e.,
+
+$$
+\operatorname* { m i n } _ { \theta } \mathcal { L } _ { d e c } \cong \operatorname* { m a x } _ { \theta } H ( Z _ { S } ) .
+$$
+
+The two propositions unveil the effects of two terms in our objective. Combining two propositions, we can further interpret Eq. $\textcircled{5}$ from an information-theoretic perspective.
+
+Theorem 1. By optimizing $E q \ ( \mathbb { 5 } )$ , we maximize the mutual information between the augmented view’s embedding $Z _ { S }$ and the input data $X$ , and minimize the mutual information between $Z _ { S }$ and the view itself $S$ , conditioned on the input data $X$ . Formally we have
+
+$$
+\operatorname* { m i n } _ { \theta } \mathcal { L } \Rightarrow \operatorname* { m a x } _ { \theta } I ( Z _ { S } , X ) a n d \operatorname* { m i n } _ { \theta } I ( Z _ { S } , S | X ) .
+$$
+
+The proof is based on the facts $I ( Z _ { S } , X ) = H ( Z _ { S } ) - H ( Z _ { S } | X )$ and $I ( Z _ { S } , S | X ) = H ( Z _ { S } | X ) +$ $H ( \bar { Z _ { S } } | S ) = H ( Z _ { S } | X )$ . Theorem $^ 1 .$ indicates that our objective Eq. $( 5 )$ learns representations that maximize the information of the input data, i.e., $I ( Z _ { S } , X )$ , and meanwhile minimize the lost information during augmentation, i.e., $I ( Z _ { S } , S | X )$ .
+
+# 4.2 Connection with the Information Bottleneck Principle
+
+The analysis in Section $\boxed { 4 . 1 }$ enables us to further build a connection between our objective Eq. $\textcircled{5}$ and the well-studied Information Bottleneck Principle [43, 44, 37, 1] under SSL settings. Recall that the supervised Information Bottleneck (IB) is defined as follows:
+
+Definition 1. The supervised $I B$ aims at maximizing an Information Bottleneck Lagrangian:
+
+$$
+\mathcal { T B } _ { s u p } = I ( Y , Z _ { X } ) - \beta I ( X , Z _ { X } ) , w h e r e \beta > 0 .
+$$
+
+As we can see, $\mathcal { T } \mathcal { B } _ { s u p }$ attempts to maximize the information between the data representation $Z _ { X }$ and its corresponding label $Y$ , and concurrently minimize the information between $Z _ { X }$ and the input data $X$ (i.e., exploiting compression of $Z _ { X }$ from $X$ ). The intuition of IB principle is that $Z _ { X }$ is expected to contain only the information that is useful for predicting $Y$ .
+
+Several recent works [9, 45, 53] propose various forms of IB under self-supervised settings. The most relevant one names Self-supervised Information Bottleneck:
+
+Definition 2. (Self-supervised Information Bottleneck $I ^ { \langle \bar { 5 } 3 | \jmath \rangle }$ . The Self-supervised IB aims at maximizing the following Lagrangian:
+
+$$
+\mathcal { T B } _ { s s l } = I ( X , Z _ { S } ) - \beta I ( S , Z _ { S } ) , \ w h e r e \ \beta > 0 .
+$$
+
+Intuitively, $\boldsymbol { \mathcal { T } } \boldsymbol { B } _ { s s l }$ posits that a desirable representation is expected to be informative to augmentation invariant features, and to be a maximally compressed representation of the input.
+
+Our objective Eq. $\textcircled{5}$ is essentially an embodiment of $\boldsymbol { \mathcal { T } } \boldsymbol { B } _ { s s l }$ :
+
+Theorem 2. Assume $0 ~ < ~ \beta ~ \leq ~ 1$ , then by minimizing Eq. $( 5 )$ , the self-supervised Information Bottleneck objective is maximized, formally:
+
+$$
+\operatorname* { m i n } _ { \theta } \mathcal { L } \Rightarrow \operatorname* { m a x } _ { \theta } \mathcal { L } B _ { s s l }
+$$
+
+Theorem $\bigtriangledown$ also shows that Eq. $( 5 )$ implicitly follows the same spirit of IB principle under selfsupervised settings. As further enlightenment, we can relate Eq. $\textcircled{5}$ with the multi-view Information Bottleneck $\pmb { \mathbb { Q } } \|$ and the minimal and sufficient representations for self-supervision [45]:
+
+Corollary 1. Let $X _ { 1 } = S$ , $X _ { 2 } = X$ and assume $0 < \beta \leq 1$ , then minimizing Eq. $\textcircled{5}$ is equivalent to minimizing the Multi-view Information Bottleneck Loss in $\pmb { \mathcal { D } } \pmb { \mathcal { J } }$ :
+
+$$
+\mathcal { L } _ { M I B } = I ( Z _ { 1 } , X _ { 1 } | X _ { 2 } ) - \beta I ( X _ { 2 } , Z _ { 1 } ) , w h e r e 0 < \beta \leq 1 .
+$$
+
+Corollary 2. When the data augmentation process is reversible, minimizing Eq. $\textcircled{5}$ is equivalent to learning the Minimal and Sufficient Representations for Self-supervision in $\pm \varTheta$ :
+
+$$
+Z _ { X } ^ { s s l } = \underset { Z _ { X } } { \operatorname { a r g m a x } } I ( Z _ { X } , S ) , Z _ { X } ^ { s s l _ { m i n } } = \underset { Z _ { X } } { \operatorname { a r g m i n } } H ( Z _ { X } | S ) ~ s . t . ~ I ( Z _ { X } , S ) ~ i s ~ m a x i m i z e d .
+$$
+
+# 4.3 Influence on Downstream Tasks
+
+We have provided a principled understanding for our new objective. Next, we discuss its effect on downstream tasks $T$ . The rationality of data augmentations in SSL is rooted in a conjecture that an ideal data augmentation approach would not change the information related to its label. We formulate this hypothesis as a building block for analysis on downstream tasks [36, 9].
+
+Assumption 2. (Task-relevant information and data augmentation). All the task-relevant information is shared across the input data $X$ and its augmentations $S$ , i.e., $I ( X , T ) = I ( S , T ) = \bar { I ( X , S , T ) }$ , or equivalently, $I ( X , T | S ) = I ( S , T | X ) = 0$ .
+
+This indicates that all the task-relevant information is contained in augmentation invariant features. We proceed to derive the following theorem which reveals the efficacy of the learned representations by our objective with respect to downstream tasks.
+
+Theorem 3. (Task-relevant/irrelevant information). By optimizing Eq. $( 5 )$ , the task-relevant information $I ( Z _ { S } , T )$ is maximized, and the task-irrelevant information $H ( Z _ { S } | \overline { { { T } } } )$ is minimized. Formally,
+
+$$
+\operatorname* { m i n } _ { \theta } \mathcal { L } \Rightarrow \operatorname* { m a x } _ { \theta } I ( Z _ { S } , T ) \ : a n d \ : \operatorname* { m i n } _ { \theta } H ( Z _ { S } | T ) .
+$$
+
+Therefore, the learned representation $Z _ { S }$ is expected to contain minimal and sufficient information about downstream tasks $\overline { { [ 4 5 ] } } , \overline { { 9 } } \overline { { ] } }$ , which further illuminates the reason why the embeddings given by SSL approaches have superior performance on various downstream tasks.
+
+Table 2: Test accuracy on citation networks. The input column highlights the data used for training. $\mathbf { X }$ for node features, A for adjacency matrix, S for diffusion matrix, and $\mathbf { Y }$ for node labels).   
+
+<table><tr><td></td><td>Methods</td><td>Input</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td></tr><tr><td rowspan="4">Supervised</td><td>MLP [47 LP [56</td><td>X,Y A,Y</td><td>55.1</td><td>46.5 45.3</td><td>71.4 63.0</td></tr><tr><td>GCN [22]</td><td>X,A,Y</td><td>68.0 81.5</td><td>70.3</td><td>79.0</td></tr><tr><td>GAT 图</td><td>X,A,Y</td><td>83.0±0.7</td><td>72.5 ± 0.7</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>79.0 ± 0.3</td></tr><tr><td rowspan="7">Unsupervised</td><td>Raw Features [48]</td><td>X</td><td>47.9 ± 0.4</td><td>49.3 ± 0.2</td><td>69.1 ± 0.3</td></tr><tr><td>Linear CCA [18]</td><td>X</td><td>58.9 ± 1.5</td><td>27.5 ±1.3</td><td>75.8 ± 0.4</td></tr><tr><td>Deep Walk [32]</td><td>A</td><td>70.7 ± 0.6</td><td>51.4 ± 0.5</td><td>74.3 ± 0.9</td></tr><tr><td>GAE 21</td><td>X,A</td><td>71.5 ± 0.4</td><td>65.8 ± 0.4</td><td>72.1 ± 0.5</td></tr><tr><td>DGI 4</td><td>X,A</td><td>82.3 ± 0.6</td><td>71.8 ± 0.7</td><td>76.8 ± 0.6</td></tr><tr><td>MVGRL1 园</td><td>X,S,A</td><td>83.5 ± 0.4</td><td>73.3 ± 0.5</td><td>80.1 ± 0.7</td></tr><tr><td>GRACE² 回 CCA-SSG (Ours)</td><td>X,A X,A</td><td>81.9 ± 0.4 84.2 ± 0.4</td><td>71.2 ± 0.5 73.1 ± 0.3</td><td>80.6 ± 0.4 81.6 ± 0.4</td></tr></table>
+
+1 Results on Cora with authors’ code is inconsistent with [15]. We adopt the results with authors’ code. 2 Results are from our reproducing with authors’ code, as [57] did not use the public splits.
+
+# 5 Experiments
+
+We assess the quality of representations after self-supervised pretraining on seven node classification benchmarks: Cora, Citeseer, Pubmed, Coauthor CS, Coauthor Physics and Amazon Computer, Amazon-Photo. We adopt the public splits for Cora, Citeseer, Pubmed, and a 1:1:9 training/validation/testing splits for the other 4 datasets. Details of the datasets are in Appendix E.
+
+Evaluation protocol. We follow the linear evaluation scheme as introduced in $\lVert \rVert \bigstar \ 8 \rVert$ : i) We first train the model on all the nodes in a graph without supervision, by optimizing the objective in Eq. $\textcircled{5}$ . ii) After that, we freeze the parameters of the encoder and obtain all the nodes’ embeddings, which are subsequently fed into a linear classifier (i.e., a logistic regression model) to generate a predicted label for each node. In the second stage, only nodes in training set are used for training the classifier, and we report the classification accuracy on testing nodes.
+
+We implement the model with PyTorch. All experiments are conducted on a NVIDIA V100 GPU with 16 GB memory. We use the Adam optimizer $\mathbb { B }$ for both stages. The graph encoder $f _ { \theta }$ is specified as a standard two-layer GCN model $\pmb { \mathbb { Z } } 2 \mathbf { l }$ for all the datasets except citeseer (where we empirically find that a one-layer GCN is better). We report the mean accuracy with a standard deviation through 20 random initialization (on Coauthor CS, Coauthor Physics and Amazon Computer, Amazon-Photo, the split is also randomly generated). Detailed hyperparameter settings are in Appendix E.
+
+# 5.1 Comparison with Peer Methods
+
+We compare CCA-SSG with classical unsupervised models, Deepwalk $\mathbb { \lVert 3 2 \rVert }$ and GAE $\left[ \left[ 2 1 \right] \right]$ , and self-supervised models, DGI [48], MVGRL [15], GRACE [57] and GCA [58]. We also compare with supervised learning models, including MLP, Label Propagation (LP) [56], and supervised baselines GCN $[ [ 2 2 ] ]$ and GAT [47]3. The results of baselines are quoted from [15, 57, 58] if not specified.
+
+We report the node classification results of citation networks and other datasets in Table 2 and Table 3 respectively. As we can see, CCA-SSG outperforms both the unsupervised competitors and the fully supervised baselines on Cora and Pubmed, despite its simple architecture. On Citeseer, CCA-SSG achieves competitive results as of the most powerful baseline MVGRL. On four larger benchmarks, CCA-SSG also achieves the best performance in four datasets except Coauther-Physics. It is worth mentioning that we empirically find that on Coauthor-CS a pure 2-layer-MLP encoder is better than GNN models. This might because the graph-structured information is much less informative than the node features, presumably providing harmful signals for classification (in fact, on Coauthor-CS, linear models using merely node features can greatly outperform DeepWalk/DeepWalk+features).
+
+Table 3: Test accuracy on co-author and co-purchase networks. We report both mean accuracy and standard deviation. Results of baseline models are from [58].   
+
+<table><tr><td></td><td>Methods</td><td>Input</td><td>Computer</td><td>Photo</td><td>CS</td><td>Physics</td></tr><tr><td></td><td>Supervised GCN 22] Supervised GAT [47]</td><td>X,A,Y X,A,Y</td><td>86.51 ± 0.54 86.93 ± 0.29</td><td>92.42 ± 0.22 92.56 ± 0.35</td><td>93.03 ± 0.31 92.31 ± 0.24</td><td>95.65 ± 0.16 95.47 ± 0.15</td></tr><tr><td>prrrrssnren</td><td>Raw Features 4 Linear CCA [18] DeepWalk [32 DeepWalk + features GAE [21 DGI MVGRL [15] GRACE1 四 GCA1 [58] X,A</td><td>X X A X,A X,A X,A X,S,A X,A</td><td>73.81 ± 0.00 79.84 ± 0.53 85.68 ± 0.06 86.28 ± 0.07 85.27 ± 0.19 83.95 ± 0.47 87.52 ± 0.11 86.25 ± 0.25</td><td>78.53 ± 0.00 86.92 ± 0.72 89.44 ± 0.11 90.05 ± 0.08 91.62 ± 0.13 91.61 ± 0.22 91.74 ± 0.07 92.15 ± 0.24</td><td>90.37 ± 0.00 93.13 ± 0.18 84.61 ± 0.22 87.70 ± 0.04 90.01 ± 0.71 92.15 ± 0.63 92.11 ± 0.12</td><td>93.58 ± 0.00 95.04 ± 0.17 91.77 ± 0.15 94.90 ± 0.09 94.92 ± 0.07 94.51 ± 0.52 95.33 ± 0.03</td></tr></table>
+
+1 GCA is essentially an enhanced version of GRACE by adopting adaptive augmentations. Both GRACE and GCA would suffer from out of memory on Coauthor-Physics using a GPU wth 16GB memory. The reported results are from authors’ papers using a 32GB GPU.
+
+# 5.2 Ablation Study and Scalability Comparison
+
+Effectiveness of invariance/decorrelation terms. We alter our loss by removing the invariance/decorrelation term respectively to study the effects of each component, with results reported in Table 4. We find that only using the invariance term will lead to merely performance drop instead of completely collapsed solutions. This is because node embeddings are normalized along the instance dimension to have a zero-mean and fixed-standard deviation, and the worst solution is no worse than dimensional collapse (i.e., all the embeddings lie in an line, and our decorrelation term can help to prevent it) instead of complete collapse (i.e., all the embeddings degenerate into a single point). As expected, only optimizing the decorrelation term will lead to poor result, as the model learns nothing meaningful but disentangled representation. In Appendix $\bigtriangledown$ we discuss the relationship between complete/dimensional collapse, when the two cases happen and how to avoid them.
+
+Effect of decorrelation intensity. We study how the intensity of feature decorrelation improves/degrades the performance by increasing the trade-off hyper-parameter $\lambda$ . Fig. $2$ shows test accuracy w.r.t. different $\lambda$ ’s on Cora, Citeseer and Pubmed. The performance benefits from a proper selection of $\lambda$ $( \mathrm { f r o m 0 . 0 0 0 5 }$ to 0.001 in our experiments). When $\lambda$ is too small, the decorrelation term does not work; if it is too large, the invariance term would be neglected, leading to serious performance degrade. An interesting finding is that even when $\lambda$ is very small or even equals to 0 (w/o $\mathcal { L } _ { d e c }$ in Table $\textcircled{4}$ , the test accuracy on Citeseer does not degrade as much as that on Cora and Citeseer. The reason is that node embeddings of Citeseer is already highly uncorrelated even without the decorrelation term. Appendix F visualizes the correlation matrices without/with decorrelations.
+
+Effect of embedding dimension. Fig. 3 shows the effect of the embedding dimension. Similar to contrastive methods [48, 15, 57, 58], CCA-SSG benefits from a large embedding dimension (compared with supervised learning), while the optimal embedding dimension of CCA-SSG (512 on most benchmarks) is a bit larger than other methods (usually 128 or 256). Yet, we notice a performance drop as the embedding dimension increases. We conjecture that the CCA is essentially a dimension-reduction method, the ideal embedding dimension ought to be smaller than the dimension of input. Hence we do not apply it on well-compressed datasets (e.g. ogbn-arXiv and ogbn-product).
+
+Scalability Comparison. Table 5 compares model size, training time (till the epoch that gives the highest evaluation accuracy) and memory cost of CCA-SSG with other methods, on Cora, Pubmed and Amazon-Computers. Overall, our method has fewer parameters, shorter training time, and fewer memory cost than MVGRL, GRACE and GCA in most cases. DGI is another simple and efficient model, but it yields much poorer performance. The results show that despite its simplicity and efficiency, our method achieves even better (or competitive) performance.
+
+Table 4: Ablation study of node classification accuracy $( \% )$ on the key components of CCA-SSG.   
+
+<table><tr><td>Variants</td><td></td><td>Cora Citeseer Pubmed</td><td></td></tr><tr><td>Baseline</td><td>84.2</td><td>73.1</td><td>81.6</td></tr><tr><td>w/o Ldec</td><td>79.1</td><td>72.2</td><td>75.3</td></tr><tr><td>w/o Linv</td><td>40.1</td><td>28.9</td><td>46.5</td></tr></table>
+
+![](images/758e83b7793b61553faea1161ee6e7d69517507bd47cf969a6ee6d5f86608746.jpg)  
+Figure 2: Effect of $\lambda$ .
+
+![](images/8f0e4054a4e9b123c9ba47e1432d090e198dd6b43293a80544a51d36e37bb474.jpg)  
+Figure 3: Effect of $D$ .
+
+Table 5: Comparison of the number of parameters, training time for achieving the best performance, and the memory cost of different methods on Cora, Pubmed and Amazon-Computer. MVGRL on Pubmed and Computer requires subgraph sampling with graph size 4000. Others are full-graph.   
+
+<table><tr><td rowspan="2">Methods</td><td colspan="3">Cora (N: 2,708)</td><td colspan="3">Pubmed (N: 19,717)</td><td colspan="3">Computer (N: 13,752)</td></tr><tr><td>#Paras</td><td>Time</td><td>Mem</td><td>#Paras</td><td>Time</td><td>Mem</td><td>#Paras</td><td>Time</td><td>Mem</td></tr><tr><td>DGI</td><td>1260K</td><td>6.4s</td><td>1.4G</td><td>782K</td><td>5.9s</td><td>1.9G</td><td>919K</td><td>14.1s</td><td>1.9G</td></tr><tr><td>MVGRL</td><td>1731K</td><td>26.9s</td><td>4.6G</td><td>775K</td><td>29s</td><td>5.4G</td><td>1049K</td><td>31.5s</td><td>5.5G</td></tr><tr><td>GRACE/GCA</td><td>997K</td><td>8.3s</td><td>1.7G</td><td>520K</td><td>756s</td><td>12.6G</td><td>273K</td><td>314s</td><td>7.6G</td></tr><tr><td>CCA-SSG(Ours)</td><td>997K</td><td>3.8s</td><td>1.6G</td><td>519K</td><td>9.6s</td><td>2.7G</td><td>656K</td><td>14.8s</td><td>2.5G</td></tr></table>
+
+# 6 Conclusion and Discussions
+
+In this paper, we have introduced CCA-SSG, a conceptually simple, efficient yet effective method for self-supervised representation learning on graphs, based on the idea of Canonical Correlation Analysis. Compared with contrastive methods, our model does not require additional components except random augmentations and a GNN encoder, whose effectiveness is justified in experiments.
+
+Limitations of the work. Despite the theoretical grounds and the promising experimental justifications, our method would suffer from several limitations. 1) The objective Eq. $\bar { ( 5 ) }$ is essentially performing dimension reduction, while SSL approach usually requires a large embedding dimension. As a result, our method might not work well on datasets where input data does not have a large feature dimension. 2) Like other augmentation based methods, CCA-SSG highly relies on a high-quality, informative and especially, label-invariant augmentations. However, the augmentations used in our model might not perfectly meet these requirements, and it remains an open problem how to generate informative graph augmentations that have non-negative impacts on the downstream tasks.
+
+Potential negative societal impacts. This work explores a simple pipeline for representation learning without large amount of labeled data. However, in industry there are many career workers whose responsibility is to label or annotate data. The proposed method might reduce the need for labeling data manually, and thus makes a few individuals unemployed (especially for developing countries and remote areas). Furthermore, our model might be biased, as it tends to pay more attention to the majority and dominant features (shared information across most of the data). The minority group whose features are scare are likely to be downplayed by the algorithm.
+
+# Acknowledgments and Disclosure of Funding
+
+This work was supported in part by NSF under grants III-1763325, III-1909323, III-2106758, and SaTC-1930941. Qitian Wu and Junchi Yan were partly supported by Shanghai Municipal Science and Technology Major Project (2021SHZDZX0102). We thank Amazon Web Services for sponsoring computation resources for this work.
+
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+
+# 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 Section 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 6.   
+(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? [Yes] See Assumption 1 and Assumption 2. (b) Did you include complete proofs of all theoretical results? [Yes] See Appendix D.
+
+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] See supplemental material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix E.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Section 5.   
+(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 Section 5.
+
+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] See Appendix E.3.   
+(b) Did you mention the license of the assets? [Yes] See Appendix E.3.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] The data used for experiments are all publicly available.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] The data we used contain no personally indentifiable information nor offensive content.
+
+5. If you used crowdsourcing or conducted research with human subjects...
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+(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]

parse/train/aRTRjVPkm-/aRTRjVPkm-.md

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+# LANGUAGE MODELS ARE OPEN KNOWLEDGE GRAPHS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+This paper shows how to construct knowledge graphs (KGs) from pre-trained language models (e.g., BERT, GPT-2/3), without human supervision. Popular KGs (e.g, Wikidata, NELL) are built in either a supervised or semi-supervised manner, requiring humans to create knowledge. Recent deep language models automatically acquire knowledge from large-scale corpora via pre-training. The stored knowledge has enabled the language models to improve downstream NLP tasks, e.g., answering questions, and writing code and articles. In this paper, we propose an unsupervised method to cast the knowledge contained within language models into KGs. We show that KGs are constructed with a single forward pass of the pretrained language models (without fine-tuning) over the corpora. We demonstrate the quality of the constructed KGs by comparing to two KGs (Wikidata, TAC KBP) created by humans. Our KGs also provide open factual knowledge that is new in the existing KGs. Our code and KGs will be made publicly available.
+
+# 1 INTRODUCTION
+
+Knowledge graphs (KGs) are an important resource for both humans and machines. Factual knowledge in KGs is injected into AI applications to imitate important skills possessed by humans, e.g., reasoning and understanding. KG construction is mainly supervised, requiring humans to handwrite every fact, such as Freebase $\{ { \mathrm { B o l l a c k e r ~ e t ~ a l . } } \} [ 2 0 0 8 \} )$ and Wikidata. KGs can also be constructed in a semi-supervised way, in which a semi-automatic extractor is used to obtain the facts from web corpora (e.g., NELL (Carlson et al., 2010) and Knowledge Vault $\textcircled { \mathrm { D o n g e t a l . } } \textcircled { 2 0 1 4 } )$ . Humans however still need to interact with the extractor to improve the quality of the discovered facts. Therefore, human supervision, which is often expensive, is required in constructing KGs.
+
+Recent progress in language models (LMs), such as BERT (Devlin et al., 2018) and GPT-2/3 (Radford et al., 2019; Brown et al., 2020), has led to superior results even outperforming humans in a wide range of tasks, e.g., sentence classification $\mathrm { ( \overline { { W a n g ~ e t ~ a l . } } ) } \mathrm { \textmu { 2 0 1 8 } } \mathrm { ) }$ question answering $\mathbf { \textregistered } \mathbf { \textmu } \mathbf { B r o w n } \mathbf { \Psi }$ et al., 2020). Pre-trained LMs are also capable to write poetry, music, and code, while such tasks often require we human to spend a significant amount of time in learning the relevant knowledge to work well. In fact, these pre-trained LMs automatically acquire factual knowledge from large-scale corpora (e.g., BookCorpus $\mathbb { ( Z h u e t a l . } ) \index { [ 2 0 1 5 ] }$ , Common Crawl $\mathrm { ( B r o w n e t a l . ] } \mathrm { \Omega } \mathrm { \stackrel { \sim } { \Sigma } } \mathrm { ) }$ via pre-training. The learned knowledge in pre-trained LMs is the key to the current success. We therefore consider the following question: instead of using the manually created knowledge, can we use the knowledge stored in pre-trained LMs to construct KGs?
+
+In this paper, we design an unsupervised approach called MAMA that successfully recovers the factual knowledge stored in LMs to build KGs from scratch. MAMA constructs a KG with a single forward pass of a pre-trained LM (without fine-tuning) over a textual corpus. As illustrated in Figure 1, MAMA has two stages: Match and Map. Match stage generates a set of candidate facts by matching the facts in the textual corpus with the knowledge in the pre-trained LM. General or world knowledge from large-scale corpora is embedded in the LM, thus candidate facts in the target corpus are often covered by the knowledge in the LM. The candidate facts are matched through an efficient beam search in the attention weight matrices of the pre-trained LM without fine-tuning. Map stage produces an open KG via mapping the matched candidate facts from Match stage to both fixed KG schema and open schema. If the schema of candidate facts exists in the KG schema, we map the candidate facts directly to the fixed KG schema. Otherwise, we reserve the unmapped candidate facts in the open schema. This results in a new type of KG, open KG, with a mixture of mapped facts in fixed KG schema and unmapped facts in the open schema.
+
+![](images/79c036ca55f633d2c72d206e9ec566aaf045e40b3f6cb0f2ca3275c4da84dac8.jpg)  
+Figure 1: Overview of the proposed approach MAMA. MAMA constructs an open knowledge graph (KG) with a single forward pass of the pre-trained language model (LM) (without fine-tuning) over the corpus. Given the input: a textual corpus containing passages and sentences, e.g., English Wikipedia, and a pre-trained LM, e.g., BERT, GPT-2/3, MAMA (1) generates a set of candidate facts via matching the knowledge in the pretrained LM with facts in the textual corpus, e.g., a candidate fact (Dylan, is, songwriter) from the sentence “Dylan is a songwriter.”, and (2) produces an open KG by mapping the matched candidate facts to both an existing KG schema, e.g., (Bob Dylan.Q392, occupation.P106, Songwriter.Q753110) in Wikidata schema, and an open schema, e.g., (Bob Dylan.Q392, sign, Albert Grossman.Q708584).
+
+Our contributions are as follows:
+
+1. We show how to construct KGs from pre-trained LMs. The KGs are constructed with a single forward pass of the pre-trained LMs without fine-tuning over the textual corpora. This helps researchers explicitly understand what the language models learn, bridging the deep LM and KG communities through enhanced model transparency.   
+2. We propose an unsupervised two-stage approach, MAMA, to first match the candidate facts in the corpora with the knowledge stored in LMs, then map the matched candidate facts to both fixed and open schema to produce a KG.   
+3. We generate a new type of KG, namely open KG, consists of mapped facts in the fixed KG schema of existing KGs (Wikidata and TAC KBP) annotated by humans; and unmapped facts in the open schema that are new in the reference KG schema. The reach of this result is broad and has downstream utility for knowledge graph construction, deep neural network interpretation, and information extraction.
+
+# 2 MAMA
+
+We introduce an unsupervised end-to-end approach Match and Map (MAMA) as illustrated in Figure 1 to construct open knowledge graphs (KGs) from language models (LMs). MAMA constructs the KGs with a single forward pass of the pre-trained LMs (without fine-tuning) over the corpora. The two stages of MAMA are:
+
+Match generates a set of candidate facts from a textual corpus. LMs contain global or world knowledge learned from large-scale corpora, which often does not perfectly match the knowledge in the target corpus. The goal of this stage is to match the knowledge stored in pre-trained LMs with facts in the corpus. Each fact is represented as a triplet (head, relation, tail) 1, in short, $( h , r , t )$ , and passed to Map stage. Match procedure is detailed in Sec. 2.1.
+
+Map produces an open KG using the matched candidate facts from Match stage. The constructed open KG has two portions: (a) mapped candidate facts that are in a fixed KG schema, e.g., (Dylan, is, songwriter) is mapped to (Bob Dylan.Q392, occupation.P106, Songwriter.Q753110) according to Wikidata schema; and (b) unmapped candidate facts that are in an open schema, e.g., a candidate fact (Dylan, signed, Albert Grossman) is partially mapped to (Bob Dylan.Q392, sign, Albert Grossman.Q708584) in the open schema. This stage is described in Sec. 2.2.
+
+![](images/55e81ba43a4a41f965e395f6b44d541eb09dc6882eef0fecd1237e966ad4ce2e.jpg)
+
+(b) Attention matrix for matching degree calculation.   
+
+<table><tr><td rowspan=1 colspan=1>Step</td><td rowspan=1 colspan=1>Action</td><td rowspan=1 colspan=1>Intermediatecandidates</td><td rowspan=1 colspan=1>Matchingdegrees</td></tr><tr><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>START</td><td rowspan=1 colspan=1>(Dylan,</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>YIELD</td><td rowspan=1 colspan=1>(Dylan, is</td><td rowspan=1 colspan=1>0.3</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>YIELD</td><td rowspan=1 colspan=1>(Dylan, is songwriter</td><td rowspan=1 colspan=1>0.7</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>STOP</td><td rowspan=1 colspan=1>(Dylan,is,songwriter)</td><td rowspan=1 colspan=1>0.7</td></tr></table>
+
+<table><tr><td colspan="6">Key:</td></tr><tr><td>Query:</td><td>Dylan</td><td>is</td><td>a</td><td>songwrite</td><td></td></tr><tr><td>Dylan</td><td>X</td><td>×</td><td>×</td><td>×</td><td></td></tr><tr><td>is</td><td>0.3</td><td>X</td><td>X</td><td>X</td><td></td></tr><tr><td>a</td><td>0.1</td><td>0.2</td><td>X</td><td>X</td><td></td></tr><tr><td>songwriter</td><td>0.1</td><td>0.4</td><td>0.2</td><td>×</td><td></td></tr></table>
+
+(a) Matching example.
+
+Figure 2: Illustration of Match stage. The upper part of (a) represents the general matching steps of generating the best matched candidate fact (Dylan, is, songwriter) from the sentence “Dylan is a songwriter.” The lower portion shows the corresponding step-by-step process. Given a head-tail pair (Dylan, songwriter), at each step, the search chooses one of the actions, i.e., START, YIELD, STOP to produce an intermediate candidate fact. The search starts by adding the head “Dylan” as an initial candidate (step 0). The matching degree of the candidate is initialized as 0. Next, a new candidate is yielded if the candidate has not reached the tail “songwriter” (step 1 and step 2), by appending the next largest attended token (with the largest score from the attention matrix (b) of the sentence) to the end of the current candidate, and the corresponding matching degrees are increased by the associated attention scores (0.3 and 0.4) to 0.3 $\left( 0 { + } 0 . 3 \right)$ and 0.7 $( 0 . 3 \substack { + 0 . 4 } )$ respectively. Otherwise, the search stops, and the candidate fact with the best matching degree is returned for the head-tail pair (step 3). The attention matrix (b) is from the forward pass of the LM without fine-tuning over the sentence. “x” marks the tokens to prevent searching backward.
+
+# 2.1 MATCH
+
+We frame the matching procedure as a search problem. To obtain the best matched candidate facts of an input sentence, the candidates with the top matching degrees are returned from a search process. The matching degree is derived from the search in the attention weight matrices of the pre-trained LM, since the attention weight matrices are one of the main containers of the knowledge in the pre-trained LM. The attention weight matrices are simply from the forward pass of the LM without fine-tuning over the sentence.
+
+# 2.1.1 BEAM SEARCH
+
+We design a simple-yet-effective beam search to find the best matched candidate facts. For every head-tail pair $( h , t )$ in a sentence, the search maintains the $k$ -best matched candidate facts of the pair. Let’s first consider the search from left to right with beam size equals to 1. An example search process is shown in Figure $\bigtriangledown$ Given a head-tail pair (Dylan, songwriter), at each step, the search performs one of the following actions:
+
+START the search from the head. The head $h$ is added as an initial candidate into the beam. For simplicity, we use ${ \mathrm { S T A R T } } ( h )$ to denote the action, which returns a candidate ( $h .$ . In Figure $\mathbf { \bar { \rho } } _ { 2 ( \mathbf { a } ) }$ , at step 0, the head “Dylan” is added as (Dylan, into the beam. The matching degree is initialized to 0.
+
+YIELD a new intermediate candidate in the beam if the current candidate has not reached the tail. The next largest attended token (with the largest score from the attention matrix) is appended to the end of the current candidate to yield the new candidate. The corresponding matching degrees are increased by the associated attention scores. At step 1 (orange arrow in Figure $2 ( \bar { \mathbf { a } } ) ,$ ), “is” is appended to the current candidate to yield (Dylan, is, , since “is” has the largest attention score with “Dylan” in the attention matrix. The attention score is 0.3 as highlighted in orange in Figure $2 ( \mathbf { b } )$ . The matching degree becomes 0.3 (i.e. $0 { + } 0 . 3$ ). The multi-head attention is reduced to a single head so that every two tokens of the sentence are associated with one attention weight. We experiment with different reduction
+
+# Algorithm 1 Beam search for matching candidate facts.
+
+Input: Head-tail pair $\overline { { ( h , t ) } }$ , sentence $s$ , attention matrix $\overline { { \mathbf { A } _ { \mathbf { s } } } }$ , action manager $\mathcal { O } = \{ \mathrm { S T A R T } , \mathrm { Y T E L D } , \mathrm { S T O P } \}$ ,   
+beam size $k$   
+Output: Candidate facts T(h,t)   
+1: $\mathbb { T } _ { ( h , t ) } \gets \{ \mathtt { S T A R T } ( h ) \}$ . Start by adding the head as a candidate in the beam   
+2: while $\exists c \in \mathbb { T } _ { ( h , t ) } [ \mathcal { O } ( c ) = \mathrm { Y } \mathrm { I } \mathrm { E } \mathrm { L } \mathrm { D } ] \ : \mathbf { d }$ o   
+3: $\widetilde { \mathbb { T } } _ { ( h , t ) } \gets \emptyset$ . Initialize a new beam   
+4: for each $c \in \mathbb { T } _ { ( h , t ) }$ do   
+5: if $\mathcal { O } ( c ) = \mathtt { Y I E L D }$ then   
+6: $\widetilde { \mathbb { T } } _ { ( h , t ) } \gets \widetilde { \mathbb { T } } _ { ( h , t ) } \cup \{ \mathrm { { Y I E L D } } ( c , s , \mathbf { A } _ { s } ) \}$ . Yield a new candidate if not reached the tail   
+7: else   
+8: $\widetilde { \mathbb { T } } _ { ( h , t ) } \gets \widetilde { \mathbb { T } } _ { ( h , t ) } \cup \{ \mathrm { S T O P } ( c , t ) \}$ . Stop then produce a valid fact if reached the tail   
+9: end if   
+10: end for   
+11: $\mathbb { T } _ { ( h , t ) } \gets \mathrm { T O P } ( k , \widetilde { \mathbb { T } } _ { ( h , t ) } )$ . Maintain $k$ -best candidates in the beam   
+12: end while   
+13: return T(h,t)
+
+setups in Sec. A.3. “x” marks the tokens (prior to the current token) that are not considered in the search to prevent searching backward. Step 2 similarly takes YIELD action to produce (Dylan, is songwriter, . The matching degree is now 0.7 (i.e. $0 . 3 \substack { + 0 . 4 ) }$ . We use YIELD $\left( c , s , \mathbf { A } _ { s } \right)$ to denote the action, where $c$ is a current candidate, $s$ represents the sentence, and ${ \bf A } _ { s }$ is the attention matrix from the forward pass of the pre-trained LM over $s$ which yields a new candidate.
+
+STOP the search step if the candidate has reached the tail, then add the candidate as a valid candidate fact into the beam. As beam size equals to 1, (Dylan, is, songwriter) is the only returned candidate fact for the given pair. The final matching degree of the candidate is 0.7. We denote this step using $\boldsymbol { \mathrm { S T O P } } ( \boldsymbol { c } , t )$ , which returns a valid fact.
+
+The details of the proposed beam search are in Algorithm 1. The inputs of the search algorithm are a head-tail pair $( h , t )$ , a sentence $s$ , an attention matrix $\mathbf { \bar { A } } _ { s }$ of $s$ . Both $h$ and $t$ are identified by the noun chunk in $s$ . ${ \bf A } _ { s }$ is the attention matrix associated with $s$ from the forward pass of LM without fine-tuning. The search gets started by adding the head $h$ as the initial candidate in the beam (line 1). While there are still new candidates waiting to be yielded (line 2), the search continues, and the top $k$ candidates sorted by the matching degrees are maintained (line 3-11) in the beam. In practice, we implement an action manager $\mathcal { O }$ to decide which action to take at each step. Given a candidate $c$ in the beam, $\begin{array} { r } { \mathcal { O } ( c ) = \bar { \mathrm { S T A R T } } } \end{array}$ always happens at the beginning of the search. If $c$ has not reached the tail $t$ yet, $\mathcal { O } ( c ) = \mathtt { Y I E L D }$ . Otherwise, $\mathcal { O } ( c ) = \mathtt { S T O P }$ . We convert the subwords to the corresponding full words. We also notice some facts are in reverse order in the sentence, e.g., said Jason Forcier , a vice president at battery maker A123 Systems Inc.” for facts of relation “org:top members employees”, thus enable bidirectionality by running the algorithm in both directions (left to right and right to left). The beam search is implemented by the breadth-first search, which is efficient as the time complexity is $O ( k \cdot d )$ , where $d$ is the maximum depth of the search tree.
+
+# 2.1.2 FILTER
+
+Although the basic functionality provided by beam search is sufficient for finding useful candidate facts, we have found a few constraints useful. Given a candidate fact $( h , r , t )$ from beam search result $\mathbb { T } _ { ( h , t ) }$ , it remains as a fact if satisfying all the following constraints.
+
+Constraint #1 The matching degree of $( h , r , t )$ is above a threshold. We compare the matching degrees corpus-wide to only reserve the facts that are matched better with the knowledge in LMs. For example, MAMA extracts a fact (Rolling Stone, wrote, pop song) from “Rolling Stone wrote: “No other pop song has so thoroughly challenged artistic conventions””, which is not an accurate fact based on the sentence. We observe the associated matching degree is below a proper threshold, while the matching degrees of high-quality facts from the same documents, e.g., (Dylan, is, songwriter), or confident facts from the other documents are beyond the threshold.
+
+Constraint #2 The distinct frequency of $r$ is above a threshold. To avoid facts to be over-specified, e.g., (Dylan, signed to Sam Peckinpah’s film, Pat Garrett and Billy the Kid), we require $r$ should take many distinct head-tail pairs in the corpus.
+
+Constraint #3 Relation $r$ is a contiguous sequence in the sentence. We can avoid $r$ that has no meaningful interpretation (Fader et al., 2011), e.g., (Rolling Stone, wrote challenged, conventions) from the above sentence.
+
+# 2.2 MAP
+
+The objective of Map stage is to generate an open KG. The open KG contains (a) mapped facts in a KG schema (Sec. 2.2.1), e.g., Wikidata schema, if the schema of the candidate facts is within the existing KG schema; and (b) unmapped facts from (a) in an open schema (Sec. 2.2.2)
+
+# 2.2.1 MAPPED FACTS IN KG SCHEMA
+
+The goal is to map a candidate fact $( h , r , t )$ to a fact $\left( h _ { k } , r _ { k } , t _ { k } \right)$ in the KG schema. The reason for mapping to an existing KG schema is to make use of the high-quality schema designed by experts (to avoid duplicated efforts of building from scratch) and enable evaluating the candidate facts with oracle KG facts contributed by human volunteers. We first map both entities $h , t$ to $h _ { k } , t _ { k }$ , then map the relation $r$ to $r _ { k }$ in the reference KG schema. Additional details are presented in Sec. A.1.
+
+Entity linking to KG schema We adapt an unsupervised entity linker based on a mention-to-entity dictionary $\left( \mathrm { S p i t k o v s k y ~ \& ~ C h a n g } \right) \left[ 2 0 1 2 \right)$ to link the entities for scalability consideration. Besides, contextual information is crucial to link the entities correctly, we use the word embedding of the context to disambiguate the entities, which means we only link the entities with high contextual similarities based on the word embedding. We adopt the entity linker to map $h , t$ to $h _ { k } , t _ { k }$ .
+
+Relation mapping with KG schema We largely follow the relation mapping method proposed by Angeli et al. (2015) to construct an offline relation map between KG relation and relation phrases of the candidate facts. The basic idea is that the more often linked head-tail pairs (i.e., entities are with type information from the entity linking step) co-occur between the candidate facts and KG facts, the more likely the corresponding relations are mapped to each other. In addition, we normalize each relation phrase of the candidate facts by lemmatization, and removing inflection, auxiliary verbs, adjectives, adverbs. After the relation mapping is constructed, one author manually checks whether the top 15 relation phrases are true mappings for each KG relation. We only reserve the true ones in the final relation mapping. This process takes approximately one day. Later, $r$ is mapped to $r _ { k }$ with an efficient look-up operation in the relation mapping.
+
+# 2.2.2 UNMAPPED FACTS IN OPEN SCHEMA
+
+An unmapped candidate fact $( h , r , t )$ means at least one of $h , r$ , and $t$ is not mapped to the KG schema based on the method in Sec. 2.2.1. There are two types of unmapped candidate facts:
+
+Partially unmapped facts represent at least one of $h , r .$ , and $t$ are mapped to the KG schema. It can be $h$ or $t$ mapped to $h _ { k }$ or $t _ { k }$ based on the entity linker in $\sec . { \boxed { 2 . 2 . 1 } }$ It can also be $r$ that mapped to $r _ { k }$ using the relation mapping in $\mathrm { S e c . } 2 . 2 . 1 .$ This actually results in unmapped facts that are in a mixture of the KG schema and the open schema. As the overall schema of the unmapped facts is not in the KG schema, we use open schema to denote such unmapped facts in the rest of the paper for simplicity. An example is (Dylan, signed, Albert Grossman) in Figure $^ { 1 , }$ where both head and tail are linked to Wikidata schema based on the entity linker in Sec. 2.2.1, but the relation cannot be mapped since there is no relation mapping from “signed” to a KG relation in Wikidata schema.
+
+Completely unmapped facts indicate all $h , r$ , and $t$ are not mapped to the KG schema. This means neither the entity linker nor the relation mapping is able to map $h , r .$ , and $t$ to $h _ { k }$ , $r _ { k }$ , $t _ { k }$ respectively. The resulting unmapped candidate facts stay in the open schema, e.g., a candidate fact (Jacob, was, A Registered Mennonite) stays the same in the open schema from a sentence “Jacob was a registered Mennonite in Amsterdam.”.
+
+The resulting open KG is a new type of KG that mixes the fixed KG schema with the flexible open schema, suggesting new directions for the next generation of KGs. The open KG not only contains existing knowledge in the reference KGs, but also extends the fixed KG schema with an additional open schema to improve the coverage of the KGs, that benefits the downstream KG based applications, e.g., QA and commonsense reasoning (Wang et al., 2019; Brown et al., 2020).
+
+<table><tr><td>KG</td><td>#of oracle facts</td><td>|#of documents</td></tr><tr><td>TACKBP</td><td>27,655 □</td><td>3,877,207</td></tr><tr><td>Wikidata</td><td>27,368,562</td><td>6,047,494</td></tr></table>
+
+Table 1: Dataset statistics of two knowledge graphs: TAC KBP and Wikidata. TAC KBP refers to TAC KBP Slot Filling 2013 challenge. # of oracle facts for TAC KBP is the number of oracle facts in the 2013 task. # of documents for TAC KBP is the number of the documents in the 2013 task. # of oracle facts for Wikidata is the total number of oracle facts in Wikidata. # of documents for Wikidata is the size of English Wikipedia.
+
+# 3 EXPERIMENTS
+
+How well can language models generate knowledge graphs? We experimentally explore how well can MAMA answer the question in the section. To measure the ability of LMs in generating KGs, we directly measure the quality of resulting open KGs. The open KG contains two types of facts: mapped facts in the fixed KG schema; and unmapped facts in the open schema. We first quantitatively evaluate MAMA by comparing the mapped facts to oracle KGs annotated by humans in Sec. 3.1, then conduct an in-depth analysis of the unmapped facts in Sec. 3.2.
+
+# 3.1 RESULTS ON MAPPED FACTS
+
+We first study the quality of the mapped facts. As the candidate facts have been mapped to the schema of oracle KGs, we are able to quantitively compare the candidate facts with the oracle facts in the reference KGs.
+
+# 3.1.1 DATASETS
+
+We compare the mapped facts from MAMA with the facts in two KGs:
+
+TAC KBP TAC Knowledge Base Population (KBP) Slot Filling is a task to search a document collection to fill in the tail/object entity for predefined relations (slots) for a given head/subject entity in a reference KG. We experiment with the reference KG in the 2013 challenge. We use the document collection and oracle facts of the 2013 task. The statistic of the dataset is shown in Table 1.
+
+Wikidata We use popular Wikidata as another KG. We use all the oracle facts in Wikidata. We use the English Wikipedia as the text corpus, since a large amount of facts in Wikidata is from English Wikipedia. The statistic is in Table 1.
+
+To evaluate the mapped facts, we first use Match stage of MAMA to run over the corresponding documents to generate the candidate facts. Then Map stage is leveraged to map the candidate facts to the schema of TAC KBP and Wikidata respectively. The parameter settings, such as beam size in Algorithm $\bigstar \bigstar$ are shared across TAC KBP and Wikidata based on the parameter study in Sec. A.3.
+
+# 3.1.2 TAC KBP
+
+To verify the ability to produce correct facts, we compare candidate facts from MAMA to the outputs of two open information systems, which also produce triplets in the form of $( h , r , t )$ . After collecting the triplets from the corresponding system, we use the same Map procedure with MAMA (Sec. 2.2.1) to map the triplets to the corresponding KG schema.
+
+Stanford OpenIE leverages POS tag and dependency parser, and generates self-contained clauses from long sentences to extract the triplets, which is the best open information extraction system (Angeli et al., 2015) on TAC KBP (Surdeanu, 2013).
+
+OpenIE 5.1 2 is one of the state-of-the-art open information extraction systems, which is the successor to Ollie (Schmitz et al., 2012), and it improves extractions from noun relations, numerical sentences, and conjunctive sentences depending on the linguistic patterns.
+
+We use two families of pre-trained LMs with MAMA. We use BERTBASE and BERTLARGE from Devlin et al. (2018) with MAMA, namely MAMA-BERTBASE and MAMA-BERTLARGE. Besides, GPT-2s from Radford et al. (2019) are used, i.e., MAMA-GPT-2, MAMA-GPT-2MEDIUM, MAMA-GPT-2LARGE, and MAMA-GPT- $2 _ { \mathrm { X L } }$ .
+
+<table><tr><td>Method</td><td>#ParamsofLM</td><td>Precision%</td><td>Recall%</td><td>F1%</td></tr><tr><td>OpenIE5.1 □</td><td></td><td>56.98</td><td>14.54</td><td>23.16</td></tr><tr><td>Stanford OpenIE Angeli et al.2015)</td><td></td><td>61.55</td><td>17.35</td><td>27.07</td></tr><tr><td>MAMA-BERTBASE (OUrS)</td><td>109M</td><td>61.57</td><td>18.79</td><td>28.79</td></tr><tr><td>MAMA-BERTLARGE (Ours)</td><td>335M</td><td>61.69</td><td>18.99</td><td>29.05</td></tr><tr><td>MAMA-GPT-2 (ours)</td><td>117M</td><td>61.62</td><td>18.17</td><td>28.07</td></tr><tr><td>MAMA-GPT-2MEDIUM (ours)</td><td>345M</td><td>62.10</td><td>18.65</td><td>28.69</td></tr><tr><td>MAMA-GPT-2LARGE (ours)</td><td>774M</td><td>62.38</td><td>19.00</td><td>29.12</td></tr><tr><td>MAMA-GPT-2xL (ours)</td><td>1558M</td><td>62.69</td><td>19.47</td><td>29.72 (+2.65)</td></tr></table>
+
+Table 2: Compare the quality of mapped facts on TAC KBP. #Params of LM refers to the number of parameters of the pre-trained LM.
+
+Table 2 shows the results on TAC KBP. We use the official scorer of TAC KBP Slot Filling 2013 to evaluate precision, recall, and F1 on TAC KBP 3.
+
+MAMA constructs improved KGs compared to open IE. From the results, we find that all our methods achieve competitive precision, which is greater than $60 \%$ , given the unsupervised nature of MAMA. All the proposed methods outperform the two open IE systems. This shows that MAMA is able to produce high-quality knowledge directly from pre-trained LMs by a single forward pass without human supervision. The results show the effectiveness of MAMA in generating candidate facts from Match stage, and producing high-quality KGs through Map stage. We also find that MAMA-GPT- $2 _ { \mathrm { X L } }$ performs the best. MAMA-GPT- $2 _ { \mathrm { X L } }$ outperforms the previous state-of-the-art Stanford OpenIE by over $2 . 6 \%$ in F1. This shows the proposed end-to-end MAMA is able to recover the knowledge stored in pre-trained LMs without relying on any extra linguistic features, such as POS tag and dependency parser used in open IE systems. The main reason leading to the moderate results of OpenIE 5.1 is that the system generates objects of the triplets with extraneous words, which hurt the performance in slot filling tasks. Even though the proposed methods all outperform the two open IE systems in the recall, however improving recall is clearly the future direction to further improve the performance of MAMA. We find that the main cause of the moderate recall is the incorrect entities caused by spaCy noun chunk as summarized in Sec. A.2.
+
+Larger/deeper LMs produce KGs of higher quality. BERTLARGE outperforms BERTBASE since the doubling parameter size. GPT-2s share similar trends, where we observe performance increases when the model size increases. This complies with our intuition on more knowledge is stored in deeper and larger models. Such increases in performance seem subtle on TAC KBP, we find this might due to the relatively small number of oracle facts by noticing a more significant improvement on Wikidata in Sec. 3.1.3. We plan to further improve the results with larger pre-trained LMs, e.g., GPT-3 (Brown et al., 2020), Megatron-LM (Shoeybi et al., 2019)
+
+BERT LMs outperform GPT-2 LMs under similar model sizes. More specifically, BERTBASE performs better than MAMA-GPT-2 in F1, and MAMA-BERTLARGE outperforms MAMA-GPT2MEDIUM in F1. BERTBASE and MAMA-GPT-2 are similar in size, while MAMA-BERTLARGE and MAMA-GPT-2MEDIUM are similar in model size as well. This is mainly because that the recall of BERT LMs is higher than that of corresponding GPT-2 LMs. The result indicates that the Cloze-style loss function (i.e., masked language model) of BERT is more effective and flexible in recovering more knowledge than the autoregressive LM objective. We also notice that the precision of GPT-2 LMs is higher than that of according BERT LMs. The reason is that the autoregressive LM objective captures more accurate knowledge than Cloze-style loss does by not introducing extra noise (e.g., masked tokens) during pre-training.
+
+# 3.1.3 WIKIDATA
+
+We select our best BERT based method MAMA-BERTLARGE, and GPT-2 based method MAMA$\mathrm { G P T } { \cdot } 2 _ { \mathrm { X L } }$ on TAC KBP to compare with Stanford OpenIE (the best open IE system on TAC KBP) for scalability experiments on Wikidata. We follow the same definition as the slot filling task to calculate precision, recall, and F1 on Wikidata. Table 3 summarizes the results.
+
+MAMA is scalable to larger corpora. Similar to the trends on TAC KBP, MAMA-GPT- $2 _ { \mathrm { X L } }$ performs the best in precision, recall, and F1. The results show the effectiveness of MAMA in generating candidate facts and high-quality KGs. We also find that MAMA-GPT- $2 _ { \mathrm { X L } }$ outperforms MAMA-BERTLARGE by over $1 \%$ in F1. This shows that the larger model $( \mathrm { G P T - } 2 _ { \mathrm { X L } }$ has $5 \mathbf { x }$ more parameters compared to BERTLARGE) contains more knowledge, and MAMA is able to restore the knowledge. When larger or deeper models (e.g., GPT-3) are used with MAMA, we can expect more gains of the KG quality. Thanks to the efficient nature of MAMA, which relies only on the forward pass of the LMs without fine-tuning, the results suggest that MAMA is scalable to large KGs.
+
+<table><tr><td>Method</td><td>#Params ofLM</td><td>Precision%</td><td>Recall%</td><td>F1%</td></tr><tr><td>Stanford OpenIE Angeli et al.2015</td><td>=</td><td>23.32</td><td>13.09</td><td>16.77</td></tr><tr><td>MAMA-BERTLARGE (ours)</td><td>335M</td><td>29.52</td><td>16.56</td><td>21.22</td></tr><tr><td>MAMA-GPT-2xL (ours)</td><td>1558M</td><td>31.32</td><td>17.42</td><td>22.39 (+5.62)</td></tr></table>
+
+Table 3: Compare the quality of mapped facts on Wikidata. #Params of LM refers to the number of parameters of the pre-trained LM.
+
+Larger corpora embed more complete KGs. In particular, MAMA-GPT- $2 _ { \mathrm { X L } }$ outperforms Stanford OpenIE by $5 . 6 \%$ in F1. MAMA-BERTLARGE outperforms Stanford OpenIE by approximately $4 . 4 \%$ in F1. Both F1 gains are larger compared to that on TAC KBP. This is because that the LMs contain world knowledge from pre-training corpora, e.g. Wikipedia and Common Crawl. The larger the textual corpora are, the more knowledge our method is able to recover and match to the knowledge stored in LMs. The finding is particularly important, since we are now able to construct larger KGs of high quality from scratch when larger datasets are used, such as WebText2 and Common Crawl (Raffel et al., 2019; Brown et al., 2020). Similar to the observations on TAC KBP, the precision is higher compared to recall. Wikidata is not fully built from Wikipedia, MAMA could improve the recall by running on those larger corpora to collect more facts.
+
+# 3.2 ANALYSIS OF UNMAPPED FACTS
+
+The open KG constructed by MAMA is a new type of KG combining the fixed KG schema with the flexible open schema. We turn to study the quality of the candidate facts that are not mapped to the above reference KG schema, but are in the open schema generated by MAMA. We manually judge such unmapped facts generated by our best method MAMA-GPT- $2 _ { \mathrm { X L } }$ from 100 sampled documents in Wikidata and TAC KBP respectively.
+
+The quality of unmapped facts is verified by human annotators. We find $3 5 . 3 \%$ of the unmapped facts are true on Wikidata. We find $8 3 . 2 \%$ of those true facts are partially unmapped facts as defined in Sec. 2.2.2, e.g., (Bob Dylan.Q392, tour with, the Grateful Dead.Q212533), whose relation is not within the schema of Wikidata, while both head and tail are in the schema. The remaining true facts are completely unmapped facts $( \mathrm { S e c . } \bigstar . \bigstar . 2 . 2 )$ e.g., a candidate fact (Jacob, was, A Registered Mennonite) stays the same in the open schema.
+
+Accurate entity detection is desired. We also notice $4 5 . 5 \%$ of the untrue unmapped facts on Wikidata are due to the incorrect entities detected by the spaCy. Incorrect or missing entity linking (to either head or tail) in Sec. 2.2.1 causes additional $9 . 1 \%$ errors in the unmapped facts. $4 . 5 \%$ o f the untrue unmapped facts are caused by the missing relation mapping in Sec. 2.2.1. The rest errors made by MAMA-GPT- $2 _ { \mathrm { X L } }$ are incorrect relation phrases, such as uninformative relation phrases, e.g., (Dylan, made, his breakthrough), which is similar to the errors made by open IE systems (Fader et al., 2011). Both entity linking and relation mapping of Map stage rely heavily on the accuracy of entity detection from the spaCy noun chunk. We conclude that the main root cause of the untrue unmapped facts is due to the errors made by the spaCy noun chunk.
+
+We observe similar trends on TAC KBP. We plan to leverage crowdsourcing platforms, e.g., Mechanical Turk, to conduct quantitative evaluations over the unmapped facts to better understand the strengths and shortage of MAMA. We plan to identify more accurate entities by relying on attention weights in LMs (Clark et al., 2019; Hewitt & Manning, 2019) instead of using extra resources. We will also investigate stronger entity linkers (Kolitsas et al., 2018) and learn a more robust relation mapping through weak or distant supervision (Mintz et al., 2009; Ratner et al., 2017). We will investigate more sophisticated approaches, such as graph neural networks (Kipf & Welling, 2016), to generate more accurate relation phrases from the attention weight matrices by considering structural information.
+
+# 4 RELATED WORK
+
+Knowledge graph construction can be generally categorized into two groups, 1) supervised approaches. Wikidata, Freebase (Bollacker et al., 2008), YAGO (Suchanek et al., 2007), YAGO2 (Hoffart et al., 2013), DBpedia (Auer et al., 2007) are built based on human supervision from Wikipedia infoboxes and other structured data sources; 2) semi-supervised approaches. Open information extraction systems, e.g., OLLIE (Schmitz et al., 2012), Reverb $\{ { \overline { { \mathrm { F a d e r ~ e t ~ a l . } } } } \} \ @ 1 1 \}$ , Stanford OpenIE (Angeli et al., 2015), and OpenIE 5.1 2 aim to leverage carefully-designed patterns based on linguistic features (e.g., dependencies and POS tags), to extract triplets from web corpora for open schema KG. Besides, NELL $\lVert \overline { { \mathbb { C } \mathrm { a r l s o n ~ e t ~ a l . } } } \rVert \overline { { 2 0 1 0 } } \rVert$ , DeepDive $\mathbb { \left( \vec { N i u \ e t \ a l . } \vec { 2 0 1 2 } \right) }$ , Knowledge Vault $\textcircled { \mathrm { D o n g ~ e t ~ a l . } } \textcircled { 2 0 1 4 }$ extract information based on a fixed schema or ontology, where humans help improve the accuracy of the extractions. Probase $\boxed { \mathrm { W u e t a l . } } \boxed { 2 0 1 2 }$ produces taxonomies instead of rich typed relations in general KGs. MAMA instead uses learned knowledge stored in pre-trained LMs without human supervision to construct an open KG, which is a mixture of fixed schema and open schema. Different from commonsense knowledge construction using Transformers (Davison et al., 2019; Bosselut et al., 2019), the proposed method is unsupervised and end-to-end, and constructs general-purpose KGs instead of commonsense knowledge.
+
+Language models, e.g., BERT (Devlin et al., 2018), GPT (Radford et al., 2018), GPT-2/3 (Radford et al., 2019; Brown et al., 2020), ELMo (Peters et al., 2018), Transformer-XL (Dai et al., 2019), ALBERT (Lan et al., 2019), RoBERTa (Liu et al., 2019), XLNet (Yang et al., 2019) and MegatronLM (Shoeybi et al., 2019) contain factual knowledge obtained via pre-training on large-scale corpora such as Wikipedia and BookCorpus (Zhu et al., 2015). Studies have leveraged the pre-trained LMs as virtual KGs, and show reasonable performance in QA tasks (Dhingra et al., 2020; Guu et al., 2020), and language modeling (Khandelwal et al., 2019). LMs are further enhanced by KGs (Peters et al., $\boxed { 2 0 1 9 }$ to improve knowledge-driven tasks. While the existing work utilizes knowledge in an implicit way, the main difference is that our approach explicitly extracts knowledgeable facts from the LMs. Compare to the joint training with knowledge base to improve shallow word embedding (Wang et al., 2014), we show that the knowledge is already stored in the deep LMs. We plan to incorporate domain knowledge into language models to construct domain-specific KGs. The main difference between LAMA $\ [ \mathrm { P e t r o n i ~ e t ~ a l . } \ ] [ 2 0 1 9 ] [ 2 0 2 0 ]$ and MAMA is mainly two-fold: (1) LAMA aims to complete Cloze-style statement, e.g., given “Dylan is a ”, LAMA predicts which words/phrases should fill the blank “ ”, which has no direct connection to KGs. There are several fundamental limitations when adapting LAMA to construct KGs, e.g., additional queries must be constructed first, and the answers for the queries must be linked to KGs. MAMA aims to solve a reasoning problem, e.g., given a passage, MAMA directly matches the fact in the form of a triplet (Dylan, is, songwriter) at the first step, then maps the fact to produce a KG. (2) The benchmark datasets used with MAMA are larger compared to the LAMA benchmark, e.g., Wikidata is 3 orders of magnitude larger compared to the largest dataset in the LAMA benchmark.
+
+Neural network interpretation here specifically refers to pre-trained deep language model analysis. There has been a lot of work to understand what the neural networks learn (Linzen et al., 2016; Adi et al., 2016; Tenney et al., 2019). With regards to analyzing Transformer (Vaswani et al., 2017) based language models (e.g., BERT and GPT-3), substantial recent work focuses on both visualizing and analyzing the attention (Vig, 2019; Jain & Wallace, 2019; Clark et al., 2019; Michel et al., 2019; Vig et al., 2020; Ramsauer et al., 2020; Hendrycks et al., 2020). Instead of analyzing or visualizing, we use LMs to generate structured KGs to directly recover what LMs learn from the corpora.
+
+# 5 CONCLUSION
+
+We show that the knowledge graphs can be constructed by a single forward pass of the language models over textual corpora. We propose a two-stage unsupervised approach MAMA to first match the facts in the corpus with the internal knowledge of the language model, and then map the matched facts to produce a knowledge graph. We demonstrate the quality of the resultant open knowledge graphs by comparing to two knowledge graphs (Wikidata and TAC KBP). The open knowledge graph also features new facts in the open schema, which could have broad implications for knowledge graphs and their downstream applications. The results also suggest that larger language models store richer knowledge than existing knowledge graphs, and generating on even larger high-quality text corpora could continue improving knowledge graphs. Additionally, the knowledge graphs generated by our approach can help researchers to look into what the language models learn, so our interpretable knowledge graphs establish a bridge between the deep learning and knowledge graph communities.
+
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+
+# A ADDITIONAL DETAILS AND ANALYSIS OF MAMA
+
+# A.1 METHOD DETAILS
+
+Map stage details To evaluate the mapped facts, we first use Match stage of MAMA to run over the corresponding documents to generate the candidate facts. For Map stage on TAC KBP, we link to the oracle annotation of the entities or spans in the TAC KBP corpus. On Wikidata, the entity linking method described in $\mathrm { S e c . } [ \underline { { 2 . 2 . 1 } } ]$ is first leveraged to link entities in the candidate facts to Wikipedia anchors. We build an enhanced mention-to-entity dictionary based on Spitkovsky & Chang (2012). In particular, we add new Wikipedia anchors to the dictionary which results in 26 million entries comparing to 21 million entries in $\mathbf { \overline { { S p i t k o v s k y \ \& \ C h a n g } } } \mathbf { \bar { \Psi } } ( \mathbf { \bar { 2 0 i 2 } } )$ . Then a Wikipedia anchor to the Wikidata item dictionary is constructed and used to further link the entities to Wikidata. If the head or tail is a pronoun, we further use neuralcoref $^ { \cdot 4 }$ for coreference resolution. We use GloVe (Pennington et al., 2014) embedding for disambiguation. The relation mapping is constructed offline for TAC KBP and Wikidata respectively using the method in Sec. 2.2.1. Besides the automatic relation mapping method proposed in Angeli et al. (2015), we manually check whether the top relation phrases are true as described in Sec. 2.2.1. For relation mapping, we randomly sampled a hold-out dataset including 2,000 documents from the TAC KBP corpus and English Wikipedia for the relation mapping construction on TAC KBP and Wikidata respectively. For oracle facts in Wikidata, we only preserve those facts describing relations between entities that could be linked to corresponding Wikipedia anchors. We rule out facts of attributes about entities and facts of auxiliary relations (such as topic’s main category.P901) and finally results in 27,368,562 oracle facts.
+
+Implementation details For Wikidata, at Match stage, we randomly split the English Wikipedia data into 20 partitions, and map the data partitions to 20 distributed servers to run. Each server is configured with four Tesla K80 12Gs. We set the max sequence length to 256, and batch size as 32 for MAMA-BERTLARGE and 4 for MAMA-GPT- $2 _ { \mathrm { X L } }$ . We use implementations of pre-trained LMs in Transformers package 5. We use spaCy sentencizer 6 to segment the documents into sentences. MAMA-BERTLARGE takes approximately 48 hours, and MAMA-GPT- $2 _ { \mathrm { X L } }$ costs around 96 hours. The resulting candidate facts of Match stage from the 20 servers are then reduced a data server, where a MongoDB database is maintained to store the oracle Wikidata and entity linking results to enable the efficient Map stage. To produce the open KGs, Map stage takes around 18 hours. The setup is similar to TAC KBP. Match stage is done within 48 hours for all the settings. The batch sizes of MAMA-BERTBASE, MAMA-GPT-2, MAMA-GPT-2MEDIUM, MAMA-GPT- $2 _ { \mathrm { L } }$ ARGE are 64, 32, 16, 8 respectively.
+
+Parameter settings The parameter settings are shared across TAC KBP and Wikidata. All the choices are based on the parameter study in Sec. A.3. The beam size of Algorithm 1 is set to 6. The matching degree threshold of Constraint #1 (Sec. 2.1.2) is set to 0.005, and the number of distinct head-tail pairs of Constraint # $2 ( \mathrm { S e c } . 2 . 1 . 2 )$ is set to 10. To generate the attention weight matrix ${ \bf A } _ { s }$ of a sentence, we reduce the weights of every attention head in the last layer of pre-trained LMs using the mean operator.
+
+# A.2 ERROR ANALYSIS
+
+There is still significant room to improve MAMA. To further understand the shortage of MAMA, we conduct an error analysis of the errors in precision (i.e., incorrect facts returned by MAMA) of Table 2 and Table 3. We choose our best method MAMA-GPT- $2 _ { \mathrm { X L } }$ for the study. We sample 100 documents from the Wikidata dataset, and manually check the reasons for the errors. We find $3 3 . 1 \%$ of the errors are caused by incorrect entities, while the relation phrases are correct. The errors are due to the incorrect noun chunk detected by the spaCy 7. $1 8 . 3 \%$ of the errors are due to the missing relation mapping created in Sec. 2.2.1. Note that we find approximately $2 3 . 8 \%$ of the errors are actually correct facts that are new in the reference KGs. e.g., (Bob Dylan.Q392, residence.P551, Nashville.Q23197) (in Figure $\boxed { 1 6 }$ is not an existing fact in Wikidata, but it is a correct mapped fact based on our annotation. The rest errors made by MAMA-GPT- $2 _ { \mathrm { X L } }$ are incorrect relation phrases, such as uninformative relation phrases. We find similar errors are made by MAMA-GPT- $2 _ { \mathrm { X L } }$ on TAC KBP. Similar to Sec. 3.2, enhancing the entity detection, entity linker, relation mapping, and relation generation are helpful. We also plan to leverage lifelong learning (Carlson et al., $\boxed { 2 0 1 0 }$ to add true facts to the reference KGs to improve the evaluation.
+
+![](images/088539de95c0dbaaa6c8507e643bcaa6a6f022db431275ac02bb4ee4b3ec6e69.jpg)  
+Figure 3: Parameter study with MAMA-BERTBASE on TAC KBP hold-out subset.
+
+# A.3 PARAMETER STUDY
+
+We study the effects of the parameters using MAMA-BERTBASE on TAC KBP. We randomly sample $20 \%$ of the oracle query entities as a hold-out dataset to tune the parameters, and use the best parameter setting achieved for both TAC KBP and Wikidata experiments. When studying the effect of a certain parameter, we keep the remaining parameters as default described in Sec. A.1. We use F1 to measure the effects.
+
+Effects of beam size Figure $3 ( \mathbf { a } )$ illustrates the effects of various beam sizes in Algorithm 1. We find that in general, the larger the beam size is, the better F1 the setting achieves. This is because that MAMA is able to reserve more potentially correct facts when more candidates are allowed in the Match stage. However, F1 improvement gradually becomes subtle, while the computation costs increase more significantly. For sake of the efficiency, we do not explore larger beam sizes. We set the beam size as 6.
+
+Effects of search constraints Figure 3(b) compares the effect of different thresholds of the matching degree of Constraint #1 in Sec. 2.1.2. We set the threshold as 0.005 since it achieves the best result. Note that the summed attention score is normalized by the length of the fact to penalize the cumbersome facts. The matching degree threshold is effective, which is mainly because of the knowledge contained in the self-attention matrix. The score in the attention matrix is representing the chance of the facts to be the true facts based on the stored knowledge. Figure $\boxed { 3 } ( \mathrm { c } )$ shows the impact of the number of distinct head-tail pairs in identifying common relations of Constraint #2 in Sec. 2.1.2. The best result is achieved when it equals 10. This shows that while MAMA mostly identifies frequent relations, it is also able to capture some rare relations for the open schema.
+
+Effects of attention weights Figure $\textcircled { 3 } ( \textcircled { \mathrm { d } } )$ shows the comparison between attention weights of the last layer and the mean of all layers. The attention weights of the last layer perform better. This is due to the attention weights in lower layers are low-level linguistic knowledge according to (Clark et al., 2019; Ramsauer et al., 2020), which are less relevant to the factual knowledge for the KG construction. Figure $\sum \limits _ { i = 1 } ^ { 3 ( \overline { { \mathbf { e } } } ) }$ compares the impact of different attention reduction, i.e., mean, max, over the attention heads of the last layer. We find the “mean” perform better. The reason is that the token often intensively attends to several specific tokens in the sequence (Michel et al., 2019), and the “mean” operator is more sensitive to such information.
+
+# B SAMPLES FROM MAMA ON TAC KBP
+
+# B.1 MAPPED FACTS
+
+We randomly sample 100 documents from TAC KBP corpus, then randomly sample sentences from those documents. The uncurated candidate facts and the corresponding mapped facts of the sampled sentences based on our best methods MAMA-BERTLARGE and MAMA-GPT- $2 _ { \mathrm { X L } }$ are shown in Figur e 4 and Figure 5 respectively. We also randomly sample several sentences in which MAMABERTLARGE differs from MAMA-GPT- $2 _ { \mathrm { X L } }$ in the resulting facts for comparison, which is illustrated in Figure $6 .$ In each table, “ID” represents the document ID of a sampled sentence in TAC KBP corpus. “Sentence” indicates the sampled sentence. “Candidate facts to mapped facts” column contains the candidate facts (on the left side of $^ { 6 6 } \to ^ { 5 9 }$ ) and their corresponding mapped facts (on the right side of $^ { 6 6 } \to ^ { 5 9 }$ ).
+
+# B.2 UNMAPPED FACTS
+
+We randomly sample 100 documents from TAC KBP corpus. From those documents, we show unmapped facts from the sampled sentences from those documents. We manually check the correctness of the unmapped facts according to $\mathrm { S e c . } \big [ 3 . 2 \big ]$ and show the correct ones. The original candidate facts with the corresponding unmapped facts of the sampled sentences generated by MAMABERTLARGE and MAMA-GPT- $2 _ { \mathrm { X L } }$ are shown in Figure 7 and Figure 8. A further comparison of the unmapped candidate facts is illustrated in Figure 9. In each table, “ID” represents the document ID of a sampled sentence in TAC KBP corpus. “Sentence” indicates the sampled sentence. “Candidate facts to unmapped facts” column contains the candidate facts (on the left side of $^ { 6 6 } \to ^ { 5 9 }$ ) and their corresponding unmapped facts (on the right side of $\ddot { \cdot } \xrightarrow { } \dot { \mathbf { \rho } }$ ).
+
+# C SAMPLES FROM MAMA ON WIKIDATA
+
+# C.1 MAPPED FACTS
+
+Similar to TAC KBP, we randomly sample 100 documents from the Wikidata corpus (i.e., English Wikipedia), then randomly sample sentences from those documents. The uncurated candidate facts and the corresponding mapped facts of the sampled sentences based on our best methods MAMABERTLARGE and MAMA-GPT- $2 _ { \mathrm { X L } }$ are shown in Figure $1 0$ and Figure $\boxed { 1 1 }$ respectively. We also randomly sample several sentences in which MAMA-BERTLARGE differs from MAMA-GPT- $2 _ { \mathrm { X L } }$ in the resulting facts for comparison, which is illustrated in Figure $\boxed { 1 2 }$ In each table, “ID” represents the Wikipedia page’s title of a sampled sentence. “Sentence” indicates the sampled sentence. “Candidate facts to mapped facts” column contains the candidate facts (on the left side of $^ { 6 6 } \to ^ { 5 9 }$ ) and their corresponding mapped facts (on the right side of $^ { 6 6 } \to ^ { 5 9 }$ ).
+
+# C.2 UNMAPPED FACTS
+
+Similar to TAC KBP, we randomly sample 100 documents from the Wikidata corpus. From those documents, we show unmapped facts from several sampled sentences from those documents. We manually check the correctness of the unmapped facts according to Sec. $\boxed { 3 . 2 }$ and show the correct ones. The original candidate facts with the corresponding unmapped facts of the sampled sentences generated by MAMA-BERTLARGE and MAMA-GPT- $2 _ { \mathrm { X L } }$ are shown in Figure 13 and Figure 14. A further comparison of the unmapped candidate facts is illustrated in Figure 15. In each table, “ID” represents the Wikipedia page’s title of a sampled sentence. “Sentence” indicates the sampled sentence. “Candidate facts to unmapped facts” column contains the candidate facts (on the left side of $^ { 6 6 } \to ^ { 5 9 }$ ) and their corresponding unmapped facts (on the right side of $^ { 6 6 } \to ^ { 5 9 }$ ).
+
+# D ADDITIONAL OPEN KG SUBGRAPHS FROM MAMA ON WIKIDATA
+
+We sample several documents from the Wikidata corpus. We visualize the mapped facts and unmapped facts from those documents as examples of subgraphs in the resulting open KGs. We show the snapshots of the subgraphs generated by MAMA-BERTLARGE from Figure 16 to Figure 24. We similarly illustrate the snapshots of the subgraphs constructed by MAMA-GPT- $2 _ { \mathrm { X L } }$ from Figure 25 to Figure $\boxed { 3 2 }$ In each figure, the blue node and arrow represent the mapped facts in the Wikidata schema, while the yellow node and arrow denote the unmapped facts in the open schema. We additionally visualize the correct facts that are new in Wikidata according to Sec. A.2 in yellow.
+
+![](images/5b5982d6e60f3925748fe91eb8efa79244d949de88fd7115168acf5f320581da.jpg)  
+Figure 4: Mapped facts: MAMA-BERTLARGE on TAC KBP.
+
+![](images/85fe39697d3ccaeddfb0bec5c3b2026d6b5ec7a264d0e1659bcde89d8ee79b02.jpg)  
+Figure 5: Mapped facts: MAMA-GPT- $2 \mathrm { x L }$ on TAC KBP.
+
+![](images/9c2044f7fc16a2192210f9394c28a9ff08184efc8053a0cbcc0c0ad7ad2a0190.jpg)  
+Figure 6: Mapped facts: MAMA-BERTLARGE vs. MAMA-GPT- $2 _ { \mathrm { X L } }$ on TAC KBP.
+
+![](images/ce101ffc28e21ac0d18e15104439f23f874fac73ffef79c58db1022f891ec2e7.jpg)
+
+![](images/9f987fe409f79d8560aa0b7601deff482a4260795c7e008af66eeab9ede6d3da.jpg)
+
+![](images/733ac3afbc874735b41766f9fe8b11ccc4abcffee4aa9b0a733be2fed370042c.jpg)
+
+![](images/c355932f0d893c0eb7985d736a24375fe256143a3aceccaa13f2bbb872cd0709.jpg)
+
+![](images/a00669933a34e9731459aa52c90f9e12e8ba0ef0e3ac3b5548a9303bcf99ddce.jpg)
+
+![](images/4c532808688c9be1f6c18c309ff4406263b8de4c66472bde77ab33bf082d9b5d.jpg)
+
+![](images/fdc4125b0cc1c11f2c15367d673fc90c74c1d7321186607dd602b42dfad40777.jpg)
+
+![](images/d0627dd72e3b25026aeaebeb03c6d1461de5c1e291131742db43f1063453adb6.jpg)  
+Figure 14: Unmapped facts: MAMA-GPT- $2 \mathrm { x L }$ on Wikidata.
+
+![](images/6e44a503389734c6971540fffffcb9ed1f99cb06c386e3874f72d6d81782c841.jpg)  
+Figure 15: Unmapped facts: MAMA-BERTLARGE vs. MAMA-GPT- $2 _ { \mathrm { X L } }$ on Wikidata.
+
+![](images/a90b829f8576129a07313b31d8786abc372e2f672a62b5ef9319043ce99a991d.jpg)  
+Figure 16: A snapshot subgraph of the open KG generated by MAMA using BERTLARGE from Wikipedia pages neighboring “Bob Dylan”. The blue node and arrow represent the mapped facts in the Wikidata schema, while the yellow node and arrow denote the unmapped facts in the open schema. We also visualize the correct facts that are new in Wikidata in yellow.
+
+![](images/9d637a85df7cc4684eb7374a42fba47e1db133850801f0bc2dcaf2cf65c15016.jpg)  
+Figure 17: A snapshot subgraph of the open KG generated by MAMA-BERTLARGE from the Wikipedia page “Douglas Bader”.
+
+![](images/5bde6ee8cc3c168c54c3d782a895155dee4d4d5018069bb544324dcd4e97786a.jpg)  
+Figure 18: A snapshot subgraph of the open KG generated by MAMA-BERTLARGE from the Wikipedia page “Helen Storrow”.
+
+![](images/e3f81488347baf7d2c283281ca51a6beae5610395a74ebf29e132ba193e6bd43.jpg)  
+Figure 19: A snapshot subgraph of the open KG generated by MAMA-BERTLARGE from the Wikipedia page “Jacob van Ruisdael”.
+
+![](images/616829635526636db0309a59dc5754c85878a4c2b1b36e01be7697859e7cc065.jpg)  
+Figure 20: A snapshot subgraph of the open KG generated by MAMA-BERTLARGE from the Wikipedia page “John Maynard Keynes”.
+
+![](images/151bcf75f55131629d25498486559ef9d5b5b3741a53d260181f1a132016161c.jpg)  
+Figure 21: A snapshot subgraph of the open KG generated by MAMA-BERTLARGE from the Wikipedia page “Liaquat Ali Khan”.
+
+![](images/f306b7214913080d4f969fdbd7735d521830b149f93b39f4e97415352dc05d84.jpg)  
+Figure 22: A snapshot subgraph of the open KG generated by MAMA-BERTLARGE from the Wikipedia page “Neville Southall”.
+
+![](images/f7a58890fbe8cae560bed832638bf962dec36d5ef9672b6d5316f13a2a7ca51d.jpg)  
+Figure 23: A snapshot subgraph of the open KG generated by MAMA-BERTLARGE from the Wikipedia page “Pauline Baynes”.
+
+![](images/dc6bb1cfebb2dcd6339e3213ed7e13ef523c1f193a1629fb3c63c36b0ae76cf9.jpg)  
+Figure 24: A snapshot subgraph of the open KG generated by MAMA-BERTLARGE from the Wikipedia page “Thor Heyerdahl’.
+
+![](images/3d12dd97bcb27d00bd75f04d25b83db84cbc6bff267a43a9a57eeeebf15bc0cf.jpg)  
+Figure 25: A snapshot subgraph of the open KG generated by MAMA-GPT- $2 \mathrm { x L }$ from the Wikipedia page “Douglas Bader”.
+
+![](images/02866b8f99cf3431803010106adb0cde6585df5bffa948f358b8e8c4ccdd889a.jpg)  
+Figure 26: A snapshot subgraph of the open KG generated by MAMA-GPT- $2 _ { \mathrm { X L } }$ from the Wikipedia page “Helen Storrow”.
+
+![](images/88035bcf9f020f008117c84b0ad413435ca9cfef13e2a847fc41b520c04827ed.jpg)  
+Figure 27: A snapshot subgraph of the open KG generated by MAMA-GPT- $2 _ { \mathrm { X L } }$ from the Wikipedia page “Jacob van Ruisdael”.
+
+![](images/11f100386b5bfb6389d63b5f325207bf9ad06048f5b2269dc4099e9f6c278a47.jpg)  
+Figure 28: A snapshot subgraph of the open KG generated by MAMA-GPT- $2 _ { \mathrm { X L } }$ from the Wikipedia page “John Maynard Keynes”.
+
+![](images/44092c52ef0629e810873e32c5ee51994a632ed0571fc76c4bcf97b40fcc3dde.jpg)  
+Figure 29: A snapshot subgraph of the open KG generated by MAMA-GPT- $2 _ { \mathrm { X L } }$ from the Wikipedia page “Liaquat Ali Khan”.
+
+![](images/b0ffebab67594db320eeb03af7f87c97c60140ce127cca946b3770b7bce7dcde.jpg)  
+Figure 30: A snapshot subgraph of the open KG generated by MAMA-GPT- $2 \mathrm { x L }$ from the Wikipedia page “Neville Southall”.
+
+![](images/27fbf314d533e9c3c8904bc6ba7d61b8354818a171e4d88846dbc10ed748bd6a.jpg)  
+Figure 31: A snapshot subgraph of the open KG generated by MAMA-GPT- $2 _ { \mathrm { X L } }$ from the Wikipedia page “Pauline Baynes”.
+
+![](images/3a3bc81504fb50ffbfdfddc8327259352d30720e3a8beab57f723a9aefa77d56.jpg)  
+Figure 32: A snapshot subgraph of the open KG generated by MAMA-GPT- $2 \mathrm { x L }$ from the Wikipedia page “Thor Heyerdahl”.
+
+![](images/a50d4d9f2dd6580d02e7c7e984c360a0e1fd3dd400604725a1321b6ceba34862.jpg)
+
+![](images/8b281dcccbe7adf85c8d75a7e97e4e409d50ebb1f1dc84782dfb70c0c97dab2c.jpg)
+
+#
+
+![](images/a04cf1229272a8aba8373755878dcf2fd9b0cae637f46d6c4c564308ba4c8caf.jpg)
+
+![](images/3351ce316d9c427b91fc7952389e1cc34debd36d8e0ab4a687569877cb3354d0.jpg)
+
+#
+
+![](images/ed3402732e79fd1a156f1191f48a4a74e8ee753c9348cf660b521a48dea6d182.jpg)
+
+![](images/3ac88b3691d5668cd2ef106332218cb6c0480568411775a2fcac190e7b7ba9b9.jpg)
+
+#
+
+#
+
+![](images/7fdee4b44974a076ed6a6f55717800258cbe4a27cdc3d10ef22062377037a5ce.jpg)
+
+![](images/6c8f14d1ad721d2a183f8d2b1ef1b2bde2650119b3e7d1a234acdd4f8cccaca7.jpg)
+
+#
+
+#
+
+<table><tr><td>Duluth.Q485708 the_Grateful_Dead.Q212533 Polar_Music_Prize.Q754841 ： award_received.P166 award_received.P166 Sara_Lownds.Q457433</td><td rowspan="2">spouse.P26 vork_location.P937 residence.P551 United_States_of_America.Q30 country_of_citizenship.P27</td><td rowspan="2">residence.P551 tour_with country_of_citizenship.P27</td><td rowspan="2">Harmonica.Q51290 instrument.P1303 BobDy an.Q392</td><td rowspan="2">tour_with</td><td rowspan="2">and_the_Heartbreakers.Q2117272 member_of.P463</td><td rowspan="2">genre.P136 record_label.P264 genre.</td><td rowspan="2">Rock_music.Q11399 Columbia_ tour_with _Records.Q183387 e.P136</td><td rowspan="2">genre.P136 the_Band.Q600344 Urban_contemporary_gospel.Q5967604 .</td></tr><tr><td>Jnited_Against_Apartheid.Q126977</td></tr><tr><td rowspan="5">New_York_City.Q60 Minneapolis.Q36091 Features new knowledge not in</td><td rowspan="2"></td><td rowspan="2">Grammy_Lifetime_Achievement_Award.Q935843</td><td rowspan="2">award_received.P166</td><td rowspan="2">genre.P136 unmarried_partner.P451</td><td>award_received.P166 Folk_music.Q235858</td><td>occupation.P106 residence.P551</td><td>songwriter.Q753110 Nashville.Q23197</td></tr><tr><td>O</td><td>/Kennedy_Center_Honors.414306</td><td>O (in blue)</td></tr><tr><td colspan="2" rowspan="3"> existing KGs (in yellow)</td><td rowspan="3">award_received.P166/ genre.P136</td><td rowspan="3">Joan_Baez.Q131725_award_received.P166 American_Folk_Music.Q1541229 ：</td><td>singer-songwriter.Q488205</td><td></td></tr><tr><td>occupation.P106 record_label.P264</td><td rowspan="2">Proper_Records.Q7250169</td><td rowspan="2">Contains knowledge in existing KGs</td></tr><tr><td>Thomas_Mertoh_Award.Q337620</td></tr></table>
+
+#

parse/train/r-gPPHEjpmw/r-gPPHEjpmw.md

@@ -0,0 +1,453 @@
+# HIERARCHICAL REINFORCEMENT LEARNING BY DISCOVERING INTRINSIC OPTIONS
+
+Jesse Zhang∗ † 1, Haonan $\mathbf { V } \mathbf { u } ^ { * 2 }$ , Wei $\mathbf { X } \mathbf { u } ^ { 2 }$ 1University of Southern California, 2Horizon Robotics
+
+# ABSTRACT
+
+We propose a hierarchical reinforcement learning method, HIDIO, that can learn task-agnostic options in a self-supervised manner while jointly learning to utilize them to solve sparse-reward tasks. Unlike current hierarchical RL approaches that tend to formulate goal-reaching low-level tasks or pre-define ad hoc lowerlevel policies, HIDIO encourages lower-level option learning that is independent of the task at hand, requiring few assumptions or little knowledge about the task structure. These options are learned through an intrinsic entropy minimization objective conditioned on the option sub-trajectories. The learned options are diverse and task-agnostic. In experiments on sparse-reward robotic manipulation and navigation tasks, HIDIO achieves higher success rates with greater sample efficiency than regular RL baselines and two state-of-the-art hierarchical RL methods. Code available at https://www.github.com/jesbu1/hidio.
+
+# 1 INTRODUCTION
+
+Imagine a wheeled robot learning to kick a soccer ball into a goal with sparse reward supervision. In order to succeed, it must discover how to first navigate in its environment, then touch the ball, and finally kick it into the goal, only receiving a positive reward at the end for completing the task. This is a naturally difficult problem for traditional reinforcement learning (RL) to solve, unless the task has been manually decomposed into temporally extended stages where each stage constitutes a much easier subtask. In this paper we ask, how do we learn to decompose the task automatically and utilize the decomposition to solve sparse reward problems?
+
+Deep RL has made great strides solving a variety of tasks recently, with hierarchical RL (hRL) demonstrating promise in solving such sparse reward tasks (Sharma et al., 2019b; Le et al., 2018; Merel et al., 2019; Ranchod et al., 2015). In hRL, the task is decomposed into a hierarchy of subtasks, where policies at the top of the hierarchy call upon policies below to perform actions to solve their respective subtasks. This abstracts away actions for the policies at the top levels of the hierarchy. hRL makes exploration easier by potentially reducing the number of steps the agent needs to take to explore its state space. Moreover, at higher levels of the hierarchy, temporal abstraction results in more aggressive, multi-step value bootstrapping when temporal-difference (TD) learning is employed. These benefits are critical in sparse reward tasks as they allow an agent to more easily discover reward signals and assign credit.
+
+Many existing hRL methods make assumptions about the task structure (e.g., fetching an object involves three stages: moving towards the object, picking it up, and combing back), and/or the skills needed to solve the task (e.g., pre-programmed motor skills) (Florensa et al., 2016; Riedmiller et al., 2018; Lee et al., 2019; Hausman et al., 2018; Lee et al., 2020; Sohn et al., 2018; Ghavamzadeh & Mahadevan, 2003; Nachum et al., 2018). Thus these methods may require manually designing the correct task decomposition, explicitly formulating the option space, or programming pre-defined options for higher level policies to compose. Instead, we seek to formulate a general method that can learn these abstractions from scratch, for any task, with little manual design in the task domain.
+
+The main contribution of this paper is HIDIO (HIerarchical RL by Discovering Intrinsic Options), a hierarchical method that discovers task-agnostic intrinsic options in a self-supervised manner while learning to schedule them to accomplish environment tasks. The latent option representation is uncovered as the option-conditioned policy is trained, both according to the same self-supervised worker objective. The scheduling of options is simultaneously learned by maximizing environment reward collected by the option-conditioned policy. HIDIO can be easily applied to new sparsereward tasks by simply re-discovering options. We propose and empirically evaluate various instantiations of the option discovery process, comparing the resulting options with respect to their final task performance. We demonstrate that HIDIO is able to efficiently learn and discover diverse options to be utilized for higher task reward with superior sample efficiency compared to other hierarchical methods.
+
+# 2 PRELIMINARIES
+
+We consider the reinforcement learning (RL) problem in a Markov Decision Process (MDP). Let $\mathbf { s } \in \mathbb { R } ^ { S }$ be the agent state. We use the terms “state” and “observation” interchangeably to denote the environment input to the agent. A state can be fully or partially observed. Without loss of generality, we assume a continuous action space $\mathbf { a } \in \mathbb { R } ^ { A }$ for the agent. Let $\pi _ { \boldsymbol { \theta } } ( \mathbf { a } | \mathbf { s } )$ be the policy distribution with learnable parameters $\theta$ , and $\mathscr { P } ( \mathbf { s } _ { t + 1 } | \mathbf { s } _ { t } , \mathbf { a } _ { t } )$ the transition probability that measures how likely the environment transitions to $\mathbf { s } _ { t + 1 }$ given that the agent samples an action by $\mathbf { a } _ { t } \sim \pi _ { \theta } ( \cdot | \mathbf { s } _ { t } )$ . After the transition to $\mathbf { s } _ { t + 1 }$ , the agent receives a deterministic scalar reward $r ( \mathbf { s } _ { t } , \mathbf { a } _ { t } , \mathbf { s } _ { t + 1 } )$ .
+
+The objective of RL is to maximize the sum of discounted rewards with respect to $\theta$
+
+$$
+\underset { \pi _ { \theta } , \mathcal { P } } { \mathbb { E } } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r ( \mathbf { s } _ { t } , \mathbf { a } _ { t } , \mathbf { s } _ { t + 1 } ) \right]
+$$
+
+where $\gamma \in [ 0 , 1 ]$ is a discount factor. We will omit $\mathcal { P }$ in the expectation for notational simplicity.
+
+In the options framework (Sutton et al., 1999), the agent can switch between different options during an episode, where an option is translated to a sequence of actions by an option-conditioned policy with a termination condition. A set of options defined over an MDP induces a hierarchy that models temporal abstraction. For a typical two-level hierarchy, a higher-level policy produces options, and the policy at the lower level outputs environment actions conditioned on the proposed options. The expectation in Eq. 1 is taken over policies at both levels.
+
+# 3 HIERARCHICAL RL BY DISCOVERING INTRINSIC OPTIONS
+
+We now introduce our hierarchical method for solving sparse reward tasks. We assume little prior knowledge about the task structure, except that it can be learned through a hierarchy of two levels. The higher-level policy (the scheduler $\pi _ { \theta }$ ), is trained to maximize environment reward, while the lower-level policy (the worker $\pi _ { \phi } ,$ ) is trained in a self-supervised manner to efficiently discover options that are utilized by $\pi _ { \theta }$ to accomplish tasks. Importantly, by self-supervision the worker gets access to dense intrinsic rewards regardless of the sparsity of the extrinsic rewards.
+
+Without loss of generality, we assume that each episode has a length of $T$ and the scheduler outputs an option every $K$ steps. The scheduled option $\mathbf { u } \in [ - 1 , \overline { { 1 } } ] ^ { D }$ (where $D$ is a pre-defined dimensionality), is a latent representation that will be
+
+![](images/776d5a7ea0dce9c72c4a10c48ac70ca2dccf8d34d2cbb0c58272a576b8f6c625.jpg)  
+Figure 1: The overall framework of HIDIO. The scheduler $\pi _ { \theta }$ samples an option $\mathbf { u } _ { h }$ every $K$ (3 in this case) time steps, which is used to guide the worker $\pi _ { \phi }$ to directly interact in the environment conditioned on $\mathbf { u } _ { h }$ and the current sub-trajectory $\overline { { \mathbf { s } } } _ { h , k } , \overline { { \mathbf { a } } } _ { h , k - 1 }$ . The scheduler receives accumulated environment rwhile the worker receives intrinsic rewards . $R _ { h }$ $r _ { h , k + 1 } ^ { \mathrm { l o } }$ fer to Eq. 2 for sampling and Eqs. 3 and 5 for training.
+
+learned from scratch given the environment task. Modulated by $\mathbf { u }$ , the worker executes $K$ steps before the scheduler outputs the next option. Let the time horizon of the scheduler be $\begin{array} { r } { H = \left\lceil \frac { T ^ { \bullet } } { K } \right\rceil } \end{array}$ . Formally, we define
+
+$$
+\begin{array} { l l } { { \mathbf { u } } _ { h } \sim \pi _ { \theta } \bigl ( \cdot | { \mathbf { s } } _ { h , 0 } \bigr ) , } & { 0 \le h < H } \\ { { \mathbf { a } } _ { h , k } \sim \pi _ { \phi } \bigl ( \cdot | { \mathbf { s } } _ { h , k } , { \mathbf { u } } _ { h } \bigr ) , } & { 0 \le k < K } \\ { { \mathbf { s } } _ { h , k + 1 } \sim \mathcal { P } \bigl ( \cdot | { \mathbf { s } } _ { h , k } , { \mathbf { a } } _ { h , k } \bigr ) , } & { 0 \le h < H , 0 \le k < K } \end{array}
+$$
+
+where we denote $\mathbf { s } _ { h , k }$ and ${ \bf a } _ { h , k }$ as the $k$ -th state and action respectively, within the $h$ -th option window of length $K$ . Note that given this sampling process, we have ${ \bf s } _ { h , K } \equiv { \bf s } _ { h + 1 , 0 }$ , namely, the last state of the current option $\mathbf { u } _ { h }$ is the initial state of the next option $\mathbf { u } _ { h + 1 }$ . The overall framework of our method is illustrated in Figure 1.
+
+# 3.1 LEARNING THE SCHEDULER
+
+Every time the scheduler issues an option $\mathbf { u } _ { h }$ , it receives an reward $R _ { h }$ computed by accumulating environment rewards over the next $K$ steps. Its objective is:
+
+$$
+\operatorname* { m a x } _ { \theta } \mathbb { E } _ { \pi _ { \theta } } \left[ \sum _ { h = 0 } ^ { H - 1 } \beta ^ { h } R _ { h } \right] , { \mathrm { w h e r e ~ } } \beta = \gamma ^ { K } { \mathrm { ~ a n d ~ } } R _ { h } = \mathbb { E } _ { \pi _ { \phi } } \left[ \sum _ { k = 0 } ^ { K - 1 } \gamma ^ { k } r ( \mathbf { s } _ { h , k } , \mathbf { a } _ { h , k } , \mathbf { s } _ { h , k + 1 } ) \right]
+$$
+
+This scheduler objective itself is not a new concept, as similar ones have been adopted by other hRL methods (Vezhnevets et al., 2017; Nachum et al., 2018; Riedmiller et al., 2018). One significant difference between our option with that of prior work is that our option $\mathbf { u }$ is simply a latent variable; there is no explicit constraint on what semantics u could represent. In contrast, existing methods usually require their options to reside in a subspace of the state space, to be grounded to the environment, or to have known structures, so that the scheduler can compute rewards and termination conditions for the worker. Note that our latent options can be easily re-trained given a new task.
+
+# 3.2 LEARNING THE WORKER
+
+The main focus of this paper is to investigate how to effectively learn the worker policy in a selfsupervised manner. Our motivation is that it might be unnecessary to make an option dictate the worker to reach some “ $\epsilon \cdot$ -space” of goals (Vezhnevets et al., 2017; Nachum et al., 2018). As long as the option can be translated to a short sequence of primitive actions, it does not need to be grounded with concrete meanings such as goal reaching. Below we will treat the option as a latent variable that modulates the worker, and propose to learn its latent representation in a hierarchical setting from the environment task.
+
+# 3.2.1 WORKER OBJECTIVE
+
+We first define a new meta MDP on top of the original task MDP so that for any $h , k$ , and $t$ :
+
+1) $\overline { { \mathbf { s } } } _ { h , k } : = ( \mathbf { s } _ { h , 0 } , \ldots , \mathbf { s } _ { h , k } ) ,$ ,   
+2) $\overline { { \mathbf { a } } } _ { h , k } : = ( \mathbf { a } _ { h , 0 } , \ldots , \mathbf { a } _ { h , k } )$ ,   
+3) $r ( \overline { { \mathbf { s } } } _ { h , k } , \overline { { \mathbf { a } } } _ { h , k } , \overline { { \mathbf { s } } } _ { h , k + 1 } ) : = r ( \mathbf { s } _ { h , k } , \mathbf { a } _ { h , k } , \mathbf { s } _ { h , k + 1 } ) ,$ , and   
+4) $\begin{array} { r } { \mathcal { P } ( \overline { { \mathbf { s } } } _ { h , k + 1 } | \overline { { \mathbf { s } } } _ { h , k } , \overline { { \mathbf { a } } } _ { h , k } ) : = \mathcal { P } ( \mathbf { s } _ { h , k + 1 } | \mathbf { s } _ { h , k } , \mathbf { a } _ { h , k } ) } \end{array}$ .
+
+This new MDP equips the worker with historical state and action information since the time $( h , 0 )$ when an option $h$ was scheduled. Specifically, each state $\overline { { \mathbf { s } } } _ { h , k }$ or action $\overline { { \mathbf { a } } } _ { h , k }$ encodes the history from the beginning $( h , 0 )$ up to $( h , k )$ within the option. In the following, we will call pairs $\{ \overline { { \mathbf { a } } } _ { h , k } , \overline { { \mathbf { s } } } _ { h , k + 1 } \}$ option sub-trajectories. The worker policy now takes option sub-trajectories as inputs: ${ \mathbf { a } } _ { h , k } \sim$ $\pi _ { \phi } \big ( \cdot | \overline { { \mathbf { s } } } _ { h , k } , \overline { { \mathbf { a } } } _ { h , k - 1 } , \mathbf { u } _ { h } \big ) , 0 \leq k < K$ , whereas the scheduler policy still operates in the original MDP.
+
+Denote Ph,k $\begin{array} { r } { \sum _ { h , k } \equiv \sum _ { h = 0 } ^ { H - 1 } \sum _ { k = 0 } ^ { K - 1 } } \end{array}$ for simplicity. The worker objective, defined on this new MDP, is to minimize the entropy of the option $\mathbf { u } _ { h }$ conditioned on the option sub-trajectory $\{ \overline { { \mathbf { a } } } _ { h , k } , \overline { { \mathbf { s } } } _ { h , k + 1 } \}$ :
+
+$$
+\operatorname* { m a x } _ { \phi } \ \underset { \pi _ { \theta } , \pi _ { \phi } } { \mathbb { E } } \sum _ { h , k } \ \underset { \mathrm { n e g a t i v e ~ c o n d i t i o n a l ~ o p t i o n ~ e n t r o p y } } { \underbrace { \log p \big ( \mathbf { u } _ { h } \big | \overline { { \mathbf { a } } } _ { h , k } , \overline { { \mathbf { s } } } _ { h , k + 1 } \big ) } } \underbrace { - \beta \log \pi _ { \phi } \big ( \mathbf { a } _ { h , k } \big | \overline { { \mathbf { s } } } _ { h , k } , \overline { { \mathbf { a } } } _ { h , k - 1 } , \mathbf { u } _ { h } \big ) } _ { \mathrm { w o r k e r ~ p o l i c y ~ e n t r o p y } }
+$$
+
+where the expectation is over the current $\pi _ { \theta }$ and $\pi _ { \phi }$ but the maximization is only with respect to $\phi$ . Intuitively, the first term suggests that the worker is optimized to confidently identify an option given a sub-trajectory. However, it alone will not guarantee the diversity of options because potentially even very similar sub-trajectories can be classified into different options if the classification model has a high capacity, in which case we say that the resulting sub-trajectory space has a very high “resolution”. As a result, the conditional entropy alone might not be able to generate useful options to be exploited by the scheduler for task solving, because the coverage of the sub-trajectory space is poor. To combat this degenerate solution, we add a second term which maximizes the entropy of the worker policy. Intuitively, while the worker generates identifiable sub-trajectories corresponding to a given option, it should act as randomly as possible to separate sub-trajectories of different options, lowering the “resolution” of the sub-trajectory space to encourage its coverage.
+
+Because directly estimating the posterior $p \big ( \mathbf { u } _ { h } \vert \overline { { \mathbf { a } } } _ { h , k } , \overline { { \mathbf { s } } } _ { h , k + 1 } \big )$ is intractable, we approximate it with a parameterized posterior $\log q _ { \psi } ( { \mathbf { u } } _ { h } | \mathbf { \overline { { a } } } _ { h , k } , \mathbf { \overline { { s } } } _ { h , k + 1 } )$ to obtain a lower bound (Barber & Agakov, 2003), where $q _ { \psi }$ is a discriminator to be learned. Then we can maximize this lower bound instead:
+
+$$
+\operatorname* { m a x } _ { \phi , \psi } \mathbb { \underline { { E } } } _ { \pi _ { \theta } , \pi _ { \phi } } \sum _ { h , k } \log q _ { \psi } ( \mathbf { u } _ { h } | \mathbf { \overline { { a } } } _ { h , k } , \mathbf { \overline { { s } } } _ { h , k + 1 } ) - \beta \log \pi _ { \phi } ( \mathbf { a } _ { h , k } | \mathbf { \overline { { s } } } _ { h , k } , \mathbf { \overline { { a } } } _ { h , k - 1 } , \mathbf { u } _ { h } ) .
+$$
+
+The discriminator $q _ { \psi }$ is trained by maximizing likelihoods of options given sampled sub-trajectories. The worker $\pi _ { \phi }$ is trained via max-entropy RL (Soft Actor-Critic (SAC) (Haarnoja et al., 2018)) with the intrinsic reward $r _ { h , k + 1 } ^ { l o } : = \log q _ { \psi } ( \cdot ) - \beta \log \pi _ { \phi } ( \cdot )$ . $\beta$ is fixed to 0.01 in our experiments.
+
+Note that there are at least four differences between Eq. 5 and the common option discovery objective in either VIC (Gregor et al., 2016) or DIAYN (Eysenbach et al., 2019):
+
+1. Both VIC and DIAYN assume that a sampled option will last through an entire episode, and the option is always sampled at the beginning of an episode. Thus their option trajectories “radiate” from the initial state set. In contrast, our worker policy learns options that initialize every $K$ steps within an episode, and they can have more diverse semantics depending on the various states $s _ { h , 0 }$ visited by the agent. This is especially helpful for some tasks where new options need to be discovered after the agent reaches unseen areas in later stages of training.   
+2. Actions taken by the worker policy under the current option will have consequences on the next option. This is because the final state $s _ { h , K }$ of the current option is defined to be the initial state $s _ { h + 1 , 0 }$ of the next option. So in general, the worker policy is trained not only to discover diverse options across the current $K$ steps, but also to make the discovery easier in the future steps. In other words, the worker policy needs to solve the credit assignment problem across options, under the expectation of the scheduler policy.   
+3. To enable the worker policy to learn from a discriminator that predicts based on option subtrajectories $\{ \overline { { \mathbf { a } } } _ { h , k } , \overline { { \mathbf { s } } } _ { h , k + 1 } \}$ instead of solely on individual states $\mathbf { s } _ { h , k }$ , we have constructed a new meta MDP where each state $\overline { { \mathbf { s } } } _ { h , k }$ encodes history from the beginning $( h , 0 )$ up to $( h , k )$ within an option $h$ . This new meta MDP is critical, because otherwise one simply cannot learn a worker policy from a reward function that is defined by multiple time steps (sub-trajectories) since the learning problem is no longer Markovian.   
+4. Lastly, thanks to the new MDP, we are able to explore various possible instantiations of the discriminator (see Section 3.3). As observed in the experiments, individual states are actually not the optimal features for identifying options.
+
+These differences constitute the major novelty of our worker objective.
+
+# 3.2.2 SHORTSIGHTED WORKER
+
+It’s challenging for the worker to accurately predict values over a long horizon, since its rewards are densely computed by a complex nonlinear function $q _ { \psi }$ . Also each option only lasts at most $K$ steps. Thus we set the discount $\eta$ for the worker in two shortsighted ways:
+
+1. Hard: setting $\eta = 0$ every $K$ -th step and $\eta = 1$ otherwise. Basically this truncates the temporal correlation (gradients) between adjacent options. Its benefit might be faster and easier value learning because the value is bootstrapped over at most $K$ steps $K \ll T ,$ ).   
+2. Soft: $\begin{array} { r } { \eta = 1 - \frac { 1 } { K } } \end{array}$ , which considers rewards of roughly $K$ steps ahead. The worker policy still needs to take into account the identification of future option sub-trajectories, but their importance quickly decays.
+
+We will evaluate both versions and compare their performance in Section 4.1.
+
+# 3.3 INSTANTIATING THE DISCRIMINATOR
+
+We explore various ways of instantiating the discriminator $q _ { \psi }$ in order to compute useful intrinsic rewards for the worker. Previous work has utilized individual states (Eysenbach et al., 2019; Jabri et al., 2019) or full observation trajectories (Warde-Farley et al., 2019; Sharma et al., 2019a; Achiam et al., 2018) for option discrimination. Thanks to the newly defined meta MDP, our discriminator is able to take option sub-trajectories instead of current individual states for prediction. In this paper, we investigate six sub-trajectory feature extractors $f _ { \psi }$ :
+
+<table><tr><td>Feature extractor</td><td>Name</td><td>Formulation</td><td>Explanation</td></tr><tr><td>fe(ah,k,Sh,k+1）=</td><td>State Action StateDiff StateAction StateConcat ActionConcat</td><td>MLP(Sh,k+1) MLP([sh,0, ah,k]) MLP(Sh,k+1- Sh,k) MLP([ah,k,Sh,k+1]) MLP([Sh,k+1]) MLP([Sh,0,ah,k])</td><td>Next state alone Action in context Difference between state pairs Action and next state Concatenation of states Concatenation of actions</td></tr></table>
+
+where the operator $[ \cdot ]$ denotes concatenation and MLP denotes a multilayer perceptron1. Our State feature extractor is most similar to DIAYN (Eysenbach et al., 2019), and StateConcat is similar to (Warde-Farley et al., 2019; Sharma et al., $2 0 1 9 \mathrm { a }$ ; Achiam et al., 2018). However we note that unlike these works, the distribution of our option sub-trajectories is also determined by the scheduler in the context of hRL. The other four feature extractors have not been evaluated before. With the extracted feature, the log-probability of predicting an option is simply computed as the negative squared L2 norm: $\begin{array} { r } { \log q _ { \psi } ( \bar { \mathbf { u } } _ { h } | \bar { \mathbf { a } } _ { h , k } , \bar { \mathbf { s } } _ { h , k + 1 } ) = - \| \bar { f } _ { \psi } ( \bar { \mathbf { a } } _ { h , k } , \bar { \mathbf { s } } _ { h , k + 1 } ) - \mathbf { \bar { u } } _ { h } \| _ { 2 } ^ { 2 } } \end{array}$ , by which we implicitly assume the discriminator’s output distribution to be a $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } _ { \mathbf { D } } )$ multivariate Gaussian.
+
+# 3.4 OFF-POLICY TRAINING
+
+The scheduler and worker objectives (Eq. 3 and Eq. 5) are trained jointly. In principle, on-policy training such as A2C (Clemente et al., 2017) is needed due to the interplay between the scheduler and worker. However, to reuse training data and improve sample efficiency, we employ off-policy training (SAC (Haarnoja et al., 2018)) for both objectives with some modifications.
+
+Modified worker objective In practice, the expectation over the scheduler $\pi _ { \theta }$ in Eq. 5 is replaced with the expectation over its historical versions. Specifically, we sample options $\mathbf { u } _ { h }$ from a replay buffer, together with sub-trajectories $\{ \overline { { \mathbf { a } } } _ { h , k } , \overline { { \mathbf { s } } } _ { h , k + 1 } \}$ . This type of data distribution modification is conventional in off-policy training (Lillicrap et al., 2016).
+
+Intrinsic reward relabeling We always recompute the rewards in Eq. 5 using the up-to-date discriminator for every update of $\phi$ , which can be trivially done without any additional interaction with the environment.
+
+Importance correction The data in the replay buffer was generated by historical worker policies. Thus a sampled option sub-trajectory will be outdated under the same option, causing confusion to the scheduler policy. To resolve this issue, when minimizing the temporal-difference (TD) error between the values of $\mathbf { s } _ { h , 0 }$ and $\mathbf { s } _ { h + 1 , 0 }$ for the scheduler, an importance ratio can be multiplied: QK−1k=0 πφ(ah,k|sh,k,ah,k−1,uh)πoldφ (ah,k|sh,k,ah,k−1,uh) . A similar correction can also be applied to the discriminator loss. However, in practice we find that this ratio has a very high variance and hinders the training. Like the similar observations made in Nachum et al. (2018); Fedus et al. (2020), even without importance correction our method is able to perform well empirically2.
+
+![](images/e9830a579e95b197386890f1f825e5908887665ad420350a809a50557da5636d.jpg)  
+Figure 2: The four tasks we evaluate on. From left to right: 7-DOF PUSHER, 7-DOF REACHER, GOALTASK, and KICKBALL. The first two tasks simulate a one-armed PR2 robot environment while the last two are in the SOCIALROBOT environment. The final picture shows a closeup of the PIONEER2DX robot used in SOCIALROBOT.
+
+# 4 EXPERIMENTS
+
+Environments We evaluate success rate and sample efficiency across two environment suites, as shown in Figure 2. Important details are presented here with more information in appendix Section B. The first suite consists of two 7-DOF reaching and pushing environments evaluated in Chua et al. (2018). They both emulate a one-armed PR2 robot. The tasks have sparse rewards: the agent gets a reward of 0 at every timestep where the goal is not achieved, and 1 upon achieved. There is also a small $L _ { 2 }$ action penalty applied. In 7-DOF REACHER, the goal is achieved when the gripper reaches a 3D goal position. In 7-DOF PUSHER, the goal is to push an object to a 3D goal position. Episodes have a fixed length of 100; a success of an episode is defined to be if the goal is achieved at the final step of the episode.
+
+We also propose another suite of environments called SOCIALROBOT 3. We construct two sparse reward robotic navigation and manipulation tasks, GOALTASK and KICKBALL. In GOALTASK, the agent gets a reward of 1 when it successfully navigates to a goal, -1 if the goal becomes too far, -0.5 every time it is too close to a distractor object, and 0 otherwise. In KICKBALL, the agent receives a reward of 1 for successfully pushing a ball into the goal, 0 otherwise, and has the same distractor object penalty. At the beginning of each episode, both the agent and the ball are spawned randomly. Both environments contain a small $L _ { 2 }$ action penalty, and terminate an episode upon a success.
+
+Comparison methods One baseline algorithm for comparison is standard SAC (Haarnoja et al., 2018), the building block of our hierarchical method. To verify if our worker policy can just be replaced with a na¨ıve action repetition strategy, we compare with SAC $^ +$ ActRepeat with an action repetition for the same length $K$ as our option interval. We also compare against HIRO (Nachum et al., 2018), a data efficient hierarchical method with importance-based option relabeling, and HiPPO (Li et al., 2020) which trains the lower level and higher level policies together with one unified PPO-based objective. Both are state-of-the-art hierarchical methods proposed to solve sparse reward tasks. Similar to our work, HiPPO makes no assumptions about options, however it utilizes a discrete option space and its options are trained with environment reward.
+
+We implement HIDIO based on an RL framework called ALF 4. A comprehensive hyperparameter search is performed for every method, with a far greater search space over HiPPO and HIRO than our method HIDIO to ensure maximum fairness in comparison; details are presented in Appendix D.
+
+Evaluation For every evaluation point during training, we evaluate the agent with current deterministic policies (by taking arg max of action distributions) for a fixed number of episodes and compute the mean success rate. We plot the mean evaluation curve over 3 randomly seeded runs with standard deviations shown as the shaded area around the curve.
+
+# 4.1 WORKER DESIGN CHOICES
+
+We ask and answer questions about the design choices in HIDIO specific to the worker policy $\pi _ { \phi }$
+
+1. What sub-trajectory feature results in good option discovery? We evaluate all six features proposed in Section 3.3 in all four environments. These features are selected to evaluate how different types of subtrajectory information affect option discovery and final performance. They encompass varying types of both local and global subtrajectory information. We plot comparisons of sample efficiency and final performance in Figure 3 across all environments (solid lines), finding that Action, StateAction, and StateDiff are generally among the top performers. StateAction includes the current action and next state, encouraging $\pi _ { \phi }$ to differentiate its options with different actions even at similar states. Similarly, Action includes the option initial state and current action, encouraging option diversity by differentiating between actions conditioned on initial states. Meanwhile StateDiff simply encodes the difference between the next and current state, encouraging $\pi _ { \phi }$ to produce options with different state changes at each step.
+
+![](images/6f1059f9e4817658d4ad848a4104cfafe8d3ebfa671a7eb08a84890142b27e44.jpg)  
+Figure 3: Comparison of all discriminator features against each other across the four environments. Solid lines indicate hard short-sighted workers (Hard), dotted lines indicated soft short-sighted workers (Soft).
+
+![](images/52b1d825e30651c1269ae68d701ab9def3baa1ad94035614165aac8397117806.jpg)  
+Figure 4: Comparisons of the mean success rates of three features of HIDIO (Action, StateAction, StateDiff; solid lines) against other methods (dashed lines).
+
+2. How do soft shortsighted workers (Soft) compare against hard shortsighted workers $( H a r d ) .$ ? In Figure 3, we plot all features with Soft in dotted lines. We can see that in general there is not much difference in performance between Hard and Soft except some extra instability of Soft in REACHER regarding the StateConcat and State features. One reason of this similar general performance could be that since our options are very short-term in Hard, the scheduler policy has the opportunity of switching to a good option before the current one leads to bad consequences. In a few cases, Hard seems better learned, perhaps due to an easier value bootstrapping for the worker.
+
+# 4.2 COMPARISON RESULTS
+
+We compare our three best sub-trajectory features of Hard, in Section 4.1, against the SAC baselines and hierarchical RL methods across all four environments in Figure 4. Generally we see that HIDIO (solid lines) achieves greater final performance with superior sample efficiency than the compared methods. Both SAC and $\mathbf { S A C + } .$ ActRepeat perform poorly across all environments, and all baseline methods perform significantly worse than HIDIO on REACHER, GOALTASK, and KICKBALL.
+
+In PUSHER, HiPPO displays competitive performance, rapidly improving from the start. However, all three HIDIO instantiations achieve nearly $100 \%$ success rates while HiPPO is unable to do so. Furthermore, HIRO and $\mathrm { { \bf S A C + } }$ ActRepeat take much longer to start performing well, but never achieve similar success rates as HIDIO. HIDIO is able to solve REACHER while HiPPO achieves only about a $60 \%$ success rate at best. Meanwhile, HIRO, SAC+ActRepeat, and SAC are unstable or non-competitive. REACHER is a difficult exploration problem as the arm starts far from the goal position, and we see that HIDIO’s automatically discovered options ease exploration for the higher level policy to consistently reach the goal. HIDIO performs well on GOALTASK, achieving $60 \%$ success rates, while the task is too challenging for every other method. In KICKBALL, the most challenging task, HIDIO achieves $3 0 { - } 4 0 \%$ success rates while every other learns poorly again, highlighting the need for the intrinsic option discovery of HIDIO in these environments.
+
+![](images/9e404a5a3abcbab1caad0eb27674aee35cb4bb040497bc271d213427eabf2d4e.jpg)  
+Figure 5: Two example options from the StateAction instantiation on KICKBALL (top) and PUSHER (bottom). The top option navigates directly to the goal by bypassing obstructions along the way and the bottom option sweeps the puck towards one direction.
+
+In summary, HIDIO demonstrates greater sample efficiency and final reward gains over all other baseline methods. Regular RL (SAC) fails on all four environments, and while HiPPO is a strong baseline on PUSHER and REACHER, it is still outperformed in both by HIDIO. All other methods fail on GOALTASK and KICKBALL, while HIDIO is able to learn and perform better in both. This demonstrates the importance of the intrinsic, short-term option discovery employed by HIDIO, where the options are diverse enough to be useful for both exploration and task completion.
+
+# 4.3 JOINT $\pi _ { \phi }$ AND $\pi _ { \theta }$ TRAINING
+
+We ask the next question: is jointly training $\pi _ { \theta }$ and $\pi _ { \phi }$ necessary? To answer this, we compare HIDIO against a pre-training baseline where we first pre-train $\pi _ { \phi }$ , with uniformly sampled options $\mathbf { u }$ for a portion $\rho$ of total numbers of training time steps, and then fix $\pi _ { \phi }$ while training $\pi _ { \theta }$ for the remaining $( 1 - \rho )$ time steps. This is essentially using pre-trained options for downstream higher-level tasks as demonstrated in DIAYN (Eysenbach et al., 2019). We conduct this experiment with the StateAction feature on both KICKBALL and PUSHER, with $\rho = \{ \textstyle { \frac { 1 } { 1 6 } } , \frac { 1 } { 8 } , \frac { 1 } { 4 } \}$ . The results are shown in Figure 6. We can see that in
+
+![](images/320650ed1efd8e0dd8649f385cb2302d4449300c74abc3898b8dc789f7b765e7.jpg)  
+at fractions Figure 6: Pretraining baseline comparison $\{ \textstyle { \frac { 1 } { 1 6 } } , \textstyle { \frac { 1 } { 8 } } , \textstyle { \frac { 1 } { 4 } } \}$ of the total number of training time steps.
+
+PUSHER, fewer pre-training time steps are more sample efficient, as the environment is simple and options can be learned from a small amount of samples. The nature of PUSHER also only requires options that can be learned independent of the scheduler policy evolution. Nevertheless, the pretraining baselines seem less stable. In KICKBALL, the optimal pre-training baseline is on $\textstyle \rho = { \frac { 1 } { 8 } }$ of the total time steps. However without the joint training scheme of HIDIO, the learned options are unable to be used as efficiently for the difficult obstacle avoidance, navigation, and ball manipulation subtasks required for performing well.
+
+# 4.4 OPTION BEHAVIORS
+
+Finally, since options discovered by HIDIO in our sparse reward environments help it achieve superior performance, we ask, what do useful options look like? To answer this question, after training, we sample options from the scheduler $\pi _ { \theta }$ to visualize their behaviors in different environments in Figure 5. For each sampled option u, we fix it until the end of an episode and use the worker $\pi _ { \phi }$ to output actions given u. We can see that the options learned by HIDIO are low-level navigation and manipulation skills useful for the respective environments. We present more visualizations in Figure 9 and more analysis in Section C.2 in the appendix. Furthermore, we present an analysis of task performance for different option lengths in appendix Section C.1 and Figures 7 and 8.
+
+# 5 RELATED WORK
+
+Hierarchical RL Much of the previous work in hRL makes assumptions about the task structure and/or the skills needed to solve the task. While obtaining promising results under specific settings, they may have difficulties with different scenarios. For example, SAC-X (Riedmiller et al., 2018) requires manually designing auxiliary subtasks as skills to solve a given downstream task. SNN4HRL (Florensa et al., 2016) is geared towards tasks with pre-training and downstream components. Lee et al. (2019; 2020) learns to modulate or compose given primitive skills that are customized for their particular robotics tasks. Ghavamzadeh & Mahadevan (2003) and Sohn et al. (2018) operate under the assumption that tasks can be manually decomposed into subtasks.
+
+The feudal reinforcement learning proposal (Dayan & Hinton, 1993) has inspired another line of works (Vezhnevets et al., 2017; Nachum et al., 2018; Levy et al., 2019; Rafati & Noelle, 2019) which make higher-level manager policies output goals for lower-level worker policies to achieve. Usually the goal space is a subspace of the state space or defined according to the task so that lower-level rewards are easy to compute. This requirement of manually “grounding” goals in the environment poses generalization challenges for tasks that cannot be decomposed into state or goal-reaching.
+
+The MAXQ decomposition (Dietterich, 2000) defines an hRL task decomposition by breaking up the target MDP into a hierarchy of smaller MDPs such that the value function in the target MDP is represented as the sum of the value functions of the smaller ones. This has inspired works that use such decompositions (Mehta et al., 2008; Winder et al., 2020; Li et al., 2017) to learn structured, hierarchical world models or policies to complete target tasks or perform transfer learning. However, building such hierarchies makes these methods limited to MDPs with discrete action spaces.
+
+Our method HIDIO makes few assumptions about the specific task at hand. It follows from the options framework (Sutton et al., 1999), which has recently been applied to continuous domains (Bacon et al., 2017), spawning a diverse set of recent hierarchical options methods (Bagaria & Konidaris, 2020; Klissarov et al., 2017; Riemer et al., 2018; Tiwari & Thomas, 2019; Jain et al., 2018). HIDIO automatically learns intrinsic options that avoids having explicit initiation or termination policies dependent on the task at hand. HiPPO (Li et al., 2020), like HIDIO, also makes no major assumptions about the task, but does not employ self-supervised learning for training the lower-level policy.
+
+Self-supervised option/skill discovery There are also plenty of prior works which attempt to learn skills or options without task reward. DIAYN (Eysenbach et al., 2019) and VIC (Gregor et al., 2016) learn skills by maximizing the mutual information between trajectory states and their corresponding skills. VALOR (Achiam et al., 2018) learns options by maximizing the probability of options given their resulting observation trajectory. DADS (Sharma et al., 2019a) learns skills that are predictable by dynamics models. DISCERN (Warde-Farley et al., 2019) maximizes the mutual information between goal and option termination states to learn a goal-conditioned reward function. Brunskill & Li (2014) learns options in discrete MDPs that are guaranteed to improve a measure of sample complexity. Portable Option Discovery (Topin et al., 2015) discovers options by merging options from source policies to apply to some target domain. Eysenbach et al. (2019); Achiam et al. (2018); Sharma et al. (2019a); Lynch et al. (2020) demonstrate pre-trained options to be useful for hRL. These methods usually pre-train options in an initial stage separate from downstream task learning; few works directly integrate option discovery into a hierarchical setting. For higher dimensional input domains, Lynch et al. (2020) learns options from human-collected robot interaction data for image-based, goal-conditioned tasks, and Chuck et al. (2020) learns a hierarchy of options by discovering objects from environment images and forming options which can manipulate them. HIDIO can also be applied to image-based environments by replacing fully-connected layers with convolutional layers in the early stages of the policy and discriminator networks. However, we leave this to future work to address possible practical challenges arising in this process.
+
+# 6 CONCLUSION
+
+Towards solving difficult sparse reward tasks, we propose a new hierarchical reinforcement learning method, HIDIO, which can learn task-agnostic options in a self-supervised manner and simultaneously learn to utilize them to solve tasks. We evaluate several different instantiations of the discriminator of HIDIO for providing intrinsic rewards for training the lower-level worker policy. We demonstrate the effectiveness of HIDIO compared against other reinforcement learning methods in achieving high rewards with better sample efficiency across a variety of robotic navigation and manipulation tasks.
+
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+
+# A PSEUDO CODE FOR HIDIO
+
+# Algorithm 1: Hierarchical RL with Intrinsic Options Discovery
+
+#
+
+$T$ Episode length $M$ Batches per iteration $\pi _ { \phi } ( \mathbf { a } _ { h , k } | \mathbf { s } _ { h , k } , \mathbf { u } _ { h } )$ Worker $B$ Batch size $\alpha$ Learning rate $q _ { \psi } \big ( { \mathbf { u } } _ { h } \big | \overline { { \mathbf { a } } } _ { h , k } , \overline { { \mathbf { s } } } _ { h , k + 1 } \big )$ Discriminator $K$ Option interval $\mathcal { P } ( \mathbf { s } _ { h , k + 1 } | \mathbf { s } _ { s , k } , \mathbf { a } _ { h , k } )$ Environment dynamics $\pi _ { \boldsymbol { \theta } } ( \mathbf { u } _ { h } | \mathbf { s } _ { h , 0 } )$ Scheduler
+
+Output: Learned parameters $\theta$ , $\phi$ , and $\psi$
+
+Initialize: Random model parameters $\theta$ , $\phi$ , and $\psi$ ; empty replay buffers $\mathcal { D } _ { \mathrm { s c h e d u l e r } }$ and $\mathcal { D } _ { \mathrm { w o r k e r } }$ . while termination not met do
+
+$/ \star$ Data collection \*/ for scheduler step $\begin{array} { r } { h = 0 . . \frac { T } { K } - 1 } \end{array}$ do Sample an option $\mathbf { u } _ { h } \sim \pi _ { \theta } \big ( \cdot | \mathbf { s } _ { h , 0 } \big )$ . for worker step $k = 0 . . K - 1$ do Sample an action $\mathbf { a } _ { h , k } \sim \pi _ { \phi } ( \cdot | \mathbf { s } _ { h , k } , \mathbf { u } _ { h } )$ . Step through the environment $\mathbf { s } _ { h , k + 1 } \sim \mathcal { P } ( \cdot | \mathbf { s } _ { h , k } , \mathbf { a } _ { h , k } )$ . $\overline { { \mathbf { a } } } _ { h , k } , \overline { { \mathbf { s } } } _ { h , k + 1 }  [ \overline { { \mathbf { a } } } _ { h , k - 1 } , \mathbf { a } _ { h , k } ] , [ \overline { { \mathbf { s } } } _ { h , k } , \mathbf { s } _ { h , k + 1 } ]$ Dworker ← Dworker $\bigcup \left( { \mathbf { u } } _ { h } , \overline { { \mathbf { a } } } _ { h , k } , \overline { { \mathbf { s } } } _ { h , k + 1 } \right)$ end $\begin{array} { r l } & { R _ { h } \gets \sum _ { k = 0 } ^ { K - 1 } r ( \mathbf s _ { h , k } , \mathbf a _ { h , k } , \mathbf s _ { h , k + 1 } ) } \\ & { \mathcal { D } _ { \mathrm { s c h e d u l e r } }  \mathcal { D } _ { \mathrm { s c h e d u l e r } } \cup ( \mathbf s _ { h , 0 } , \mathbf u _ { h } , \mathbf s _ { h + 1 , 0 } , R _ { h } ) } \end{array}$ end $/ \star$ Model training \*/ for batch $m = 0 . . M - 1$ do $/ \star$ Scheduler training \*/ Uniformly sample transitions $\{ ( \mathbf { s } _ { t } , \mathbf { u } _ { t } , \mathbf { s } _ { t + 1 } ) \} _ { b = 1 } ^ { B } \sim \mathcal { D } _ { \mathrm { s c h e d u l e r } }$ . Compute gradient $\Delta \theta$ according to Eq. 3. Update models $\theta  \theta + \alpha \Delta \theta$ . $/ \star$ Worker training \*/ Uniformly sample transitions $\{ ( \mathbf { u } _ { h } , \overline { { \mathbf { a } } } _ { h , k } , \overline { { \mathbf { s } } } _ { h , k + 1 } ) \} _ { b = 1 } ^ { B } \sim \mathcal { D } _ { \mathrm { w o r k e r } } .$ . Compute intrinsic rewards $r _ { h , k } ^ { l o }  q _ { \psi } ( \mathbf { u } _ { h } | \overline { { \mathbf { a } } } _ { h , k } , \overline { { \mathbf { s } } } _ { h , k + 1 } )$ . Compute gradient $\Delta \psi$ and $\Delta \phi$ according to Eq. 5. Update models $\phi  \phi + \alpha \Delta \phi$ and $\psi  \psi + \alpha \Delta \psi$ . end end
+
+![](images/71e3c1eaf556e7c079f58da2aa841d9ebc60a7f5284d6b70a702b296e2c3918f.jpg)  
+Figure 7: Comparisons of the mean success rates of three features of HIDIO (Action, StateAction, StateDiff at different option lengths $K$ . Dotted lines indicate $K = 1$ , solid lines indicate $K = 3$ , and dashed lines indicate $K = 5$ . $K = 3$ was used across all environments for the results in the main text.
+
+# B MORE ENVIRONMENT DETAILS
+
+# B.0.1 PUSHER AND REACHER
+
+These environments both have a time horizon of 100 with no early termination: each episode always runs for 100 steps regardless of goal achievement. For both, a success is when the agent achieves the goal at the final step of an episode. In REACHER, observations are 17-dimensional, including the positions, angles, and velocities of the robot arm, and in PUSHER observations also include the 3D object position. Both include the goal position in the observation space. Actions are 7-dimensional vectors for joint velocity control. The action range is $[ - 2 0 , 2 0 ]$ in REACHER and $[ - 2 , 2 ]$ in PUSHER.
+
+There is an action penalty in both environments: at every timestep the squared $L _ { 2 }$ norm of the agent action is subtracted from the reward. In PUSHER, this penalty is multiplied by a coefficient of 0.001. In REACHER, it’s multiplied by 0.0001.
+
+# B.0.2 GOALTASK AND KICKBALL
+
+For both SOCIALROBOT environments, an episode terminates early when either a success is reached or the goal is out of range. For each episode, the positions of all objects (including the agent) are randomly picked. Observations are 18-dimensional. In GOALTASK, these observations include egocentric positions, distances, and directions from the agent to different objects while in KICKBALL, they are absolute positions and directions. In KICKBALL, the agent receives a reward of 1 for successfully pushing a ball into the goal (episode termination) and 0 otherwise. At the beginning of each episode, the ball is spawned randomly inside the neighborhood of the agent. Three distractor objects are included on the ground to increase task difficulty. In GOALTASK, the number of distractor objects increases to 5. Both environments contain a small $L _ { 2 }$ action penalty: at every time step the squared $L _ { 2 }$ norm of the agent action, multiplied by 0.01, is subtracted from the reward. GOALTASK has a time horizon of 100 steps, while KICKBALL’s horizon is 200. Observations are 30-dimensional, including absolute poses and velocities of the goal, the ball, and the agent. Both GOALTASK and KICKBALL use the same navigation robot PIONEER2DX which has 2-dimensional actions that control the angular velocities (scaled to $[ - 1 , 1 ] ,$ ) of the two wheels.
+
+# C OPTION DETAILS
+
+# C.1 OPTION LENGTH ABLATION
+
+We ablate the option length $K$ in all four environments on the three best HIDIO instantiations in Figure 7. $K = \{ 1 , 3 , 5 \}$ timesteps per option are shown, with $K = 3$ and $K = 5$ performing similarly across all environments, but $K = 1$ performing very poorly in comparison. $K = 1$ provides no temporal abstraction, resulting in worse sample efficiency in PUSHER and REACHER, and failing to learn in GOALTASK and KICKBALL. Although $K = 5$ and $K = 3$ are generally similar, we see in GOALTASK that $K = 5$ results in better performance than $K = 3$ across all three instantiations, demonstrating the potential benefit of longer temporal abstraction lengths.
+
+![](images/aa1519bbbeedb6ac3c4b2839713bc3657e7d111d82e15a4176f449dd4cd03606.jpg)  
+Figure 8: Trajectory distributions compared for different option lengths $K$ for the StateAction HIDIO instantiation in both SOCIALROBOT environments. These are obtained by randomly sampling an option uniformly in $[ - 1 , 1 ] ^ { D }$ and keeping it fixed for the entire trajectory. 100 trajectories from each option are visualized and plotted in different colors.
+
+We also plot the distribution of $( x , y )$ velocities5 in GOALTASK and $( x , y )$ coordinates in KICKBALL of randomly sampled options of different lengths in Figure 8. Despite the fact that these two dimensions only represent a small subspace of the entire (30-dimensional) state space, they still demonstrate a difference in option behavior at different option lengths. We can see that as the option length $K$ increases, the option behaviors become more consistent within a trajectory. Meanwhile regarding coverage, $K = 1$ ’s (blue) trajectory distribution in both environments is less concentrated near the center, while $K = 5$ (green) is the most concentrated at the center. $K = 3$ (orange) lies somewhere in between. We believe that this difference in behavior signifies a trade off between the coverage of the state space and how consistent the learned options can be depending on the option length. Given the same entropy coefficient ( $\beta$ in Eq 5), with longer option lengths, it is likely that the discriminator can more easily discriminate the sub-trajectories created by these options, so that their coverage does not have to be as wide for the worker policy to obtain high intrinsic rewards. Meanwhile, with shorter option lengths, the shorter sub-trajectories have to be more distinct for the discriminator to be able to successfully differentiate between the options.
+
+# C.2 OPTION VISUALIZATIONS
+
+We visualize more option behaviors in Figure 9, produced in the same way as in Figure 5 and as detailed in Section 4.4. The top 4 picture reels are from KICKBALL. We see that KICKBALL options lead to varied directional driving behaviors that can be utilized for efficient navigation. For example, the second, third, and fourth highlight options that produce right turning behavior, however at different speeds and angles. The option in the third reel is a quick turn that results in the robot tumbling over into an unrecoverable state, but the options in the second and fourth reels turn more slowly and do not result in the robot flipping. The first option simply proceeds forward from the robot starting position, kicking the ball into the goal.
+
+The bottom 4 reels are from PUSHER. Each option results in different sweeping behaviors with varied joint positioning and arm height. These sweeping and arm folding behaviors, when utilized in short sub-trajectories, are useful for controlling where and how to move the arm to push the puck into the goal.
+
+# D HYPERPARAMETERS
+
+To ensure a fair comparison across all methods, we perform a hyperparameter search over the following values for each algorithm and suite of environments.
+
+![](images/22e9daa21f3191cefd35de2d0b8c0dfc7bb37cf6a0d4fb66dec47f7641654313.jpg)  
+Figure 9: Eight example options from the StateAction instantiation on KICKBALL (top 4) and PUSHER (bottom 4).
+
+# D.1 PUSHER AND REACHER
+
+Shared hyperparameters across all methods are listed below (where applicable, and except when overridden by hyperparameters listed for each individual method). For all methods, we take the hyperparameters that perform best across 3 random seeds in terms of the area under the evaluation success curve (AUC) in the PUSHER environment.
+
+• Number of parallel actors/environments per rollout: 20   
+• Steps per episode: 100   
+• Batch size: 2048   
+• Learning rate: $1 0 ^ { - 4 }$ for all network modules   
+• Policy/Q network hidden layers: (256, 256, 256) with ReLU non-linearities   
+Polyak averaging coefficient for target Q: 0.999   
+• Target Q update interval (training iterations): 1   
+Training batches per iteration: 100   
+• Episodes per evaluation: 50   
+• Initial environment steps for data collection before training: 10000
+
+Rollouts and training iterations are performed alternatively, one after the other. The rollout length searched below refers to how many time steps in each environment are taken per rollout/training iteration, effectively controlling the ratio of gradient steps to environment steps. A smaller rollout length corresponds to a higher ratio. This ratio is also searched over for HIPPO and HIRO. Other hyperparameters searched separately for each algorithm are listed below, and selected ones are bolded.
+
+# D.1.1 SAC
+
+• Target entropy min prob $\Delta \mathbf { \Psi }$ : {0.1, 0.2, 0.3} • Replay buffer length per parallel actor: $\{ 5 0 0 0 0 , 2 0 0 0 0 0 \}$ • Rollout Length: $\{ 1 2 , 2 5 , 5 0 , 1 0 0 \}$
+
+D.1.2 SAC W/ ACTION REPETITION
+
+• Action repetition length7: 3 • Rollout Length: {4, 8, 16, 33}
+
+Other hyperparameters are kept the same as the optimal SAC ones.
+
+# D.1.3 HIDIO
+
+The hyperparameters of HIDIO were mostly heuristically chosen due to the hyperparameter search space being too large.
+
+• Latent option u vector dimension $( D ) \colon \{ 8 , 1 2 \}$   
+• Policy/Q network hidden layers for $\pi _ { \phi } : ( 1 2 8 , 1 2 8 , 1 2 8 )$   
+• Steps per option $( K )$ : 3   
+• $\pi _ { \phi }$ has a fixed entropy coefficient $\alpha$ of 0.01. Target entropy min prob $\Delta$ for $\pi _ { \theta }$ is 0.2. • Discriminator network hidden layers: (64, 64)   
+• Replay buffer length per parallel actor: $\{ 5 0 0 0 0 , 2 0 0 0 0 0 \}$   
+• Rollout Length: $\{ 2 5 , 5 0 , 1 0 0 \}$
+
+D.1.4 HIRO
+
+• Steps per option: {3, 5, 8}   
+• Replay buffer size (total): $\{ 5 0 0 0 0 0 , 2 0 0 0 0 0 0 \}$   
+• Meta action space (actions are relative, e.g., meta-action is current obs $^ +$ action): (-np.ones(obs space - 3D goal pos) $^ { \star 2 }$ , np.ones(obs space - 3D goal pos) $^ { \star 2 }$ )   
+• Policy stddev noise: $\{ \mathbf { 0 . 1 } , 0 . 3 , 0 . 5 \}$   
+• Number of gradient updates per training iteration: $\{ 1 0 0 , 2 0 0 , 4 0 0 \}$
+
+D.1.5 HIPPO
+
+For most hyperparameters, the search ranges chosen were derived after discussion with the first author of HiPPO.
+
+• Learning rate: $3 \times 1 0 ^ { - 4 }$   
+• Policy network hidden layers: (256, 256)   
+• Skill selection network hidden layers: $\{ ( 3 2 , 3 2 ) , ( 1 2 8 , 6 4 ) \}$   
+• Latent skill vector size: $\{ 5 , 1 0 , 1 5 \}$   
+• PPO clipping parameter: $\{ 0 . 0 5 , \mathbf { 0 . 1 } \}$   
+• Time commitment range: $\{ ( 2 , 5 ) , ( 3 , 7 ) \}$   
+• Policy training steps per epoch: $\{ 2 5 , 5 0 , 1 0 0 \}$
+
+# D.2 SOCIALROBOT
+
+For all methods, we select the hyperparameters with the best area under the evaluation success curve (AUC) in the KICKBALL environment, and apply them to both KICKBALL and GOALTASK. The shared hyperparameters are as follows (if applicable to the algorithm, and except when overridden by the respective algorithm’s list of hyperparameters):
+
+• Number of parallel actors/environments per rollout: 10   
+• Steps per episode: 100 (GOALTASK), 200 (KICKBALL)   
+• Batch size: 1024   
+• Learning rate: $5 \times 1 0 ^ { - 4 }$ for all network modules   
+• Policy/Q network hidden layers: (256, 256, 256) with ReLU non-linearities   
+• Polyak averaging coefficient for target Q: 0.95   
+• Target Q update interval (training iterations): 1   
+• Training batches per iteration: 100   
+• Episodes per evaluation: 100   
+• Evaluation interval (training iterations): 100   
+• Initial environment steps for data collection before training: 100000
+
+The training terminology here generally follows section D.1.
+
+# D.2.1 SAC
+
+• Target entropy min prob $\Delta$ $\mathbf { \partial } : \{ 0 . 1 , \mathbf { 0 . 2 } , 0 . 3 \}$ • Replay buffer length per parallel actor: $\{ 2 0 0 0 0 , 1 0 0 0 0 \}$ • Rollout length: $\{ 1 2 , 2 5 , 5 0 , 1 0 0 \}$
+
+# D.2.2 SAC W/ ACTION REPETITION
+
+• Action repetition length8: 3 • Rollout Length: $\{ 4 , 8 , 1 6 , 3 3 \}$
+
+Other hyperparameters are kept the same as the optimal SAC ones.
+
+# D.2.3 HIDIO
+
+Due to the large hyperparameter search space, we only search over the option vector size and rollout length, and select everything else heuristically.
+
+• Latent option u vector dimension $( D ) \colon \{ 4 , 6 \}$   
+• Policy/Q network hidden layers for $\pi _ { \phi }$ (128, 128, 128)   
+• Steps per option $( K )$ : 3   
+• $\pi _ { \phi }$ has a fixed entropy coefficient $\alpha$ of 0.01. Target entropy min prob $\Delta$ for $\pi _ { \theta }$ is 0.2. • Discriminator network hidden layers: (32, 32)   
+• Replay buffer length per parallel actor: 20000   
+• Rollout Length: $\{ 5 0 , 1 0 0 \}$
+
+D.2.4 HIRO • Learning rate: $3 \times 1 0 ^ { - 4 }$ • Steps per option: $\{ 3 , 5 , 8 \}$ • Replay buffer size (total): $\{ \pmb { 5 0 0 0 0 0 0 } , 2 0 0 0 0 0 0 \}$ • Meta action space (actions are relative, e.g., meta-action is current obs $^ +$ action):
+
+– GOALTASK: (-np.ones(obs space) $\star$ 2, np.ones(obs space) $\star$ 2)   
+– KICKBALL: (-np.ones(obs space - goal space) $\star \_ 2$ , np.ones(obs space - goal space) $\star \_ 2$ ) (because the goal position is given but will not change in the observation space)
+
+• Policy stddev noise $\{ \mathbf { 0 . 1 } , 0 . 3 , 0 . 5 \}$ • Number of gradient updates per training iteration: $\{ 1 0 0 , 2 0 0 , 4 0 0 \}$
+
+D.2.5 HIPPO
+
+• Learning rate: $3 \times 1 0 ^ { - 4 }$   
+• Policy network hidden layers: $\{ ( 6 4 , 6 4 ) , ( 2 5 6 , 2 5 6 ) \}$   
+• Skill selection network hidden layers: $\{ ( 3 2 , 3 2 ) , ( 1 2 8 , 6 4 ) \}$   
+• Latent skill vector size: $\{ 4 , 8 \}$   
+• PPO clipping parameter: $\{ \mathbf { 0 . 0 5 , 0 . 1 } \}$   
+• Time commitment range: $\{ ( { \bf 2 } , { \bf 5 } ) , ( { 3 } , { 7 } ) \}$   
+• Policy training steps per epoch: $\{ 2 5 , 5 0 , 1 0 0 \}$