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+# ALFWORLD: ALIGNING TEXT AND EMBODIED ENVIRONMENTS FOR INTERACTIVE LEARNING
+
+Mohit Shridhar† Xingdi Yuan♡ Marc-Alexandre Côté♡ Yonatan Bisk‡ Adam Trischler♡ Matthew Hausknecht †University of Washington ♡Microsoft Research, Montréal ‡Carnegie Mellon University ♣Microsoft Research
+
+ALFWorld.github.io
+
+# ABSTRACT
+
+Given a simple request like Put a washed apple in the kitchen fridge, humans can reason in purely abstract terms by imagining action sequences and scoring their likelihood of success, prototypicality, and efficiency, all without moving a muscle. Once we see the kitchen in question, we can update our abstract plans to fit the scene. Embodied agents require the same abilities, but existing work does not yet provide the infrastructure necessary for both reasoning abstractly and executing concretely. We address this limitation by introducing ALFWorld, a simulator that enables agents to learn abstract, text-based policies in TextWorld (Côté et al., 2018) and then execute goals from the ALFRED benchmark (Shridhar et al., 2020) in a rich visual environment. ALFWorld enables the creation of a new BUTLER agent whose abstract knowledge, learned in TextWorld, corresponds directly to concrete, visually grounded actions. In turn, as we demonstrate empirically, this fosters better agent generalization than training only in the visually grounded environment. BUTLER’s simple, modular design factors the problem to allow researchers to focus on models for improving every piece of the pipeline (language understanding, planning, navigation, and visual scene understanding).
+
+# 1 INTRODUCTION
+
+Consider helping a friend prepare dinner in an unfamiliar house: when your friend asks you to clean and slice an apple for an appetizer, how would you approach the task? Intuitively, one could reason abstractly: (1) find an apple (2) wash the apple in the sink (3) put the clean apple on the cutting board (4) find a knife (5) use the knife to slice the apple (6) put the slices in a bowl. Even in an unfamiliar setting, abstract reasoning can help accomplish the goal by leveraging semantic priors. Priors like locations of objects – apples are commonly found in the kitchen along with implements for cleaning and slicing, object affordances – a sink is useful for washing an apple unlike a refrigerator, pre-conditions – better to wash an apple before slicing it, rather than the converse. We hypothesize that, learning to solve tasks using abstract language, unconstrained by the particulars of the physical world, enables agents to complete embodied tasks in novel environments by leveraging the kinds of semantic priors that are exposed by abstraction and interaction.
+
+![](images/9854d9e9d416ac34dea87005dfca14a8905f21203d7793f93ebd7ee35f0b8879.jpg)  
+Figure 1: ALFWorld: Interactive aligned text and embodied worlds. An example with high-level text actions (left) and low-level physical actions (right).
+
+To test this hypothesis, we have created the novel ALFWorld framework, the first interactive, parallel environment that aligns text descriptions and commands with physically embodied robotic simulation. We build ALFWorld by extending two prior works: TextWorld (Côté et al., 2018) - an engine for interactive text-based games, and ALFRED (Shridhar et al., 2020) - a large scale dataset for visionlanguage instruction following in embodied environments. ALFWorld provides two views of the same underlying world and two modes by which to interact with it: TextWorld, an abstract, text-based environment, generates textual observations of the world and responds to high-level text actions; ALFRED, the embodied simulator, renders the world in high-dimensional images and responds to low-level physical actions as from a robot (Figure 1).1 Unlike prior work on instruction following (MacMahon et al., 2006; Anderson et al., 2018a), which typically uses a static corpus of cross-modal expert demonstrations, we argue that aligned parallel environments like ALFWorld offer a distinct advantage: they allow agents to explore, interact, and learn in the abstract environment of language before encountering the complexities of the embodied environment.
+
+While fields such as robotic control use simulators like MuJoCo (Todorov et al., 2012) to provide infinite data through interaction, there has been no analogous mechanism – short of hiring a human around the clock – for providing linguistic feedback and annotations to an embodied agent. TextWorld addresses this discrepancy by providing programmatic and aligned linguistic signals during agent exploration. This facilitates the first work, to our knowledge, in which an embodied agent learns the meaning of complex multi-step policies, expressed in language, directly through interaction.
+
+Empowered by the ALFWorld framework, we introduce BUTLER (Building Understanding in Textworld via Language for Embodied Reasoning), an agent that first learns to perform abstract tasks in TextWorld using Imitation Learning (IL) and then transfers the learned policies to embodied tasks in ALFRED. When operating in the embodied world, BUTLER leverages the abstract understanding gained from TextWorld to generate text-based actions; these serve as high-level subgoals that facilitate physical action generation by a low-level controller. Broadly, we find that BUTLER is capable of generalizing in a zero-shot manner from TextWorld to unseen embodied tasks and settings. Our results show that training first in the abstract text-based environment is not only $7 \times$ faster, but also yields better performance than training from scratch in the embodied world. These results lend credibility to the hypothesis that solving abstract language-based tasks can help build priors that enable agents to generalize to unfamiliar embodied environments.
+
+Our contributions are as follows:
+
+$\ S 2$ ALFWorld environment: The first parallel interactive text-based and embodied environment.
+
+$\ S \ O 3$ BUTLER architecture: An agent that learns high-level policies in language that transfer to low-level embodied executions, and whose modular components can be independently upgraded.
+
+$\ S 4$ Generalization: We demonstrate empirically that BUTLER, trained in the abstract text domain, generalizes better to unseen embodied settings than agents trained from corpora of demonstrations or from scratch in the embodied world.
+
+# 2 ALIGNING ALFRED AND TEXTWORLD
+
+The ALFRED dataset (Shridhar et al., 2020), set in the THOR simulator (Kolve et al., 2017), is a benchmark for learning to complete embodied household tasks using natural language instructions and egocentric visual observations. As shown in Figure 1 (right), ALFRED tasks pose challenging interaction and navigation problems to an agent in a high-fidelity simulated environment. Tasks are annotated with a goal description that describes the objective (e.g., “put a pan on the dining table”). We consider both template-based and human-annotated goals; further details on goal specification can be found in Appendix H. Agents observe
+
+<table><tr><td rowspan=1 colspan=1>Task type</td><td rowspan=1 colspan=2># train # seen# unseen</td></tr><tr><td rowspan=7 colspan=1>Pick&amp;PlaceExamine in LightClean&amp;PlaceHeat &amp;PlaceCool &amp;PlacePick Two &amp;PlaceAll</td><td rowspan=1 colspan=1>790</td><td rowspan=1 colspan=1>35   24</td></tr><tr><td rowspan=1 colspan=1>308</td><td rowspan=1 colspan=1>18</td></tr><tr><td rowspan=1 colspan=1>650</td><td rowspan=1 colspan=1>31</td></tr><tr><td rowspan=1 colspan=1>459</td><td rowspan=1 colspan=1>16   23</td></tr><tr><td rowspan=1 colspan=1>533</td><td rowspan=1 colspan=1>25   21</td></tr><tr><td rowspan=1 colspan=1>813</td><td rowspan=1 colspan=1>17</td></tr><tr><td rowspan=1 colspan=1>3,553</td><td rowspan=1 colspan=1>134</td></tr></table>
+
+Table 1: Six ALFRED task types with heldout seen and unseen evaluation sets.
+
+the world through high-dimensional pixel images and interact using low-level action primitives: MOVEAHEAD, ROTATELEFT/RIGHT, LOOKUP/DOWN, PICKUP, PUT, OPEN, CLOSE, and TOGGLEON/OFF.
+
+The ALFRED dataset also includes crowdsourced language instructions like “turn around and walk over to the microwave” that explain how to complete a goal in a step-by-step manner. We depart from the ALFRED challenge by omitting these step-by-step instructions and focusing on the more diffcult problem of using only on goal descriptions specifying what needs to be achieved.
+
+Our aligned ALFWorld framework adopts six ALFRED task-types (Table 1) of various difficulty levels.2 Tasks involve first finding a particular object, which often requires the agent to open and search receptacles like drawers or cabinets. Subsequently, all tasks other than Pick & Place require some interaction with the object such as heating (place object in microwave and start it) or cleaning (wash object in a sink). To complete the task, the object must be placed in the designated location.
+
+Within each task category there is significant variation: the embodied environment includes 120 rooms (30 kitchens, 30 bedrooms, 30 bathrooms, 30 living rooms), each dynamically populated with a set of portable objects (e.g., apple, mug), and static receptacles (e.g., microwave, fridge). For each task type we construct a larger train set, as well as seen and unseen validation evaluation sets: (1): seen consists of known task instances {task-type, object, receptacle, room} in rooms seen during training, but with different instantiations of object locations, quantities, and visual appearances (e.g. two blue pencils on a shelf instead of three red pencils in a drawer seen in training). (2): unseen consists of new task instances with possibly known object-receptacle pairs, but always in unseen rooms with different receptacles and scene layouts than in training tasks.
+
+The seen set is designed to measure in-distribution generalization, whereas the unseen set measures out-of-distribution generalization. The scenes in ALFRED are visually diverse, so even the same task instance can lead to very distinct tasks, e.g., involving differently colored apples, shaped statues, or textured cabinets. For this reason, purely vision-based agents such as the unimodal baselines in Section 5.2 often struggle to generalize to unseen environments and objects.
+
+The TextWorld framework (Côté et al., 2018) procedurally generates text-based environments for training and evaluating language-based agents. In order to extend TextWorld to create text-based analogs of each ALFRED scene, we adopt a common latent structure representing the state of the simulated world. ALFWorld uses PDDL - Planning Domain Definition Language (McDermott et al., 1998) to describe each scene from ALFRED and to construct an equivalent text game using the TextWorld engine. The dynamics of each game are defined by the PDDL domain (see Appendix C for additional details). Textual observations shown in Figure 1 are generated with templates sampled from a context-sensitive grammar designed for the ALFRED environments. For interaction, TextWorld environments use the following high-level actions:
+
+goto {recep}   
+open {recep}   
+clean {obj} with {recep}
+
+take {obj} from {recep} put {obj} in/on {recep} close {recep} toggle {obj}{recep} heat {obj} with {recep} cool {obj} with {recep} where {obj} and {recep} correspond to objects and receptacles. Note that heat, cool, clean, and goto are high-level actions that correspond to several low-level embodied actions.
+
+ALFWorld, in summary, is an cross-modal framework featuring a diversity of embodied tasks with analogous text-based counterparts. Since both components are fully interactive, agents may be trained in either the language or embodied world and evaluated on heldout test tasks in either modality. We believe the equivalence between objects and interactions across modalities make ALFWorld an ideal framework for studying language grounding and cross-modal learning.
+
+# 3 INTRODUCING BUTLER: AN EMBODIED MULTI-TASK AGENT
+
+We investigate learning in the abstract language modality before generalizing to the embodied setting. The BUTLER agent uses three components to span the language and embodied modalities: BUTLER::BRAIN – the abstract text agent, BUTLER::VISION – the language state estimator, and BUTLER::BODY – the low-level controller. An overview of BUTLER is shown in Figure 2 and each component is described below.
+
+![](images/4368898b196b72dd2abee45309ce8de1a6bc71e49c9d2de3b9446d7c421e699e.jpg)  
+Figure 2: BUTLER Agent consists of three modular components. 1) BUTLER::BRAIN: a text agent pre-trained with the TextWorld engine (indicated by the dashed yellow box) which simulates an abstract textual equivalent of the embodied world. When subsequently applied to embodied tasks, it generates high-level actions that guide the controller. 2) BUTLER::VISION: a state estimator that translates, at each time step, the visual frame $v _ { t }$ from the embodied world into a textual observation $o _ { t }$ using a pre-trained Mask R-CNN detector. The generated observation $o _ { t }$ , the initial observation $o _ { 0 }$ , and the task goal $g$ are used by the text agent the to predict the next high-level action $a _ { t }$ . 3) BUTLER::BODY: a controller that translates the high-level text action $a _ { t }$ into a sequence of one or more low-level embodied actions.
+
+# 3.1 BUTLER::BRAIN (TEXT AGENT) $\mathbf { \partial } : o _ { 0 } , o _ { t } , g \to a _ { t }$
+
+BUTLER::BRAIN is a novel text-based game agent that generates high-level text actions in a token-by-token fashion akin to Natural Language Generation (NLG) approaches for dialogue (Sharma et al., 2017) and summarization (Gehrmann et al., 2018). An overview of the agent’s architecture is shown in Figure 3. At game step $t$ , the encoder takes the initial text observation $o _ { 0 }$ , current observation $o _ { t }$ , and the goal description $g$ as input and generates a context
+
+![](images/f6f21b0399ec723fc196115b32388fc2e796af496bdf3ffad10ea3e11b8b03f4.jpg)  
+Figure 3: BUTLER::BRAIN: The text agent takes the initial/current observations $o _ { 0 } / o _ { t }$ , and goal $g$ to generate a textual action $a _ { t }$ token-by-token.
+
+aware representation of the current observable game state. The observation $o _ { 0 }$ explicitly lists all the navigable receptacles in the scene, and goal $g$ is sampled from a set of language templates (see Appendix $\mathrm { H }$ ). Since the games are partially observable, the agent only has access to the observation describing the effects of its previous action and its present location. Therefore, we incorporate two memory mechanisms to imbue the agent with history: (1) a recurrent aggregator, adapted from Yuan et al. (2018), combines the encoded state with recurrent state $h _ { t - 1 }$ from the previous game step; (2) an observation queue feeds in the $k$ most recent, unique textual observations. The decoder generates an action sentence $a _ { t }$ token-by-token to interact with the game. The encoder and decoder are based on a Transformer Seq2Seq model with pointer softmax mechanism (Gulcehre et al., 2016). We leverage pre-trained BERT embeddings (Sanh et al., 2019), and tie output embeddings with input embeddings (Press and Wolf, 2016). The agent is trained in an imitation learning setting with DAgger (Ross et al., 2011) using expert demonstrations. See Appendix A for complete details.
+
+When solving a task, an agent might get stuck at certain states due to various failures (e.g., action is grammatically incorrect, wrong object name). The observation for a failed action does not contain any useful feedback, so a fully deterministic actor tends to repeatedly produce the same incorrect action. To address this problem, during evaluation in both TextWorld and ALFRED, BUTLER::BRAIN uses Beam Search (Reddy et al., 1977) to generate alternative action sentences in the event of a failed action. But otherwise greedily picks a sequence of best words for efficiency. Note that Beam Search is not used to optimize over embodied interactions like prior work (Wang et al., 2019). but rather to simply improve the generated action sentence during failures.
+
+# 3.2 BUTLER::VISION (STATE ESTIMATOR) ∶ vt → ot
+
+At test time, agents in the embodied world must operate purely from visual input. To this end, BUTLER::VISION’s language state estimator functions as a captioning module that translates visual observations $v _ { t }$ into textual descriptions $o _ { t }$ . Specifically, we use a pre-trained Mask R-CNN detector (He et al., 2017) to identify objects in the visual frame. The detector is trained separately in a supervised setting with random frames from ALFRED training scenes (see Appendix D). For each frame $v _ { t }$ , the detector generates $N$ detections $\{ ( c _ { 1 } , m _ { 1 } ) , ( c _ { 2 } , \bar { m _ { 2 } } ) , \ldots , ( c _ { N } , \bar { m _ { N } } ) \}$ , where $c _ { n }$ is the predicted object class, and $m _ { n }$ is a pixel-wise object mask. These detections are formatted into a sentence using a template e.g., On table 1, you see a mug 1, a tomato 1, and a tomato 2. To handle multiple instances of objects, each object is associated with a class $c _ { n }$ and a number ID e.g., tomato 1. Commands goto, open, and examine generate a list of detections, whereas all other commands generate affirmative responses if the action succeeds e.g., $a _ { t }$ : put mug 1 on desk $2  o _ { t + 1 }$ : You put mug 1 on desk 2, otherwise produce Nothing happens to indicate failures or no state-change. See Appendix G for a full list of templates. While this work presents preliminary results with template-based descriptions, future work could generate more descriptive observations using pre-trained image-captioning models (Johnson et al., 2016), video-action captioning frameworks (Sun et al., 2019), or scene-graph parsers (Tang et al., 2020).
+
+$$
+\mathrm { U T L E R } { \mathrel { : } } \mathrm { B o D Y } \left( \mathrm { C O N T R O L L E R } \right) : v _ { t } , a _ { t } \to \left\{ \hat { a } _ { 1 } , \hat { a } _ { 2 } , \dots , \hat { a } _ { L } \right\}
+$$
+
+The controller translates a high-level text action $a _ { t }$ into a sequence of $L$ low-level physical actions $\{ \hat { a } _ { 1 } , \hat { a } _ { 2 } , \dots , \hat { a } _ { L } \}$ that are executable in the embodied environment. The controller handles two types of commands: manipulation and navigation. For manipulation actions, we use the ALFRED API to interact with the simulator by providing an API action and a pixel-wise mask based on Mask R-CNN detections $m _ { n }$ that was produced during state-estimation. For navigation commands, each episode is initialized with a pre-built grid-map of the scene, where each receptacle instance is associated with a receptacle class and an interaction viewpoint $( x , y , \theta , \phi )$ with $x$ and $y$ representing the 2D position, $\theta$ and $\phi$ representing the agent’s yaw rotation and camera tilt. The goto command invokes an $\mathbf { A } ^ { * }$ planner to find the shortest path between two viewpoints. The planner outputs a sequence of $L$ displacements in terms of motion primitives: MOVEAHEAD, ROTATERIGHT, ROTATELEFT, LOOKUP, and LOOKDOWN, which are executed in an open-loop fashion via the ALFRED API. We note that a given pre-built grid-map of receptacle locations is a strong prior assumption, but future work could incorporate existing models from the vision-language navigation literature (Anderson et al., 2018a; Wang et al., 2019) for map-free navigation.
+
+# 4 EXPERIMENTS
+
+We design experiments to answer the following questions: (1) How important is an interactive language environment versus a static corpus? (2) Do policies learnt in TextWorld transfer to embodied environments? (3) Can policies generalize to human-annotated goals? (4) Does pre-training in an abstract textual environment enable better generalization in the embodied world?
+
+# 4.1 IMPORTANCE OF INTERACTIVE LANGUAGE
+
+The first question addresses our core hypothesis that training agents in interactive TextWorld environments leads to better generalization than training agents with a static linguistic corpus. To test this hypothesis, we use DAgger (Ross et al., 2011) to train the BUTLER::BRAIN agent in TextWorld and compare it against Seq2Seq, an identical agent trained with Behavior Cloning from an equivalentlysized corpus of expert demonstrations. The demonstrations come from the same expert policies and we control the number of episodes to ensure a fair comparison. Table 2 presents results for agents trained in TextWorld and subsequently evaluated in embodied environments in a zero-shot manner. The agents are trained independently on individual tasks and also jointly on all six task types. For each task category, we select the agent with best evaluation performance in TextWorld (from 8 random seeds); this is done separately for each split: seen and unseen. These best-performing agents are then evaluated on the heldout seen and unseen embodied ALFRED tasks. For embodied evaluations, we also report goal-condition success rates, a metric proposed in ALFRED (Shridhar et al., 2020) to measure partial goal completion.
+
+<table><tr><td rowspan="2">task-type</td><td colspan="2">TextWorld</td><td colspan="2">Seq2Seq</td><td colspan="2">BUTLER</td><td colspan="2">BUTLER-ORACLE</td><td colspan="2">Human Goals</td></tr><tr><td>seen</td><td>unseen</td><td>seen</td><td>unseen</td><td>seen</td><td>unseen</td><td>seen</td><td>unseen</td><td>seen</td><td>unseen</td></tr><tr><td>Pick&amp;Place</td><td>69</td><td>50</td><td>28(28)</td><td>17 (17)</td><td>30 (30)</td><td>24 (24)</td><td>53 (53)</td><td>31 (31)</td><td>20 (20)</td><td>10 (10)</td></tr><tr><td>Examine in Light</td><td>69</td><td>39</td><td>5(13)</td><td>0 (6)</td><td>10 (26)</td><td>0 (15)</td><td>22 (41)</td><td>12 (37)</td><td>2 (9)</td><td>0 (8)</td></tr><tr><td>Clean &amp; Place</td><td>67</td><td>74</td><td>32 (41)</td><td>12 (31)</td><td>32 (46)</td><td>22 (39)</td><td>44 (57)</td><td>41 (56)</td><td>18 (31)</td><td>22 (39)</td></tr><tr><td>Heat&amp;Place</td><td>88</td><td>83</td><td>10 (29)</td><td>12 (33)</td><td>17 (38)</td><td>16 (39)</td><td>60 (66)</td><td>60 (72)</td><td>8 (29)</td><td>5 (30)</td></tr><tr><td>Cool &amp; Place</td><td>76</td><td>91</td><td>2(19)</td><td>21 (34)</td><td>5 (21)</td><td>19 (33)</td><td>41 (49)</td><td>27 (44)</td><td>7(26)</td><td>17 (34)</td></tr><tr><td>Pick Two &amp; Place</td><td>54</td><td>65</td><td>12 (23)</td><td>0 (26)</td><td>15 (33)</td><td>8 (30)</td><td>32 (42)</td><td>29 (44)</td><td>6(16)</td><td>0 (6)</td></tr><tr><td>All Tasks</td><td>40</td><td>35</td><td>6(15)</td><td>5(14)</td><td>19 (31)</td><td>10 (20)</td><td>37 (46)</td><td>26 (37)</td><td>8(17)</td><td>3 (12)</td></tr></table>
+
+Table 2: Zero-shot Domain Transfer. Left: Success percentages of the best BUTLER::BRAIN agents evaluated purely in TextWorld. Mid-Left: Success percentages after zero-shot transfer to embodied environments. Mid-Right: Success percentages of BUTLER with an oracle state-estimator and controller, an upper-bound. Right: Success percentages of BUTLER with human-annotated goal descriptions, an additional source of generalization difficulty. All successes are averaged across three evaluation runs. Goal-condition success rates (Shridhar et al., 2020) are given in parentheses. The Seq2Seq baseline is trained in TextWorld from pre-recorded expert demonstrations using standard supervised learning. BUTLER is our main model using the Mask R-CNN detector and $\mathbf { A } ^ { * }$ navigator. BUTLER-ORACLE uses an oracle state-estimator with ground-truth object detections and an oracle controller that directly teleports between locations.
+
+Comparing BUTLER to Seq2Seq, we see improved performance on all types of seen tasks and five of the seven types of unseen tasks, supporting the hypothesis that interactive TextWorld training is a key component in generalizing to unseen embodied tasks. Interactive language not only allows agents to explore and build an understanding of successful action patterns, but also to recover from mistakes. Through trial-and-error the BUTLER agent learns task-guided heuristics, e.g., searching all the drawers in kitchen to look for a knife. As Table 2 shows, these heuristics are subsequently more capable of generalizing to the embodied world. More details on TextWorld training and generalization performance can be found in Section 5.1.
+
+# 4.2 TRANSFERRING TO EMBODIED TASKS
+
+Since TextWorld is an abstraction of the embodied world, transferring between modalities involves overcoming domain gaps that are present in the real world but not in TextWorld. For example, the physical size of objects and receptacles must be respected – while TextWorld will allow certain objects to be placed inside any receptacle, in the embodied world it might be impossible to put a larger object into a small receptacle (e.g. a large pot into a microwave).
+
+Subsequently, a TextWorld-trained agent’s ability to solve embodied tasks is hindered by these domain gaps. So to study the transferability of the text agent in isolation, we introduce BUTLER-ORACLE in Table 2, an oracle variant of BUTLER which uses perfect state-estimation, object-detection, and navigation. Despite these advantages, we nevertheless observe a notable drop in performance from TextWorld to BUTLER-ORACLE. This performance gap results from the domain gaps described above as well as misdetections from Mask R-CNN and navigation failures caused by collisions. Future work might address this issue by reducing the domain gap between the two environments, or performing additional fine-tuning in the embodied setting.
+
+The supplementary video contains qualitative examples of the BUTLER agent solving tasks in unseen environments. It showcases 3 successes and 1 failure of a TextWorld-only agent trained on All Tasks. In “put a watch in the safe”, the agent has never seen the ‘watch’-‘safe’ combination as a goal.
+
+# 4.3 GENERALIZING TO HUMAN-ANNOTATED GOALS
+
+BUTLER is trained with templated language, but in realistic scenarios, goals are often posed with open-ended natural language. In Table 2, we present Human Goals results of BUTLER evaluated on human-annotated ALFRED goals, which contain 66 unseen verbs (e.g., ‘wash’, ‘grab’, ‘chill’) and 189 unseen nouns (e.g., ‘rag’, ‘lotion’, ‘disc’; see Appendix H for full list). Surprisingly, we find non-trivial goal-completion rate indicating that certain categories of task, such as pick and place, are quite generalizable to human language. While these preliminary results with natural language are encouraging, we expect future work could augment the templated language with synthetic-to-real transfer methods (Marzoev et al., 2020) for better generalization.
+
+4.4 TO PRETRAIN OR NOT TO PRETRAIN IN TEXTWORLD?
+
+Given the domain gap between TextWorld and the embodied world, Why not eliminate this gap by training from scratch in the embodied world? To answer this question, we investigate three training strategies: (i) EMBODIED-ONLY: pure embodied training, (ii) TW-ONLY: pure TextWorld training followed by zero-shot embodied transfer and
+
+<table><tr><td rowspan=2 colspan=1>Training Strategy</td><td rowspan=2 colspan=1> train(succ %)</td><td rowspan=1 colspan=1>seen</td><td rowspan=2 colspan=1>unseen  train speed(succ %)   (eps/s)</td></tr><tr><td rowspan=1 colspan=1>(succ %)</td></tr><tr><td rowspan=3 colspan=1>EMBODIED-ONLYTW-ONLYHYBRID</td><td rowspan=1 colspan=1>21.6</td><td rowspan=1 colspan=1>33.6</td><td rowspan=2 colspan=1>23.1      0.934.3      6.1</td></tr><tr><td rowspan=2 colspan=1>23.111.9</td><td rowspan=1 colspan=1>27.1</td></tr><tr><td rowspan=1 colspan=1>21.4</td><td rowspan=1 colspan=1>23.1      0.7</td></tr></table>
+
+Table 3: Training Strategy Success. Trained on All Tasks for 50K episodes and evaluated in embodied scenes using an oracle state-estimator and controller.
+
+(iii) HYBRID training that switches between the two environments with $7 5 \%$ probability for TextWorld and $2 5 \%$ for embodied world. Table 3 presents success rates for these agents trained and evaluated on All Tasks. All evaluations were conducted with an oracle state-estimator and controller. For a fair comparison, each agent is trained for 50K episodes and the training speed is recorded for each strategy. We report peak performance for each split.
+
+Results indicate that TW-ONLY generalizes better to unseen environments while EMBODIED-ONLY quickly overfits to seen environments (even with a perfect object detector and teleport navigator). We hypothesize that the abstract TextWorld environment allows the agent to focus on quickly learning tasks without having to deal execution-failures and expert-failures caused by physical constraints inherent to embodied environments. TextWorld training is also $7 \times$ faster4 since it does not require running a rendering or physics engine like in the embodied setting. See Section F for more quantitative evaluations on the benefits of training in TextWorld.
+
+# 5 ABLATIONS
+
+We conduct ablation studies to further investigate: (1) The generalization performance of BUTLER::BRAIN within TextWorld environments, (2) The ability of unimodal agents to learn directly through visual observations or action history, (3) The importance of various hyper-parameters and modeling choices for the performance of BUTLER::BRAIN.
+
+# 5.1 GENERALIZATION WITHIN TEXTWORLD
+
+We train and evaluate BUTLER::BRAIN in abstract TextWorld environments spanning the six tasks in Table 1, as well as All Tasks. Similar to the zero-shot results presented in Section 4.1, the All Tasks setting shows the extent to which a single policy can learn and generalize on the large set of 3,553 different tasks, but here without having to deal with failures from embodied execution.
+
+We first experimented with training BUTLER::BRAIN through reinforcement learning (RL) where the agent is rewarded after completing a goal. Due to the infesibility of using candidate commands or command templates as discussed in Section I, the RL agent had to generate actions token-by-token. Since the probability of randomly stumbling upon a grammatically correct and contextually valid action is very low (7.02e-44 for sequence length 10), the RL agent struggled to make any meaningful progress towards the tasks.
+
+After concluding that current reinforcement learning approaches were not successful on our set of training tasks, we turned to DAgger (Ross et al., 2011) assisted by a rule-based expert (detailed in Appendix E). BUTLER::BRAIN is trained for 100K episodes using data collected by interacting with the set of training games.
+
+Results in Table 4 show (i) Training success rate varies from $1 6 \mathrm { - } 6 0 \%$ depending on the category of tasks, illustrating the challenge of solving hundreds to thousands of training tasks within each category. (ii) Transferring from training to heldout test games typically reduces performance, with the unseen rooms leading to the largest performance drops. Notable exceptions include heat and cool tasks where unseen performance exceeds training performance. (iii) Beam search is a key contributor to test performance; its ablation causes a performance drop of $21 \%$ on the seen split of All Tasks. (iv) Further ablating the DAgger strategy and directly training a Sequence-to-Sequence (Seq2Seq) model
+
+<table><tr><td></td><td colspan="3">Pick &amp;Place</td><td colspan="3">Examine in Light</td><td colspan="3">Clean&amp; Place</td><td colspan="3">Heat &amp; Place</td><td colspan="3">Cool&amp; Place</td><td colspan="3">Pick Two&amp; Place</td><td colspan="3">All Tasks</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>tn</td><td>sn</td><td>un</td><td>tn</td><td>sn</td><td>un</td><td>tn</td><td>sn</td><td>un</td><td>tn</td><td>sn</td><td>un</td><td>tn</td><td>sn</td><td>un</td><td>tn</td><td>sn</td><td>un</td><td>tn</td><td>sn</td><td>un</td></tr><tr><td>BUTLER</td><td>54 54</td><td>61</td><td>46</td><td>59</td><td>39</td><td>22</td><td>37</td><td>44</td><td>39</td><td>60</td><td>81</td><td>74</td><td>46</td><td>60</td><td>100</td><td>27</td><td>29</td><td>24</td><td>16</td><td>40</td><td>37</td></tr><tr><td>BUTLERg Seq2Seq</td><td>31</td><td>43 26</td><td>33 8</td><td>59</td><td>31 31</td><td>17</td><td>37</td><td>30</td><td>26 42</td><td>60 36</td><td>69 50</td><td>70 30</td><td>46 27</td><td>50 32</td><td>76 33</td><td>27 17</td><td>38 8</td><td>12 6</td><td>16 9</td><td>19 10</td><td>22</td></tr><tr><td></td><td></td><td></td><td></td><td>44</td><td></td><td>11</td><td>34</td><td>30</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>9</td></tr></table>
+
+Table 4: Generalization within TextWorld environments: We independently train BUTLER::BRAIN on each type of TextWorld task and evaluate on heldout scenes of the same type. Respectively, tn/sn/un indicate success rate on train/seen/unseen tasks. All sn and un scores are computed using the random seeds (from 8 in total) producing the best final training score on each task type. BUTLER is trained with DAgger and performs beam search during evaluation. Without beam search, ${ \bf B U T L E R } _ { g }$ decodes actions greedily and gets stuck repeating failed actions. Further removing DAgger and training the model in a Seq2Seq fashion leads to worse generalization. Note that tn scores for BUTLER are lower than sn and un as they were computed without beam search.
+
+with pre-recorded expert demonstrations causes a bigger performance drop of $30 \%$ on seen split of All Tasks. These results suggest that online interaction with the environment, as facilitated by DAgger learning and beam search, is essential for recovering from mistakes and sub-optimal behavior.
+
+# 5.2 UNIMODAL BASELINES
+
+Table 5 presents results for unimodal baseline comparisons to BUTLER. For all baselines, the action space and controller are fixed, but the state space is substituted with different modalities. To study the agents’ capability of learning a single policy that generalizes across various tasks, we train and evaluate on All Tasks. In VISION (RESNET18), the textual observation from the state-estimator is replaced with ResNet-18 fc7 features (He et al., 2016) from the visual frame. Similarly, VISION (MCNN-FPN) uses the pre-trained Mask R-CNN from the state-estimator to extract FPN layer features for the whole image. ACTION-ONLY acts without any visual or textual feedback. We report peak performance for each split.
+
+Table 5: Unimodal Baselines. Trained on All Tasks with 50K episodes and evaluated in the embodied environment.   
+
+<table><tr><td>Agent</td><td>seen (succ %)</td><td>unseen (succ %)</td></tr><tr><td>BUTLER</td><td>18.8</td><td>10.1</td></tr><tr><td>VISION (RESNET18)</td><td>10.0</td><td>6.0</td></tr><tr><td>VISION (MCNN-FPN)</td><td>11.4</td><td>4.5</td></tr><tr><td>ACTION-ONLY</td><td>0.0</td><td>0.0</td></tr></table>
+
+The visual models tend to overfit to seen environments and generalize poorly to unfamiliar environments. Operating in text-space allows better transfer of policies without needing to learn state representations that are robust to visually diverse environments. The zero-performing ACTION-ONLY baseline indicates that memorizing action sequences is an infeasible strategy for agents.
+
+# 5.3 MODEL ABLATIONS
+
+Figure 4 illustrates more factors that affect the performance of BUTLER::BRAIN. The three rows of plots show training curves, evaluation curves in seen and unseen settings, respectively. All experiments were trained and evaluated on All Tasks with 8 random seeds.
+
+In the first column, we show the effect of using different observation queue lengths $k$ as described in Section 3.1, in which size 0 refers to not providing any observation information to the agent. In the second column, we examine the effect of explicitly keeping the initial observation $o _ { 0 }$ , which lists all the receptacles in the
+
+![](images/cafa440a954abe0d2b3d3483bd299ea55169f1739ab65bb72e98e401220dd06f.jpg)  
+Figure 4: Model ablations on All Tasks. $\mathbf { X }$ -axis: 0 to $5 0 \mathrm { k }$ episodes; y-axis: normalized success from 0 to $7 5 \%$ .
+
+scene. Keeping the initial observation $o _ { 0 }$ facilitates the decoder to generate receptacle words more accurately for unseen tasks, but may be unnecessary in seen environments. The third column suggests that the recurrent component in our aggregator is helpful in making history-based decisions particularly in seen environments where keeping track of object locations is useful. Finally, in the fourth column, we see that using more training games can lead to better generalizability in both seen and unseen settings. Fewer training games achieve high training scores by quickly overfitting, which lead to zero evaluation scores.
+
+# 6 RELATED WORK
+
+The longstanding goal of grounding language learning in embodied settings (Bisk et al., 2020) has lead to substantial work on interactive environments. ALFWorld extends that work with fully-interactive aligned environments that parallel textual interactions with photo-realistic renderings and physical interactions.
+
+Interactive Text-Only Environments: We build on the work of text-based environments like TextWorld (Côté et al., 2018) and Jericho (Hausknecht et al., 2020). While these environment allow for textual interactions, they are not grounded in visual or physical modalities.
+
+Vision and language: While substantial work exists on vision-language representation learning e.g., MAttNet (Yu et al., 2018b), CMN (Hu et al., 2017), VQA (Antol et al., 2015), CLEVR (Johnson et al., 2017), ViLBERT (Lu et al., 2019), they lack embodied or sequential decision making.
+
+Embodied Language Learning: To address language learning in embodied domains, a number of interactive environments have been proposed: BabyAI (Chevalier-Boisvert et al., 2019), Room2Room (Anderson et al., 2018b), ALFRED (Shridhar et al., 2020), InteractiveQA (Gordon et al., 2018), EmbodiedQA (Das et al., 2018), and NetHack (Küttler et al., 2020). These environments use language to communicate instructions, goals, or queries to the agent, but not as a fully-interactive textual modality.
+
+Language for State and Action Representation: Others have used language for more than just goal-specification. Schwartz et al. (2019) use language as an intermediate state to learn policies in VizDoom. Similarly, Narasimhan et al. (2018) and Zhong et al. (2020) use language as an intermediate representation to transfer policies across different environments. Hu et al. (2019) use a natural language instructor to command a low-level executor, and Jiang et al. (2019) use language as an abstraction for hierarchical RL. However these works do not feature an interactive text environment for pre-training the agent in an abstract textual space. Zhu et al. (2017) use high-level commands similar to ALFWorld to solve tasks in THOR with IL and RL-finetuning methods, but the policy only generalizes to a small set of tasks due to the vision-based state representation. Using symbolic representations for state and action is also an inherent characteristic of works in task-and-motionplanning (Kaelbling and Lozano-Pérez, 2011; Konidaris et al., 2018) and symbolic planning (Asai and Fukunaga, 2017).
+
+World Models: The concept of using TextWorld as a “game engine” to represent the world is broadly related to inverse graphics (Kulkarni et al., 2015) and inverse dynamics (Wu et al., 2017) where abstract visual or physical models are used for reasoning and future predictions. Similarly, some results in cognitive science suggest that humans use language as a cheaper alternative to sensorimotor simulation (Banks et al., 2020; Dove, 2014).
+
+# 7 CONCLUSION
+
+We introduced ALFWorld, the first interactive text environment with aligned embodied worlds. ALFWorld allows agents to explore, interact, and learn abstract polices in a textual environment. Pre-training our novel BUTLER agent in TextWorld, we show zero-shot generalization to embodied tasks in the ALFRED dataset. The results indicate that reasoning in textual space allows for better generalization to unseen tasks and also faster training, compared to other modalities like vision.
+
+BUTLER is designed with modular components which can be upgraded in future work. Examples include the template-based state-estimator and the $\mathbf { A } ^ { * }$ navigator which could be replaced with learned modules, enabling end-to-end training of the full pipeline. Another avenue of future work is to learn “textual dynamics models” through environment interactions, akin to vision-based world models (Ha and Schmidhuber, 2018). Such models would facilitate construction of text-engines for new domains, without requiring access to symbolic state descriptions like PDDL. Overall, we are excited by the challenges posed by aligned text and embodied environments for better cross-modal learning.
+
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+
+# A DETAILS OF BUTLER::BRAIN
+
+In this section, we use $o _ { t }$ to denote text observation at game step $t , g$ to denote the goal description provided by a game.
+
+We use $L$ to refer to a linear transformation and $L ^ { f }$ means it is followed by a non-linear activation function $f$ . Brackets $[ \cdot ; \cdot ]$ denote vector concatenation, $\odot$ denotes element-wise multiplication.
+
+# A.1 OBSERVATION QUEUE
+
+As mentioned in Section 3.1, we utilize an observation queue to cache the text observations that have been seen recently. Since the initial observation $o _ { 0 }$ describes the high level layout of a room, including receptacles present in the current game, we it visible to BUTLER::BRAIN at all game steps, regardless of the length of the observation queue. Specifically, the observation queue has an extra space storing $o _ { 0 }$ , at any game step, we first concatenate all cached observations in the queue, then prepend the $o _ { 0 }$ to form the input to the encoder. We find this helpful because it facilitates the pointer softmax mechanism in the decoder (described below) by guiding it to point to receptacle words in the observation. An ablation study on this is provided in Section 5.
+
+# A.2 ENCODER
+
+We use a transformer-based encoder, which consists of an embedding layer and a transformer block (Vaswani et al., 2017). Specifically, embeddings are initialized by pre-trained 768-dimensional BERT embeddings (Sanh et al., 2019). The embeddings are fixed during training in all settings.
+
+The transformer block consists of a stack of 5 convolutional layers, a self-attention layer, and a 2-layer MLP with a ReLU non-linear activation function in between. In the block, each convolutional layer has 64 filters, each kernel’s size is 5. In the self-attention layer, we use a block hidden size $H$ of 64, as well as a single head attention mechanism. Layernorm (Ba et al., 2016) is applied after each component inside the block. Following standard transformer training, we add positional encodings into each block’s input.
+
+At every game step $t$ , we use the same encoder to process text observation $o _ { t }$ and goal description g. The resulting representations are hot ∈ RLot ×H and $h _ { g } \in \mathbb { R } ^ { L _ { g } \times H }$ , where $L _ { o _ { t } }$ is the number of tokens in $o _ { t }$ , $L _ { g }$ denotes the number of tokens in $g$ $\phantom { } _ { l } , H = 6 \dot { 4 }$ is the hidden size.
+
+# A.3 AGGREGATOR
+
+We adopt the context-query attention mechanism from the question answering literature (Yu et al., 2018a) to aggregate the two representations $h _ { o _ { t } }$ and $h _ { g }$ .
+
+Specifically, a tri-linear similarity function is used to compute the similarity between each token in $h _ { o _ { t } }$ with each token in $h _ { g }$ . The similarity between $i$ -th token in $h _ { o }$ and $j$ -th token in $h _ { g }$ is thus computed by (omitting game step $t$ for simplicity):
+
+$$
+\mathrm { S i m } ( i , j ) = W ( h _ { o _ { i } } , h _ { g _ { j } } , h _ { o _ { i } } \odot h _ { g _ { j } } ) ,
+$$
+
+where $W$ is a trainable parameter in the tri-linear function. By applying the above computation for each ho and hg pair, we get a similarity matrix S ∈ RLo×Lg .
+
+By computing the softmax of the similarity matrix $S$ along both dimensions (number of tokens in goal description $L _ { g }$ and number of tokens in observation $L _ { o . }$ ), we get $S _ { g }$ and $S _ { o }$ , respectively. The two representations are then aggregated by:
+
+$$
+\begin{array} { r l r } & { } & { h _ { o g } = [ h _ { o } ; P ; h _ { o } \odot P ; h _ { o } \odot Q ] , } \\ & { } & { P = S _ { g } h _ { g } ^ { \top } , \qquad } \\ & { } & { Q = S _ { g } S _ { o } ^ { \top } h _ { o } ^ { \top } , \qquad } \end{array}
+$$
+
+where $h _ { o g } \in \mathbb { R } ^ { L _ { o } \times 4 H }$ is the aggregated observation representation.
+
+Next, a linear transformation projects the aggregated representations to a space with size $H = 6 4$ :
+
+$$
+h _ { o g } = L ^ { \mathrm { t a n h } } ( h _ { o g } ) .
+$$
+
+To incorporate history, we use a recurrent neural network. Specifically, we use a GRU (Cho et al., 2014):
+
+$$
+\begin{array} { r } { h _ { \mathrm { R N N } } = \mathrm { M e a n } ( h _ { o g } ) , \qquad } \\ { h _ { t } = \mathrm { G R U } ( h _ { \mathrm { R N N } } , h _ { t - 1 } ) , \qquad } \end{array}
+$$
+
+in which, the mean pooling is performed along the dimension of number of tokens, i.e., $h _ { \mathrm { R N N } } \in \mathbb { R } ^ { H }$ .   
+$h _ { t - 1 }$ is the output of the GRU cell at game step $t - 1$ .
+
+# A.4 DECODER
+
+Our decoder consists of an embedding layer, a transformer block and a pointer softmax mechanism (Gulcehre et al., 2016). We first obtain the source representation by concatenating $h _ { o g }$ and $h _ { t }$ , resulting hsrc ∈ RLo×2H .
+
+Similar to the encoder, the embedding layer is frozen after initializing it with pre-trained BERT embeddings. The transformer block consists of two attention layers and a 3-layer MLP with ReLU non-linear activation functions inbetween. The first attention layer computes the self attention of the input embeddings $h _ { \mathrm { s e l f } }$ as a contextual encoding for the target tokens. The second attention layer then computes the attention $\boldsymbol { \alpha } _ { \mathrm { s r c } } ^ { i } \in \mathbb { R } ^ { L _ { o } }$ between the source representation $h _ { \mathrm { { s r c } } }$ and the $i$ -th token in $h _ { \mathrm { s e l f } }$ . The $i \cdot$ -th target token is consequently represented by the weighted sum of $h _ { \mathrm { { s r c } } }$ , with the weights $\alpha _ { \mathrm { s r c } } ^ { i }$ . This generates a source information-aware target representation $h _ { \mathrm { t g t } } ^ { \prime } \in \mathbb { R } ^ { L _ { \mathrm { t g t } } \times H }$ , where $L _ { \mathrm { t g t } }$ denotes the number of tokens in the target sequence. Next, $h _ { \mathrm { t g t } } ^ { \prime }$ is fed into the 3-layer MLP with ReLU activation functions inbetween, resulting $h _ { \mathrm { t g t } } \in \mathbb { R } ^ { L _ { \mathrm { t g t } } \times H }$ . The block hidden size of this transformer is $H = 6 4$ .
+
+Taking $h _ { \mathrm { t g t } }$ as input, a linear layer with tanh activation projects the target representation into the same space as the embeddings (with dimensionality of 768), then the pre-trained embedding matrix $E$ generates output logits (Press and Wolf, 2016), where the output size is same as the vocabulary size. The resulting logits are then normalized by a softmax to generate a probability distribution over all tokens in vocabulary:
+
+$$
+p _ { a } ( y ^ { i } ) = E ^ { \mathrm { S o f t m a x } } ( L ^ { \mathrm { t a n h } } ( h _ { \mathrm { t g t } } ) ) ,
+$$
+
+in which, $p _ { a } ( y ^ { i } )$ is the generation (abstractive) probability distribution.
+
+We employ the pointer softmax (Gulcehre et al., 2016) mechanism to switch between generating a token $y ^ { i }$ (from a vocabulary) and pointing (to a token in the source text). Specifically, the pointer softmax module computes a scalar switch $s ^ { i }$ at each generation time-step $i$ and uses it to interpolate the abstractive distribution $p _ { a } ( y ^ { i } )$ over the vocabulary (Equation 5) and the extractive distribution $p _ { x } ( y ^ { i } ) = \alpha _ { \mathrm { s r c } } ^ { i }$ over the source text tokens:
+
+$$
+p ( y ^ { i } ) = s ^ { i } \cdot p _ { a } ( y ^ { i } ) + ( 1 - s ^ { i } ) \cdot p _ { x } ( y ^ { i } ) ,
+$$
+
+where $s ^ { i }$ is conditioned on both the attention-weighted source representation $\sum _ { j } \alpha _ { \mathrm { s r c } } ^ { i , j } \cdot h _ { \mathrm { s r c } } ^ { j }$ and the decoder state $h _ { \mathrm { t g t } } ^ { i }$ :
+
+$$
+s ^ { i } = L _ { 1 } ^ { \mathrm { s i g m o i d } } ( \operatorname { t a n h } ( L _ { 2 } ( \sum _ { j } \alpha _ { \mathrm { s r c } } ^ { i , j } \cdot h _ { \mathrm { s r c } } ^ { j } ) + L _ { 3 } ( h _ { \mathrm { t g t } } ^ { i } ) ) ) .
+$$
+
+In which, $\boldsymbol { L } _ { 1 } \in \mathbb { R } ^ { H \times 1 }$ , $L _ { 2 } \in \mathbb { R } ^ { 2 H \times H }$ and L3 ∈ RH×H are linear layers, $H = 6 4$ .
+
+# B TRAINING AND IMPLEMENTATION DETAILS
+
+In this section, we provide hyperparameters and other implementation details.
+
+For all experiments, we use Adam (Kingma and Ba, 2014) as the optimizer. The learning rate is set to 0.001 with a clip gradient norm of 5.
+
+During training with DAgger, we use a batch size of 10 to collect transitions (tuples of $\{ o _ { 0 } , o _ { t } , g , \hat { a } _ { t } \} )$ at each game step $t$ , where $\hat { a } _ { t }$ is the ground-truth action provided by the rule-based expert (see Section E). We gather a sequence of transitions from each game episode, and push each sequence into a replay buffer, which has a capacity of 500K episodes. We set the max number of steps per episode to be 50. If the agent uses up this budget, the game episode is forced to terminate. We linearly anneal the fraction of the expert’s assistance from $100 \%$ to $1 \%$ across a window of 50K episodes.
+
+The agent is updated after every 5 steps of data collection. We sample a batch of 64 data points from the replay buffer. In the setting with the recurrent aggregator, every sampled data point is a sequence of 4 consecutive transitions. Following the training strategy used in the recurrent DQN literature (Hausknecht and Stone, 2015; Yuan et al., 2018), we use the first 2 transitions to estimate the recurrent states, and the last 2 transitions for updating the model parameters.
+
+BUTLER::BRAIN learns to generate actions token-by-token, where we set the max token length to be 20. The decoder stops generation either when it generates a special end-of-sentence token [EOS], or hits the token length limit.
+
+When using the beam search heuristic to recover from failed actions (see Figure 5), we use a beam width of 10, and take the top-5 ranked outputs as candidates. We iterate through the candidates in the rank order until one of them succeeds. This heuristic is not always guaranteed to succeed, however, we find it helpful in most cases. Note that we do not employ beam search when we evaluate during the training process for efficiency, e.g., in the seen and unseen curves shown in Figure 4. We take the best performing checkpoints and then apply this heuristic during evaluation and report the resulting scores in tables (e.g., Table 2).
+
+![](images/89b5335ffe53ec6f8b1ec9b1693a7b18a6a70e03080d0b23ed7deaeeaf3202b9.jpg)  
+Figure 5: Beam search for recovery actions.
+
+By default unless mentioned otherwise (ablations), we use all available training games in each of the task types. We use an observation queue length of 5 and use a recurrent aggregator. The model is trained with DAgger, and during evaluation, we apply the beam search heuristic to produce the reported scores. All experiment settings in TextWorld are run with 8 random seeds. All text agents are trained for 50,000 episodes.
+
+# C TEXTWORLD ENGINE
+
+Internally, the TextWorld Engine is divided into two main components: a planner and text generator.
+
+Planner TextWorld Engine uses Fast Downward (Helmert, 2006), a domain-independent classical planning system to maintain and update the current state of the game. A state is represented by a set of predicates which define the relations between the entities (objects, player, room, etc.) present in the game. A state can be modified by applying production rules corresponding to the actions listed in Table 6. All variables, predicates, and rules are defined using the PDDL language.
+
+For instance, here is a simple state representing a player standing next to a microwave which is closed and contains a mug:
+
+$$
+\begin{array} { r } { \begin{array} { r l } { s _ { t } = } & { \mathsf { a t } \big ( p \mathcal { I } a y e r , m i c r o w a v e \big ) \otimes \mathrm { i n } \big ( m u g , m i c r o w a v e \big ) } \\ & { \otimes \mathrm { c l o s e d } \big ( m i c r o w a v e \big ) \otimes \mathrm { o p e n a b l e } \big ( m i c r o w a v e \big ) , } \end{array} } \end{array}
+$$
+
+where the symbol $\otimes$ is the linear logic multiplicative conjunction operator. Given that state, a valid action could be open microwave, which would essentially transform the state by replacing $\mathsf { c l o s e d } ( m i c r o w a v e )$ with open(microwave).
+
+Text generator The other component of the TextWorld Engine, the text generator, uses a contextsensitive grammar designed for the ALFRED environments. The grammar consists of text templates similar to those listed in Table 6. When needed, the engine will sample a template given some context,
+
+i.e., the current state and the last action. Then, the template gets realized using the predicates found in the current state.
+
+# D MASK R-CNN DETECTOR
+
+We use a Mask R-CNN detector (He et al., 2017) pre-trained on MSCOCO (Lin et al., 2014) and fine-tune it with additional labels from ALFRED training scenes. To generate additional labels, we replay the expert demonstrations from ALFRED and record ground-truth image and instance segmentation pairs from the simulator (THOR) after completing each high-level action e.g., goto, pickup etc. We generate a dataset of 50K images, and fine-tune the detector for 4 epochs with a batch size of 8 and a learning rate of 5e-4. The detector recognizes 73 object classes where each class could vary up to 1-10 instances. Since demonstrations in the kitchen are often longer as they involve complex sequences like heating, cleaning etc., the labels are slightly skewed towards kitchen objects. To counter this, we balance the number of images sampled from each room (kitchen, bedroom, livingroom, bathroom) so the distribution of object categories is uniform across the dataset.
+
+# E RULE-BASED EXPERT
+
+To train text agents in an imitation learning (IL) setting, we use a rule-based expert for supervision. A given task is decomposed into sequence of subgoals (e.g., for heat & place: find the object, pick the object, find the microwave, heat the object with the microwave, find the receptacle, place the object in the receptacle), and a closed-loop controller tries to sequentially execute these goals. We note that while designing rule-based experts for ALFWorld is relatively straightforward, experts operating directly in embodied settings like the PDDL planner used in ALFRED are prone to failures due to physical infeasibilities and non-deterministic behavior in physics-based environments.
+
+# F BENEFITS OF TRAINING IN TEXTWORLD OVER EMBODIED WORLD
+
+Pre-training in TextWorld offers several benefits over directly training in embodied environments. Figure 6 presents the performance of an expert (that agents are trained to imitate) across various environments. The abstract textual space leads to higher goal success rates resulting from successful navigation and manipulation subroutines. TextWorld agents also do not suffer from object misdetections and slow execution speed.
+
+![](images/551c5791f186d6727bf3786990a371ae20e8639ef9b39d7574ef6763f75493b1.jpg)  
+Figure 6: Domain Analysis: The performance of an expert across various environments.
+
+# G OBSERVATION TEMPLATES
+
+The following templates are used by the state-estimator to generate textual observations $o _ { t }$ . The object IDs {obj id} correspond to Mask R-CNN objects detection or ground-truth instance IDs. The receptacle IDs {recep id} are based on the receptacles listed in the initial observation $o _ { 0 }$ . Failed actions and actions without any state-changes result in Nothing happens.
+
+Table 6: High-level text actions supported in ALFWorld along with their observation templates.   
+
+<table><tr><td>Actions</td><td>Templates</td></tr><tr><td>goto</td><td>(a)You arrive at {loc id}. On the {recep id}, you see a {obj1 id}， ..． and a {objN id}. (b)You arrive at {loc id}. The {recep id} is closed. (c) You arrive at {loc id}. The {recep id} is open. On it，you see a {obji id}，... and a {objN id}.</td></tr><tr><td>take</td><td>You pick up the {obj id} from the {recep id}.</td></tr><tr><td>put</td><td>You put the {obj id} on the {recep id}.</td></tr><tr><td>open</td><td>(a) You open the {recep id}. In it， you see a {obj1 id}，... and a {objN id}. (b)You open the {recep id}. The {recep id} is empty.</td></tr><tr><td>close</td><td>You close the {recep id}.</td></tr><tr><td>toggle</td><td>You turn the {obj id} on.</td></tr><tr><td>heat</td><td>You heat the {obj id} with the {recep id}.</td></tr><tr><td>cool</td><td>You cool the {obj id} with the {recep id}.</td></tr><tr><td>clean</td><td>You clean the {obj id} with the {recep id}.</td></tr><tr><td>inventory</td><td>(a) You are carrying: {obj id}. (b)You are not carrying anything.</td></tr><tr><td>examine</td><td>(a) On the {recep id}，you see a {obj1 id}，.. and a {objN id}. (b)This is a hot/cold/clean {obj).</td></tr></table>
+
+# H GOAL DESCRIPTIONS
+
+# H.1 TEMPLATED GOALS
+
+The goal instructions for training games are generated with following templates. Here obj, recep, lamp refer to object, receptacle, and lamp classes, respectively, that pertain to a particular task. For each task, the two corresponding templates are sampled with equal probability.
+
+Table 7: Task-types and the corresponding goal description templates.   
+
+<table><tr><td>task-type</td><td>Templates</td></tr><tr><td>Pick&amp;Place</td><td>(a)put a {obj} in {recep}. (b) put some {obj} on{recep}.</td></tr><tr><td>Examine in Light</td><td>(a)look at {obj} under the {lamp}. (b)examine ethe {obj} with the {lamp}.</td></tr><tr><td>Clean&amp;Place</td><td>(a)put a clean {obj} in {recep}. (b) clean some {obj} and put it in {recep}.</td></tr><tr><td>Heat&amp;Place</td><td>(a)puta hot {obj} in {recep}. (b)heat some {obj} and put it in {recep}.</td></tr><tr><td>Cool&amp;Place</td><td>(a)put a cool {obj} in {recep}. (b)cool some {obj} and put it in {recep}.</td></tr><tr><td>Pick Two&amp;Place</td><td>(a)put two {obj} in {recep}. (b) find two {obj} and put them {recep}.</td></tr></table>
+
+# H.2 HUMAN ANNOTATED GOALS
+
+The human goal descriptions used during evaluation contain 66 unseen verbs and 189 unseen nouns with respect to the templated goal instructions used during training.
+
+Unseen Verbs: acquire, arrange, can, carry, chill, choose, cleaning, clear, cook, cooked, cooled, dispose, done, drop, end, fill, filled, frying, garbage, gather, go, grab, handled, heated, heating, hold, holding, inspect, knock, left, lit, lock, microwave, microwaved, move, moving, pick, picking, place, placed, placing, putting, read, relocate, remove, retrieve, return, rinse, serve, set, soak, stand, standing, store, take, taken, throw, transfer, turn, turning, use, using, walk, warm, wash, washed.
+
+Unseen Nouns: alarm, area, back, baisin, bar, bars, base, basin, bathroom, beat, bed, bedroom, bedside, bench, bin, books, bottle, bottles, bottom, box, boxes, bureau, burner, butter, can, canteen, card, cardboard, cards, cars, cds, cell, chair, chcair, chest, chill, cistern, cleaning, clock, clocks, coffee, container, containers, control, controllers, controls, cooker, corner, couch, count, counter, cover, cream, credit, cupboard, dining, disc, discs, dishwasher, disks, dispenser, door, drawers, dresser, edge, end, floor, food, foot, freezer, game, garbage, gas, glass, glasses, gold, grey, hand, head, holder, ice, inside, island, item, items, jars, keys, kitchen, knifes, knives, laddle, lamp, lap, left, lid, light, loaf, location, lotion, machine, magazine, maker, math, metal, microwaves, move, nail, newsletters, newspapers, night, nightstand, object, ottoman, oven, pans, paper, papers, pepper, phone, piece, pieces, pillows, place, polish, pot, pullout, pump, rack, rag, recycling, refrigerator, remote, remotes, right, rinse, roll, rolls, room, safe, salt, scoop, seat, sets, shaker, shakers, shelves, side, sink, sinks, skillet, soap, soaps, sofa, space, spatulas, sponge, spoon, spot, spout, spray, stand, stool, stove, supplies, table, tale, tank, television, textbooks, time, tissue, tissues, toaster, top, towel, trash, tray, tv, vanity, vases, vault, vegetable, wall, wash, washcloth, watches, water, window, wine.
+
+# I ACTION CANDIDATES VS ACTION GENERATION
+
+BUTLER::BRAIN generates actions in a token-by-token fashion. Prior text-based agents typically use a list of candidate commands from the game engine (Adhikari et al., 2020) or populate a list of command templates (Ammanabrolu and Hausknecht, 2020). We initially trained our agents with candidate commands from the TextWorld Engine, but they quickly ovefit without learning affordances, commonsense, or pre-conditions, and had zero performance on embodied transfer. In the embodied setting, without access to a TextWorld Engine, it is difficult to generate candidate actions unless a set of heuristics is handcrafted with strong priors and commonsense knowledge. We also experimented with populating a list of command templates, but found this to be infeasible as some scenarios involved 1000s of populated actions per game step.
+
+# J ALFRED TASK DESCRIPTIONS
+
+The following descriptions describe the processes involved in each of six task-types:
+
+• Pick & Place (e.g., “put a plate on the coffee table”) - the agent must find an object of the desired type, pick it up, find the correct location to place it, and put it down there.   
+• Examine in Light (e.g., “examine a book under the lamp”) - the agent must find an object of the desired type, locate and turn on a light source with the desired object in-hand.   
+• Clean & Place (e.g., “clean the knife and put in the drawer”) - the agent must find an object of the desired type, pick it up, go to a sink or a basin, wash the object by turning on the faucet, then find the correct location to place it, and put it down there.   
+• Heat & Place (e.g., “heat a mug and put on the coffee table”) - the agent must find an object of the desired type, pick it up, go to a microwave, heat the object turning on the microwave, then find the correct location to place it, and put it down there.   
+• Cool & Place (e.g., “put a cool bottle on the countertop”) - the agent must find an object of the desired type, pick it up, go to a fridge, put the object inside the fridge and cool it, then find the correct location to place it, and put it down there.   
+• Pick Two & Place (e.g., “put two pencils in the drawer”) - the agent must find an object of the desired type, pick it up, find the correct location to place it, put it down there, then look for another object of the desired type, pick it up, return to previous location, and put it down there with the other object.
+
+# K ALFWORLD TEXT GAME EXAMPLES
+
+Three examples of ALFWorld text game walkthroughs:
+
+You are in the middle of a room. Looking quickly around you, you see a towelholder 1, a toilet 1, a bathtubbasin 1, a drawer 4, a handtowelholder 2, a drawer 6, a drawer 1, a countertop 1, a sinkbasin 1, a drawer 2, a drawer 3, a toiletpaperhanger 1, a drawer 5, a handtowelholder 1, a towelholder 2, a sinkbasin 2, and a garbagecan 1.
+
+Your task is to: put a clean cloth in bathtubbasin.
+
+$>$ go to drawer 5   
+You arrive at loc 9. The drawer 5 is closed.   
+> open drawer 5   
+You open the drawer 5. The drawer 5 is open. In it, you see a candle 1.   
+> close drawer 5   
+You close the drawer 5.   
+$>$ go to countertop 1   
+You arrive at loc 25. On the countertop 1, you see a cloth 2, a soapbottle 1, a cloth 1, and a candle 3.   
+> take cloth 1 from countertop 1   
+You pick up the cloth 1 from the countertop 1.   
+$>$ go to sinkbasin 1   
+You arrive at loc 20. On the sinkbasin 1, you see nothing.   
+> clean cloth 1 with sinkbasin 1   
+You clean the cloth 1 using the sinkbasin 1.   
+> go to bathtubbasin 1   
+You arrive at loc 14. On the bathtubbasin 1, you see a soapbar 1.   
+> put cloth 1 in/on bathtubbasin 1   
+You won!
+
+You are in the middle of a room. Looking quickly around you, you see a drawer 15, a drawer 19, a drawer 7, a armchair 1, a drawer 3, a drawer 16, a drawer 2, a drawer 17, a sidetable 3, a sidetable 5, a drawer 4, a drawer 20, a sofa 1, a drawer 13, a drawer 12, a drawer 9, a drawer 21, a sidetable 4, a sidetable 2, a drawer 5, a drawer 11, a drawer 1, a garbagecan 1, a dresser 1, a cabinet 1, a drawer 8, a drawer 18, a sidetable 1, a drawer 14, a drawer 6, and a drawer 10.
+
+Your task is to: find two remotecontrol and put them in armchair.
+
+$>$ go to sidetable 2   
+You arrive at loc 34. On the sidetable 2, you see a remotecontrol 1.   
+$>$ take remotecontrol 1 from sidetable 2   
+You pick up the remotecontrol 1 from the sidetable 2.   
+$>$ go to armchair 1   
+You arrive at loc 1. On the armchair 1, you see nothing.   
+$>$ put remotecontrol 1 in/on armchair 1   
+You put the remotecontrol 1 in/on the armchair 1.   
+$>$ go to dresser 1   
+You arrive at loc 18. On the dresser 1, you see a creditcard 1, a television 1, a laptop 1, a book 1, a statue 2, and a cellphone 2.   
+> go to sidetable 5   
+You arrive at loc 17. On the sidetable 5, you see a box 1, and a creditcard 2.   
+$>$ go to sofa 1   
+You arrive at loc 2. On the sofa 1, you see a newspaper 1, a pillow 1, and a remotecontrol 2. $>$ take remotecontrol 2 from sofa 1   
+You pick up the remotecontrol 2 from the sofa 1.   
+> go to armchair 1   
+You arrive at loc 1. On the armchair 1, you see a remotecontrol 1.   
+> put remotecontrol 2 in/on armchair 1   
+You won!
+
+You are in the middle of a room. Looking quickly around you, you see a safe 1, a shelf 4, a drawer 2, a bed 1, a drawer 1, a shelf 5, a shelf 2, a sidetable 2, a shelf 3, a drawer 3, a shelf 1, a sidetable 1, a desk 1, and a garbagecan 1.

parse/train/CZ8Y3NzuVzO/CZ8Y3NzuVzO.md

@@ -0,0 +1,251 @@
+# WHAT SHOULD NOT BE CONTRASTIVE IN CONTRASTIVE LEARNING
+
+Tete Xiao UC Berkeley
+
+Xiaolong Wang UC San Diego
+
+Alexei A. Efros UC Berkeley
+
+Trevor Darrell UC Berkeley
+
+# ABSTRACT
+
+Recent self-supervised contrastive methods have been able to produce impressive transferable visual representations by learning to be invariant to different data augmentations. However, these methods implicitly assume a particular set of representational invariances (e.g., invariance to color), and can perform poorly when a downstream task violates this assumption (e.g., distinguishing red vs. yellow cars). We introduce a contrastive learning framework which does not require prior knowledge of specific, task-dependent invariances. Our model learns to capture varying and invariant factors for visual representations by constructing separate embedding spaces, each of which is invariant to all but one augmentation. We use a multi-head network with a shared backbone which captures information across each augmentation and alone outperforms all baselines on downstream tasks. We further find that the concatenation of the invariant and varying spaces performs best across all tasks we investigate, including coarse-grained, fine-grained, and few-shot downstream classification tasks, and various data corruptions.
+
+# 1 INTRODUCTION
+
+Self-supervised learning, which uses raw image data and/or available pretext tasks as its own supervision, has become increasingly popular as the inability of supervised models to generalize beyond their training data has become apparent. Different pretext tasks have been proposed with different transformations, such as spatial patch prediction (Doersch et al., 2015; Noroozi & Favaro, 2016), colorization (Zhang et al., 2016; Larsson et al., 2016; Zhang et al., 2017), rotation (Gidaris et al., 2018). Whereas pretext tasks aim to recover the transformations between different “views” of the same data, more recent contrastive learning methods (Wu et al., 2018; Tian et al., 2019; He et al., 2020; Chen et al., 2020a) instead try to learn to be invariant to these transformations, while remaining discriminative with respect to other data points. Here, the transformations are generated using classic data augmentation techniques which correspond to common pretext tasks, e.g., randomizing color, texture, orientation and cropping.
+
+Yet, the inductive bias introduced through such augmentations is a double-edged sword, as each augmentation encourages invariance to a transformation which can be beneficial in some cases and harmful in others: e.g., adding rotation may help with view-independent aerial image recognition, but significantly downgrade the capacity of a network to solve tasks such as detecting which way is up in a photograph for a display application. Current self-supervised contrastive learning methods assume implicit knowledge of downstream task invariances. In this work, we propose to learn visual representations which capture individual factors of variation in a contrastive learning framework without presuming prior knowledge of downstream invariances.
+
+Instead of mapping an image into a single embedding space which is invariant to all the handcrafted augmentations, our model learns to construct separate embedding sub-spaces, each of which is sensitive to a specific augmentation while invariant to other augmentations. We achieve this by optimizing multiple augmentation-sensitive contrastive objectives using a multi-head architecture with a shared backbone. Our model aims to preserve information with regard to each augmentation in a unified representation, as well as learn invariances to them. The general representation trained with these augmentations can then be applied to different downstream tasks, where each task is free to selectively utilize different factors of variation in our representation. We consider transfer of either the shared backbone representation, or the concatenation of all the task-specific heads; both outperform all baselines; the former uses same embedding dimensions as typical baselines, while the latter provides greatest overall performance in our experiments. In this paper, we experiment with three types of augmentations: rotation, color jittering, and texture randomization, as visualized in Figure 1. We evaluate our approach across a variety of diverse tasks including large-scale classification (Deng et al., 2009), fine-grained classification (Wah et al., 2011; Van Horn et al., 2018), few-shot classification (Nilsback & Zisserman, 2008), and classification on corrupted data (Barbu et al., 2019; Hendrycks & Dietterich, 2019). Our representation shows consistent performance gains with increasing number of augmentations. Our method does not require hand-selection of data augmentation strategies, and achieves better performance against state-of-the-art MoCo baseline (He et al., 2020; Chen et al., 2020b), and demonstrates superior transferability, generalizability and robustness across tasks and categories. Specifically, we obtain around $1 0 \%$ improvement over MoCo in classification when applied on the iNaturalist (Van Horn et al., 2018) dataset.
+
+![](images/489934d534cfeaf7066c2a579b0b58a5ee066fdcaa831c70b5d6d9dc6dfd313b.jpg)  
+Figure 1: Self-supervised contrastive learning relies on data augmentations as depicted in (a) to learn visual representations. However, current methods introduce inductive bias by encouraging neural networks to be less sensitive to information w.r.t. augmentation, which may help or may hurt. As illustrated in (b), rotation invariant embeddings can help on certain flower categories, but may hurt animal recognition performance; conversely color invariance generally seems to help coarse grained animal classification, but can hurt many flower categories and bird categories. Our method, shown in the following figure, overcomes this limitation.
+
+# 2 BACKGROUND: CONTRASTIVE LEARNING FRAMEWORK
+
+Contrastive learning learns a representation by maximizing similarity and dissimilarity over data samples which are organized into similar and dissimilar pairs, respectively. It can be formulated as a dictionary look-up problem (He et al., 2020), where a given reference image $\mathcal { T }$ is augmented into two views, query and key, and the query token $q$ should match its designated key $k ^ { + }$ over a set of sampled negative keys $\{ k ^ { - } \}$ from other images. In general, the framework can be summarized as the following components: (i) A data augmentation module $\tau$ constituting $n$ atomic augmentation operators, such as random cropping, color jittering, and random flipping. We denote a pre-defined atomic augmentation as random variable $X _ { i }$ . Each time the atomic augmentation is executed by sampling a specific augmentation parameter from the random variable, i.e., $x _ { i } \sim X _ { i }$ . One sampled data augmentation module transforms image $\mathcal { T }$ into a random view $\widetilde { \boldsymbol { \tau } }$ , denoted as $\widetilde { \mathcal { T } } = \mathcal { T } [ x _ { 1 } , x _ { 2 } , \dots , x _ { n } ] \left( \mathcal { T } \right)$ . Positive pair $( q , k ^ { + } )$ is generated by applying two randomly sampled data augmentation on the same reference image. (ii) An encoder network $f$ which extracts the feature $\textbf {  { v } }$ of an image $\mathcal { T }$ by mapping it into a $d$ -dimensional space $\mathbb { R } ^ { d }$ . (iii) A projection head $h$ which further maps extracted representations into a hyper-spherical (normalized) embedding space. This space is subsequently used for a specific pretext task, i.e., contrastive loss objective for a batch of positive/negative pairs. A common choice is InfoNCE (Oord et al., 2018):
+
+$$
+\mathcal { L } _ { q } = - \log \frac { \exp { ( q \cdot k ^ { + } / \tau ) } } { \exp { ( q \cdot k ^ { + } / \tau ) } + \sum _ { k ^ { - } } \exp { ( q \cdot k ^ { - } / \tau ) } } ,
+$$
+
+where $\tau$ is a temperature hyper-parameter scaling the distribution of distances.
+
+As a key towards learning a good feature representation (Chen et al., 2020a), a strong augmentation policy prevents the network from exploiting na¨ıve cues to match the given instances. However, inductive bias is introduced through the selection of augmentations, along with their hyper-parameters defining the strength of each augmentation, manifested in Equation 1 that any views by the stochastic augmentation module $\tau$ of the same instance are mapped onto the same point in the embedding space. The property negatively affects the learnt representations: 1) Generalizability and transferability are harmed if they are applied to the tasks where the discarded information is essential, e.g., color plays an important role in fine-grained classification of birds; 2) Adding an extra augmentation is complicated as the new operator may be helpful to certain classes while harmful to others, e.g., a rotated flower could be very similar to the original one, whereas it does not hold for a rotated car; 3) The hyper-parameters which control the strength of augmentations need to be carefully tuned for each augmentation to strike a delicate balance between leaving a short-cut open and completely invalidate one source of information.
+
+![](images/d810d82d1477f353bd4e21496d9e7c0d7369cd3fbde5d059d8aa21840585f046.jpg)  
+Figure 2: Framework of the Leave-one-out Contrastive Learning approach, illustrated with two types of augmentations, i.e., random rotation and color jittering. We generate multiple views with leave-one-out strategy, then project their representations into separate embedding spaces with contrastive objective, where each embedding space is either invariant to all augmentations, or invariant to all but one augmentation. The learnt representation can be the general embedding space $\nu$ (blue region), or the concatenation of embedding sub-spaces $\mathcal { Z }$ (grey region). Our results show that either of our proposed representations are able to outperform baseline contrastive embeddings and do not suffer from decreased performance when adding augmentations to which the task is not invariant (i.e., the red X’s in Figure 1).
+
+# 3 LOOC: LEAVE-ONE-OUT CONTRASTIVE LEARNING
+
+We propose Leave-one-out Contrastive Learning (LooC), a framework for multi-augmentation contrastive learning. Our framework can selectively prevent information loss incurred by an augmentation. Rather than projecting every view into a single embedding space which is invariant to all augmentations, in our LooC method the representations of input images are projected into several embedding spaces, each of which is not invariant to a certain augmentation while remaining invariant to others, as illustrated in Figure 2. In this way, each embedding sub-space is specialized to a single augmentation, and the shared layers will contain both augmentation-varying and invariant information. We learn a shared representation jointly with the several embedding spaces; we transfer either the shared representation alone, or the concatenation of all spaces, to downstream tasks.
+
+View Generation. Given a reference image and $n$ atomic augmentations, we first augment the reference image with two sets of independently sampled augmentation parameters into the query view $\mathcal { T } _ { q }$ and the first key view $\mathcal { T } _ { k _ { 0 } }$ , i.e., $\dot { \mathcal { T } } _ { \{ q , k _ { 0 } \} } = \mathcal { T } [ x _ { 1 } ^ { \{ q , k _ { 0 } \} } , x _ { 2 } ^ { \{ q , k _ { 0 } \} } , \ldots , \dot { x } _ { n } ^ { \{ q , k _ { 0 } \} } ] ( \mathbb { Z } ) .$ x{q,k0}n ] (I ). Additionally, we generate $n$ views from the reference image as extra key views, denoted as ${ \mathcal { T } } _ { k _ { i } }$ , $\forall i \in \left\{ 1 , \ldots , n \right\}$ . For the $i ^ { t h }$ additional key view, the parameter of $i ^ { t h }$ atomic augmentation is copied from it of the query view, i.e., $x _ { i } ^ { k _ { i } } \equiv x _ { i } ^ { q }$ , $\forall i \in \left\{ 1 , \ldots , n \right\}$ ; whereas the parameter of other atomic augmentations are still independently sampled, i.e., $x _ { j } ^ { k _ { i } } \sim X _ { j }$ , $\forall j \ne i$ . For instance, assume that we have a set of two atomic augmentations {random_rotation, color_jitter}, $\mathcal { T } _ { q }$ and $\mathcal { T } _ { k _ { 1 } }$ are always augmented by the same rotation angle but different color jittering; $\mathcal { T } _ { q }$ and $\mathcal { T } _ { k _ { 2 } }$ are always augmented by the same color jittering but different rotation angle; $\mathcal { T } _ { q }$ and ${ \mathcal { T } } _ { k _ { 0 } }$ are augmented independently, as illustrated in the left part of Figure 2.
+
+Contrastive Embedding Space. The augmented views are encoded by a neural network encoder $f ( \cdot )$ into feature vectors $\hat { v ^ { q } } , v ^ { k _ { 0 } } , \cdot \cdot \cdot , v ^ { k _ { n } }$ in a joint embedding space $\gamma \in \mathbb { R } ^ { d }$ . Subsequently, they are projected into $n { + 1 }$ normalized embedding spaces $\mathcal { Z } _ { 0 } , \mathcal { Z } _ { 1 } , \cdot \cdot \cdot , \mathcal { Z } _ { n } \in \mathbb { R } ^ { d ^ { \prime } }$ by projection heads $h : \mathcal { V } \mapsto \mathcal { Z }$ , among which $\mathcal { Z } _ { 0 }$ is invariant to all types of augmentations, whereas $\mathcal { Z } _ { i }$ $( \forall i \in \{ 1 , 2 , \cdots , n \} )$ is dependent on the $i ^ { t h }$ type of augmentation but invariant to other types of augmentations. In other words, in $\mathcal { Z } _ { 0 }$ all features $\textbf {  { v } }$ should be mapped to a single point, whereas in $\mathcal { Z } _ { i }$ $( \forall i \in \{ 1 , 2 , \cdot \cdot \cdot , n \} )$ only $v ^ { q }$ and $\mathbf { \boldsymbol { v } } ^ { k _ { i } }$ should be mapped to a single point while $\pmb { v } ^ { k _ { j } } \ \forall j \neq i$ should be mapped to $n { - } 1$ separate points, as only $\mathcal { T } _ { q }$ and $\mathcal { T } _ { k _ { i } }$ share the same $i ^ { t h }$ augmentation.
+
+We perform contrastive learning in all normalized embedding spaces based on Equation 1, as shown in the right part of Figure 2. For each query $z ^ { q }$ , denote $z ^ { k ^ { + } }$ as the keys from the same instance, and $z ^ { k ^ { - } }$ as the keys from other instances. Since all views should be mapped to the single point in $\mathcal { Z } _ { 0 }$ , the positive pair for the query $z _ { 0 } ^ { q }$ is $z _ { 0 } ^ { k _ { 0 } ^ { + } }$ , and the negative pairs are embeddings of other instances in this embedding space $\{ z _ { 0 } ^ { k _ { 0 } ^ { - } } \}$ ; for embedding spaces $\mathcal { Z } _ { 1 } , \cdots , \mathcal { Z } _ { n }$ , the positive pair for the query $z _ { i } ^ { q }$ is $z _ { i } ^ { k _ { i } ^ { + } }$ , while the negative pairs are embeddings of other instances in this embedding spac e {z k ii } , and $\{ \boldsymbol { z } _ { i } ^ { k _ { j } ^ { + } } \mid \forall j \in \{ 0 , 1 , \cdots , n \}$ and $j \neq i \}$ , which are the embeddings of the same instance with different $i ^ { t h }$ augmentation. The network then learns to be sensitive to one type of augmentation while insensitive to other types of augmentations in one embedding space. Denote $\bar { E _ { i , j } ^ { \{ + , - \} } } = \exp { ( z _ { i } ^ { q } \cdot z _ { i } ^ { k _ { j } ^ { \{ + , - \} } } / \tau ) }$ . The overall training objective for $q$ is:
+
+$$
+\mathcal { L } _ { q } = - \frac { 1 } { n + 1 } \left( \log \frac { E _ { 0 , 0 } ^ { + } } { E _ { 0 , 0 } ^ { + } + \sum _ { k ^ { - } } E _ { 0 , 0 } ^ { - } } + \sum _ { i = 1 } ^ { n } \log \frac { E _ { i , i } ^ { + } } { \sum _ { j = 0 } ^ { n } E _ { i , j } ^ { + } + \sum _ { k ^ { - } } E _ { i , i } ^ { - } } \right) ,
+$$
+
+The network must preserve information w.r.t. all augmentations in the general embedding space $\nu$ in order to optimize the combined learning objectives of all normalized embedding spaces.
+
+Learnt representations. The representation for downstream tasks can be from the general embedding space $\nu$ (Figure 2, blue region), or the concatenation of all embedding sub-spaces (Figure 2, grey region). LooC method returns $\nu$ ; we term the implementation using the concatenation of all embedding sub-spaces as $\mathrm { L o o C + + }$ .
+
+# 4 EXPERIMENTS
+
+Methods. We adopt Momentum Contrastive Learning (MoCo) (He et al., 2020) as the backbone of our framework for its efficacy and efficiency, and incorporate the improved version from (Chen et al., 2020b). We use three types of augmentations as pretext tasks for static image data, namely color jittering (including random gray scale), random rotation $9 0 ^ { \circ }$ , $1 8 0 ^ { \circ }$ , or $2 7 0 ^ { \circ }$ ), and texture randomization (Gatys et al., 2016; Geirhos et al., 2018) (details in the Appendix). We apply random-resized cropping, horizontal flipping and Gaussian blur as augmentations without designated embedding spaces. Note that random rotation and texture randomization are not utilized in state-of-the-art contrastive learning based methods (Chen et al., 2020a; He et al., 2020; Chen et al., 2020b) and for good reason, as we will empirically show that na¨ıvely taking these augmentations negatively affects the performance on some specific benchmarks. For $\mathrm { L o o C + + }$ , we include Conv5 block into the projection head $h$ , and use the concatenated features at the last layer of Conv5, instead of the last layer of $h$ , from each head. Note than for both LooC and $_ { \mathrm { L o o C + + } }$ the augmented additional keys are only fed into the key encoding network, which is not back-propagated, thus it does not much increase computation or GPU memory consumption.
+
+Datasets and evaluation metrics. We train our model on the 100-category ImageNet (IN-100) dataset, a subset of the ImageNet (Deng et al., 2009) dataset, for fast ablation studies of the proposed framework. We split the subset following (Tian et al., 2019). The subset contains ${ \sim } 1 2 5 \mathrm { k }$ images, sufficiently large to conduct experiments of statistical significance. After training, we adopt linear classification protocol by training a supervised linear classifier on frozen features of feature space $\nu$ for LooC, or concatenated feature spaces $\mathcal { Z }$ for $\mathrm { L o o C + + }$ . This allows us to directly verify the quality of features from a variation of models, yielding more interpretable results. We test the models on various downstream datasets (more information included in the Appendix): 1) IN-100 validation set; 2) The iNaturalist 2019 (iNat-1k) dataset (Van Horn et al., 2018), a large-scale classification dataset containing 1,010 species. Top-1 and top-5 accuracy on this dataset are reported; 3) The CaltechUCSD Birds 2011 (CUB-200) dataset (Wah et al., 2011), a fine-grained classification dataset of 200 bird species. Top-1 and top-5 classification accuracy are reported. 4) VGG Flowers (Flowers-102) dataset (Nilsback & Zisserman, 2008), a consistent of 102 flower categories. We use the dataset for few-shot classification and report 5-shot and 10-shot classification accuracy over 10 trials within $9 5 \%$ confidence interval. Unlike many few-shot classification methods which conduct evaluation on a subset of categories, we use all 102 categories in our study; 5) ObjectNet dataset (Barbu et al., 2019), a test set collected to intentionally show objects from new viewpoints on new backgrounds with different rotations of real-world images. We only use the 13 categories which overlap with IN-100, termed as ON-13; 6) ImageNet-C dataset (Hendrycks & Dietterich, 2019), a benchmark for model robustness of image corruptions. We use the 100 categories as IN-100, termed as IN-C-100. Note that ON and IN-C are test sets, so we do not train a supervised linear classifier exclusively while directly benchmark the linear classifier trained on IN-100 instead.
+
+Table 1: Classification accuracy on 4-class rotation and IN-100 under linear evaluation protocol. Adding rotation augmentation into baseline MoCo significantly reduces its capacity to classify rotation angles while downgrades its performance on IN-100. In contrast, our method better leverages the information gain of the new augmentation.   
+
+<table><tr><td>model</td><td>Rotation Acc.</td><td>IN-100</td><td>top-5</td></tr><tr><td>Supervised</td><td>72.3</td><td>top-1 83.7</td><td>95.7</td></tr><tr><td>MoCo</td><td>61.1</td><td>81.0</td><td>95.2</td></tr><tr><td>MoCo+Rotation</td><td>43.3</td><td>79.4</td><td>94.1</td></tr><tr><td>MoCo +Rotation (same for q and k) LooC +Rotation [ours]</td><td>45.5 65.2</td><td>78.1 80.2</td><td>94.3 95.5</td></tr></table>
+
+Table 2: Evaluation on multiple downstream tasks. Our method demonstrates superior generalizability and transferability with increasing number of augmentations.   
+
+<table><tr><td>model</td><td>Augmentation ColorRotation</td><td></td><td>iNat-1k top-1 top-5</td><td>CUB-200 top-1</td><td>top-5 5-shot</td><td>Flowers-102 10-shot</td><td>top-1</td><td>IN-100 top-5</td></tr><tr><td>MoCo</td><td>√</td><td></td><td>36.2</td><td>62.0</td><td>36.7 64.7</td><td>67.9 (± 0.5)</td><td>77.3 (± 0.1)</td><td>81.0 95.2</td></tr><tr><td rowspan="3">LooC</td><td>√</td><td></td><td>41.2</td><td>67.0 40.1</td><td>69.7</td><td>68.2 (± 0.6)</td><td>77.6(± 0.1)</td><td>81.1 95.3</td></tr><tr><td></td><td>√</td><td>40.0</td><td>65.4</td><td>38.8 67.0</td><td>70.1 (± 0.4)</td><td>79.3 (± 0.1)</td><td>80.2 95.5</td></tr><tr><td>√</td><td>√</td><td>44.0</td><td>69.3</td><td>39.6 69.2</td><td>70.9(± 0.3)</td><td>80.8(±0.2)</td><td>79.2 94.7</td></tr><tr><td>LooC++</td><td>√</td><td>√</td><td>46.1</td><td>71.5</td><td>39.3 69.3</td><td>68.1(± 0.4)</td><td>78.8(±0.2)</td><td>81.2 95.2</td></tr></table>
+
+Implementation details. We closely follow (Chen et al., 2020b) for most training hyperparameters. We use a ResNet-50 (He et al., 2016) as our feature extractor. We use a two-layer MLP head with a 2048-d hidden layer and ReLU for each individual embedding space. We train the network for 500 epochs, and decrease the learning rate at 300 and 400 epochs. We use separate queues (He et al., 2020) for individual embedding space and set the queue size to 16,384. Linear classification evaluation details can be found in the Appendix. The batch size during training of the backbone and the linear layer is set to 256.
+
+Study on augmentation inductive biases. We start by designing an experiment which allows us to directly measure how much an augmentation affects a downstream task which is sensitive to the augmentation. For example, consider two tasks which can be defined on IN-100: Task A is 4- category classification of rotation degrees for an input image; Task B is 100-category classification of ImageNet objects. We train a supervised linear classifier for task A with randomly rotated IN-100 images, and another classifier for task B with unrotated images. In Table 1 we compare the accuracy of the original MoCo (w/o rotation augmentation), MoCo w/ rotation augmentation, and our model w/ rotation augmentation. A priori, with no data labels to perform augmentation selection, we have no way to know if rotation should be utilized or not. Adding rotation into the set of augmentations for MoCo downgrades object classification accuracy on IN-100, and significantly reduces the capacity of the baseline model to distinguish the rotation of an input image. We further implement a variation enforcing the random rotating angle of query and key always being the same. Although it marginally increases rotation accuracy, IN-100 object classification accuracy further drops, which is inline with our hypothesis that the inductive bias of discarding certain type of information introduced by adopting an augmentation into contrastive learning objective is significant and cannot be trivially resolved by tuning the distribution of input images. On the other hand, our method with rotation augmentation not only sustains accuracy on IN-100, but also leverages the information gain of the new augmentation. We can include all augmentations with our LooC multi-self-supervised method and obtain improved performance across all condition without any downstream labels or a prior knowledged invariance.
+
+Table 3: Evaluation on datasets of real-world corruptions. Rotation augmentation is beneficial for ON-13, and texture augmentation if beneficial for IN-C-100.   
+
+<table><tr><td rowspan="2">model</td><td rowspan="2">Aug Rot. Tex.</td><td rowspan="2">ON-13 top-1 top-5</td><td colspan="7">IN-C-100 (top-1)</td><td rowspan="2">IN-100</td></tr><tr><td>Noise</td><td>Blur</td><td>Weather Digital</td><td></td><td>All</td><td>d≥3</td><td>top-1 top-5</td></tr><tr><td>Supervised</td><td></td><td>30.9</td><td>54.8 28.4</td><td>47.1</td><td>44.9</td><td>58.5</td><td>47.2</td><td>36.5</td><td>83.7</td><td>95.7</td></tr><tr><td>MoCo</td><td></td><td>29.2 54.2</td><td>37.9</td><td>38.5</td><td>47.7</td><td>60.1</td><td>48.2</td><td>37.2</td><td>81.0</td><td>95.2</td></tr><tr><td rowspan="3">LooC</td><td>√</td><td>34.2</td><td>59.6</td><td>31.3</td><td>33.1</td><td>42.4</td><td>54.9</td><td>42.7</td><td>31.8</td><td>80.2 95.5</td></tr><tr><td></td><td>30.1</td><td>54.1</td><td>42.4</td><td>39.6 54.0</td><td>61.9</td><td>51.3</td><td>41.9</td><td>81.0</td><td>94.7</td></tr><tr><td>√</td><td>? 33.3</td><td>59.2</td><td>37.0</td><td>35.2</td><td>50.2</td><td>56.9</td><td>46.5</td><td>37.2 79.4</td><td>94.3</td></tr><tr><td>LooC++</td><td>√ √</td><td>32.6</td><td>57.3</td><td>38.3</td><td>37.6</td><td>52.0</td><td>60.0</td><td>48.8</td><td>38.9</td><td>82.1 95.1</td></tr></table>
+
+Table 4: Comparisons of LooC vs. MoCo trained with all augmentations.   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>IN-100top-1top-5</td><td rowspan=1 colspan=1>iNat-1ktop-1top-5</td><td rowspan=1 colspan=1>Flowers-1025-shot     10-shot</td><td rowspan=1 colspan=1>IN-C-100all-top-1</td></tr><tr><td rowspan=1 colspan=1>MoCoLooC</td><td rowspan=1 colspan=1>77.9 93.778.594.0</td><td rowspan=1 colspan=1>39.5 65.141.7 67.5</td><td rowspan=1 colspan=1>72.1 (± 0.4)81.1 (± 0.2)72.1(± 0.7)81.4(± 0.2)</td><td rowspan=1 colspan=1>47.445.4</td></tr><tr><td rowspan=1 colspan=1>MoCo++LooC++</td><td rowspan=1 colspan=1>80.8 94.682.2 95.3</td><td rowspan=1 colspan=1>43.4 68.545.9 71.4</td><td rowspan=1 colspan=1>70.0 (±0.8)80.5 (±0.3)71.0 (± 0.7)81.9 (± 0.3)</td><td rowspan=1 colspan=1>48.348.0</td></tr></table>
+
+Table 5: Comparisons of concatenating features from different embedding spaces in $\mathbf { L o o C + + }$ jointly trained on color, rotation and texture augmentations. Different downstream tasks show nonidentical preferences for augmentation-dependent or invariant representations.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Col.</td><td rowspan="2">Variance Head Rot.</td><td rowspan="2">Tex.</td><td colspan="2">IN-100</td><td rowspan="2">iNat-1k top-1 top-5</td><td colspan="2">Flowers-102</td><td rowspan="2">IN-C-100 all-top-1</td></tr><tr><td>top-1</td><td>top-5</td><td>5-shot</td><td>10-shot</td></tr><tr><td rowspan="5">LooC++</td><td></td><td></td><td></td><td>78.5</td><td>94.3</td><td>38.5</td><td>64.7</td><td>68.6(± 0.6) 77.6 (± 0.1)</td><td>48.0</td></tr><tr><td>Y</td><td></td><td></td><td>79.7</td><td>94.4</td><td>42.9 68.7</td><td></td><td>69.1 (± 0.7) 79.5 (± 0.2)</td><td>47.1</td></tr><tr><td></td><td>√</td><td></td><td>81.5</td><td>94.9</td><td>41.4 67.4</td><td>70.5 (±0.6)8</td><td>80.0 (± 0.2)</td><td>52.6</td></tr><tr><td></td><td></td><td>√</td><td>80.3</td><td>94.9</td><td>43.0 68.6</td><td>70.4 (± 0.5)</td><td>80.5 (± 0.2)</td><td>44.1</td></tr><tr><td>厂</td><td>√</td><td>√</td><td>82.2</td><td>95.3</td><td>45.9 71.4</td><td>71.0 (± 0.7)</td><td>81.9 (± 0.3)</td><td>48.0</td></tr></table>
+
+Fine-grained recognition results. A prominent application of unsupervised learning is to learn features which are transferable and generalizable to a variety of downstream tasks. To fairly evaluate this, we compare our method with original MoCo on a diverse set of downstream tasks. Table 2 lists the results on iNat-1k, CUB-200 and Flowers-102. Although demonstrating marginally superior performance on IN-100, the original MoCo trails our LooC counterpart on all other datasets by a noticeable margin. Specifically, applying LooC on random color jiterring boosts the performance of the baseline which adopts the same augmentation. The comparison shows that our method can better preserve color information. Rotation augmentation also boosts the performance on iNat-1k and Flowers-102, while yields smaller improvements on CUB-200, which supports the intuition that some categories benefit from rotation-invariant representations while some do not. The performance is further boosted by using LooC with both augmentations, demonstrating the effectiveness in simultaneously learning the information w.r.t. multiple augmentations.
+
+Interestingly, $\mathrm { L o o C + + }$ brings back the slight performance drop on IN-100, and yields more gains on iNat-1k, which indicates the benefits of explicit feature fusion without hand-crafting what should or should not be contrastive in the training objective.
+
+Robustness learning results. Table 3 compares our method with MoCo and supervised model on ON-13 and IN-C-100, two testing sets for real-world data generalization under a variety of noise conditions. The linear classifier is trained on standard IN-100, without access to the testing distribution. The fully supervised network is most sensitive to perturbations, albeit it has highest accuracy on the source dataset IN-100. We also see that rotation augmentation is beneficial for ON-13, but significantly downgrades the robustness to data corruptions in IN-C-100. Conversely, texture randomization increases the robustness on IN-C-100 across all corruption types, particularly significant on “Blur” and “Weather”, and on the severity level above or equal to 3, as the representations must be insensitive to local noise to learn texture-invariant features, but its improvement on ON-13 is marginal. Combining rotation and texture augmentation yields improvements on both datasets, and $\mathrm { L o o C + + }$ further improves its performance on IN-C-100.
+
+![](images/a5123be19ac2689c00e96088a424265c4e0766e416856edbc22e6f3a0914b7c0.jpg)  
+Figure 3: Top nearest-neighbor retrieval results of LooC vs. corresponding invariant MoCo baseline with color (left) and rotation (right) augmentations on IN-100 and iNat-1k. The results show that our model can better preserve information dependent on color and rotation despite being trained with those augmentations.
+
+Qualitative results. In Figure 3 we show nearest-neighbor retrieval results using features learnt with LooC vs. corresponding MoCo baseline. The top retrieval results demonstrate that our model can better preserve information which is not invariant to the transformations presented in the augmentations used in contrastive learning.
+
+Ablation: MoCo w/ all augmentations vs. LooC. We compare our method and MoCo trained with all augmentations. We also add multiple Conv5 heads to MoCo, termed as $\mathrm { M o C o + + }$ , for a fair comparison with $\mathrm { L o o C + + }$ . The results are listed in Table 4. Using multiple heads boosts the performance of baseline MoCo, nevertheless, our method achieves better or comparable results compared with its baseline counterparts.
+
+Note that the results in Table 2 to 5 should be interpreted in the broader context of Table 1. Table 1 illustrates the catastrophic consequences of not separating the varying and invariant factors of an augmentation (in this case, rotation). It can be imagined that if we add “rotation classification” as one downstream task in Table 4, $\mathrm { M o C o + + }$ will perform as poorly as in Table 1. The key of our work is to avoid what has happened in Table 1 and simultaneously boosts performance.
+
+Ablation: Augmentation-dependent embedding spaces vs. tasks. We train a $_ { \mathrm { L o o C + + } }$ with all types of augmentations, and subsequently train multiple linear classifiers with concatenated features from different embedding spaces: all-invariant, color, rotation and texture. Any additional variance features boost the performance on IN-100, iNat-1k and Flowers-102. Adding texture-dependent features decreases the performance on IN-C-100: Textures are (overly) strong cues for ImageNet classification (Geirhos et al., 2018), thus the linear classifier is prone to use texture-dependent features, loosing the gains of texture invariance. Adding rotation-dependent features increases the performance on IN-C-100: Rotated objects of most classes in IN-100 are rare, thus the linear classifier is prone to use rotation-dependent features, so that drops on IN-C-100 triggered by rotation-invariant augmentation are re-gained. Using all types of features yields best performance on IN-100, iNat1k and Flowers-102; the performance on IN-C-100 with all augmentations remains comparable to MoCo, which does not suffer from loss of robustness introduced by rotation invariance.
+
+In Figure 4 we show the histogram of correct predictions (activations $\times$ weights of classifier) by each augmentation-dependent head of a few instances from IN-100 and iNat-1k. The classifier prefers texture-dependent information over other kinds on an overwhelmingly majority of samples from IN-100, even for classes where shape is supposed to be the dominant factor, such as “pickup” and “mixing bowl” ((a), top row). This is consistent with the findings from (Geirhos et al., 2018) that ImageNet-trained CNNs are strongly biased towards texture-like representations. Interestingly, when human or animal faces dominant an image ((a), bottom-left), $\mathrm { L o o C + + }$ sharply prefers rotation-dependent features, which also holds for face recognition of humans. In contrast, on iNat1k $\mathrm { L o o C + + }$ prefers a more diverse set of features, such as color-dependent feature for a dragonfly species, rotation and texture-dependent features for birds, as well as rotation-invariant features for flowers. Averaged over the datasets, the distribution of classifier preferences is more balanced on iNat-1k than IN-100, as can be seen from the entropy that the distribution on iNat-1k is close to 2 bits, whereas it is close to 1 bit on IN-100, as it is dominated by only two elements. It corroborates the large improvements on iNat-1k gained from multi-dependent features learnt by our method.
+
+![](images/6b847cc1eef4f9a685453af1d89d23117753f1e3a21c2d22b3cfe1662516a3c9.jpg)  
+Figure 4: Histograms of correct predictions (activations $\times$ weights of classifier) by each augmentation-dependent head from IN-100 and iNat-1k. The classifier on IN-100 heavily relies on texture-dependent information, whereas it is much more balanced on iNat-1k. This is consistent with the improvement gains observed when learning with multiple augmentations.
+
+# 5 RELATED WORK
+
+Pretext Tasks. In computer vision, feature design and engineering used to be a central topic before the wide application of deep learning. Researchers have proposed to utilize cue combination for image retrieval and recognition tasks (Martin et al., 2004; Frome et al., 2007a;b; Malisiewicz & Efros, 2008; Rabinovich et al., 2006). For example, the local brightness, color, and texture features are combined together to represent an image and a simple linear model can be trained to detect boundaries (Martin et al., 2004). Interestingly, the recent development of unsupervised representation learning in deep learning is also progressed by designing different self-supervised pretext tasks (Wang & Gupta, 2015; Doersch et al., 2015; Pathak et al., 2016; Noroozi & Favaro, 2016; Zhang et al., 2016; Gidaris et al., 2018; Owens et al., 2016). For example, relative patch prediction (Doersch et al., 2015) and rotation prediction (Gidaris et al., 2018) are designed to discover the underlined structure of the objects; image colorization task (Zhang et al., 2016) is used to learn representations capturing color information. The inductive bias introduced by each pretext task can often be associated with a corresponding hand-crafted descriptor.
+
+Multi-Task Self-Supervised Learning. Multi-task learning has been widely applied in image recognition (Kokkinos, 2017; Teichmann et al., 2018; He et al., 2017). However, jointly optimizing multiple tasks are not always beneficial. As shown in Kokkinos (2017), training with two tasks can yield better performance than seven tasks together, as some tasks might be conflicted with each other. This phenomenon becomes more obvious in multi-task self-supervised learning (Doersch & Zisserman, 2017; Wang et al., 2017; Pinto & Gupta, 2017; Piergiovanni et al., 2020; Alwassel et al., 2019) as the optimization goal for each task can be very different depending on the pretext task. To solve this problem, different weights for different tasks are learned to optimize for the downstream tasks (Piergiovanni et al., 2020). However, searching the weights typically requires labels, and is time-consuming and does not generalize to different tasks. To train general representations, researchers have proposed to utilize sparse regularization to factorize the network representations to encode different information from different tasks (Doersch & Zisserman, 2017; Misra et al., 2016). In this paper, we also proposed to learn representation which can factorize and unify information from different augmentations. Instead of using sparse regularization, we define different contrastive learning objective in a multi-head architecture.
+
+Contrastive Learning. Instead of designing different pretext tasks, recent work on contrastive learning (Wu et al., 2018; Oord et al., 2018; Tian et al., 2019; He et al., 2020; Misra & van der Maaten, 2020; Chen et al., 2020a) trained networks to be invariant to various corresponding augmentations. Researchers (Chen et al., 2020a) elaborated different augmentations and pointed out which augmentations are helpful or harmful for ImageNet classification. It is also investigated in Tian et al. (2019) that different augmentations can be beneficial to different downstream tasks. Instead of enumerating all the possible selections of augmentations, we proposed a unified framework which captures different factors of variation introduced by different augmentations.
+
+# 6 CONCLUSIONS
+
+Current contrastive learning approaches rely on specific augmentation-derived transformation invariances to learn a visual representation, and may yield suboptimal performance on downstream tasks if the wrong transformation invariances are presumed. We propose a new model which learns both transformation dependent and invariant representations by constructing multiple embeddings, each of which is not contrastive to a single type of transformation. Our framework outperforms baseline contrastive method on coarse-grained, fine-grained, few-shot downstream classification tasks, and demonstrates better robustness of real-world data corruptions.
+
+# ACKNOWLEDGEMENT
+
+Prof. Darrell’s group was supported in part by DoD, NSF, BAIR, and BDD. Prof. Wang’s group was supported, in part, by gifts from Qualcomm and TuSimple. We would like to thank Allan Jabri, Colorado Reed and Ilija Radosavovic for helpful discussions.
+
+# REFERENCES
+
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+
+# A AUGMENTATION DETAILS
+
+Following (Chen et al., 2020b), we set the probability of color jittering to 0.8, with (brightness, contrast, saturation, hue) as (0.4, 0.4, 0.4, 0.1), and probability of random scale to 0.2. We set the probability of random rotation and texture randomization as 0.5.
+
+# B DATASETS
+
+iNat-1k, a large-scale classification dataset containing 1,010 species with a combined training and validation set of 268,243 images. We randomly reallocate $10 \%$ of training images into the validation set as the original validation set is relatively small.
+
+CUB-200, which contains 5,994 training and 5,794 testing images of 200 bird species.
+
+Flowers-102, which contains 102 flower categories consisting of between 40 and 258 images.
+
+ObjectNet, a test set collected to intentionally show objects from new viewpoints on new backgrounds with different rotations of real-world images. It originally has 313-category. We only use the 13 categories which overlap with IN-100.
+
+ImageNet-C, which consists of 15 diverse corruption types applied to validation images of ImageNet.
+
+# C LINEAR CLASSIFICATION
+
+We train the linear layer for 200 epochs for IN-100 and CUB-200, 100 epochs for iNat-1k, optimized by momentum SGD with a learning rate of 30 decreased by 0.1 at $60 \%$ and $80 \%$ of training schedule; for Flowers-102 we train the linear layer with Adam optimizer for 250 iterations with a learning rate of 0.03.
+
+# D LEAVE-ONE-OUT VS. ADD-ONE AUGMENTATION
+
+Table 6: Leave-one-out vs. add-one Augmentation. \*: Default (none add-one) augmentation strategy.   
+
+<table><tr><td rowspan="2">model</td><td colspan="2">Augmentation</td><td colspan="2">IN-100</td></tr><tr><td>Color</td><td>Rotation</td><td>top-1</td><td>top-5</td></tr><tr><td rowspan="2">MoCo</td><td>√</td><td></td><td>81.0</td><td>95.2</td></tr><tr><td>√</td><td>√</td><td>79.4</td><td>94.1</td></tr><tr><td rowspan="2">MoCo+AddOne</td><td>√</td><td></td><td>74.9</td><td>92.5</td></tr><tr><td>*</td><td>√</td><td>79.3</td><td>94.4</td></tr><tr><td rowspan="2">LooC[ours]</td><td>√</td><td></td><td>81.1</td><td>95.3</td></tr><tr><td>*</td><td>√</td><td>80.2</td><td>95.5</td></tr></table>
+
+A straight-forward alternative for our leave-one-out augmentation strategy is add-one augmentation. Instead of applying all augmentations and augmenting two views in the same manner, add-one strategy keeps the query image unaugmentated, while in each augmentation-specific view the designated type of augmentation is applied. The results are shown in Table 6. Add-one strategy oversimplifies the instance discrimination task, e.g., leaving color augmentation out of query view makes it very easy for the network to spot the same instance out of a set of candidates. Our leave-one-out strategy does not suffer such degeneration.
+
+# E IMAGENET-1K EXPERIMENTS
+
+Table 7: Results of models trained on 1000 category ImageNet and fine-tuned on iNat-1k following linear classification protocol.
+
+<table><tr><td rowspan="2">model</td><td colspan="2">iNat-1k</td></tr><tr><td>top-1</td><td>top-5</td></tr><tr><td>MoCo</td><td>47.8</td><td>74.3</td></tr><tr><td>LooC++ [ours]</td><td>51.2</td><td>76.5</td></tr></table>
+
+We conduct experiments on 1000 category full ImageNet dataset. The models are trained by selfsupervised learning on IN-1k, and fine-tuned on iNat-1k following linear classification protocol. Our model is trained with all augmentations, i.e., color, rotation and texture. Results are reported in Table 7.
+
+# F DISCUSSIONS
+
+F.1 THE DIMENSIONS OF MOCO, LOOC, LOOC++
+
+The representations of MoCo and LooC are of exactly the same dimension (2048); same for $\mathrm { M o C o + + }$ and $\mathrm { L o o C + + }$ (2048 \* # augmentations). It is specifically designed for fair comparisons.
+
+F.2 ARE THE HYPER-PARAMETERS TUNED SPECIFICALLY FOR OUR SUBSETS?
+
+No, except that we increase the number of training epochs as the amount of data increases. We did not specifically tune the baseline so that our method can outperform it most; on the contrary, we first made baseline as strong as possible, then directly applied the same hyper-parameters to our method. The subset of ImageNet100 behaviors similarly as ImageNet1k; our baseline already significantly outperforms the best method on the same subset from previous literature ( $7 5 . 8 \%$ CMC vs. $8 1 . 0 \%$ top1 [ours]), and since our method is derived from MoCo, they are directly comparable.

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+        "text": "WHAT SHOULD NOT BE CONTRASTIVE IN CONTRASTIVE LEARNING ",
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+        "text": "Tete Xiao UC Berkeley ",
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+        "text": "Xiaolong Wang UC San Diego ",
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+    {
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+        "text": "Alexei A. Efros UC Berkeley ",
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+        "text": "Trevor Darrell UC Berkeley ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
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+        "text": "Recent self-supervised contrastive methods have been able to produce impressive transferable visual representations by learning to be invariant to different data augmentations. However, these methods implicitly assume a particular set of representational invariances (e.g., invariance to color), and can perform poorly when a downstream task violates this assumption (e.g., distinguishing red vs. yellow cars). We introduce a contrastive learning framework which does not require prior knowledge of specific, task-dependent invariances. Our model learns to capture varying and invariant factors for visual representations by constructing separate embedding spaces, each of which is invariant to all but one augmentation. We use a multi-head network with a shared backbone which captures information across each augmentation and alone outperforms all baselines on downstream tasks. We further find that the concatenation of the invariant and varying spaces performs best across all tasks we investigate, including coarse-grained, fine-grained, and few-shot downstream classification tasks, and various data corruptions. ",
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+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    },
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+        "text": "Self-supervised learning, which uses raw image data and/or available pretext tasks as its own supervision, has become increasingly popular as the inability of supervised models to generalize beyond their training data has become apparent. Different pretext tasks have been proposed with different transformations, such as spatial patch prediction (Doersch et al., 2015; Noroozi & Favaro, 2016), colorization (Zhang et al., 2016; Larsson et al., 2016; Zhang et al., 2017), rotation (Gidaris et al., 2018). Whereas pretext tasks aim to recover the transformations between different “views” of the same data, more recent contrastive learning methods (Wu et al., 2018; Tian et al., 2019; He et al., 2020; Chen et al., 2020a) instead try to learn to be invariant to these transformations, while remaining discriminative with respect to other data points. Here, the transformations are generated using classic data augmentation techniques which correspond to common pretext tasks, e.g., randomizing color, texture, orientation and cropping. ",
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+        "text": "Yet, the inductive bias introduced through such augmentations is a double-edged sword, as each augmentation encourages invariance to a transformation which can be beneficial in some cases and harmful in others: e.g., adding rotation may help with view-independent aerial image recognition, but significantly downgrade the capacity of a network to solve tasks such as detecting which way is up in a photograph for a display application. Current self-supervised contrastive learning methods assume implicit knowledge of downstream task invariances. In this work, we propose to learn visual representations which capture individual factors of variation in a contrastive learning framework without presuming prior knowledge of downstream invariances. ",
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+        "text": "Instead of mapping an image into a single embedding space which is invariant to all the handcrafted augmentations, our model learns to construct separate embedding sub-spaces, each of which is sensitive to a specific augmentation while invariant to other augmentations. We achieve this by optimizing multiple augmentation-sensitive contrastive objectives using a multi-head architecture with a shared backbone. Our model aims to preserve information with regard to each augmentation in a unified representation, as well as learn invariances to them. The general representation trained with these augmentations can then be applied to different downstream tasks, where each task is free to selectively utilize different factors of variation in our representation. We consider transfer of either the shared backbone representation, or the concatenation of all the task-specific heads; both outperform all baselines; the former uses same embedding dimensions as typical baselines, while the latter provides greatest overall performance in our experiments. In this paper, we experiment with three types of augmentations: rotation, color jittering, and texture randomization, as visualized in Figure 1. We evaluate our approach across a variety of diverse tasks including large-scale classification (Deng et al., 2009), fine-grained classification (Wah et al., 2011; Van Horn et al., 2018), few-shot classification (Nilsback & Zisserman, 2008), and classification on corrupted data (Barbu et al., 2019; Hendrycks & Dietterich, 2019). Our representation shows consistent performance gains with increasing number of augmentations. Our method does not require hand-selection of data augmentation strategies, and achieves better performance against state-of-the-art MoCo baseline (He et al., 2020; Chen et al., 2020b), and demonstrates superior transferability, generalizability and robustness across tasks and categories. Specifically, we obtain around $1 0 \\%$ improvement over MoCo in classification when applied on the iNaturalist (Van Horn et al., 2018) dataset. ",
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+        "image_caption": [
+            "Figure 1: Self-supervised contrastive learning relies on data augmentations as depicted in (a) to learn visual representations. However, current methods introduce inductive bias by encouraging neural networks to be less sensitive to information w.r.t. augmentation, which may help or may hurt. As illustrated in (b), rotation invariant embeddings can help on certain flower categories, but may hurt animal recognition performance; conversely color invariance generally seems to help coarse grained animal classification, but can hurt many flower categories and bird categories. Our method, shown in the following figure, overcomes this limitation. "
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+        "text": "2 BACKGROUND: CONTRASTIVE LEARNING FRAMEWORK ",
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+        "text": "Contrastive learning learns a representation by maximizing similarity and dissimilarity over data samples which are organized into similar and dissimilar pairs, respectively. It can be formulated as a dictionary look-up problem (He et al., 2020), where a given reference image $\\mathcal { T }$ is augmented into two views, query and key, and the query token $q$ should match its designated key $k ^ { + }$ over a set of sampled negative keys $\\{ k ^ { - } \\}$ from other images. In general, the framework can be summarized as the following components: (i) A data augmentation module $\\tau$ constituting $n$ atomic augmentation operators, such as random cropping, color jittering, and random flipping. We denote a pre-defined atomic augmentation as random variable $X _ { i }$ . Each time the atomic augmentation is executed by sampling a specific augmentation parameter from the random variable, i.e., $x _ { i } \\sim X _ { i }$ . One sampled data augmentation module transforms image $\\mathcal { T }$ into a random view $\\widetilde { \\boldsymbol { \\tau } }$ , denoted as $\\widetilde { \\mathcal { T } } = \\mathcal { T } [ x _ { 1 } , x _ { 2 } , \\dots , x _ { n } ] \\left( \\mathcal { T } \\right)$ . Positive pair $( q , k ^ { + } )$ is generated by applying two randomly sampled data augmentation on the same reference image. (ii) An encoder network $f$ which extracts the feature $\\textbf {  { v } }$ of an image $\\mathcal { T }$ by mapping it into a $d$ -dimensional space $\\mathbb { R } ^ { d }$ . (iii) A projection head $h$ which further maps extracted representations into a hyper-spherical (normalized) embedding space. This space is subsequently used for a specific pretext task, i.e., contrastive loss objective for a batch of positive/negative pairs. A common choice is InfoNCE (Oord et al., 2018): ",
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+        "text": "$$\n\\mathcal { L } _ { q } = - \\log \\frac { \\exp { ( q \\cdot k ^ { + } / \\tau ) } } { \\exp { ( q \\cdot k ^ { + } / \\tau ) } + \\sum _ { k ^ { - } } \\exp { ( q \\cdot k ^ { - } / \\tau ) } } ,\n$$",
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+        "text": "As a key towards learning a good feature representation (Chen et al., 2020a), a strong augmentation policy prevents the network from exploiting na¨ıve cues to match the given instances. However, inductive bias is introduced through the selection of augmentations, along with their hyper-parameters defining the strength of each augmentation, manifested in Equation 1 that any views by the stochastic augmentation module $\\tau$ of the same instance are mapped onto the same point in the embedding space. The property negatively affects the learnt representations: 1) Generalizability and transferability are harmed if they are applied to the tasks where the discarded information is essential, e.g., color plays an important role in fine-grained classification of birds; 2) Adding an extra augmentation is complicated as the new operator may be helpful to certain classes while harmful to others, e.g., a rotated flower could be very similar to the original one, whereas it does not hold for a rotated car; 3) The hyper-parameters which control the strength of augmentations need to be carefully tuned for each augmentation to strike a delicate balance between leaving a short-cut open and completely invalidate one source of information. ",
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+            "Figure 2: Framework of the Leave-one-out Contrastive Learning approach, illustrated with two types of augmentations, i.e., random rotation and color jittering. We generate multiple views with leave-one-out strategy, then project their representations into separate embedding spaces with contrastive objective, where each embedding space is either invariant to all augmentations, or invariant to all but one augmentation. The learnt representation can be the general embedding space $\\nu$ (blue region), or the concatenation of embedding sub-spaces $\\mathcal { Z }$ (grey region). Our results show that either of our proposed representations are able to outperform baseline contrastive embeddings and do not suffer from decreased performance when adding augmentations to which the task is not invariant (i.e., the red X’s in Figure 1). "
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+        "text": "3 LOOC: LEAVE-ONE-OUT CONTRASTIVE LEARNING ",
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+        "text": "We propose Leave-one-out Contrastive Learning (LooC), a framework for multi-augmentation contrastive learning. Our framework can selectively prevent information loss incurred by an augmentation. Rather than projecting every view into a single embedding space which is invariant to all augmentations, in our LooC method the representations of input images are projected into several embedding spaces, each of which is not invariant to a certain augmentation while remaining invariant to others, as illustrated in Figure 2. In this way, each embedding sub-space is specialized to a single augmentation, and the shared layers will contain both augmentation-varying and invariant information. We learn a shared representation jointly with the several embedding spaces; we transfer either the shared representation alone, or the concatenation of all spaces, to downstream tasks. ",
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+        "text": "View Generation. Given a reference image and $n$ atomic augmentations, we first augment the reference image with two sets of independently sampled augmentation parameters into the query view $\\mathcal { T } _ { q }$ and the first key view $\\mathcal { T } _ { k _ { 0 } }$ , i.e., $\\dot { \\mathcal { T } } _ { \\{ q , k _ { 0 } \\} } = \\mathcal { T } [ x _ { 1 } ^ { \\{ q , k _ { 0 } \\} } , x _ { 2 } ^ { \\{ q , k _ { 0 } \\} } , \\ldots , \\dot { x } _ { n } ^ { \\{ q , k _ { 0 } \\} } ] ( \\mathbb { Z } ) .$ x{q,k0}n ] (I ). Additionally, we generate $n$ views from the reference image as extra key views, denoted as ${ \\mathcal { T } } _ { k _ { i } }$ , $\\forall i \\in \\left\\{ 1 , \\ldots , n \\right\\}$ . For the $i ^ { t h }$ additional key view, the parameter of $i ^ { t h }$ atomic augmentation is copied from it of the query view, i.e., $x _ { i } ^ { k _ { i } } \\equiv x _ { i } ^ { q }$ , $\\forall i \\in \\left\\{ 1 , \\ldots , n \\right\\}$ ; whereas the parameter of other atomic augmentations are still independently sampled, i.e., $x _ { j } ^ { k _ { i } } \\sim X _ { j }$ , $\\forall j \\ne i$ . For instance, assume that we have a set of two atomic augmentations {random_rotation, color_jitter}, $\\mathcal { T } _ { q }$ and $\\mathcal { T } _ { k _ { 1 } }$ are always augmented by the same rotation angle but different color jittering; $\\mathcal { T } _ { q }$ and $\\mathcal { T } _ { k _ { 2 } }$ are always augmented by the same color jittering but different rotation angle; $\\mathcal { T } _ { q }$ and ${ \\mathcal { T } } _ { k _ { 0 } }$ are augmented independently, as illustrated in the left part of Figure 2. ",
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+        "text": "Contrastive Embedding Space. The augmented views are encoded by a neural network encoder $f ( \\cdot )$ into feature vectors $\\hat { v ^ { q } } , v ^ { k _ { 0 } } , \\cdot \\cdot \\cdot , v ^ { k _ { n } }$ in a joint embedding space $\\gamma \\in \\mathbb { R } ^ { d }$ . Subsequently, they are projected into $n { + 1 }$ normalized embedding spaces $\\mathcal { Z } _ { 0 } , \\mathcal { Z } _ { 1 } , \\cdot \\cdot \\cdot , \\mathcal { Z } _ { n } \\in \\mathbb { R } ^ { d ^ { \\prime } }$ by projection heads $h : \\mathcal { V } \\mapsto \\mathcal { Z }$ , among which $\\mathcal { Z } _ { 0 }$ is invariant to all types of augmentations, whereas $\\mathcal { Z } _ { i }$ $( \\forall i \\in \\{ 1 , 2 , \\cdots , n \\} )$ is dependent on the $i ^ { t h }$ type of augmentation but invariant to other types of augmentations. In other words, in $\\mathcal { Z } _ { 0 }$ all features $\\textbf {  { v } }$ should be mapped to a single point, whereas in $\\mathcal { Z } _ { i }$ $( \\forall i \\in \\{ 1 , 2 , \\cdot \\cdot \\cdot , n \\} )$ only $v ^ { q }$ and $\\mathbf { \\boldsymbol { v } } ^ { k _ { i } }$ should be mapped to a single point while $\\pmb { v } ^ { k _ { j } } \\ \\forall j \\neq i$ should be mapped to $n { - } 1$ separate points, as only $\\mathcal { T } _ { q }$ and $\\mathcal { T } _ { k _ { i } }$ share the same $i ^ { t h }$ augmentation. ",
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+        "text": "We perform contrastive learning in all normalized embedding spaces based on Equation 1, as shown in the right part of Figure 2. For each query $z ^ { q }$ , denote $z ^ { k ^ { + } }$ as the keys from the same instance, and $z ^ { k ^ { - } }$ as the keys from other instances. Since all views should be mapped to the single point in $\\mathcal { Z } _ { 0 }$ , the positive pair for the query $z _ { 0 } ^ { q }$ is $z _ { 0 } ^ { k _ { 0 } ^ { + } }$ , and the negative pairs are embeddings of other instances in this embedding space $\\{ z _ { 0 } ^ { k _ { 0 } ^ { - } } \\}$ ; for embedding spaces $\\mathcal { Z } _ { 1 } , \\cdots , \\mathcal { Z } _ { n }$ , the positive pair for the query $z _ { i } ^ { q }$ is $z _ { i } ^ { k _ { i } ^ { + } }$ , while the negative pairs are embeddings of other instances in this embedding spac e {z k ii } , and $\\{ \\boldsymbol { z } _ { i } ^ { k _ { j } ^ { + } } \\mid \\forall j \\in \\{ 0 , 1 , \\cdots , n \\}$ and $j \\neq i \\}$ , which are the embeddings of the same instance with different $i ^ { t h }$ augmentation. The network then learns to be sensitive to one type of augmentation while insensitive to other types of augmentations in one embedding space. Denote $\\bar { E _ { i , j } ^ { \\{ + , - \\} } } = \\exp { ( z _ { i } ^ { q } \\cdot z _ { i } ^ { k _ { j } ^ { \\{ + , - \\} } } / \\tau ) }$ . The overall training objective for $q$ is: ",
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+    {
+        "type": "equation",
+        "img_path": "images/61c5c926587c677f125bd308a70e9cd050cf835e213abccef2ec4fa687a35954.jpg",
+        "text": "$$\n\\mathcal { L } _ { q } = - \\frac { 1 } { n + 1 } \\left( \\log \\frac { E _ { 0 , 0 } ^ { + } } { E _ { 0 , 0 } ^ { + } + \\sum _ { k ^ { - } } E _ { 0 , 0 } ^ { - } } + \\sum _ { i = 1 } ^ { n } \\log \\frac { E _ { i , i } ^ { + } } { \\sum _ { j = 0 } ^ { n } E _ { i , j } ^ { + } + \\sum _ { k ^ { - } } E _ { i , i } ^ { - } } \\right) ,\n$$",
+        "text_format": "latex",
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+        "text": "The network must preserve information w.r.t. all augmentations in the general embedding space $\\nu$ in order to optimize the combined learning objectives of all normalized embedding spaces. ",
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+        "text": "Learnt representations. The representation for downstream tasks can be from the general embedding space $\\nu$ (Figure 2, blue region), or the concatenation of all embedding sub-spaces (Figure 2, grey region). LooC method returns $\\nu$ ; we term the implementation using the concatenation of all embedding sub-spaces as $\\mathrm { L o o C + + }$ . ",
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+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "Methods. We adopt Momentum Contrastive Learning (MoCo) (He et al., 2020) as the backbone of our framework for its efficacy and efficiency, and incorporate the improved version from (Chen et al., 2020b). We use three types of augmentations as pretext tasks for static image data, namely color jittering (including random gray scale), random rotation $9 0 ^ { \\circ }$ , $1 8 0 ^ { \\circ }$ , or $2 7 0 ^ { \\circ }$ ), and texture randomization (Gatys et al., 2016; Geirhos et al., 2018) (details in the Appendix). We apply random-resized cropping, horizontal flipping and Gaussian blur as augmentations without designated embedding spaces. Note that random rotation and texture randomization are not utilized in state-of-the-art contrastive learning based methods (Chen et al., 2020a; He et al., 2020; Chen et al., 2020b) and for good reason, as we will empirically show that na¨ıvely taking these augmentations negatively affects the performance on some specific benchmarks. For $\\mathrm { L o o C + + }$ , we include Conv5 block into the projection head $h$ , and use the concatenated features at the last layer of Conv5, instead of the last layer of $h$ , from each head. Note than for both LooC and $_ { \\mathrm { L o o C + + } }$ the augmented additional keys are only fed into the key encoding network, which is not back-propagated, thus it does not much increase computation or GPU memory consumption. ",
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+        "text": "Datasets and evaluation metrics. We train our model on the 100-category ImageNet (IN-100) dataset, a subset of the ImageNet (Deng et al., 2009) dataset, for fast ablation studies of the proposed framework. We split the subset following (Tian et al., 2019). The subset contains ${ \\sim } 1 2 5 \\mathrm { k }$ images, sufficiently large to conduct experiments of statistical significance. After training, we adopt linear classification protocol by training a supervised linear classifier on frozen features of feature space $\\nu$ for LooC, or concatenated feature spaces $\\mathcal { Z }$ for $\\mathrm { L o o C + + }$ . This allows us to directly verify the quality of features from a variation of models, yielding more interpretable results. We test the models on various downstream datasets (more information included in the Appendix): 1) IN-100 validation set; 2) The iNaturalist 2019 (iNat-1k) dataset (Van Horn et al., 2018), a large-scale classification dataset containing 1,010 species. Top-1 and top-5 accuracy on this dataset are reported; 3) The CaltechUCSD Birds 2011 (CUB-200) dataset (Wah et al., 2011), a fine-grained classification dataset of 200 bird species. Top-1 and top-5 classification accuracy are reported. 4) VGG Flowers (Flowers-102) dataset (Nilsback & Zisserman, 2008), a consistent of 102 flower categories. We use the dataset for few-shot classification and report 5-shot and 10-shot classification accuracy over 10 trials within $9 5 \\%$ confidence interval. Unlike many few-shot classification methods which conduct evaluation on a subset of categories, we use all 102 categories in our study; 5) ObjectNet dataset (Barbu et al., 2019), a test set collected to intentionally show objects from new viewpoints on new backgrounds with different rotations of real-world images. We only use the 13 categories which overlap with IN-100, termed as ON-13; 6) ImageNet-C dataset (Hendrycks & Dietterich, 2019), a benchmark for model robustness of image corruptions. We use the 100 categories as IN-100, termed as IN-C-100. Note that ON and IN-C are test sets, so we do not train a supervised linear classifier exclusively while directly benchmark the linear classifier trained on IN-100 instead. ",
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+    {
+        "type": "table",
+        "img_path": "images/6ff400ffd238513b0be0294a2ad59c1d8355fb33e6bddce8be84d98c759ac415.jpg",
+        "table_caption": [
+            "Table 1: Classification accuracy on 4-class rotation and IN-100 under linear evaluation protocol. Adding rotation augmentation into baseline MoCo significantly reduces its capacity to classify rotation angles while downgrades its performance on IN-100. In contrast, our method better leverages the information gain of the new augmentation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>model</td><td>Rotation Acc.</td><td>IN-100</td><td>top-5</td></tr><tr><td>Supervised</td><td>72.3</td><td>top-1 83.7</td><td>95.7</td></tr><tr><td>MoCo</td><td>61.1</td><td>81.0</td><td>95.2</td></tr><tr><td>MoCo+Rotation</td><td>43.3</td><td>79.4</td><td>94.1</td></tr><tr><td>MoCo +Rotation (same for q and k) LooC +Rotation [ours]</td><td>45.5 65.2</td><td>78.1 80.2</td><td>94.3 95.5</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 2: Evaluation on multiple downstream tasks. Our method demonstrates superior generalizability and transferability with increasing number of augmentations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>model</td><td>Augmentation ColorRotation</td><td></td><td>iNat-1k top-1 top-5</td><td>CUB-200 top-1</td><td>top-5 5-shot</td><td>Flowers-102 10-shot</td><td>top-1</td><td>IN-100 top-5</td></tr><tr><td>MoCo</td><td>√</td><td></td><td>36.2</td><td>62.0</td><td>36.7 64.7</td><td>67.9 (± 0.5)</td><td>77.3 (± 0.1)</td><td>81.0 95.2</td></tr><tr><td rowspan=\"3\">LooC</td><td>√</td><td></td><td>41.2</td><td>67.0 40.1</td><td>69.7</td><td>68.2 (± 0.6)</td><td>77.6(± 0.1)</td><td>81.1 95.3</td></tr><tr><td></td><td>√</td><td>40.0</td><td>65.4</td><td>38.8 67.0</td><td>70.1 (± 0.4)</td><td>79.3 (± 0.1)</td><td>80.2 95.5</td></tr><tr><td>√</td><td>√</td><td>44.0</td><td>69.3</td><td>39.6 69.2</td><td>70.9(± 0.3)</td><td>80.8(±0.2)</td><td>79.2 94.7</td></tr><tr><td>LooC++</td><td>√</td><td>√</td><td>46.1</td><td>71.5</td><td>39.3 69.3</td><td>68.1(± 0.4)</td><td>78.8(±0.2)</td><td>81.2 95.2</td></tr></table>",
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+        "text": "Implementation details. We closely follow (Chen et al., 2020b) for most training hyperparameters. We use a ResNet-50 (He et al., 2016) as our feature extractor. We use a two-layer MLP head with a 2048-d hidden layer and ReLU for each individual embedding space. We train the network for 500 epochs, and decrease the learning rate at 300 and 400 epochs. We use separate queues (He et al., 2020) for individual embedding space and set the queue size to 16,384. Linear classification evaluation details can be found in the Appendix. The batch size during training of the backbone and the linear layer is set to 256. ",
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+        "text": "Study on augmentation inductive biases. We start by designing an experiment which allows us to directly measure how much an augmentation affects a downstream task which is sensitive to the augmentation. For example, consider two tasks which can be defined on IN-100: Task A is 4- category classification of rotation degrees for an input image; Task B is 100-category classification of ImageNet objects. We train a supervised linear classifier for task A with randomly rotated IN-100 images, and another classifier for task B with unrotated images. In Table 1 we compare the accuracy of the original MoCo (w/o rotation augmentation), MoCo w/ rotation augmentation, and our model w/ rotation augmentation. A priori, with no data labels to perform augmentation selection, we have no way to know if rotation should be utilized or not. Adding rotation into the set of augmentations for MoCo downgrades object classification accuracy on IN-100, and significantly reduces the capacity of the baseline model to distinguish the rotation of an input image. We further implement a variation enforcing the random rotating angle of query and key always being the same. Although it marginally increases rotation accuracy, IN-100 object classification accuracy further drops, which is inline with our hypothesis that the inductive bias of discarding certain type of information introduced by adopting an augmentation into contrastive learning objective is significant and cannot be trivially resolved by tuning the distribution of input images. On the other hand, our method with rotation augmentation not only sustains accuracy on IN-100, but also leverages the information gain of the new augmentation. We can include all augmentations with our LooC multi-self-supervised method and obtain improved performance across all condition without any downstream labels or a prior knowledged invariance. ",
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+    {
+        "type": "table",
+        "img_path": "images/1b111872d2e23353a3db9b23c4c510fcf9a456d0a406088ea42209d37d8781c7.jpg",
+        "table_caption": [
+            "Table 3: Evaluation on datasets of real-world corruptions. Rotation augmentation is beneficial for ON-13, and texture augmentation if beneficial for IN-C-100. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">model</td><td rowspan=\"2\">Aug Rot. Tex.</td><td rowspan=\"2\">ON-13 top-1 top-5</td><td colspan=\"7\">IN-C-100 (top-1)</td><td rowspan=\"2\">IN-100</td></tr><tr><td>Noise</td><td>Blur</td><td>Weather Digital</td><td></td><td>All</td><td>d≥3</td><td>top-1 top-5</td></tr><tr><td>Supervised</td><td></td><td>30.9</td><td>54.8 28.4</td><td>47.1</td><td>44.9</td><td>58.5</td><td>47.2</td><td>36.5</td><td>83.7</td><td>95.7</td></tr><tr><td>MoCo</td><td></td><td>29.2 54.2</td><td>37.9</td><td>38.5</td><td>47.7</td><td>60.1</td><td>48.2</td><td>37.2</td><td>81.0</td><td>95.2</td></tr><tr><td rowspan=\"3\">LooC</td><td>√</td><td>34.2</td><td>59.6</td><td>31.3</td><td>33.1</td><td>42.4</td><td>54.9</td><td>42.7</td><td>31.8</td><td>80.2 95.5</td></tr><tr><td></td><td>30.1</td><td>54.1</td><td>42.4</td><td>39.6 54.0</td><td>61.9</td><td>51.3</td><td>41.9</td><td>81.0</td><td>94.7</td></tr><tr><td>√</td><td>? 33.3</td><td>59.2</td><td>37.0</td><td>35.2</td><td>50.2</td><td>56.9</td><td>46.5</td><td>37.2 79.4</td><td>94.3</td></tr><tr><td>LooC++</td><td>√ √</td><td>32.6</td><td>57.3</td><td>38.3</td><td>37.6</td><td>52.0</td><td>60.0</td><td>48.8</td><td>38.9</td><td>82.1 95.1</td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/f45d67f662cabca11f229c391fd7d665423aa779798e0a4a0265d4efbc430139.jpg",
+        "table_caption": [
+            "Table 4: Comparisons of LooC vs. MoCo trained with all augmentations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>IN-100top-1top-5</td><td rowspan=1 colspan=1>iNat-1ktop-1top-5</td><td rowspan=1 colspan=1>Flowers-1025-shot     10-shot</td><td rowspan=1 colspan=1>IN-C-100all-top-1</td></tr><tr><td rowspan=1 colspan=1>MoCoLooC</td><td rowspan=1 colspan=1>77.9 93.778.594.0</td><td rowspan=1 colspan=1>39.5 65.141.7 67.5</td><td rowspan=1 colspan=1>72.1 (± 0.4)81.1 (± 0.2)72.1(± 0.7)81.4(± 0.2)</td><td rowspan=1 colspan=1>47.445.4</td></tr><tr><td rowspan=1 colspan=1>MoCo++LooC++</td><td rowspan=1 colspan=1>80.8 94.682.2 95.3</td><td rowspan=1 colspan=1>43.4 68.545.9 71.4</td><td rowspan=1 colspan=1>70.0 (±0.8)80.5 (±0.3)71.0 (± 0.7)81.9 (± 0.3)</td><td rowspan=1 colspan=1>48.348.0</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/36c49cea0e414e6b1d16e03bddece5452b189acd2ca08d53141886333db1a36b.jpg",
+        "table_caption": [
+            "Table 5: Comparisons of concatenating features from different embedding spaces in $\\mathbf { L o o C + + }$ jointly trained on color, rotation and texture augmentations. Different downstream tasks show nonidentical preferences for augmentation-dependent or invariant representations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td rowspan=\"2\">Col.</td><td rowspan=\"2\">Variance Head Rot.</td><td rowspan=\"2\">Tex.</td><td colspan=\"2\">IN-100</td><td rowspan=\"2\">iNat-1k top-1 top-5</td><td colspan=\"2\">Flowers-102</td><td rowspan=\"2\">IN-C-100 all-top-1</td></tr><tr><td>top-1</td><td>top-5</td><td>5-shot</td><td>10-shot</td></tr><tr><td rowspan=\"5\">LooC++</td><td></td><td></td><td></td><td>78.5</td><td>94.3</td><td>38.5</td><td>64.7</td><td>68.6(± 0.6) 77.6 (± 0.1)</td><td>48.0</td></tr><tr><td>Y</td><td></td><td></td><td>79.7</td><td>94.4</td><td>42.9 68.7</td><td></td><td>69.1 (± 0.7) 79.5 (± 0.2)</td><td>47.1</td></tr><tr><td></td><td>√</td><td></td><td>81.5</td><td>94.9</td><td>41.4 67.4</td><td>70.5 (±0.6)8</td><td>80.0 (± 0.2)</td><td>52.6</td></tr><tr><td></td><td></td><td>√</td><td>80.3</td><td>94.9</td><td>43.0 68.6</td><td>70.4 (± 0.5)</td><td>80.5 (± 0.2)</td><td>44.1</td></tr><tr><td>厂</td><td>√</td><td>√</td><td>82.2</td><td>95.3</td><td>45.9 71.4</td><td>71.0 (± 0.7)</td><td>81.9 (± 0.3)</td><td>48.0</td></tr></table>",
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+        "text": "Fine-grained recognition results. A prominent application of unsupervised learning is to learn features which are transferable and generalizable to a variety of downstream tasks. To fairly evaluate this, we compare our method with original MoCo on a diverse set of downstream tasks. Table 2 lists the results on iNat-1k, CUB-200 and Flowers-102. Although demonstrating marginally superior performance on IN-100, the original MoCo trails our LooC counterpart on all other datasets by a noticeable margin. Specifically, applying LooC on random color jiterring boosts the performance of the baseline which adopts the same augmentation. The comparison shows that our method can better preserve color information. Rotation augmentation also boosts the performance on iNat-1k and Flowers-102, while yields smaller improvements on CUB-200, which supports the intuition that some categories benefit from rotation-invariant representations while some do not. The performance is further boosted by using LooC with both augmentations, demonstrating the effectiveness in simultaneously learning the information w.r.t. multiple augmentations. ",
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+        "type": "text",
+        "text": "Interestingly, $\\mathrm { L o o C + + }$ brings back the slight performance drop on IN-100, and yields more gains on iNat-1k, which indicates the benefits of explicit feature fusion without hand-crafting what should or should not be contrastive in the training objective. ",
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+        "type": "text",
+        "text": "Robustness learning results. Table 3 compares our method with MoCo and supervised model on ON-13 and IN-C-100, two testing sets for real-world data generalization under a variety of noise conditions. The linear classifier is trained on standard IN-100, without access to the testing distribution. The fully supervised network is most sensitive to perturbations, albeit it has highest accuracy on the source dataset IN-100. We also see that rotation augmentation is beneficial for ON-13, but significantly downgrades the robustness to data corruptions in IN-C-100. Conversely, texture randomization increases the robustness on IN-C-100 across all corruption types, particularly significant on “Blur” and “Weather”, and on the severity level above or equal to 3, as the representations must be insensitive to local noise to learn texture-invariant features, but its improvement on ON-13 is marginal. Combining rotation and texture augmentation yields improvements on both datasets, and $\\mathrm { L o o C + + }$ further improves its performance on IN-C-100. ",
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+        "img_path": "images/a5123be19ac2689c00e96088a424265c4e0766e416856edbc22e6f3a0914b7c0.jpg",
+        "image_caption": [
+            "Figure 3: Top nearest-neighbor retrieval results of LooC vs. corresponding invariant MoCo baseline with color (left) and rotation (right) augmentations on IN-100 and iNat-1k. The results show that our model can better preserve information dependent on color and rotation despite being trained with those augmentations. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Qualitative results. In Figure 3 we show nearest-neighbor retrieval results using features learnt with LooC vs. corresponding MoCo baseline. The top retrieval results demonstrate that our model can better preserve information which is not invariant to the transformations presented in the augmentations used in contrastive learning. ",
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+        "type": "text",
+        "text": "Ablation: MoCo w/ all augmentations vs. LooC. We compare our method and MoCo trained with all augmentations. We also add multiple Conv5 heads to MoCo, termed as $\\mathrm { M o C o + + }$ , for a fair comparison with $\\mathrm { L o o C + + }$ . The results are listed in Table 4. Using multiple heads boosts the performance of baseline MoCo, nevertheless, our method achieves better or comparable results compared with its baseline counterparts. ",
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+    {
+        "type": "text",
+        "text": "Note that the results in Table 2 to 5 should be interpreted in the broader context of Table 1. Table 1 illustrates the catastrophic consequences of not separating the varying and invariant factors of an augmentation (in this case, rotation). It can be imagined that if we add “rotation classification” as one downstream task in Table 4, $\\mathrm { M o C o + + }$ will perform as poorly as in Table 1. The key of our work is to avoid what has happened in Table 1 and simultaneously boosts performance. ",
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+    {
+        "type": "text",
+        "text": "Ablation: Augmentation-dependent embedding spaces vs. tasks. We train a $_ { \\mathrm { L o o C + + } }$ with all types of augmentations, and subsequently train multiple linear classifiers with concatenated features from different embedding spaces: all-invariant, color, rotation and texture. Any additional variance features boost the performance on IN-100, iNat-1k and Flowers-102. Adding texture-dependent features decreases the performance on IN-C-100: Textures are (overly) strong cues for ImageNet classification (Geirhos et al., 2018), thus the linear classifier is prone to use texture-dependent features, loosing the gains of texture invariance. Adding rotation-dependent features increases the performance on IN-C-100: Rotated objects of most classes in IN-100 are rare, thus the linear classifier is prone to use rotation-dependent features, so that drops on IN-C-100 triggered by rotation-invariant augmentation are re-gained. Using all types of features yields best performance on IN-100, iNat1k and Flowers-102; the performance on IN-C-100 with all augmentations remains comparable to MoCo, which does not suffer from loss of robustness introduced by rotation invariance. ",
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+    {
+        "type": "text",
+        "text": "In Figure 4 we show the histogram of correct predictions (activations $\\times$ weights of classifier) by each augmentation-dependent head of a few instances from IN-100 and iNat-1k. The classifier prefers texture-dependent information over other kinds on an overwhelmingly majority of samples from IN-100, even for classes where shape is supposed to be the dominant factor, such as “pickup” and “mixing bowl” ((a), top row). This is consistent with the findings from (Geirhos et al., 2018) that ImageNet-trained CNNs are strongly biased towards texture-like representations. Interestingly, when human or animal faces dominant an image ((a), bottom-left), $\\mathrm { L o o C + + }$ sharply prefers rotation-dependent features, which also holds for face recognition of humans. In contrast, on iNat1k $\\mathrm { L o o C + + }$ prefers a more diverse set of features, such as color-dependent feature for a dragonfly species, rotation and texture-dependent features for birds, as well as rotation-invariant features for flowers. Averaged over the datasets, the distribution of classifier preferences is more balanced on iNat-1k than IN-100, as can be seen from the entropy that the distribution on iNat-1k is close to 2 bits, whereas it is close to 1 bit on IN-100, as it is dominated by only two elements. It corroborates the large improvements on iNat-1k gained from multi-dependent features learnt by our method. ",
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+    {
+        "type": "image",
+        "img_path": "images/6b847cc1eef4f9a685453af1d89d23117753f1e3a21c2d22b3cfe1662516a3c9.jpg",
+        "image_caption": [
+            "Figure 4: Histograms of correct predictions (activations $\\times$ weights of classifier) by each augmentation-dependent head from IN-100 and iNat-1k. The classifier on IN-100 heavily relies on texture-dependent information, whereas it is much more balanced on iNat-1k. This is consistent with the improvement gains observed when learning with multiple augmentations. "
+        ],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Pretext Tasks. In computer vision, feature design and engineering used to be a central topic before the wide application of deep learning. Researchers have proposed to utilize cue combination for image retrieval and recognition tasks (Martin et al., 2004; Frome et al., 2007a;b; Malisiewicz & Efros, 2008; Rabinovich et al., 2006). For example, the local brightness, color, and texture features are combined together to represent an image and a simple linear model can be trained to detect boundaries (Martin et al., 2004). Interestingly, the recent development of unsupervised representation learning in deep learning is also progressed by designing different self-supervised pretext tasks (Wang & Gupta, 2015; Doersch et al., 2015; Pathak et al., 2016; Noroozi & Favaro, 2016; Zhang et al., 2016; Gidaris et al., 2018; Owens et al., 2016). For example, relative patch prediction (Doersch et al., 2015) and rotation prediction (Gidaris et al., 2018) are designed to discover the underlined structure of the objects; image colorization task (Zhang et al., 2016) is used to learn representations capturing color information. The inductive bias introduced by each pretext task can often be associated with a corresponding hand-crafted descriptor. ",
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+    {
+        "type": "text",
+        "text": "Multi-Task Self-Supervised Learning. Multi-task learning has been widely applied in image recognition (Kokkinos, 2017; Teichmann et al., 2018; He et al., 2017). However, jointly optimizing multiple tasks are not always beneficial. As shown in Kokkinos (2017), training with two tasks can yield better performance than seven tasks together, as some tasks might be conflicted with each other. This phenomenon becomes more obvious in multi-task self-supervised learning (Doersch & Zisserman, 2017; Wang et al., 2017; Pinto & Gupta, 2017; Piergiovanni et al., 2020; Alwassel et al., 2019) as the optimization goal for each task can be very different depending on the pretext task. To solve this problem, different weights for different tasks are learned to optimize for the downstream tasks (Piergiovanni et al., 2020). However, searching the weights typically requires labels, and is time-consuming and does not generalize to different tasks. To train general representations, researchers have proposed to utilize sparse regularization to factorize the network representations to encode different information from different tasks (Doersch & Zisserman, 2017; Misra et al., 2016). In this paper, we also proposed to learn representation which can factorize and unify information from different augmentations. Instead of using sparse regularization, we define different contrastive learning objective in a multi-head architecture. ",
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+    {
+        "type": "text",
+        "text": "Contrastive Learning. Instead of designing different pretext tasks, recent work on contrastive learning (Wu et al., 2018; Oord et al., 2018; Tian et al., 2019; He et al., 2020; Misra & van der Maaten, 2020; Chen et al., 2020a) trained networks to be invariant to various corresponding augmentations. Researchers (Chen et al., 2020a) elaborated different augmentations and pointed out which augmentations are helpful or harmful for ImageNet classification. It is also investigated in Tian et al. (2019) that different augmentations can be beneficial to different downstream tasks. Instead of enumerating all the possible selections of augmentations, we proposed a unified framework which captures different factors of variation introduced by different augmentations. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Current contrastive learning approaches rely on specific augmentation-derived transformation invariances to learn a visual representation, and may yield suboptimal performance on downstream tasks if the wrong transformation invariances are presumed. We propose a new model which learns both transformation dependent and invariant representations by constructing multiple embeddings, each of which is not contrastive to a single type of transformation. Our framework outperforms baseline contrastive method on coarse-grained, fine-grained, few-shot downstream classification tasks, and demonstrates better robustness of real-world data corruptions. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENT ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Prof. Darrell’s group was supported in part by DoD, NSF, BAIR, and BDD. Prof. Wang’s group was supported, in part, by gifts from Qualcomm and TuSimple. We would like to thank Allan Jabri, Colorado Reed and Ilija Radosavovic for helpful discussions. ",
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+        "text": "Yonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. arXiv preprint arXiv:1906.05849, 2019. 1, 4, 8 ",
+        "bbox": [
+            174,
+            219,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Grant Van Horn, Oisin Mac Aodha, Yang Song, Yin Cui, Chen Sun, Alex Shepard, Hartwig Adam, Pietro Perona, and Serge Belongie. The inaturalist species classification and detection dataset. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 8769–8778, 2018. 2, 4 ",
+        "bbox": [
+            173,
+            257,
+            825,
+            314
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Catherine Wah, Steve Branson, Peter Welinder, Pietro Perona, and Serge Belongie. The caltech-ucsd birds-200-2011 dataset. 2011. 2, 5 ",
+        "bbox": [
+            171,
+            323,
+            825,
+            352
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Xiaolong Wang and Abhinav Gupta. Unsupervised learning of visual representations using videos. In Proceedings of the IEEE International Conference on Computer Vision, pp. 2794–2802, 2015. 8 ",
+        "bbox": [
+            173,
+            361,
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+            402
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Xiaolong Wang, Kaiming He, and Abhinav Gupta. Transitive invariance for self-supervised visual representation learning. In ICCV, 2017. 8 ",
+        "bbox": [
+            174,
+            412,
+            823,
+            441
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Zhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via nonparametric instance discrimination. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3733–3742, 2018. 1, 8 ",
+        "bbox": [
+            173,
+            450,
+            823,
+            493
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In European conference on computer vision, pp. 649–666. Springer, 2016. 1, 8 ",
+        "bbox": [
+            169,
+            501,
+            823,
+            531
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Richard Zhang, Phillip Isola, and Alexei A Efros. Split-brain autoencoders: Unsupervised learning by cross-channel prediction. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1058–1067, 2017. 1 ",
+        "bbox": [
+            176,
+            540,
+            825,
+            583
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A AUGMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            102,
+            426,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Following (Chen et al., 2020b), we set the probability of color jittering to 0.8, with (brightness, contrast, saturation, hue) as (0.4, 0.4, 0.4, 0.1), and probability of random scale to 0.2. We set the probability of random rotation and texture randomization as 0.5. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            176
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            195,
+            295,
+            212
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "iNat-1k, a large-scale classification dataset containing 1,010 species with a combined training and validation set of 268,243 images. We randomly reallocate $10 \\%$ of training images into the validation set as the original validation set is relatively small. ",
+        "bbox": [
+            174,
+            227,
+            825,
+            270
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "CUB-200, which contains 5,994 training and 5,794 testing images of 200 bird species. ",
+        "bbox": [
+            174,
+            276,
+            738,
+            291
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Flowers-102, which contains 102 flower categories consisting of between 40 and 258 images. ",
+        "bbox": [
+            173,
+            297,
+            785,
+            313
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "ObjectNet, a test set collected to intentionally show objects from new viewpoints on new backgrounds with different rotations of real-world images. It originally has 313-category. We only use the 13 categories which overlap with IN-100. ",
+        "bbox": [
+            174,
+            319,
+            825,
+            362
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "ImageNet-C, which consists of 15 diverse corruption types applied to validation images of ImageNet. ",
+        "bbox": [
+            173,
+            368,
+            823,
+            397
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C LINEAR CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            416,
+            419,
+            433
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We train the linear layer for 200 epochs for IN-100 and CUB-200, 100 epochs for iNat-1k, optimized by momentum SGD with a learning rate of 30 decreased by 0.1 at $60 \\%$ and $80 \\%$ of training schedule; for Flowers-102 we train the linear layer with Adam optimizer for 250 iterations with a learning rate of 0.03. ",
+        "bbox": [
+            173,
+            448,
+            825,
+            503
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D LEAVE-ONE-OUT VS. ADD-ONE AUGMENTATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            525,
+            614,
+            541
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/3c89422c252b420821ca2a949129261f3f4cebbdda06a024a2ed191244773c24.jpg",
+        "table_caption": [
+            "Table 6: Leave-one-out vs. add-one Augmentation. \\*: Default (none add-one) augmentation strategy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">model</td><td colspan=\"2\">Augmentation</td><td colspan=\"2\">IN-100</td></tr><tr><td>Color</td><td>Rotation</td><td>top-1</td><td>top-5</td></tr><tr><td rowspan=\"2\">MoCo</td><td>√</td><td></td><td>81.0</td><td>95.2</td></tr><tr><td>√</td><td>√</td><td>79.4</td><td>94.1</td></tr><tr><td rowspan=\"2\">MoCo+AddOne</td><td>√</td><td></td><td>74.9</td><td>92.5</td></tr><tr><td>*</td><td>√</td><td>79.3</td><td>94.4</td></tr><tr><td rowspan=\"2\">LooC[ours]</td><td>√</td><td></td><td>81.1</td><td>95.3</td></tr><tr><td>*</td><td>√</td><td>80.2</td><td>95.5</td></tr></table>",
+        "bbox": [
+            316,
+            599,
+            678,
+            712
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A straight-forward alternative for our leave-one-out augmentation strategy is add-one augmentation. Instead of applying all augmentations and augmenting two views in the same manner, add-one strategy keeps the query image unaugmentated, while in each augmentation-specific view the designated type of augmentation is applied. The results are shown in Table 6. Add-one strategy oversimplifies the instance discrimination task, e.g., leaving color augmentation out of query view makes it very easy for the network to spot the same instance out of a set of candidates. Our leave-one-out strategy does not suffer such degeneration. ",
+        "bbox": [
+            173,
+            727,
+            825,
+            825
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "E IMAGENET-1K EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            455,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 7: Results of models trained on 1000 category ImageNet and fine-tuned on iNat-1k following linear classification protocol. ",
+        "bbox": [
+            173,
+            137,
+            823,
+            165
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/e55a608fb1b0d8b01b321a9e54cda6f73115a5065f6c3d6e9203f1d9e948b4f7.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">model</td><td colspan=\"2\">iNat-1k</td></tr><tr><td>top-1</td><td>top-5</td></tr><tr><td>MoCo</td><td>47.8</td><td>74.3</td></tr><tr><td>LooC++ [ours]</td><td>51.2</td><td>76.5</td></tr></table>",
+        "bbox": [
+            383,
+            176,
+            611,
+            234
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We conduct experiments on 1000 category full ImageNet dataset. The models are trained by selfsupervised learning on IN-1k, and fine-tuned on iNat-1k following linear classification protocol. Our model is trained with all augmentations, i.e., color, rotation and texture. Results are reported in Table 7. ",
+        "bbox": [
+            174,
+            251,
+            825,
+            306
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "F DISCUSSIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            327,
+            321,
+            343
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "F.1 THE DIMENSIONS OF MOCO, LOOC, LOOC++ ",
+        "bbox": [
+            174,
+            358,
+            534,
+            373
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The representations of MoCo and LooC are of exactly the same dimension (2048); same for $\\mathrm { M o C o + + }$ and $\\mathrm { L o o C + + }$ (2048 \\* # augmentations). It is specifically designed for fair comparisons. ",
+        "bbox": [
+            176,
+            385,
+            825,
+            412
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "F.2 ARE THE HYPER-PARAMETERS TUNED SPECIFICALLY FOR OUR SUBSETS? ",
+        "bbox": [
+            176,
+            430,
+            722,
+            444
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "No, except that we increase the number of training epochs as the amount of data increases. We did not specifically tune the baseline so that our method can outperform it most; on the contrary, we first made baseline as strong as possible, then directly applied the same hyper-parameters to our method. The subset of ImageNet100 behaviors similarly as ImageNet1k; our baseline already significantly outperforms the best method on the same subset from previous literature ( $7 5 . 8 \\%$ CMC vs. $8 1 . 0 \\%$ top1 [ours]), and since our method is derived from MoCo, they are directly comparable. ",
+        "bbox": [
+            174,
+            455,
+            825,
+            540
+        ],
+        "page_idx": 12
+    }
+]

parse/train/H1goBoR9F7/H1goBoR9F7.md

@@ -0,0 +1,421 @@
+# DYNAMIC SPARSE GRAPH FOR EFFICIENT DEEP LEARNING
+
+Liu $\mathbf { L i u ^ { 1 2 * } }$ , Lei Deng2∗, Xing $\mathbf { H } \mathbf { u } ^ { 2 }$ , Maohua $\mathbf { Z } \mathbf { h } \mathbf { u } ^ { 2 }$ , Guoqi $\mathbf { L i ^ { 3 } }$ , Yufei $\mathbf { D i n g ^ { 2 } }$ , Yuan Xie1   
+1Department of Electrical and Computer Engineering, University of California, Santa Barbara   
+2Department of Computer Science, University of California, Santa Barbara   
+3Center for Brain Inspired Computing Research,   
+Department of Precision Instrument, Tsinghua University   
+∗Equal contribution   
+{liu liu, leideng, huxing, maohua, yuanxie}@ece.ucsb.edu   
+yufeiding@cs.ucsb.edu   
+liguoqi@mail.tsinghua.edu.cn
+
+# ABSTRACT
+
+We propose to execute deep neural networks (DNNs) with dynamic and sparse graph (DSG) structure for compressive memory and accelerative execution during both training and inference. The great success of DNNs motivates the pursuing of lightweight models for the deployment onto embedded devices. However, most of the previous studies optimize for inference while neglect training or even complicate it. Training is far more intractable, since (i) the neurons dominate the memory cost rather than the weights in inference; (ii) the dynamic activation makes previous sparse acceleration via one-off optimization on fixed weight invalid; (iii) batch normalization (BN) is critical for maintaining accuracy while its activation reorganization damages the sparsity. To address these issues, DSG activates only a small amount of neurons with high selectivity at each iteration via a dimensionreduction search and obtains the BN compatibility via a double-mask selection. Experiments show significant memory saving (1.7-4.5x) and operation reduction (2.3-4.4x) with little accuracy loss on various benchmarks.
+
+# 1 INTRODUCTION
+
+Deep Neural Networks (DNNs) (LeCun et al., 2015) have been achieving impressive progress in a wide spectrum of domains (Simonyan & Zisserman, 2014; He et al., 2016; Abdel-Hamid et al., 2014; Redmon & Farhadi, 2016; Wu et al., 2016), while the models are extremely memory- and compute-intensive. The high representational and computational costs motivate many researchers to investigate approaches on improving the execution performance, including matrix or tensor decomposition (Xue et al., 2014; Novikov et al., 2015; Garipov et al., 2016; Yang et al., 2017; Alvarez & Salzmann, 2017), data quantization (Courbariaux et al., 2016; Zhou et al., 2016; Deng et al., 2018; Leng et al., 2017; Wen et al., 2017; Wu et al., 2018; McKinstry et al., 2018), and network pruning (Ardakani et al., 2016; Han et al., 2015b;a; Liu et al., 2017; Li et al., 2016; He et al., 2017; Luo et al., 2017; Wen et al., 2016; Molchanov et al., 2016; Sun et al., 2017; Spring & Shrivastava, 2017; Lin et al., 2017a; Zhang et al., 2018; He et al., 2018a; Chin et al., 2018; Ye et al., 2018; Luo & Wu, 2018; Hu et al., 2018; He et al., 2018b). However, most of the previous work aim at inference while the challenges for reducing the representational and computational costs of training are not well-studied. Although some works demonstrate acceleration in the distributed training (Lin et al., 2017b; Goyal et al., 2017; You et al., 2017), we target at the single-node optimization, and our method can also boost training in a distributed fashion.
+
+DNN training, which demands much more hardware resources in terms of both memory capacity and computation volume, is far more challenging than inference. Firstly, activation data in training will be stored for backpropagation, significantly increasing the memory consumption. Secondly, training iteratively updates model parameters using mini-batched stochastic gradient descent. We almost always expect larger mini-batches for higher throughput (Figure 1(a)), faster convergence, and better accuracy (Smith et al., 2017). However, memory capacity is often the limitation factor (Figure 1(b))
+
+![](images/b5ec57a2c7023e119f70dd82613c22c8e876d2108bb909dd72295cd59db58c12.jpg)  
+Figure 1: Comprehensive motivation illustration. (a) Using larger mini-batch size helps improve throughput until it is compute-bound; (b) Limited memory capacity on a single computing node prohibits the use of large mini-batch size; (c) Neuronal activation dominates the representational cost when mini-batch size becomes large; (d) BN is indispensable for maintaining accuracy; (e) Upper and lower one are the feature maps before and after BN, respectively. However, using BN damages the sparsity through information fusion; (f) There exists such great representational redundancy that more than $80 \%$ of activations are close to zero.
+
+that may cause performance degradation or even make large models with deep structures or targeting high-resolution vision tasks hard to train (He et al., 2016; Wu & He, 2018).
+
+It is difficult to apply existing sparsity techniques towards inference phase to training phase because of the following reasons: 1) Prior arts mainly compress the pre-trained and fixed weight parameters to reduce the off-chip memory access in inference (Han et al., 2016; 2017), while instead, the dynamic neuronal activations turn out to be the crucial bottleneck in training (Jain et al., 2018), making the prior inference-oriented methods inefficient. Besides, during training we need to stash a vast batched activation space for the backward gradient calculation. Therefore, neuron activations creates a new memory bottleneck (Figure 1(c)). In this paper, we will sparsify the neuron activations for training compression. 2) The existing inference accelerations usually add extra optimization problems onto the critical path (Wen et al., 2016; Molchanov et al., 2016; Liu et al., 2017; Luo et al., 2017; Liang et al., 2018; Zhang et al., 2018; Hu et al., 2018; Luo & Wu, 2018; Ye et al., 2018), i.e., ‘complicated training $\Rightarrow$ simplified inference’, which embarrassingly complicates the training phase. 3) Moreover, previous studies reveal that batch normalization (BN) is crucial for improving accuracy and robustness (Figure 1(d)) through activation fusion across different samples within one mini-batch for better representation (Morcos et al., 2018; Ioffe & Szegedy, 2015). BN almost becomes a standard training configuration; however, inference-oriented methods seldom discuss BN and treat BN parameters as scaling and shift factors in the forward pass. We further find that BN will damage the sparsity due to the activation reorganization (Figure 1(e)). Since this work targets both training and inference, the BN compatibility problem should be addressed.
+
+From the view of information representation, the activation of each neuron reflects its selectivity to the current stimulus sample (Morcos et al., 2018), and this selectivity dataflow propagates layer by layer forming different representation levels. Fortunately, there is much representational redundancy, for example, lots of neuron activations for each stimulus sample are so small and can be removed (Figure 1(f)). Motivated by above comprehensive analysis regarding memory and compute, we propose to search critical neurons for constructing a sparse graph at every iteration. By activating only a small amount of neurons with a high selectivity, we can significantly save memory and simplify computation with tolerable accuracy degradation. Because the neuron response dynamically changes under different stimulus samples, the sparse graph is variable. The neuronaware dynamic and sparse graph (DSG) is fundamentally distinct from the static one in previous work on permanent weight pruning since we never prune the graph but activate part of them each time. Therefore, we maintain the model expressive power as much as possible. A graph selection method, dimension-reduction search, is designed for both compressible activations with elementwise unstructured sparsity and accelerative vector-matrix multiplication (VMM) with vector-wise structured sparsity. Through double-mask selection design, it is also compatible with BN. We can use the same selection pattern and extend our method to inference. In a nutshell, we propose a compressible and accelerative DSG approach supported by dimension-reduction search and doublemask selection. It can achieve $1 . 7 – 4 . 5 \mathrm { x }$ memory compression and $2 . 3 – 4 . 4 \mathrm { x }$ computation reduction with minimal accuracy loss. This work simultaneously pioneers the approach towards efficient online training and offline inference, which can benefit the deep learning in both the cloud and the edge.
+
+# 2 APPROACH
+
+Our method forms DSGs for different inputs, which are accelerative and compressive, as shown in Figure2(a). On the one hand, choosing a small number of critical neurons to participate in computation, DSG can reduce the computational cost by eliminating calculations of non-critical neurons. On the other hand, it can further reduce the representational cost via compression on sparsified activations. Different from previous methods using permanent pruning, our approach does not prune any neuron and the associated weights; instead, it activates a sparse graph according to the input sample at each iteration. Therefore, DSG does not compromise the expressive power of the model.
+
+![](images/9971a8d6c19ae652448d717a55dedbef3ed040412c37d341d83953a76e1df034.jpg)  
+Figure 2: (a) Illustration of dynamic and sparse graph (DSG); (b) Dimension-reduction search for construction of DSG; (c) Double-mask selection for BN compatibility. ‘DRS’ denotes dimensionreduction search.
+
+Constructing DSG needs to determine which neurons are critical. A naive approach is to select critical neurons according to the output activations. If the output neurons have a small or negative activation value, i.e., not selective to current input sample, they can be removed for saving representational cost. Because these activations will be small or absolute zero after the following ReLU non-linear function (i.e., $\mathrm { R e L U } ( x ) = \operatorname* { m a x } ( 0 , x ) )$ , it’s reasonable to set all of them to be zero. However, this naive approach requires computations of all VMM operations within each layer before the selection of critical neurons, which is very costly.
+
+# 2.1 DIMENSION-REDUCTION SEARCH
+
+To avoid the costly VMM operations in the mentioned naive selection, we propose an efficient method, i.e., dimension reduction search, to estimate the importance of output neurons. As shown in Figure2(b), we first reduce the dimensions of $\mathbf { X }$ and $\mathbf { W }$ , and then execute the lightweight VMM operations in a low-dimensional space with minimal cost. After that, we estimate the neuron importance according to the virtual output activations. Then, a binary selection mask can be produced in which the zeros represent the non-critical neurons with small activations that are removable. We use a top- $k$ search method that only keeps largest $k$ neurons, where an inter-sample threshold sharing mechanism is leveraged to greatly reduce the search cost 1. Note that $k$ is determined by the output size and a pre-configured sparsity parameter $\gamma$ . Then we can just compute the accurate activations of the critical neurons in the original high-dimensional space and avoid the calculation of the noncritical neurons. Thus, besides the compressive sparse activations, the dimension-reduction search can further save a significant amount of expensive operations in the high-dimensional space.
+
+![](images/d069229bab257e713a80a78c36c7a4fe761f3b5c6287b71ca20f4ccfd52880ac.jpg)  
+Figure 3: Compressive and accelerative DSG. (a) Original dense convolution; (b) Converted accelerative VMM operation; (c) Zero-value compression.
+
+In this way, a vector-wise structured sparsity can be achieved, as shown in Figure 3(b). The ones in the selection mask (marked as colored blocks) denote the critical neurons, and the non-critical ones can bypass the memory access and computation of their corresponding columns in the weight matrix. Furthermore, the generated sparse activations can be compressed via the zero-value compression (Zhang et al., 2000; Vijaykumar et al., 2015; Rhu et al., 2018) (Figure 3(c)). Consequently, it is critical to reduce the vector dimension but keep the activations calculated in the low-dimensional space as accurate as possible, compared to the ones in the original high-dimensional space.
+
+# 2.2 SPARSE RANDOM PROJECTION FOR EFFICIENT DIMENSION-REDUCTION SEARCH
+
+Notations: Each CONV layer has a four dimensional weight tensor $( n _ { K } , n _ { C } , n _ { R } , n _ { S } )$ , where $n _ { K }$ is the number of filters, i.e., the number of output feature maps (FMs); $n _ { C }$ is the number of input FMs; $( n _ { R } , n _ { S } )$ represents the kernel size. Thus, the CONV layer in Figure 3(a) can be converted to many VMM operations, as shown in Figure 3(b). Each row in the matrix of input FMs is the activations from a sliding window across all input FMs $( n _ { C R S } = n _ { C } \times n _ { R } \times n _ { S } )$ , and after the VMM operation with the weight matrix $( n _ { C R S } \times n _ { K } )$ it can generate $n _ { K }$ points at the same location across all output FMs. Further considering the $n _ { P Q } = n _ { P } \times n _ { Q }$ size of each output FM and the mini-batch size of $m$ , the whole $n _ { P Q } \times m$ rows of VMM operations has a computational complexity of $O ( m \times n _ { P Q } \times$ $n _ { C R S } \times n _ { K } )$ . For the FC layer with $n _ { C }$ input neurons and $n _ { K }$ output neurons, this complexity is $O ( m \times n _ { C } \times n _ { K } )$ . Note that here we switch the order of BN and ReLU layer from ‘CONV/FCBN-ReLU’ to ‘CONV/FC-ReLU-BN’, because it’s hard to determine the activation value of the non-critical neurons if the following layer is BN (this value is zero for ReLU). As shown in previous work, this reorganization could bring better accuracy (Mishkin & Matas, 2015).
+
+For the sake of simplicity, we just consider the operation for each sliding window in the CONV layer or the whole FC layer under one single input sample as a basic optimization problem. The generation of each output activation $y _ { j }$ requires an inner product operation, as follows:
+
+$$
+y _ { j } = \varphi ( \langle \mathbf { X } _ { i } , \mathbf { W } _ { j } \rangle )
+$$
+
+where $\mathbf { X } _ { i }$ is the $i$ -th row in the matrix of input FMs (for the FC layer, there is only one $\mathbf { X }$ vector), $\mathbf { W } _ { j }$ is the $j$ -th column of the weight matrix $W$ , and $\varphi ( \cdot )$ is the neuronal transformation (e.g., ReLU function, here we abandon bias). Now, according to equation (1), the preservation of the activation is equivalent to preserve the inner product.
+
+We introduce a dimension-reduction lemma, named Johnson-Lindenstrauss Lemma (JLL) (Johnson & Lindenstrauss, 1984), to implement the dimension-reduction search with inner product preservation. This lemma states that a set of points in a high-dimensional space can be embedded into a low-dimensional space in such a way that the Euclidean distances between these points are nearly preserved. Specifically, given $0 < \epsilon < 1$ , a set of $N$ points in $\mathbb { R } ^ { d }$ (i.e., all $\mathbf { X } _ { i }$ and $\mathbf { W } _ { j }$ ), and a number of $\begin{array} { r } { k > O \bigl ( \frac { l o g ( N ) } { \epsilon ^ { 2 } } \bigr ) } \end{array}$ , there exists a linear map $f : \mathbb { R } ^ { d } \Rightarrow \mathbb { R } ^ { k }$ such that
+
+$$
+( 1 - \epsilon ) \| { \mathbf { X } } _ { i } - { \mathbf { W } } _ { j } \| ^ { 2 } \leq \| { f ( { \mathbf { X } } _ { i } ) - f ( { \mathbf { W } } _ { j } ) } \| ^ { 2 } \leq ( 1 + \epsilon ) \| { \mathbf { X } } _ { i } - { \mathbf { W } } _ { j } \| ^ { 2 }
+$$
+
+for any given $\mathbf { X } _ { i }$ and $\mathbf { W } _ { j }$ pair, where $\epsilon$ is a hyper-parameter to control the approximation error, i.e., larger $\epsilon \Rightarrow$ larger error. When $\epsilon$ is sufficiently small, one corollary from JLL is the following norm preservation (Vu, 2016; Kakade & Shakhnarovich, 2009):
+
+$$
+P [ ( 1 - \epsilon ) \| \mathbf { Z } \| ^ { 2 } \leq \| f ( \mathbf { Z } ) \| ^ { 2 } \leq ( 1 + \epsilon ) \| \mathbf { Z } \| ^ { 2 } ] \geq 1 - O ( \epsilon ^ { 2 } )
+$$
+
+where $\mathbf { Z }$ could be any $\mathbf { X } _ { i }$ or $\mathbf { W } _ { j }$ , and $P$ denotes a probability. It means the vector norm can be preserved with a high probability controlled by $\epsilon$ . Given these basics, we can further get the inner product preservation:
+
+$$
+P [ | \langle f ( \mathbf { X } _ { i } ) , f ( \mathbf { W } _ { j } ) \rangle - \langle \mathbf { X } _ { i } , \mathbf { W } _ { j } \rangle | \leq \epsilon ] \geq 1 - O ( \epsilon ^ { 2 } ) .
+$$
+
+The detailed proof can be found in Appendix A.
+
+Random projection (Vu, 2016; Ailon $\&$ Chazelle, 2009; Achlioptas, 2001) is widely used to construct the linear map $f ( \cdot )$ . Specifically, the original $d$ -dimensional vector is projected to a $k$ - dimensional $( k \ll d )$ one, using a random $k \times d$ matrix $\mathbf { R }$ . Then we can reduce the dimension of all $\mathbf { X } _ { i }$ and $\mathbf { W } _ { j }$ by
+
+$$
+f ( \mathbf { X } _ { i } ) = \frac { 1 } { \sqrt { k } } \mathbf { R } \mathbf { X } _ { i } \in \mathbb { R } ^ { k } , f ( \mathbf { W } _ { j } ) = \frac { 1 } { \sqrt { k } } \mathbf { R } \mathbf { W } _ { j } \in \mathbb { R } ^ { k } .
+$$
+
+The random projection matrix $\mathbf { R }$ can be generated from Gaussian distribution (Ailon $\&$ Chazelle, 2009). In this paper, we adopt a simplified version, termed as sparse random projection (Achlioptas, 2001; Bingham $\&$ Mannila, 2001; Li et al., 2006) with
+
+$$
+P ( \mathbf { R } _ { p q } = { \sqrt { s } } ) = { \frac { 1 } { 2 s } } ; P ( \mathbf { R } _ { p q } = 0 ) = 1 - { \frac { 1 } { s } } ; P ( \mathbf { R } _ { p q } = - { \sqrt { s } } ) = { \frac { 1 } { 2 s } }
+$$
+
+for all elements in $\mathbf { R }$ . This $\mathbf { R }$ only has ternary values that can remove the multiplications during projection, and the remained additions are very sparse. Therefore, the projection overhead is negligible compared to other high-precision operations involving multiplication. Here we set $s = 3$ with $67 \%$ sparsity in statistics.
+
+![](images/545ff4038b829098413785a7efda66efb7c6413129479d88c3f814d2922a3210.jpg)  
+Figure 4: Structured selection via dynamic dimension-reduction search for producing sparse pattern of neuronal activations.
+
+Equation (4) indicates the low-dimensional inner product $\left. f ( \mathbf { X } _ { i } ) , f ( \mathbf { W } _ { j } ) \right.$ can still approximate the original high-dimensional one $\langle \mathbf { X } _ { i } , \mathbf { W } _ { j } \rangle$ in equation (1) if the reduced dimension is sufficiently high. Therefore, it is possible to calculate equation (1) in a low-dimensional space for activation estimation, and select the important neurons. As shown in Figure 3(b), each sliding window dynamically selects its own important neurons for the calculation in high-dimensional space, marked in red and blue as two examples. Figure 4 visualizes two sliding windows in a real network to help understand the dynamic process of dimension-reduction search. Here the neuronal activation vector $\lceil n _ { K }$ length) is reshaped to a matrix for clarity. Now For the CONV layer, the computational complexity is only $O [ m \times n _ { P Q } \times n _ { K } \times ( k + ( 1 - \gamma ) \times n _ { C R S } ) ] ,$ , which is less than the original high-dimensional computation with $O ( m \times n _ { P Q } \times n _ { C R S } \times n _ { K } )$ ) complexity because we usually have $[ \ k + ( 1 - \gamma ) \times n _ { C R S } \ ] \ \ll \ n _ { C R S }$ . For the FC layer, we also have ${ \cal O } [ m \times n _ { K } \times ( k + ( 1 - \dot { \gamma } ) \times n _ { C } ) ] \ll { \cal O } ( m \times n _ { C } \times n _ { K } ) .$
+
+# 2.3 DOUBLE-MASK SELECTION FOR BN COMPATIBILITY
+
+To deal with the important but intractable BN layer, we propose a double-mask selection method presented in Figure 2(c). After the dimension-reduction search based importance estimation, we produce a sparsifying mask that removes the unimportant neurons. The ReLU activation function can maintain this mask by inhibiting the negative activation (actually all the activations of the CONV layer or FC layer after the selection mask are positive with reasonably large sparsity). However, the BN layer will damage this sparsity through inter-sample activation fusion. To address this issue, we copy the same selection mask before the BN layer and directly use it on the BN output. It is straightforward but reasonable because we find that although BN causes the zero activation to be non-zero (Figure 1(f)), these non-zero activations are still very small and can also be removed. This is because BN just scales and shifts the activations that won’t change the relative sort order. In this way, we can achieve fully sparse activation dataflow. The back propagated gradients will also be forcibly sparsified every time they pass a mask layer.
+
+# 3 EXPERIMENTAL RESULTS
+
+# 3.1 EXPERIMENT SETUP
+
+The overall training algorithm is presented in Appendices B. Going through the dataflow where the red color denotes the sparse tensors, a widespread sparsity in both the forward and backward passes is demonstrated. The projection matrices are fixed after a random initialization at the beginning of training. We just update the projected weights in the low-dimensional space every 50 iterations to reduce the projection overhead. The detailed search method and the computational complexity of the dimension-reduction search are provided in Appendix B. Regarding the evaluation network models, we use LeNet (LeCun et al., 1998) and a multi-layered perceptron (MLP) on small-scale FASHION dataset (Xiao et al., 2017), VGG8 (Courbariaux et al., 2016; Deng et al., 2018)/ResNet8 (a customized ResNet-variant with 3 residual blocks and 2 FC layers)/ResNet20/WRN-8-2 (Zagoruyko & Komodakis, 2016) on medium-scale CIFAR10 dataset (Krizhevsky & Hinton, 2009), VGG8/WRN8-2 on another medium-scale CIFAR100 dataset (Krizhevsky & Hinton, 2009), and AlexNet (Krizhevsky et al., 2012)/VGG16 (Simonyan & Zisserman, 2014)/ResNet18, ResNet152 (He et al., 2016)/WRN-18-2 (Zagoruyko & Komodakis, 2016) on large-scale ImageNet dataset (Deng et al., 2009) as workloads. The programming framework is PyTorch and the training platform is based on NVIDIA Titan Xp GPU. We adopt the zero-value compression method (Zhang et al., 2000; Vijaykumar et al., 2015; Rhu et al., 2018) for memory compression and MKL compute library (Wang et al., 2014) on Intel Xeon CPU for acceleration evaluation.
+
+# 3.2 ACCURACY ANALYSIS
+
+In this section, we provide a comprehensive analysis regarding the influence of sparsity on accuracy and explore the robustness of MLP and CNN, the graph selection strategy, the BN compatibility, and the importance of width and depth.
+
+Accuracy using DSG. Figure 5(a) presents the accuracy curves on small and medium scale models by using DSG under different sparsity levels. Three conclusions are observed: 1) The proposed DSG affects little on the accuracy when the sparsity is ${ < } 6 0 \%$ , and the accuracy will present an abrupt descent with sparsity larger than $80 \%$ . 2) Usually, the ResNet model family is more sensitive to the sparsity increasing due to fewer parameters than the VGG family. For the VGG8 on CIFAR10, the accuracy loss is still within $0 . 5 \%$ when sparsity reaches $80 \%$ . 3) Compared to MLP, CNN can tolerate more sparsity. Figure 5(b) further shows the results on large scale models on ImageNet. Because training large model is time costly, we only present several experimental points. Consistently, the VGG16 shows better robustness compared to the ResNet18, and the WRN with wider channels on each layer performs much better than the other two models. We will discuss the topic of width and depth later.
+
+Graph Selection Strategy. To investigate the influence of graph selection strategy, we repeat the sparsity vs. accuracy experiments on CIFAR10 under different selection methods. Two baselines are used here: the oracle one that keeps the neurons with top-k activations after the whole VMM computation at each layer, and the random one that randomly selects neurons to keep. The results are shown in Figure 5(c), in which we can see that our dimension-reduction search and the oracle one perform much better than the random selection under high sparsity condition. Moreover, dimension-reduction search achieves nearly the same accuracy with the oracle top-k selection, which indicates the proposed random projection method can find an accurate activation estimation in the low-dimensional space. In detail, Figure 5(d) shows the influence of parameter $\epsilon$ that reflects the degree of dimension reduction. Lower $\epsilon$ can approach the original inner product more accurately, that brings higher accuracy but at the cost of more computation for graph selection since less dimension reduction. With $\epsilon = 0 . 5$ , the accuracy loss is within $1 \%$ even if the sparsity reaches $80 \%$ .
+
+BN Compatibility. Figure 5(e) focuses the BN compatibility issue. Here we use dimensionreduction search for the graph sparsifying, and compare three cases: 1) removing the BN operation and using single mask; 2) keeping BN and using only single mask (the first one in Figure 2(c)); 3) keeping BN and using double masks (i.e. double-mask selection). The one without BN is very sensitive to the graph ablation, which indicates the importance of BN for training. Comparing the two with BN, the double-mask selection even achieves better accuracy since the regularization effect.
+
+![](images/930580bee800c61d64f86375fa56ac5219ab2abdb523ea833843e32ed1b6a231.jpg)  
+Figure 5: Comprehensive analysis on sparsity v.s. accuracy. (a) & (b) Accuracy using DSG; (c) Influence of the graph selection strategy; (d) Influence of the dimension-reduction degree; (e) Influence of the double-mask selection for BN compatibility; (f) Influence of the network depth and width. ‘DRS’ denotes dimension-reduction search.
+
+This observation indicates the effectiveness of the proposed double-mask selection for simultaneously recovering the sparsity damaged by the BN layer and maintaining the accuracy.
+
+Width or Depth. Furthermore, we investigate an interesting comparison regarding the network width and depth, as shown in Figure 5(f). On the training set, WRN with fewer but wider layers demonstrates more robustness than the deeper one with more but slimmer layers. On the validation set, the results are a little more complicated. Under small and medium sparsity, the deeper ResNet performs better $( 1 \% )$ than the wider one. While when the sparsity increases substantial $( > 7 5 \% )$ , WRN can maintain the accuracy better. This indicates that, in medium-sparse space, the deeper network has stronger representation ability because of the deep structure; however, in ultra-highsparse space, the deeper structure is more likely to collapse since the accumulation of the pruning error layer by layer. In reality, we can determine which type of model to use according to the sparsity requirement. In Figure 5(b) on ImageNet, the reason why WRN-18-2 performs much better is that it has wider layers without reducing the depth.
+
+Convergence. DSG does not slow down the convergence speed, which can be seen from Figure 10(a)-(b) in Appendix C. This owes to the high fidelity of inner product when we use random projection to reduce the data dimension, as shown in Figure 10(c). Interestingly, Figure 11 (also in Appendix C) reveals that the selection mask for each sample also converges as training goes on, however, the selection pattern varies across samples. To save the selection patterns of all samples is memory consuming, which is the reason why we do not directly suspend the selection patterns after training but still do on-the-fly dimension-reduction search in inference.
+
+# 3.3 REPRESENTATIONAL COST REDUCTION
+
+This section presents the benefits from DSG on representational cost. We measure the memory consumption over five CNN benchmarks on both the training and inference phases. For data compression, we use zero-value compression algorithm (Zhang et al., 2000; Vijaykumar et al., 2015; Rhu et al., 2018). Figure 6 shows the memory optimization results, where the model name, mini-batch size, and the sparsity are provided. In training, besides the parameters, the activations across all layers should be stashed for the backward computation. Consistent with the observation mentioned above that the neuron activation beats weight to dominate memory overhead, which is different from the previous work on inference. We can reduce the overall representational cost by average $1 . 7 \mathrm { x }$ (2.72 GB), $3 . 2 \mathbf { x }$ (4.51 GB), and $4 . 2 \mathrm { x }$ (5.04 GB) under $50 \%$ , $80 \%$ and $90 \%$ sparsity, respectively. If only considering the neuronal activation, these ratios could be higher up to 7.1x. The memory overhead for the selection masks is minimal $( < 2 \% )$ .
+
+![](images/4793b6d2be7a636fd9820372b4ac9d6a7203f32965cb088e01e7ec7eb462652b.jpg)  
+Figure 6: Memory footprint comparisons for (a) training and (b) inference.
+
+During inference, only memory space to store the parameters and the activations of the layer with maximum neuron amount is required. The benefits in inference are relatively smaller than that in training since weight is the dominant memory. On ResNet152, the extra mask overhead even offsets the compression benefit under $50 \%$ sparsity, whereas, we can still achieve up to $7 . 1 \mathrm { x }$ memory reduction for activations and $1 . 7 \mathrm { x }$ for overall memory. Although the compression is limited for inference, it still can achieve noticeable acceleration that will be shown in the next section. Moreover, reducing costs for both training and inference is our major contribution.
+
+# 3.4 COMPUTATIONAL COST REDUCTION
+
+We assess the results on reducing the computational cost for both training and inference. As shown in Figure 7, both the forward and backward pass consume much fewer operations, i.e., multiplyand-accumulate (MAC). On average, $1 . 4 \mathbf { x }$ (5.52 GMACs), $1 . 7 \mathrm { x }$ (9.43 GMACs), and $2 . 2 \mathbf { x }$ (10.74 GMACs) operation reduction are achieved in training under $50 \%$ , $80 \%$ and $90 \%$ sparsity, respectively. For inference with only forward pass, the results increase to $1 . 5 \mathrm { x }$ (2.26 GMACs), $2 . 8 \mathrm { x }$ (4.22 GMACs), and $3 . 9 \mathbf { X }$ (4.87 GMACs), respectively. The overhead of the dimension-reduction search in the low-dimensional space is relatively larger ( $( < 6 . 5 \%$ in training and ${ < } 1 9 . 5 \%$ in inference) compared to the mask overhead in memory cost. Note that the training demonstrates less improvement than the inference, which is because the acceleration of the backward pass is partial. The error propagation is accelerative, but the weight gradient generation is not because of the irregular sparsity that is hard to obtain practical acceleration. Although the computation of this part is also very sparse with much fewer operations 2, we do not include its GMACs reduction for practical concern.
+
+![](images/79b3bf15e029f303a15757df281291c717dc822fea0a4110d6e0e667e4dc0fba.jpg)  
+Figure 7: Computational complexity comparisons for (a) training and (b) inference. ‘DRS’ denotes dimension-reduction search.
+
+Finally, we evaluate the execution time on CPU using Intel MKL kernels (Wang et al. (2014)). As shown in Figure 8(a), we evaluate the execution time of these layers after the dimension-reduction search on VGG8. Comparing to VMM baselines, our approach can achieve 2.0x, 5.0x, and $8 . 5 \mathrm { x }$ average speedup under $50 \%$ , $80 \%$ , and $90 \%$ sparsity, respectively. When the baselines change to
+
+GEMM (general matrix multiplication), the average speedup decreases to 0.6x, 1.6x, and $2 . 7 \mathbf { x }$ , respectively. The reason is that DSG generates dynamic vector-wise sparsity, which is not well supported by GEMM. A potential way to improve GEMM-based implementation, at workload mapping and tiling time, is reordering executions at the granularity of vector inner-product and grouping non-redundant executions to the same tile to improve local data reuse.
+
+On the same network, we further compare our approach with smaller dense models which could be another way to reduce the computational cost. As shown in Figure 8(b), comparing with dense baseline, our approach can reduce training time with little accuracy loss. Even though the equivalent smaller dense models with the same effective nodes, i.e., reduced MACs, save more training time, the accuracy is much worse than our DSG approach. Figure 12 in Appendix D gives more results on ResNet8 and AlexNet.
+
+![](images/4a30a1ebe352b54e425bedaa39cf33ab7e5c7ce433aa02f6e1472b8c30a393bf.jpg)  
+Figure 8: On VGG8: (a) Layer-wise execution time comparison; (b) Validation accuracy v.s. training time of different models: large-sparse ones and smaller-dense ones with equivalent MACs.
+
+# 4 RELATED WORK
+
+DNN Compression (Ardakani et al., 2016) achieved up to $90 \%$ weight sparsity by randomly removing connections. (Han et al., 2015b;a) reduced the weight parameters by pruning the unimportant connections. The compression is mainly achieved on FC layers, which makes it ineffective for CONV layer-dominant networks, e.g., ResNet. To improve the pruning performance, Y. He et al. (He et al., 2018b) leveraged reinforcement learning to optimize the sparsity configuration across layers. However, it is difficult to obtain practical speedup due to the irregularity of the element-wise sparsity (Han et al., 2015b;a). Even if designing ASIC from scratch (Han et al., 2016; 2017), the index overhead is enormous and it only works under high sparsity. These methods usually require a pre-trained model, iterative pruning, and fine-tune retraining, that targets inference optimization.
+
+DNN Acceleration Different from compression, the acceleration work consider more on the sparse pattern. In contrast to the fine-grain compression, coarse-grain sparsity was further proposed to optimize the execution speed. Channel-level sparsity was gained by removing unimportant weight filters (He et al., 2018a; Chin et al., 2018), training penalty coefficients (Liu et al., 2017; Ye et al., 2018; Luo & Wu, 2018), or solving optimization problem (Luo et al., 2017; He et al., 2017; Liang et al., 2018; Hu et al., 2018). Wen et al. (2016) introduced a L2-norm group-lasso optimization for both medium-grain sparsity (row/column) and coarse-grain weight sparsity (channel/filter/layer). Molchanov et al. (2016) introduced the Taylor expansion for neuron pruning. However, they just benefit the inference acceleration, and the extra solving of the optimization problem usually makes the training more complicated. Lin et al. (2017a) demonstrated predicting important neurons then bypassed the unimportant ones via low-precision pre-computation with less cost. Spring & Shrivastava (2017) leveraged the randomized hashing to predict the important neurons. However, the hashing search aims at finding neurons whose weight bases are similar to the input vector, which cannot estimate the inner product accurately thus will probably cause significant accuracy loss on large models. Sun et al. (2017) used a straightforward top- $\mathbf { \nabla } \cdot \mathbf { k }$ pruning on the back propagated errors for training acceleration. But they only simplified the backward pass and presented the results on tiny FC models. Furthermore, the BN compatibility problem that is very important for large-model training still remains untouched. Lin et al. (2017b) pruned the gradients for accelerating distributed training, but the focus is on multi-node communication rather than the single-node scenario discussed in this paper.
+
+# 5 CONCLUSION
+
+In this work, we propose DSG (dynamic and sparse graph) structure for efficient DNN training and inference through a dimension-reduction search based sparsity forecast for compressive memory and accelerative execution and a double-mask selection for BN compatibility without sacrificing model’s expressive power. It can be easily extended to the inference by using the same selection pattern after training. Our experiments over various benchmarks demonstrate significant memory saving (up to $4 . 5 \mathrm { x }$ for training and $1 . 7 \mathrm { x }$ for inference) and computation reduction (up to $2 . 3 \mathbf { x }$ for training and $4 . 4 \times$ for inference). Through significantly boosting both forward and backward passes in training, as well as in inference, DSG promises efficient deep learning in both the cloud and edge.
+
+# ACKNOWLEDGMENT
+
+This work was partially supported by the National Science Foundations(NSF) under Grant No. 1725447 and 1730309, the National Natural Science Foundation of China under Grant No. 61603209 and 61876215. Financial support from the Beijing Innovation Center for Future Chip is also gratefully acknowledged.
+
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+
+# APPENDIX A PROOF OF THE DIMENSION-REDUCTION SEARCH FOR INNER PRODUCT PRESERVATION
+
+Theorem 1. Given a set of $N$ points in $\mathbb { R } ^ { d }$ (i.e. all $\mathbf { X } _ { i }$ and $\mathbf { W } _ { j }$ ), and a number of $\boldsymbol { k } > O ( \frac { l o g ( N ) } { \epsilon ^ { 2 } } )$ there exist a linear map $f : \mathbb { R } ^ { d } \Rightarrow \mathbb { R } ^ { k }$ and a $\epsilon _ { 0 } \in ( 0 , 1 )$ , for $0 < \epsilon \le \epsilon _ { 0 }$ we have
+
+$$
+P [ | \langle f ( \mathbf { X } _ { i } ) , f ( \mathbf { W } _ { j } ) \rangle - \langle \mathbf { X } _ { i } , \mathbf { W } _ { j } \rangle | \leq \epsilon ] \geq 1 - O ( \epsilon ^ { 2 } ) .
+$$
+
+for all $\mathbf { X } _ { i }$ and $\mathbf { W } _ { j }$
+
+Proof. According to the definition of inner product and vector norm, any two vectors a and $\mathbf { b }$ satisfy
+
+$$
+\left\{ \begin{array} { l l } { \langle \mathbf { a } , \mathbf { b } \rangle = ( \| \mathbf { a } \| ^ { 2 } + \| \mathbf { b } \| ^ { 2 } - \| \mathbf { a } - \mathbf { b } \| ^ { 2 } ) / 2 } \\ { \langle \mathbf { a } , \mathbf { b } \rangle = ( \| \mathbf { a } + \mathbf { b } \| ^ { 2 } - \| \mathbf { a } \| ^ { 2 } - \| \mathbf { b } \| ^ { 2 } ) / 2 } \end{array} \right. .
+$$
+
+It is easy to further get
+
+$$
+( \mathbf { a } , \mathbf { b } ) = ( \| \mathbf { a } + \mathbf { b } \| ^ { 2 } - \| \mathbf { a } - \mathbf { b } \| ^ { 2 } ) / 4 .
+$$
+
+Therefore, we can transform the target in equation (7) to
+
+$$
+\begin{array} { r l r } & { } & { \mid \langle f ( \mathbf { X } _ { i } ) , f ( \mathbf { W } _ { j } ) \rangle - \langle \mathbf { X } _ { i } , \mathbf { W } _ { j } \rangle \mid } \\ & { } & { = \mid \| f ( \mathbf { X } _ { i } ) + f ( \mathbf { W } _ { j } ) \| ^ { 2 } - \| f ( \mathbf { X } _ { i } ) - f ( \mathbf { W } _ { j } ) \| ^ { 2 } - \| \mathbf { X } _ { i } + \mathbf { W } _ { j } \| ^ { 2 } + \| \mathbf { X } _ { i } - \mathbf { W } _ { j } \| ^ { 2 } \mid / 4 } \\ & { } & { \le \mid \| f ( \mathbf { X } _ { i } ) + f ( \mathbf { W } _ { j } ) \| ^ { 2 } - \| \mathbf { X } _ { i } + \mathbf { W } _ { j } \| ^ { 2 } \mid / 4 + \mid \| f ( \mathbf { X } _ { i } ) - f ( \mathbf { W } _ { j } ) \| ^ { 2 } - \| \mathbf { X } _ { i } - \mathbf { W } _ { j } \| ^ { 2 } \mid / 4 , } \end{array}
+$$
+
+which is also based on the fact that $| u - v | \leq | u | + | v |$ . Now recall the definition of random projection in equation (5) of the main text
+
+$$
+f ( \mathbf { X } _ { i } ) = \frac { 1 } { \sqrt { k } } \mathbf { R } \mathbf { X } _ { i } \in \mathbb { R } ^ { k } , f ( \mathbf { W } _ { j } ) = \frac { 1 } { \sqrt { k } } \mathbf { R } \mathbf { W } _ { j } \in \mathbb { R } ^ { k } .
+$$
+
+Substituting equation (11) into equation (10), we have
+
+$$
+\begin{array} { r l r } & { } & { \quad \big | \langle f ( \mathbf { X } _ { i } ) , f ( \mathbf { W } _ { j } ) \rangle - \langle \mathbf { X } _ { i } , \mathbf { W } _ { j } \rangle \big | } \\ & { } & { \quad \le \mid \| \frac { 1 } { \sqrt { k } } \mathbf { R } \mathbf { X } _ { i } + \frac { 1 } { \sqrt { k } } \mathbf { R } \mathbf { W } _ { j } \| ^ { 2 } - \| \mathbf { X } _ { i } + \mathbf { W } _ { j } \| ^ { 2 } \mid / 4 + \mid \| \frac { 1 } { \sqrt { k } } \mathbf { R } \mathbf { X } _ { i } - \frac { 1 } { \sqrt { k } } \mathbf { R } \mathbf { W } _ { j } \| ^ { 2 } - \| \mathbf { X } _ { i } - \mathbf { W } _ { j } \| ^ { 2 } \mid / 4 } \\ & { } & { \quad = \mid \| \frac { 1 } { \sqrt { k } } \mathbf { R } ( \mathbf { X } _ { i } + \mathbf { W } _ { j } ) \| ^ { 2 } - \| \mathbf { X } _ { i } + \mathbf { W } _ { j } \| ^ { 2 } \mid / 4 + \mid \| \frac { 1 } { \sqrt { k } } \mathbf { R } ( \mathbf { X } _ { i } - \mathbf { W } _ { j } ) \| ^ { 2 } - \| \mathbf { X } _ { i } - \mathbf { W } _ { j } \| ^ { 2 } \mid / 4 } \\ & { } & { \quad = \mid \| f ( \mathbf { X } _ { i } + \mathbf { W } _ { j } ) \| ^ { 2 } - \| \mathbf { X } _ { i } + \mathbf { W } _ { j } \| ^ { 2 } \mid / 4 + \mid \| f ( \mathbf { X } _ { i } - \mathbf { W } _ { j } ) \| ^ { 2 } - \| \mathbf { X } _ { i } - \mathbf { W } _ { j } \| ^ { 2 } \mid / 4 . } \end{array} .
+$$
+
+Further recalling the norm preservation in equation (3) of the main text: there exist a linear map $f : \mathbb { R } ^ { d } \Rightarrow \mathbb { R } ^ { k }$ and a $\epsilon _ { 0 } \in ( 0 , 1 )$ , for $0 < \epsilon \le \epsilon _ { 0 }$ we have
+
+$$
+P [ ( 1 - \epsilon ) \| \mathbf { Z } \| ^ { 2 } \leq \| f ( \mathbf { Z } ) \| ^ { 2 } \leq ( 1 + \epsilon ) \| \mathbf { Z } \| ^ { 2 } ] \geq 1 - O ( \epsilon ^ { 2 } ) .
+$$
+
+Substituting the equation (13) into equation (12) yields
+
+$$
+\begin{array} { r l } &  P [ \mathbf { \lvert \lvert \lvert \delta f ( \mathbf { X } } _ { i } + \mathbf { W } _ { j } ) \rvert ] ^ { 2 } - \lVert \mathbf { X } _ { i } + \mathbf { W } _ { j } \rVert ^ { 2 } \mathbf { \lvert \langle 4 + \lvert \lvert \phi ( \mathbf { X } _ { i } - \mathbf { W } _ { j } ) \rvert \rvert ^ { 2 } - \lVert \mathbf { X } _ { i } - \mathbf { W } _ { j } \rVert ^ { 2 } \mathbf { \lvert \langle 4 . . . } } \\ & { \qquad \leq \frac { \epsilon } { 4 } ( \lVert \mathbf { X } _ { i } + \mathbf { W } _ { j } \rVert ^ { 2 } + \lVert \mathbf { X } _ { i } - \mathbf { W } _ { j } \rVert ^ { 2 } ) = \frac { \epsilon } { 2 } ( \lVert \mathbf { X } _ { i } \rVert ^ { 2 } + \lVert \mathbf { W } _ { j } \rVert ^ { 2 } ) ] . . . } \\ & { \qquad \geq P \bigl ( \mathbf { \lvert \lvert \delta f ( \mathbf { X } } _ { i } + \mathbf { W } _ { j } ) \rVert ^ { 2 } - \lVert \mathbf { X } _ { i } + \mathbf { W } _ { j } \rVert ^ { 2 } \mathbf { \lvert \langle 4 \leq \frac { \epsilon } { 4 } \lVert \mathbf { X } _ { i } + \mathbf { W } _ { j } \rVert ^ { 2 } \rangle } . . . } \\ & { \qquad \times P \bigl ( \mathbf { \lvert \delta f ( \mathbf { X } } _ { i } - \mathbf { W } _ { j } ) \rvert \rvert ^ { 2 } - \lVert \mathbf { X } _ { i } - \mathbf { W } _ { j } \rVert ^ { 2 } \mathbf { \lvert \langle 4 \leq \frac { \epsilon } { 4 } \lVert \mathbf { X } _ { i } - \mathbf { W } _ { j } \rVert ^ { 2 } \rangle } . . . } \\ & { \qquad \geq \mathrm { [ 1 } - O ( \epsilon ^ { 2 } ) \ ] \cdot \left[ \mathrm { 1 } - O ( \epsilon ^ { 2 } ) \right] = 1 - O ( \epsilon ^ { 2 } ) . } \end{array} .
+$$
+
+Combining equation (12) and (14), finally we have
+
+$$
+\begin{array} { r } { P [ \mathbf { \lvert \langle f ( \mathbf { X } _ { i } ) , f ( \mathbf { W } _ { j } ) \rangle } - \langle \mathbf { X } _ { i } , \mathbf { W } _ { j } \rangle \mathbf { \lvert \leq } \frac { \epsilon } { 2 } ( \lVert \mathbf { X } _ { i } \rVert ^ { 2 } + \lVert \mathbf { W } _ { j } \rVert ^ { 2 } ) ] \geq 1 - O ( \epsilon ^ { 2 } ) \ . } \end{array}
+$$
+
+It can be seen that, for any given $\mathbf { X } _ { i }$ and $\mathbf { W } _ { j }$ pair, the inner product can be preserved if the $\epsilon$ is sufficiently small. Actually, previous work (Achlioptas, 2001; Bingham & Mannila, 2001; Vu, 2016) discussed a lot on the random projection for various big data applications, here we re-organize these supporting materials to form a systematic proof. We hope this could help readers to follow this paper. In practical experiments, there exists a trade-off between the dimension reduction degree and the recognition accuracy. Smaller $\epsilon$ usually brings more accurate inner product estimation and better recognition accuracy while at the cost of higher computational complexity with larger $k$ , and vice versa. Because the $\| \mathbf { X } _ { i } \| ^ { 2 }$ and $\| \mathbf { W } _ { j } \| ^ { 2 }$ are not strictly bounded, the approximation may suffer from some noises. Anyway, from the abundant experiments in this work, the effectiveness of our approach for training dynamic and sparse neural networks has been validated.
+
+Data: A mini-batch of inputs $\&$ targets $( \mathbf { X } _ { 0 } , \mathbf { X } ^ { * } )$ , previous weights $\mathbf { W } ^ { t }$ , previous BN parameters $\theta ^ { t }$ .   
+Result: Update weights $\dot { \mathbf { W } } ^ { t + 1 }$ , update BN parameters $\theta ^ { t + 1 }$ .
+
+Random projection: $f ( \mathbf { W } _ { k } ^ { t } ) \Leftarrow \mathbf { W } _ { k } ^ { t }$ ;
+
+Step 1. Forward Computation;   
+for $k { = } l$ to $L$ do if $k { < } L$ then Projection: $f ( \mathbf { X } _ { k - 1 } ) \Leftarrow \mathbf { X } _ { k - 1 }$ ; Generating $M a s k _ { k }$ via dimension-reduction search according to $f ( \mathbf { X } _ { k - 1 } )$ and $f ( \mathbf { W } _ { k } ^ { t } )$ ; $\mathbf { S } _ { k } \Leftarrow \varphi [ M a s k _ { k } ( \mathbf { X } _ { k - 1 } \mathbf { W } _ { k } ^ { t } ) ]$ ; $\left. \begin{array} { l l } { \langle } & { M a s k _ { k } [ B N ( \mathrm { ~  ~ \kappa ~ } , \theta _ { k } ^ { t } ) ] } \end{array} \right.$ ; else $\begin{array} { r l } { | } & { { } \mathbf { X } _ { L } \Leftarrow l i n e a r ( \mathbf { X } _ { L - 1 } \mathbf { W } _ { L } ^ { t } ) ; } \end{array}$ ; end   
+end
+
+Step 2. Backward Computation;
+
+Compute the gradient of the output layer $\begin{array} { r } { \mathbf { G } _ { \mathbf { X } _ { L } } = \frac { \partial C ( \mathbf { X } _ { L } , \mathbf { X } ^ { * } ) } { \partial \mathbf { X } _ { L } } } \end{array}$ ;   
+for $k { = } L$ to $^ { l }$ do if $k { = } { = } L$ then $\begin{array} { r l } { } & { { } \Leftarrow M a s k _ { k - 1 } ( { \bf G } _ { { \bf X } _ { L } } ( { \bf W } _ { L } ^ { t } ) ^ { T } ) ; } \end{array}$ $\mathbf { G } _ { \mathbf { W } _ { L } } \Leftarrow \mathbf { G } _ { \mathbf { X } _ { L } } ^ { T } \mathbf { X } _ { L - 1 }$ else $( \mathbf { G } _ { \mathbf { S } _ { k } } , \mathbf { G } _ { \theta _ { k } } ) \Leftarrow M a s k _ { k } [ B N _ { - } g r a d ( \mathbf { G } _ { \mathbf { X } _ { k } } , \mathbf { S } _ { k } , \theta _ { k } ^ { t } ) ]$ $\mathbf { G } _ { \mathbf { W } _ { k } } \Leftarrow ( \mathbf { G } _ { \mathbf { S } _ { k } } \odot \varphi \lrcorner g r a d ) ^ { T } \mathbf { X } _ { k - 1 }$ ; if $k { > } I$ then $\begin{array} { r } { \big \vert \mathbf { G } _ { \mathbf { X } _ { k - 1 } } \Leftarrow M a s k _ { k - 1 } \big [ \left( \mathbf { G } _ { \mathbf { S } _ { k } } \odot \varphi _ { - } g r a d \right) ( \mathbf { W } _ { k } ^ { t } ) ^ { T } \big ] ; } \end{array}$ end end   
+end   
+Step 3. Parameter Update;   
+for $k { = } l$ to $L$ do $\mathbf { W } _ { k } ^ { t + 1 } \Leftarrow O p t i m i z e r ( \mathbf { W } _ { k } ^ { t } , \mathbf { G } _ { \mathbf { W } _ { k } } ) ;$ $\theta _ { k } ^ { t + 1 } \Leftarrow O p t i m i z e r ( \theta _ { k } ^ { t } , { \bf G } \theta _ { k } )$ ;   
+end
+
+# APPENDIX B IMPLEMENTATION AND OVERHEAD
+
+The training algorithm for generating DSG is presented in Algorithm 1. The generation procedure of the critical neuron mask based on the virtual activations estimated in the low-dimensional space is presented in Figure 9, which is a typical top- $k$ search. The $k$ value is determined by the activation size and the desired sparsity $\gamma$ . To reduce the search cost, we calculate the first input sample $X ( 1 )$ within the current mini-batch and then conduct a top- $k$ search over the whole virtual activation matrix for obtaining the top- $k$ threshold under this sample. The remaining samples share the top- $k$ threshold from the first sample to avoid costly searching overhead. At last, the overall activation mask is generated by setting the mask element to one if the estimated activation is larger than the top- $k$ threshold and setting others to zero. In this way, we greatly reduce the search cost. Note that, for the FC layer, each sample $X ( i )$ is a vector.
+
+![](images/3b6571becf35a4ecb2fe6c8e7a1383fd3f91d754850520cd154a5ae82c1aa74a.jpg)  
+Figure 9: Selection mask generation: using a top- $k$ search on the first input sample $X ( 1 )$ within each mini-batch to obtain a top- $k$ threshold which is shared by the following samples. Then, we apply thresholding on the whole output activation tensor to generate the importance mask for the same mini-batch.
+
+Table 1: Computational complexity of dimension-reduction search. MMACs denotes mega-MACs and BL denotes baseline.   
+
+<table><tr><td>Layers</td><td colspan="4">Dimension</td><td colspan="4">Operations (MMACs)</td></tr><tr><td>npQ,ncRs,nK</td><td>BL 0.3</td><td>0.5</td><td>0.7</td><td>0.9</td><td>BL 0.3</td><td>0.5</td><td>0.7</td><td>0.9</td></tr><tr><td>1024,1152, 128</td><td>1152 539</td><td>232</td><td>148</td><td>119</td><td>144 67.37</td><td>29</td><td>18.5</td><td>14.88</td></tr><tr><td>256, 1152,256</td><td>1152 616</td><td>266</td><td>169</td><td>136</td><td>72 38.5</td><td>16.63</td><td>10.56</td><td>8.5</td></tr><tr><td>256,2304,256</td><td>2304 616</td><td>266</td><td>169</td><td>136</td><td>144 38.5</td><td>16.63</td><td>10.56</td><td>8.5</td></tr><tr><td>64,2304,512</td><td>2304 693</td><td>299</td><td>190</td><td>154</td><td>72 21.65</td><td>9.34</td><td>5.94</td><td>4.81</td></tr><tr><td>64, 4608, 512</td><td>4608 693</td><td>299</td><td>190</td><td>154</td><td>144 21.65</td><td>9.34</td><td>5.94</td><td>4.81</td></tr></table>
+
+Furthermore, we investigate the influence of the $\epsilon$ on the computation cost of dimension-reduction search for importance estimation. We take several layers from the VGG8 on CIFAR10 as a case study, as shown in Table 1. With $\epsilon$ larger, the dimension-reduction search can achieve lower dimension with much fewer operations. The average reduction of the dimension is $3 . 6 \mathbf { x }$ $( \epsilon = 0 . 3$ ), $8 . 5 \mathrm { x }$ $\epsilon = 0 . 5$ ), $1 3 . 3 \mathrm { x }$ $\epsilon = 0 . 7 )$ , and $1 6 . 5 \mathrm { x }$ $\mathit { \check { \epsilon } } = 0 . 9$ ). The resulting operation reduction is 3.1x, 7.1x, $1 1 . 1 \mathbf { x }$ , and $1 3 . 9 \mathrm { X }$ , respectively.
+
+# APPENDIX C CONVERGENCE ANALYSIS
+
+One interesting question is that whether DSG slows down the training convergence or not, which is answered by Figure 10. According to Figure 10(a)-(b), the convergence speed under DSG constraints varies little from the vanilla model training. This probably owes to the high fidelity of inner product when we use random projection to reduce the data dimension. Figure 10(c) visualizes the distribution of the pairwise difference between the original high-dimensional inner product and the low-dimensional one for the CONV5 layer of VGG8 on CIFAR10. Most of the inner product differences are around zero, which implies an accurate approximation capability of the proposed dimension-reduction search. This helps reduce the training variance and avoid training deceleration.
+
+![](images/38a2bf7866fa4b8412441903891151ff0068c6ed7194e483fb35233524074a98.jpg)  
+Figure 10: Accuracy convergence. (a) Training curve with validation accuracy of VGG8 on CIFAR10; (b) Training curve with top-5 validation accuracy of ResNet-18 on ImageNet; (c) Distribution of pairwise difference between the original high-dimensional inner product and the lowdimensional one for the CONV5 layer in VGG8.
+
+Another question in DSG is that whether the selection masks converge during training or not. To explore the answer, we did an additional experiment as shown in the Figure 11. We select a minibatch of training samples as a case study for data recording. Each curve presents the results of one layer (CONV2-CONV6). For each sample at each layer, we recorded the change of binary selection mask between two adjacent training epochs. Here the change is obtained by calculating the $L 1$ - norm value of the difference tensor of two mask tensors at two adjacent epochs, i.e., change $=$ batch avg L1norm(maski+1 − maski). Here the batch avg L1norm(·) indicates the average $L 1$ -norm value across all samples in one mini-batch. As shown in Figure 11(a), the selection mask for each sample converges as training goes on.
+
+![](images/2da28a1a1895d8b68fe16750c2a687694e117ca98f05613112ebb52186576eeb.jpg)  
+Figure 11: Selection mask convergence. (a) Average $L 1$ -norm value of the difference mask tensors between adjacent training epochs across all samples in one mini-batch; (b) Average $L 1$ -norm value of the difference mask tensors between adjacent samples after training.
+
+In our implementation we inherit the random projection matrix from training and perform the same on-the-fly dimension-reduction search in inference. We didn’t try to directly suspend the selection masks, because the selection mask might vary across samples even if we observe convergence for each sample. This can be seen from Figure 11(b), where the difference mask tensors between adjacent samples in one mini-batch present significant differences (large $L 1$ -norm value) after training. Therefore, it will consume lot of memory space to save these trained masks for all samples, which is less efficient than conducting on-the-fly search during inference.
+
+![](images/f53624729091ed931ce31895daaac487a79bddc694443d28a1217ba5da650b6e.jpg)  
+Figure 12: Comparison with smaller-dense models with equivalent MACs using ResNet8 on CIFAR10 and AlexNet on ImageNet.
+
+# APPENDIX D COMPARISON WITH OTHER METHODS
+
+Figure 12 extends Figure 8(b) in the main text to more network structures, including ResNet8 on CIFAR10 and AlexNet on ImageNet. The similar observation can be achieved: the equivalent smaller dense models with the same effective MACs are able to save more training time but the accuracy degradation will be increased. Note that in this figure, the DSG training uses a warm-up training with dense model for the first 10 epochs. The overhead of the warm-up training has been taken account into the entire training cost. To make the accuracy results on CIFAR10 and ImageNet comparable for figure clarity, AlexNet reports the top-5 accuracy.
+
+Our work targets at both the training and inference phases while most of previous work focused on the inference compression. In prior methods, the training usually becomes more complicated with various regularization constraints or iterative fine-tuning/retraining. Therefore, it is not very fair to compare with them during training. For this reason, we just compare with them on the inference pruning. Different from doing DSG training from scratch, here we utilize DSG for fine-tuning based on pre-trained models.
+
+Table 2: Comparison with other structured sparsification methods for inference. All the results are from VGG16 on ImageNet, and the default accuracy is top-1 accuracy. The baseline methods are Taylor Expansion (Molchanov et al., 2016), ThiNet (Luo et al., 2017), Channel Pruning (Hu et al., 2018), AutoPrunner (Luo & Wu, 2018), and AMC (He et al., 2018b).   
+
+<table><tr><td>Methods</td><td>Taylor Expansion</td><td>ThiNet</td><td>Channel Pruning</td><td>AutoPrunner</td><td>AMC</td><td>DSG</td></tr><tr><td>Operation Sparsity</td><td>62.86%</td><td>69.81%</td><td>69.32%</td><td>73.6%</td><td>80%</td><td>62.92%</td></tr><tr><td>Accuracy</td><td>87%(top-5)</td><td>67.34%</td><td>70.42%</td><td>68.43%</td><td>69.1%</td><td>71.44%(top-1) 90.56%(top-5)</td></tr></table>
+
+To guarantee the fairness, all the results are from the same network (VGG16) on the same dataset (ImageNet). Since our DSG produces structured sparsity, we also select structured sparsity work as comparison baselines. Different from the previous experiments in this paper, we further take the input sparsity at each layer into account rather than only count the output sparsity. This is due to the fact that the baselines consider all zero operands. The results are listed in Table 2, from which we can see that DSG is able to achieve a good balance between the operation amount and model accuracy.

parse/train/KAV7BDCcN6/KAV7BDCcN6.md

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+# What Breaks the Curse of Dimensionality in Deep Learning?
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Although learning in high dimensions is commonly believed to suffer from the   
+2 curse of dimensionality, modern machine learning methods often exhibit an as  
+3 tonishing power to tackle a wide range of challenging real-world learning prob  
+4 lems without using abundant amounts of data. How exactly these methods break   
+5 this curse remains a fundamental open question in the theory of deep learning.   
+6 While previous efforts have investigated this question by studying the data (D),   
+7 model (M), and inference algorithm (I) as independent modules, in this paper   
+8 we analyzes the triple (D, M, I) as an integrated system. We examine the basic   
+9 symmetries of such systems, focusing on four of the main architectures in deep   
+10 learning: fully-connected networks (FCN), locally-connected networks (LCN), and   
+11 convolutional networks with and without pooling (GAP/VEC). By computing an   
+12 eigen-decomposition of the infinite-width limits (aka Neural Kernels) of these   
+13 architectures, we characterize how inductive biases (locality, weight-sharing, pool  
+14 ing, etc) and the breaking of spurious symmetries can affect the performance of   
+15 these learning systems. Our theoretical analysis shows that for many real-world   
+16 tasks it is locality rather than symmetry that provides the first-order remedy to the   
+17 curse of dimensionality. Empirical results on state-of-the-art models on ImageNet   
+18 corroborate our results.
+
+# 19 1 Introduction
+
+20 Statistical problems with high-dimensional data are frequently plagued by the curse of dimensionality,   
+21 in which the number of samples required to solve the problem grows rapidly with the dimensionality   
+22 of the input. Classical theory explains this phenomenon as the consequence of basic geometric and   
+23 algebraic properties of high-dimensional spaces; for example, the number of $\epsilon$ -cubes inside a unit   
+24 cube in $\mathbb { R } ^ { \hat { d } }$ grows exponentially like $\epsilon ^ { - d }$ , and the number of degree $r$ polynomials in $\mathbb { R } ^ { d }$ grows like a   
+25 power-law $d ^ { r }$ . Since for real-world problems $d$ is typically in the hundreds or thousands, classical   
+26 wisdom suggests that learning is likely to be infeasible. However, starting from the groundbreaking   
+27 work AlexNet [1], practitioners in deep learning have tackled a wide range of difficult real-world   
+28 learning problems ([2–6]) in high dimensions, once believed by many to be out-of-scope of current   
+29 techniques. The astonishing success of modern machine learning methods clearly contradicts the   
+30 curse of dimensinonality and therefore poses the fundamental question: mathematically, how do   
+31 modern machine learning methods break the curse of dimensionality?   
+32 To answer this question, we must trace back to the most fundamental ingredients of machine learning   
+33 methods. They are the data $( \mathcal { D } )$ , the model $( \mathcal { M } )$ , and the inference algorithm $( \mathcal { T } )$ .   
+34 Data $( \mathcal { D } )$ is of course central in machine learning. In the classical learning theory setting, the learning   
+35 objective usually has a power-law decay $m ^ { - \bar { \beta } }$ as the function of the number of training samples   
+36 $m$ . The theoretical bound on $\beta$ is usually tiny, owing to the curse of dimensionality, and is of   
+37 limited practical utility for high-dimensional data. On the other hand, empirical measurements of   
+38 $\beta$ in state-of-the-art deep learning models typically reveal values of $\beta$ that are not at all small (e.g.   
+39 $\beta = 0 . 4 3$ for ResNet in Fig.S2) even though $d$ is quite large (e.g. $\dot { d } \sim 1 0 ^ { 5 }$ for ImageNet). This   
+40 example suggests that the learning curve must have important functional dependence on $\mathcal { M }$ and $\mathcal { T }$ .   
+41 Indeed, as we will observe later, many of the best performing methods exhibit learning curves for   
+42 which $\beta = \beta ( m )$ actually increases as $m$ becomes larger, i.e. data makes the usage of data more   
+43 efficient. We call this phenomenon DIDE, for data improves data efficiency.   
+44 Designing machine learning models $( \mathcal { M } )$ that maximize data-efficiency is critical to the success   
+45 of solving real-world tasks. Indeed, breakthroughs in machine learning are often driven by novel   
+46 architectures LeNet [7], AlexNet[1], Transformer [2], etc. While some of the inductive biases of these   
+47 methods are clear (e.g. translation symmetries of CNNs), others tend to build off of prior empirical   
+48 success and are less well-understood (e.g. the implicit bias of SGD). To build our understanding of   
+49 these biases and how they affect learning, we conduct a theoretical analysis of them in the infinite  
+50 width setting [8–12], which preserves most salient aspects of the architecture while enabling tractable   
+51 calculations. We classify all phenomena that could be explained by infinite networks alone as the   
+52 consequences of inductive biases.   
+53 The inference procedure $( \mathcal { T } )$ is what enables learning in machine learning methods. It is widely   
+54 believed that modern inference methods, specifically gradient descent and variants, ‘implicitly‘ bias   
+55 the solutions of the networks towards those that generalize well and away from those that generalize   
+56 poorly [13–15]. The effects of the inference algorithm are intimately tied to the specifics of the model   
+57 (e.g. weight-sharing) and the data (e.g. augmentation), and might not be fully understood with a   
+58 fixed-data, fixed-model analysis. Indeed, good performance may derive from interactions between   
+59 $( { \mathcal { M } } , { \mathcal { T } } )$ , or $( \mathcal { D } , \mathcal { I } )$ , or even $( \mathcal { D } , \mathcal { M } , \mathcal { T } )$ . In Sec. 3.1, we demonstrate the DIDE effect for a particular   
+60 choice of $( \mathcal { D } , \mathcal { M } , \mathcal { T } )$ and show that this effect disappears if any one of $\mathcal { D }$ , $\mathcal { M }$ , or $\mathcal { T }$ is altered.
+
+The above discussion highlights the insufficiency of treating $\mathcal { D }$ , $\mathcal { M }$ , and $\mathcal { T }$ as separate non-interacting modules. They must be considered as an integrated system. Throughout this paper, we will refer to the triplet $( \mathcal { D } , \mathcal { M } , \mathcal { T } )$ as a (machine) learning system and the tuple $( { \mathcal { M } } , { \mathcal { T } } )$ as the learning algorithm of the system that operates on $\mathcal { D }$ . We summarize our contributions below.
+
+1. We surface the basic symmetries of various $( \mathcal { D } , \mathcal { M } , \mathcal { T } )$ associated to four of the main architectures in deep learning $\mathsf { F C N } _ { n }$ (fully-connected networks), $\mathsf { L C N } _ { n }$ (locally-connected networks), ${ \mathsf { V E C } } _ { n } / { \mathsf { G A P } } _ { n }$ (convolution networks with a flattening /a global average pooling readout layer), their infinite width counterparts $\mathsf { F C N } _ { \infty } / \mathsf { L C N } _ { \infty } / \mathsf { V E C } _ { \infty } / \mathsf { G A P } _ { \infty }$ . Treating $\mathsf { F C N } _ { n / \infty }$ as the baseline model, we show that the locality from $\mathsf { L C N } _ { n }$ and the weight-sharing from $\mathsf { \dot { V } E C } _ { n } / \mathsf { G A P } _ { n }$ break spurious symmetries and lead to better systems. Empirically, we examine the relation between the symmetries and the performance of the systems in the infinite width setting and finite width setting with various of interventions. Surprisingly, we observe that state-of-the-art learning system (EfficientNet[16]) on ImageNet can learn almost equally well even the coordinate of the data are transformed by the symmetry group defined by $\mathsf { L C N } _ { n }$ .
+
+2. We show that although the weight-sharing from ${ \mathsf { V E C } } _ { n }$ provides coordinate information of the data to the system, as the width gets larger, it becomes harder for the learning algorithm to explore such information and at infinite width, the system restores the symmetry group that is identical to $\mathsf { L C N } _ { n }$ , and is completely unaware of the coordinate information. As a consequence, the performance of the network, as a function of width, monotonically decays [12]. This is in stark contrast to recent finding that the performance of network is positively correlated to its width. We show that this phenomenon continues to hold even with various interventions (larger learning rate and l2 regularization) to the training procedures. However, with more data (e.g. data augmentation) ${ \mathsf { V E C } } _ { n }$ can be on par with ${ \mathsf { G A P } } _ { n }$ .
+
+3. The function space defined by $\mathsf { L C N } _ { n }$ is a super set of that defined by ${ \mathsf { V E C } } _ { n }$ . We prove the opposite is true. Therefore, ${ \mathsf { V E C } } _ { n }$ is able to express functions in the space with a stronger inductive bias ${ \mathsf { G A P } } _ { n }$ (translation invariance) and functions in a seemingly much larger class $\mathsf { L C N } _ { n }$ . We hypothesize that as the dataset grows, the learned functions using ${ \mathsf { V E C } } _ { n }$ is transitioned away from those learned using $\mathsf { L C N } _ { n }$ and become closer to those learned using ${ \mathsf { G A P } } _ { n }$ . This suggests, even though the prior (provided by human) is not $100 \%$ correct, with the help of more data, gradient descent might be able to correct it, a possible explanation of DIDE.
+
+4. When the input space is the product of hyperspheres, we eigendecompose the kernels associated to one-hidden layer infinite width network, $\mathsf { F C N } _ { \infty }$ , $\mathsf { V } \bar { \mathsf { E } } \mathsf { C } _ { \infty } = \mathsf { L } \bar { \mathsf { C } } \mathsf { N } _ { \infty }$ and $\mathsf { G A P } _ { \infty }$ . We treat $\mathsf { F C N } _ { \infty }$ as the baseline, whose order $r$ eigenspace has dimension of order $d ^ { r }$ and eigenvalues of order $d ^ { - r }$ for $r \geq 0$ [17]. We show that locality alone (i.e. $\mathsf { V E C } _ { \infty , \mathsf { \Lambda } }$ ) dramatically reduces the dimension of the $r$ -eigenspace for $r \geq 2$ and the spectral gap between all $r$ -eigenspaces but $r = 0$ and $r = 1$ , making learning of higher order eigenspaces feasible with dramatically fewer samples and gradient steps. In addition, pooling (i.e. $\mathsf { G A P } _ { \infty } \mathrm { \Gamma } _ { \infty } .$ ) reduces the dimension of $r$ -eigenspace for $r \geq 1$ by a factor equal to the size of the pooling window, but it does not change the spectra in an essential way.
+
+02 Our empirical and theoretical results surface the importance of locality which, we believe, provides   
+03 the first-order remedy to the curse of dimensionality for many real-world tasks and which has been   
+04 largely overlooked.
+
+# 2 Preliminary and Notation
+
+# 2.1 Neural Networks
+
+107 We focus our presentation on the supervised learning setting and more concretely, on image   
+108 recognition. Let $\mathcal { D } \subseteq ( \mathbb { R } ^ { d } ) ^ { 3 } \times \mathbb { R } ^ { k } \overset { \cdot } { \equiv } \mathbb { R } ^ { 3 d } \times \mathbb { R } ^ { k }$ denote the data set (training and test) and   
+109 $\mathcal { X } = \{ x : ( x , y ) \in \mathcal { D } \}$ and $\mathcal { V } = \{ y : ( x , y ) \in \mathcal { D } \}$ denote the input space (images) and label space,   
+110 respectively. Here $d$ is the spatial dimension (e.g. $d = 3 2 \times 3 2$ for CIFAR-10) of the images and 3 is   
+111 the total number of channels (i.e. RGB). We use $\mathsf { F C N } _ { n }$ to denote a $L$ -hidden layer fully-connected   
+112 network with identical hidden widths $n _ { l } = n \in \mathbb { N }$ for $l = 1 , . . . , L$ and with readout width $n _ { L + 1 } = k$   
+113 (the number of logits). For each $x \in \mathbb { R } ^ { 3 d } = ( \mathbb { R } ^ { d } ) ^ { 3 }$ , we use $h ^ { l } ( x ) , x ^ { l } ( x ) \in \mathbb R ^ { n _ { l } }$ to represent the pre  
+114 and post-activation functions at layer $l$ with input $x$ . The recurrence relation FCN is given by
+
+$$
+\left\{ \begin{array} { l l } { h ^ { l + 1 } } & { = x ^ { l } W ^ { l + 1 } } \\ { x ^ { l + 1 } } & { = \phi \left( h ^ { l + 1 } \right) } \end{array} \right. \mathrm { a n d } \ W _ { i , j } ^ { l } = \frac { 1 } { \sqrt { n _ { l } } } \omega _ { i j } ^ { l } , \quad \omega _ { i j } ^ { l } \sim \mathcal { N } ( 0 , 1 )
+$$
+
+115 where $\phi$ is a point-wise activation function, $W ^ { l + 1 } \in \mathbb { R } ^ { n _ { l } \times n _ { l + 1 } }$ are the weights and $\omega _ { i j } ^ { l }$ are the   
+116 trainable parameters, drawn i.i.d. from a standard Gaussian $\sim \mathcal { N } ( 0 , 1 )$ at initialization. For simplicity   
+117 of the presentation, the bias terms and the hyperparameters (the variances of the weights) are omitted.   
+118 Adding them back won’t affect the conclusion of the paper.   
+119 For convolutional networks or locally-connected networks, the inputs are treated as tensors in $( \mathbb { R } ^ { d } ) ^ { 3 }$ .   
+120 The recurrent relation of convolutional networks can be written as
+
+$$
+x _ { \alpha , j } ^ { l + 1 } = \phi ( h _ { \alpha , j } ^ { l + 1 } ) \quad \mathrm { a n d } \quad h _ { \alpha , j } ^ { l + 1 } \equiv { \frac { 1 } { \sqrt { ( 2 k + 1 ) n ^ { l } } } } \sum _ { j = 1 } ^ { n ^ { l } } \sum _ { \beta = - k } ^ { k } x _ { \alpha + \beta , i } ^ { l } \omega _ { i j , \beta } ^ { l }
+$$
+
+121 Here $\alpha \in [ d ]$ denote the spatial location, $i / j \in [ n ]$ denotes the fanin/fanout channel indices. For   
+122 notational convenience, we assume circular padding and stride equal to 1 for all layers. The features   
+123 of the penultimate layer are 2D tensors and there are two commonly used approaches to map them   
+124 to the logit layer: stack a dense layer after either vectorizing the 2D tensor to a 1D vector or   
+125 applying a global average pooling layer to each channel. We use ${ \mathsf { V E C } } _ { n } / { \mathsf { G A P } } _ { n }$ to denote the network   
+126 obtain from the former/latter, which are known to be equipped with the inductive biases translation   
+127 equivariant/invariant. The readout layer of ${ \mathsf { V E C } } _ { n } / { \mathsf { G A P } } _ { n } ^ { - }$ could be written as
+
+$$
+x _ { j } ^ { L + 1 } = \frac { 1 } { \sqrt { d n } } \sum _ { \alpha \in [ d ] } x _ { \alpha , i } ^ { L } w _ { \alpha , i j } ^ { L + 1 } , x _ { j } ^ { L + 1 } = \frac { 1 } { \sqrt { n } } \sum _ { i \in [ n ] } \left( \frac { 1 } { d } \sum _ { \alpha \in [ d ] } x _ { \alpha , i } ^ { L } \right) w _ { i j } ^ { L + 1 }
+$$
+
+128 We briefly remark the the key difference between the two. In ${ \mathsf { V E C } } _ { n }$ , each pixel in the penultimate   
+129 layer has its own (independent random) variable while pixels within the same channel shared the   
+130 same (random) variable in $\mathsf { G A P } _ { n }$ . It is clear that the function space of ${ \mathsf { V E C } } _ { n }$ contains that of ${ \mathsf { G A P } } _ { n }$ .   
+131 Locally Connected Networks $\mathsf { L C N } _ { n }$ [18, 19] are convolutional network without weight sharing   
+132 between spatial locations. $\mathsf { L C N } _ { n }$ preserve the connectivity pattern, and thus topology, of a convnet.   
+133 Mathematically, the current formula is defind as in Equation 2 with all the shared parameters $\omega _ { i j , \beta } ^ { l }$   
+134 replaced by unshared $\omega _ { i j , \alpha , \beta } ^ { l } \sim \mathcal { N } ( 0 , 1 )$   
+135 In this note, we assume that the $\mathsf { L C N } _ { n }$ are always associated with a vectorization readout layer and it   
+136 is clear, as a function space, $\mathsf { L C N } _ { n }$ is a super set of ${ \mathsf { V E C } } _ { n }$ . Interestingly,the opposite is also true.   
+37 Theorem 2.1 (Sec. B). Let $\mathsf { V E C } _ { n } / \mathsf { L C N } _ { n } / \mathsf { G A P } _ { n }$ denote the set of functions that can be represented   
+138 by $L$ -hidden layer $\mathsf { V E C } _ { n } / \mathsf { L C N } _ { n } / \mathsf { G A P } _ { n }$ networks with hidden width $n$ . Then
+
+$$
+{ \mathsf { G A P } } _ { n } \subseteq { \mathsf { V E C } } _ { n } \subseteq { \mathsf { L C N } } _ { n } \subseteq { \mathsf { V E C } } _ { d n }
+$$
+
+139 The significance of this theorem is that if we consider the function space ${ \mathsf { V E C } } _ { n }$ as a soft prior,   
+140 gradient descent could move it closer to a better prior ${ \mathsf { G A P } } _ { n }$ (translation invariance) if the average   
+141 pooling is (approximately) learned in the readout layer or it might remain close to $\mathsf { L C N } _ { n }$ .
+
+# 2.2 Gradient Descent Training
+
+143 We use $f$ to denote any functions defined by the architectures above and $\theta$ to denote the collection   
+144 of all parameters. Denote by $\theta _ { t }$ the time-dependence of the parameters and by $\theta _ { 0 }$ their initial   
+145 values. We use $f _ { t } ( x ) \equiv f ( \dot { x } , \theta _ { t } ) \in \mathbb R ^ { k }$ to denote the output (or logits) of the neural network at   
+146 time $t$ . Let $\ell ( \hat { y } , y ) : \mathbb { R } ^ { k } \times \mathbb { R } ^ { k } \to \mathbb { R }$ denote the loss function where the first/second argument is   
+147 the prediction/true label. By applying continuous time gradient descent to minimize the objective   
+148 $\begin{array} { r } { \mathcal { L } = \sum _ { ( x , y ) \in \mathcal { D } } \ell ( f _ { t } ( x , \theta ) , y ) } \end{array}$ , the evolution of the parameters $\theta$ and the logits $f$ can be written as
+
+$$
+\begin{array} { r } { \dot { \theta } _ { t } = - \nabla _ { \theta } f _ { t } ( \mathcal { X } _ { T } ) ^ { T } \nabla _ { f _ { t } ( \mathcal { X } _ { T } ) } \mathcal { L } , \qquad \dot { f } _ { t } ( \mathcal { X } _ { T } ) = \nabla _ { \theta } f _ { t } ( \mathcal { X } _ { T } ) \dot { \theta } _ { t } = - \hat { \Theta } _ { t } ( \mathcal { X } _ { T } , \mathcal { X } _ { T } ) \nabla _ { f _ { t } ( \mathcal { X } _ { T } ) } \mathcal { L } } \end{array}
+$$
+
+149 where $f _ { t } ( \mathcal { X } _ { T } ) = \operatorname { v e c } \left( [ f _ { t } \left( x \right) ] _ { x \in \mathcal { X } _ { T } } \right)$ , the $k | \mathcal { D } | \times 1$ vector of concatenated logits for all examples, and   
+150 $\nabla _ { f _ { t } ( \mathcal { X } _ { T } ) } \mathcal { L }$ is the gradient of the loss with respect to the model’s output, $f _ { t } ( \mathcal { X } _ { T } )$ . $\hat { \Theta } _ { t } \equiv \hat { \Theta } _ { t } ( \mathcal { X } _ { T } , \mathcal { X } _ { T } )$   
+151 is the tangent kernel at time $t$ , which is a $k | \mathcal { D } | \times k | \mathcal { D } |$ kernel matrix
+
+$$
+\hat { \Theta } _ { t } = \nabla _ { \theta } f _ { t } ( \mathcal { X } _ { T } ) \nabla _ { \theta } f _ { t } { ( \mathcal { X } _ { T } ) } ^ { T }
+$$
+
+152 One can define the tangent kernel for general arguments, e.g. $\hat { \Theta } _ { t } ( x , \mathcal { X } _ { T } )$ where $x$ is test input. At   
+153 finite-width, $\hat { \Theta }$ will depend on the specific random draw of the parameters and evolve with time. As   
+154 such, for a test point $x$ the prediction $f _ { t } ( x )$ depends on the random initalization and is also stochastic.   
+155 Note that the parameters are initialized randomly and the randomness will be carried out through the   
+156 training procedure. As a consequence, the prediction functions are stochastic.
+
+# 57 2.3 Infinite Network: Gaussian Processes and the Neural Tangent Kernels
+
+158 Neural Networks as Gaussian Processes (NNGP). As the width $n  \infty$ , at initialization the   
+159 output $f _ { 0 } ( \mathcal { X } )$ forms a Gaussian Process $f _ { 0 } ( \mathcal { X } ) \sim \mathcal { G P } ( 0 , \mathcal { K } ( \mathcal { X } , \mathcal { X } ) )$ , known as the NNGP [8, 20, 21].   
+160 Here $\kappa$ is the GP kernel and can be computed in closed form for a variety of architectures. By treating   
+161 this infinite width network as a Bayesian model (aka Bayesian Neural Networks) and applying   
+162 Bayesian inference, the posterior is also a GP
+
+$$
+\mathcal { N } \left( \mathcal { K } ( \mathcal { X } _ { * } , \mathcal { X } _ { T } ) \mathcal { K } ^ { - 1 } ( \mathcal { X } _ { T } , \mathcal { X } _ { T } ) \mathcal { Y } _ { T } , \mathcal { K } ( \mathcal { X } _ { * } , \mathcal { X } _ { * } ) - \mathcal { K } ( \mathcal { X } _ { * } , \mathcal { X } ) \mathcal { K } ( \mathcal { X } , \mathcal { X } ) ^ { - 1 } \mathcal { K } ( \mathcal { X } _ { * } , \mathcal { X } ) ^ { T } \right)
+$$
+
+163 Neural Tangent Kernelss(NTK). Recent advance in global convergence theory of over  
+164 parameterized networks [22–25, 12] has shown that under certain assumptions, the tangent kernels is   
+165 almost stationary over the course of training and is concentrated on its infinite width limit $\Theta$ in the   
+166 sense there is a constant $C$ independent of $t$ and the network’s width $n$ such that
+
+$$
+\operatorname* { s u p } _ { t \geq 0 } \| \hat { \Theta } _ { t } ( \mathcal { X } _ { T } , \mathcal { X } _ { T } ) - \Theta ( \mathcal { X } _ { T } , \mathcal { X } _ { T } ) \| _ { F } + \| \hat { \Theta } _ { t } ( \mathcal { X } _ { T } , \mathcal { X } _ { * } ) - \Theta ( \mathcal { X } _ { T } , \mathcal { X } _ { * } ) \| _ { F } \leq \frac { C } { \sqrt { n } } .
+$$
+
+167 where is the infinite width limit of $\Theta$ at initialization, whose existence has been proved in [22, 26].   
+168 As such, when the loss is the mean squared error (MSE), the mean prediction (marginarized over   
+169 random initialization) has the following closed form
+
+$$
+\begin{array} { r } { f ( \mathcal { X } _ { * } ) = \Theta \left( \mathcal { X } _ { * } , \mathcal { X } _ { T } \right) \Theta ^ { - 1 } ( \mathcal { X } _ { T } , \mathcal { X } _ { T } ) \left( I - e ^ { - \eta \Theta ( \mathcal { X } _ { T } , \mathcal { X } _ { T } ) t } \right) \mathcal { V } , } \end{array}
+$$
+
+170 Letting $t \to \infty$ , the above solution is the same as that of the kernel ridgeless regression using the   
+171 infinite width tangent kernel $\Theta$ . We use $\mathsf { F C N } _ { \infty } ( x ) , \mathsf { L C N } _ { \infty } ( x ) , \mathsf { V E C } _ { \infty } ( \bar { x } )$ and ${ \mathsf { G A P } } _ { \infty } ( x )$ to denote   
+172 the infinite width solutions (either the GP inference or the NTK regression) for the corresponding   
+173 architectures, where we have suppressed the dependence on the training data $( \mathcal { X } _ { T } , \mathcal { Y } _ { T } )$ .   
+175 Symmetry is fundamental in physical systems. So is it in machine learning systems. We explore   
+176 symmetries of various machine learning systems in this section. Given $\mathcal { D } = ( \mathcal { X } , \mathcal { Y } )$ and a transforma  
+177 tion on the input space $\tau : \mathbb { R } ^ { 3 d }  \mathbb { R } ^ { 3 d }$ , we set $\boldsymbol { \tau } ( \mathcal { D } ) = ( \boldsymbol { \tau } ( \boldsymbol { \mathcal { X } } ) , \boldsymbol { \mathcal { Y } } )$ . Let ${ \mathrm { O } } ( 3 d )$ denote the orthogonal   
+178 group on the flatten input space $\mathbb { R } ^ { 3 d }$ . The subgroup $0 ( 3 ) ^ { d } \leq 0 ( 3 d )$ operates on the un-flattened   
+179 input $( \mathbb { R } ^ { d } ) ^ { 3 }$ , whose element rotates each pixel $x _ { \alpha } \in \mathbb { R } ^ { 3 }$ by an independent element $\tau _ { \alpha } \in \mathbf { O } ( 3 )$ . The   
+180 smaller subgroup ${ \mathbf O } ( 3 ) \otimes { \mathbf I } _ { d } \le { \mathbf O } ( 3 ) ^ { d }$ applies the shared rotation (i.e. $\tau _ { \alpha } = \tau$ to all $x _ { \alpha }$ for $\alpha \in [ d ] )$ .   
+181 We use $ { \mathbf { P } } ( 3 d )$ to denote the permutation group on $\mathbb { R } ^ { 3 d }$ and $\mathsf { P } ( 3 ) ^ { d }$ and $\mathbf { P ( 3 ) } \otimes \mathbf { I } _ { d }$ are defined similarly.   
+182 Note that rotating $\mathcal { X }$ by $\tau$ is equivalent to transfer the original coordinate system by the adjoint   
+183 tranformation $\tau ^ { * } = \tau ^ { - 1 }$ .   
+184 For a deterministic (stochastic) learning algorithm $\mathbf { \mathcal { A } } = \left( \mathcal { M } , \mathcal { T } \right)$ , we use $\boldsymbol { \mathcal { A } } ( \mathcal { D } _ { T } )$ to denote the learned   
+185 function (distribution of the learned functions) using training set $\mathcal { D } _ { T }$ . We use $\mathcal { A } ^ { \tau } ( \mathcal { D } _ { T } )$ to denote   
+186 the learned function(s) using $\tau ( \mathcal { D } _ { T } )$ and makes prediction on the transformed test point $\tau ( \mathcal { X } _ { * } )$ . In   
+187 another word, the learning algorithm is conducted in the input space whose coordinate system is   
+188 transformed by τ −1.
+
+Definition 1. Let $\mathcal { G }$ be a group of transformations $\mathbb { R } ^ { 3 d } \to \mathbb { R } ^ { 3 d }$ . We say a deterministic (stochastic) learning algorithm $\mathbf { \mathcal { A } } = \left( \mathcal { M } , \mathcal { T } \right)$ is $g$ -invariant if $\mathcal { A } = \mathcal { A } ^ { g }$ $\boldsymbol { \mathcal { A } } = ^ { d } \_ { A } \boldsymbol { g }$ ). In this case, we say the system $( \mathcal { D } , \mathcal { M } , \mathcal { T } )$ is $g$ -invariant and use the notation $( \mathcal { D } , \mathcal { M } , \mathcal { Z } ) = ( g \mathcal { D } , \mathcal { M } , \mathcal { Z } )$ . If this holds for all $g \in { \mathcal { G } }$ , then we say the algorithm and the system are $\mathcal { G }$ -invariant.
+
+193 If $( { \mathcal { M } } , { \mathcal { T } } )$ is the algorithm of minimum norm linear regressor, then $( \mathcal { D } , \mathcal { M } , \mathcal { T } )$ is $\mathrm { O } ( 3 ) ^ { d }$ -invariant;   
+194 see Sec.G for more details. Note that the symmetry (invariance) in our definition is a property of a   
+195 system and is different from the notion of symmetry that are commonly used in the machine learning   
+196 community, which is a property of a function (e.g. translation invariance).
+
+197 Theorem 3.1 (Sec.C). If the parameters of the networks are initialized with iid $\mathcal { N } ( 0 , 1 )$ , then • $\mathsf { F C N } _ { n / \infty }$ are ${ \bf O } ( 3 d )$ -invariant. 200 • ${ \mathsf { V E C } } _ { n }$ is $\mathbf { O ( 3 ) } \otimes \mathbf { I } _ { d }$ -invariant and ${ \mathsf { V E C } } _ { \infty }$ 201 is $O ( 3 ) ^ { d }$ -invariant. • $\mathsf { L C N } _ { n / \infty }$ are O(3)d-invariant. 202 • ${ \mathsf { G A P } } _ { n / \infty }$ are ${ \mathbf { O } } ( 3 ) \otimes { \mathbf { I } } _ { d }$ -invariant.
+
+203 The ${ \bf O } ( 3 d )$ -invariant of $\mathsf { F C N } _ { \infty }$ is because the NTK/NNGP kernel is an inner product kernel, namely,   
+204 there is a function $k$ such that the kernels have the form $k ( \langle x , x ^ { \prime } \rangle )$ . The ${ \bf O } ( 3 d )$ -invariant of finite   
+205 width $\mathsf { F C N } _ { n }$ is due to the Gaussian initialization of the first layer which was first observed and   
+206 proved in [27]. Rotating the input by $\tau \in \mathrm { O } ( 3 d )$ is equivalent to rotating the weight matrix $\omega$ of   
+207 the first layer by $\tau ^ { * }$ . Since for $\omega \in \mathcal { N } ( 0 , 1 ) ^ { 3 d } \tau ^ { * } \omega = ^ { d } \omega$ , at random initialization, the distribution   
+208 of the output functions (the prior) are unchanged if all inputs are rotated by the same element in   
+209 ${ \mathrm { O } } ( 3 d )$ . This property continues to hold throughout the course of (continue/discrete) gradient descent   
+210 training with/without $L ^ { 2 }$ -regularization and Bayesian posterior inference. For the same reason, $\mathsf { L C N } _ { n }$   
+211 is $O ( 3 ) ^ { d }$ -invariant because each patch of the image uses independent Gaussian random variables.   
+212 However, weight-sharing in ${ \mathsf { V E C } } _ { n }$ and ${ \mathsf { G A P } } _ { n }$ breaks the $\mathrm { O } ( 3 ) ^ { d }$ symmetry, reducing it to ${ \bf O ( 3 ) } \otimes { \bf I } _ { d }$ .
+
+13 For infinite networks, $\mathsf { L C N } _ { \infty } = \mathsf { V E C } _ { \infty }$ [28–31]. The kernels of ${ \mathsf { V E C } } _ { \infty }$ and $\mathsf { G A P } _ { \infty }$ are of the forms
+
+$$
+\Theta _ { \mathsf { V E C } } ( x , x ^ { \prime } ) = k \bigl ( \{ \langle x _ { \alpha } , x _ { \alpha } ^ { \prime } \rangle \} _ { \alpha \in [ d ] } \bigr ) \quad \mathrm { a n d } \quad \Theta _ { \mathsf { G A P } } ( x , x ^ { \prime } ) = k \bigl ( \{ \langle x _ { \alpha } , x _ { \alpha ^ { \prime } } ^ { \prime } \rangle \} _ { \alpha , \alpha ^ { \prime } \in [ d ] } \bigr ) ,
+$$
+
+214 resp. The former depends only on the inner product between pixels in the same spatial location,   
+215 breaking the ${ \mathrm { O } } ( 3 d )$ symmetry and reducing it to $O ( 3 ) ^ { d }$ . In addition, the latter depends also on the   
+216 inner products of pixels across different spatial locations due to pooling, which breaks the $O ( 3 ) ^ { d }$   
+217 symmetry and reduces it to ${ \mathbf { O } } ( 3 ) \otimes { \mathbf { I } } _ { d }$ .   
+218 Note that $\dim ( \mathrm { O } ( 3 d ) ) = 3 d ( 3 d - 1 ) / 2$ , $\dim ( \mathbf { O } ( 3 ) ^ { d } ) = 3 d$ and $\mathrm { d i m } ( \mathrm { O } ( 3 ) \otimes \mathbf { I } _ { d } ) = 3$ . $\mathsf { L C N } _ { n } / \mathsf { V E C } _ { \infty }$   
+219 dramatically reduces the dimension of the symmetry group. It is worth mentioning that while   
+220 $\mathrm { d i m } ( \mathrm { O } ( 3 d ) )$ many pairs of rotated and unrotated images are needed to recover the exact rotation in   
+221 ${ \mathrm { O } } ( 3 d )$ , only 3 pairs are sufficient for $O ( 3 ) ^ { d }$ , same as that of ${ \mathbf { O } } ( 3 ) \otimes { \mathbf { I } } _ { d }$ . The results of the paper   
+222 are presented in the most vanilla setting. Our methods can easily extend to more complicated   
+223 architectures like ResNet[32], MLP-Mixer[33] and etc. The symmetry groups of such systems   
+224 need to be computed in a case-by-case manner by identifying the invariant group of the random   
+225 initialization and training procedures.
+
+![](images/7dffd65fd9bab817e33f302ca23bfe7a49196818e65d6d61bb370de067db4c5a.jpg)  
+Figure 1: Performance vs Symmetry. Machine learning systems are equipped with various kinds of symmetries. Transforming the system by the associated symmetry does not affect the performance of the system. However, injecting spurious symmetries beyond the associated symmetries could dramatically degrade their performance for both finite and infinite networks.
+
+![](images/f9c643da52d95b664c033422d176c91a9dded01c5708e8de19b86b1c76ad7b26.jpg)  
+Figure 2: Even in the $\mathrm { N N } +$ setting, ${ \mathsf { V E C } } _ { n }$ is closer to ${ \mathsf { G A P } } _ { n }$ for small $n$ and moves towards $\mathsf { V E C } _ { \infty }$ with more symmetries and/or larger $n$ and accuracy drops.
+
+# 226 3.1 Empirical Supports and Observations
+
+Performance under Rotations. We examinate the performance of: FCN, VEC, LCN, GAP and $\mathsf { L A P ^ { 4 / 8 } }$ , when the coordinates of the data are transformed by six different groups ( $x$ -axis in Fig.1) using the standard dataset CIFAR-10. , Here $\mathsf { L A P ^ { 4 / 8 } }$ is the same as GAP except the readout layer is replaced by the Local Average Pooling with window size $4 \times 4 / 8 \times 8$ . We consider 4 types of training methods: (1) NTK, i.e. infinite networks (2)NN, our baseline for finite width neural network which is trained with momemtum using a small learning rate and without $L ^ { 2 }$ regularizer and the network is centered $( + \mathsf C )$ to reduce the variance from random initialization $( 3 ) \mathrm { N N + = N N + L R + L 2 - C }$ , i.e. using a larger learning rate $( + \mathsf { L R } )$ , adding $L ^ { 2 }$ regularization $( + \mathsf { L } 2 ) ]$ and removing the centering $( - C )$ (4) $\mathsf { N N + + = N N + + D A }$ , adding MixUp[34] data augmentation $( + \mathsf { D A } )$ to $\mathrm { N N } +$ . Overall, we observe that, for most of the cases in $\mathrm { N T K / N N / N N } +$ , adding spurious symmetry to a system $( \mathcal { D } , \mathcal { M } , \mathcal { Z } )$ degrades the performance towards that of the system invariant to that symmetry. Surprisingly, in the baseline NN, performance of $\mathsf { V E C } _ { n } + \mathsf { O } ( 3 ) \otimes \mathbf { I } _ { d }$ rotation is slightly worse than that of ${ \mathsf { V E C } } _ { n } + \mathbf { O } ( 3 ) ^ { d }$ and than that of $\mathsf { L C N } _ { n }$ , indicating that the system with $\mathcal { M } = \mathsf { V E C } _ { n }$ is likely operating closely on the $\mathrm { O } ( 3 ) ^ { d }$ symmetry. The interventions $- C + \mathsf { L } 2 + \mathsf { L } \mathsf { R }$ in $\mathrm { N N } +$ distinguishes the performance of $\mathsf { V E C } _ { n } + \mathsf { O } ( 3 ) \otimes \mathbf { I } _ { d }$ from ${ \mathsf { V E C } } _ { n } + \mathbf { O } ( 3 ) ^ { d }$ and $+ \mathsf { D } \mathsf { A }$ eventually closes the performance gap between $\mathsf { V E C } _ { n } + \mathsf { O } ( 3 ) \otimes \mathbf { I } _ { d }$ and ${ \mathsf { G A P } } _ { n } + { \mathsf { O } } ( 3 ) \otimes \mathbf { I } _ { d }$ , helping the system to be aware of the smaller symmetry ${ \mathbf { O } } ( 3 ) \otimes { \mathbf { I } } _ { d }$ , escaping from the $\mathrm { O } ( 3 ) ^ { d }$ symmetry.
+
+244 Symmetry Breaking of ${ \mathsf { V E C } } _ { n }$ . Assuming Equation 8, namely, the network is in the NTK regime,
+
+$$
+\operatorname* { l i m } _ { n \to \infty } | \mathbb { E } \mathsf { V } \mathsf { E } \mathsf { C } _ { n } ( x ) - \mathsf { V } \mathsf { E } \mathsf { C } _ { \infty } ( x ) | + \operatorname* { l i m } _ { n \to \infty } | \mathbb { E } \mathsf { V } \mathsf { E } \mathsf { C } _ { n } ( x ) - \mathbb { E } \mathsf { V } \mathsf { E } \mathsf { C } _ { n } ^ { \tau } ( x ) | \leq C n ^ { - \frac { 1 } { 2 } }
+$$
+
+245 where the expectation $\mathbb { E }$ is over random initialization and ${ \mathsf { V E C } } _ { n } ( x )$ is the prediction of the test point   
+246 $x$ when $t = \infty$ , i.e. training loss is $0 . \mathsf { V E C } _ { n } ^ { \tau }$ is the prediction of the $\tau$ -rotated system, $\tau \in { \mathrm { O } } ( 3 ) ^ { d }$ .   
+247 The $O ( 3 ) ^ { d }$ symmetry is restored as $n \to \infty$ . As such, for large $n$ , the system is approximately $\mathrm { O } ( 3 ) ^ { d }$ -   
+248 invariant. We randomly sample $\tau \in { \mathrm { O } } ( 3 ) ^ { d }$ and use the exponential map to construct a continuous   
+249 interpolation $\tau _ { t } \in \mathbf { O } ( 3 ) ^ { d }$ with $\tau _ { 0 } = \mathbf { I d }$ and $\tau _ { 1 } = \tau$ . We train the network as in $\Nu \mathrm { N + + } ( + \mathsf { L R } + \mathsf { L } 2 - \mathsf { C } )$   
+250 using different $n$ and $\tau _ { t }$ and average the predictions over 10 random initialization as an approximation   
+251 of $\bar { \mathbb { E } } \bar { \mathsf { V } } \bar { \mathsf { E } } \mathsf { C } _ { n } ^ { \tau _ { t } } ( x )$ . Not surprisingly, as $n$ increases and/or $t$ increases, (1) test performance decays   
+252 monotonically (left panel in Fig.2), (2) the distance to $\mathbb { E } G \mathsf { A P } _ { n }$ increases monotonically (middle   
+253 panel) and (3) distance to ${ \mathsf { V E C } } _ { \infty }$ decrease monotonically (right panel). Clearly, the coordinate   
+254 information from the data is utilized by smaller width ${ \mathsf { V E C } } _ { n }$ .   
+255 DIDE for ${ \mathsf { V E C } } _ { n }$ . To understand the role of data, we vary the training set size of Cifar10 from about   
+256 $2 ^ { 6 }$ to $5 0 \mathrm { k }$ (the whole un-augmented training set) and to $1 0 0 \mathrm { k }$ (adding left-right flip augmentation) and   
+257 plot the learning curves in Fig.3. We observe dramatic speedup of learning for ${ \mathsf { V E C } } _ { n }$ in the larger   
+258 data set regime, which isn’t the case for ${ \mathsf { V E C } } _ { \infty }$ (kernel), $\mathsf { L C N } _ { n }$ , $\mathsf { G A P } _ { \infty }$ and even for ${ \mathsf { G A P } } _ { n }$ after   
+259 $m = 2 ^ { 1 2 }$ . We argue that this is due to the prior (the function space defined by the model) is too large   
+260 (and not optimal) for the task and the coupled effect of more data together with inference procedures   
+261 corrects the prior, as it is suggested by Theorem 2.1.
+
+![](images/619a9e725b2cd9e8a196f30b6c9e3ebd296f7f511e5e5af86bf84b2a003236a6.jpg)  
+Figure 3: Data Bends Learning Curve of ${ \mathsf { V E C } } _ { n }$ . We study the effect of training set size to the network’s performance for various models. In the small dataset regime, the slope of the learning curve (in the log-log plot) of ${ \mathsf { V E C } } _ { n }$ is similar to that of $\mathsf { V E C } _ { \infty }$ and $\mathsf { F C N } _ { n }$ . However, as the dataset gets larger, the slope increases significantly. This is hinted by Theorem 2.1.   
+Figure 4: With coordinate of the input data rotated by $O ( 3 ) ^ { d }$ , state of the art models learn as good as without rotation. middle/right: slopes of the learning curves increases due to more data. DIDE
+
+DIDE for SOTA models. In the middle and right panels of Fig.S2, we provide additional evidence in a larger scale setting. We generate learning curves of ImageNet using ResNet50 and MLP-Mixer, a very recent architecture that contains no convolution layers except the first layer, which is a convolution with filter size and stride equal to (16, 16) (patches are disjoint). The symmetry group associated to ResNet is similar to that of ${ \mathsf { G A P } } _ { n }$ which is relatively small. However, the symmetry group induced by the first layer of the Mixer is $\mathbf { O } ( 3 \times 1 6 ^ { 2 } ) \otimes \mathbf { I } _ { 1 4 ^ { 2 } }$ , where $\mathrm { 3 \times 1 6 ^ { 2 } }$ is number of entries in the $( 1 6 , 1 6 , 3 )$ patch (RGB channels) and $1 4 ^ { 2 } = 2 2 4 ^ { 2 } / 1 6 ^ { 2 }$ is the number of patches. Although the dimension of ${ \mathrm { O } } ( 3 \times 1 6 ^ { 2 } ) \otimes { \mathbf { I } } _ { 1 4 ^ { 2 } }$ is quite large (about $( 3 \times 1 6 ^ { 2 } ) ^ { 2 } / 2 $ ), it is still dramatically smaller than that of applying a fully-connected layer to the flatten images, which ${ \mathrm { O } } ( 3 \times 2 2 4 ^ { 2 } )$ (about $( 3 \times 2 2 4 ^ { 2 } ) ^ { 2 } / 2 )$ . In the middle panel of Fig.S2, we observe an almost perfect power-law scaling for the learning curve for the ResNet50 system with unrotated images. When the images are rotated by $O ( 3 ) ^ { d }$ $( d \bar { = } 2 2 4 ^ { 2 }$ ), the learning curve is relatively flat in the smaller data regime (green dashed line). However, the data set grows, it eventually catches up (purple dashed line) as that of the unrotated setting; see Sec.E for ResNet34/101. In the third panel, we see the learning curves are much flatter (red) for the Mixer and even more so for the rotated images (green). Again, these curves are bent towards that of ResNet50 with unrorated images as data increases, indicating the prior was being corrected.
+
+Finally, in the left panel of Fig.S2, we compare the accuracy of state-of-the-art models trained on both unrotated and $O ( 3 { \bar { ) } } ^ { d }$ rotated images. Surprisingly, the gap between the two are not large and becomes smaller for better performant models. For EfficientNet B7 1, the top-1 accuracy of the rotated system is only $1 . 2 \%$ off from the unroated one.
+
+# 4 Eigenecomposition of Neural Kernels
+
+To get insights into the inductive biases, we eigendecompose the kernels using spherical harmonics. We assume the input space ${ \mathcal { X } } \ = \ \{ \xi \ = \ ( \xi _ { 0 } , \ldots , \cdot \xi _ { p - 1 } ) \in \ ( { \sqrt { d _ { 0 } } } \mathbb { S } ^ { ( d _ { 0 } - { \bar { 1 } } ) } ) ^ { p } \} \ \subseteq \ \mathbb { R } ^ { d _ { 0 } p } ,$ , i.e.
+
+286 the $p$ -product of $( d _ { 0 } - 1 )$ -sphere with radius $\sqrt { d _ { 0 } }$ . We call $\xi _ { i } \in \sqrt { d _ { 0 } } \mathbb { S } ^ { ( d _ { 0 } - 1 ) }$ a mini-patch and   
+287 $( \xi _ { i } , \xi _ { i + 1 } , \ldots , \xi _ { i + s - 1 } ) \in ( \sqrt { d _ { 0 } } \mathbb { S } ^ { ( d _ { 0 } - 1 ) } ) ^ { s } \}$ a patch for $i \in [ p ]$ , where circular boundary condition is   
+288 assumed. We consider the asymptotic limit when $d _ { 0 } = \bar { d ^ { \alpha } } \bar { , } p = d ^ { 1 - \alpha }$ and $d = p d _ { 0 }  \infty$ and treat   
+289 $0 \textless \alpha \textless 1$ and $s$ as fixed constant. The input space $\mathcal { X }$ is associated with the product measure√   
+290 $\mu \equiv \sigma _ { d _ { 0 } } ^ { p }$ , where $\sigma _ { d _ { 0 } }$ is the normalized uniform measure on $\sqrt { d _ { 0 } } \mathbb { S } ^ { ( d _ { 0 } - 1 ) }$ . The kernels associated to   
+291 the one-hidden layer infinite networks (either NNGP or NTK) has the following general forms
+
+$$
+k \left( \frac { 1 } { p } \sum _ { i \in [ p ] } \xi _ { i } ^ { T } \eta _ { i } / d _ { 0 } \right) \quad \frac { 1 } { p } \sum _ { i \in [ p ] } k \left( \frac { 1 } { s } \sum _ { b \in [ s ] } \xi _ { i + b } ^ { T } \eta _ { i + b } / d _ { 0 } \right) \quad \frac { 1 } { p ^ { 2 } } \sum _ { i , j \in [ p ] } k \left( \frac { 1 } { s } \sum _ { b \in [ s ] } \xi _ { i + b } ^ { T } \eta _ { j + b } / d _ { 0 } \right) ,
+$$
+
+292 although that exact form of the (positive definite) kernel function $k : \mathbb { R }  \mathbb { R }$ depends on the kernel   
+293 types (NNGP vs NTK), activations, hyperparameters and etc. We assume the kernel is sufficiently   
+294 smooth in $( - 1 , 1 )$ and the Tayor expansion of $k ^ { ( r ) }$ converges uniformly in $[ - 1 , 1 ]$ for sufficiently   
+295 many $r \in \mathbb N$ . We use the notation that $A \sim B$ if there are positive constants $c$ and $C$ such that   
+296 $c A \leq B \leq C A$ for $d$ sufficiently large. We use $\kappa$ to represent any kernels above and consider it as a   
+297 Hilbert–Schmidt operator on $L ^ { \tilde { 2 } } ( \chi , \breve { \mu } )$
+
+$$
+\mathcal K f ( \xi ) = \int _ { \mathcal K } { \mathcal K } ( \xi , \eta ) f ( \eta ) d \mu , \quad f \in L ^ { 2 } ( \mathcal { X } , \mu ) ,
+$$
+
+which is well-defined since 298 $\mu$ is a probability measure and $k$ is bounded. Let $\vec { r } = ( r _ { 0 } , \ldots , r _ { p - 1 } ) \in \mathbb { N } ^ { p }$ , 299 $\tau$ the shifting operator $\tau \vec { r } = \left( r _ { p - 1 } , r _ { 0 } , \ldots , r _ { p - 2 } \right)$ . The $s$ -banded subset of $\mathbb { N } ^ { p }$ is defined to be
+
+$$
+B ( \mathbb { N } ^ { p } , s ) = \{ { \vec { r } } \in \mathbb { N } ^ { p } : \operatorname { d i s t } ( \operatorname { a r g m a x } _ { j } r _ { j } \neq 0 , \operatorname { a r g m i n } _ { j } r _ { j } \neq 0 ) \leq s - 1 \}
+$$
+
+300 which is a quantifier used to restrict the support of a function on a patch. Here $\mathrm { d i s t } ( i , j ) = \operatorname* { m i n } \{ | i -$   
+301 $j | , p - | i - \bar { j } | \}$ , a distance defined on the cyclic group $[ p ] = \mathbb { Z } / p \bar { \mathbb { Z } }$ . The quotient space $B ( \mathbb { N } ^ { p } , s ) / \tau$   
+302 denotes a subset of $B ( \mathbb { N } ^ { p } , s )$ by identifying $\vec { v } = \vec { v } ^ { \prime }$ as the same element if ${ \vec { v } } = \tau ^ { a } { \vec { r } } ^ { \prime }$ for some $a \in [ p ]$   
+303 Finally, $Y _ { r _ { j } , l _ { j } } ( \xi _ { j } )$ is used to denote the $l _ { j }$ -th spherical harmonic of degree $r _ { j }$ in the unit sphere   
+304 $\mathbb { S } ^ { ( d _ { 0 } - 1 ) }$ and has unit norm under the normalized measure on $\mathbb { S } ^ { ( d _ { 0 } - 1 ) }$ . As such $Y _ { r _ { j } , l _ { j } } ( \xi _ { j } / \sqrt { d _ { 0 } } ) \in$   
+305 $L ^ { 2 } ( \sqrt { d _ { 0 } } \mathbb { S } ^ { ( d _ { 0 } - 1 ) } , \sigma _ { d _ { 0 } } )$ has unit norm. Recall that the total number of spherical harmonic of degree   
+306 $r _ { j }$ in $\mathbb { S } ^ { ( d _ { 0 } - 1 ) }$ is $\begin{array} { r } { N ( \bar { d } _ { 0 } , r _ { j } ) = ( 2 r _ { j } + d _ { 0 } - 2 ) \binom { d _ { 0 } + r _ { j } - 3 } { r _ { j } - 1 } \sim d _ { 0 } ^ { r _ { j } } / r _ { j } ! } \end{array}$ as $d _ { 0 } \to \infty$ . We use $N ( d _ { 0 } , \vec { r } ) =$   
+307 $\textstyle \prod _ { j \in [ p ] } N ( d _ { 0 } , r _ { j } )$ and $\begin{array} { r } { [ N ( d _ { 0 } , \vec { r } ) ] = \prod _ { j \in [ p ] } [ N ( \vec { d _ { 0 } } , r _ { j } ) ] } \end{array}$ , resp. Let
+
+$$
+\vec { Y } _ { \vec { r } , \vec { l } } ( \xi ) = \prod _ { j \in [ p ] } Y _ { r _ { j } , l _ { j } } ( \xi _ { j } )
+$$
+
+308 The following theorem shows that locality $\mathrm { ( V E C _ { \infty , } }$ ) dramatically reduces both the dimensions of   
+309 Eigendecomposition $r \geq 1$ eigenspaces and the spectral gap between them. In addition, pooling   
+310 (i.e. translation symmetry of $\mathsf { G A P } _ { n } \mathrm { ~ . ~ }$ ) reduces their dimensions by a factor of $p$ . See Sec.E for the   
+311 implication of this theorem to learning.
+
+312 Theorem 4.1. [Sec.D] We have the following eigendecomposition for the integral operator $\kappa$
+
+$$
+\mathsf { H } = \bigcup _ { r \in \mathbb { N } } \mathsf { H } ^ { ( r ) } = \bigcup _ { r \in \mathbb { N } } \bigcup _ { \vec { r } \in Q ( \mathcal { K } , r ) } \mathsf { H } ^ { ( \vec { r } ) } ,
+$$
+
+where 313 $Q ( K , r )$ is a quantifier defined below. If ${ \mathfrak { r } } = 0 , \mathsf { H } ^ { ( 0 ) }$ is the space of constant functions and the 314 eigenvalue is $\sim k ( 0 )$ . For $r \geq 1$ ,
+
+315
+
+1. $i f \mathcal { K } = \mathcal { K } _ { \sf F C N }$ , then $Q ( \mathcal { K } , r ) = \{ \vec { r } \in \mathbb { N } ^ { p } : | \vec { r } | = r \}$ and the unit eigenfunctions are
+
+$$
+\{ \mathsf { H } ^ { ( \vec { r } ) } = \mathrm { s p a n } \{ Y _ { \vec { r } , \vec { l } } \} _ { \vec { l } \in [ B ( d _ { 0 } , \vec { r } ) ] }
+$$
+
+316
+
+2. $i f K = \mathcal { K } _ { \mathsf { V E C } }$ , ${ \cal Q } ( { \cal K } , r ) = \{ \vec { r } \in { \cal B } ( \mathbb { N } ^ { p } , s ) : | \vec { r } | = r \}$ the unit eigenfunctions are
+
+$$
+\left\{ \begin{array} { l l } { { \displaystyle { \sf H } _ { \sf V E C } ^ { ( \vec { r } ) } = \mathrm { s p a n } \left\{ Y _ { \vec { r } , \vec { l } } \right\} _ { \vec { l } \in [ B ( d _ { 0 } , \vec { r } ) ] } } _ { a \mathrm { r } } } \\ { { \mathrm { d i m } ( { \sf H } _ { \sf V E C } ^ { ( \vec { r } ) } ) \sim p ( s d _ { 0 } ) ^ { r } = s ^ { r } d ^ { 1 - \alpha + r \alpha } } } & { { a n d \quad \lambda ( { \sf H } _ { \sf V E C } ^ { ( \vec { r } ) } ) \sim p ^ { - 1 } ( s d _ { 0 } ) ^ { - r } \delta ( k ^ { ( r ) } ( { 0 } ) ) } } \end{array} \right.
+$$
+
+![](images/f333434682c669b1e856367975aa3b0a045c441b329742338d697f73dd0ac551.jpg)  
+Figure 5: Eigenvalue Decay of Relu NTK of $\mathsf { F C N } _ { \infty }$ , ${ \mathsf { V E C } } _ { \infty }$ and $\mathsf { G A P } _ { \infty }$ . $d _ { 0 } = s = 3$ . The eigenvalues of $\mathsf { G A P } _ { \infty }$ decays faster because with $m = 1 5 k$ many samples, higher order eigenspace can be covered by $\mathsf { G A P } _ { \infty }$ but not $\mathsf { F C N } _ { \infty } / \mathsf { V E C } _ { \infty }$ due to Theorem 4.1.
+
+317   
+318
+
+3. and finally, $i f K = \mathcal { K } _ { \mathsf { G A P } }$ , then $Q ( \mathcal { K } , r ) = \{ \vec { r } \in B ( \mathbb { N } ^ { p } , s ) / \tau : | \vec { r } | = r \}$ , the unit eigenfunctions are
+
+$$
+\left\{ \begin{array} { l } { { \sf H } _ { \sf G A P } ^ { ( \vec { r } ) } = \mathrm { s p a n } \left\{ \frac { 1 } { \sqrt { p } } \sum _ { \tau \in [ p ] } Y _ { \vec { r } , \vec { l } } ( \tau \xi ) \right\} _ { \vec { l } \in [ B ( d _ { 0 } , \vec { r } ) ] } } \\ { \dim ( { \sf H } _ { \sf G A P } ^ { ( r ) } ) \sim ( s d _ { 0 } ) ^ { r } = s ^ { r } d ^ { r \alpha } \quad a n d \quad \lambda ( { \sf H } _ { \sf G A P } ^ { ( \vec { r } ) } ) \sim p ^ { - 1 } ( s d _ { 0 } ) ^ { - r } \delta ( k ^ { ( r ) } ( 0 ) ) } \end{array} \right.
+$$
+
+# 319 5 Related Work
+
+The study of infinite networks dates back to seminal work by Neal [8] who showed the convergence of single hidden-layer networks to Gaussian Processes (GPs). Recently, there has been renewed interest in studying random, infinite, networks starting with concurrent work on “conjugate kernels” [10, 35] and “mean-field theory” [9, 36], taking a statistical learning and statistical physics view of points, resp. Since then this analysis has been extended to include a wide range for architectures [20, 21, 37, 29, 26, 38]. The inducing kernel is often referred to as the Neural Network Gaussian Process (NNGP) kernel. The neural tangent kernel (NTK), first introduced in Jacot et al. [22], along with followup work [12, 39] showed that the distribution of functions induced by gradient descent for infinite-width networks is a Gaussian Process with NTK as the kernel.
+
+329 The study of implicit bias (regularization) of gradient descent has received considerable interests.   
+330 The work [15, 40–43] demonstrate the convergence of SGD to the maximal margin solution for   
+331 logistic-type of losses during late time training. [44–50] study the early-time SGD dynamics, spectral   
+332 biases of neural networks. These results aim to explain the order of learning of neural networks:   
+333 functions of less complexity are usually learned before more complex functions.   
+334 [27] is the first to show that the prediction functions obtained from training FCN depend, in addition   
+335 on the labels, only on the covariance of the input data. This implies our result regarding the ${ \mathrm { O } } ( 3 d )$   
+336 invariance of FCN. By utilizing this symmetry, recent work [51] constructs a particular task where   
+337 the label function is a second order polynomial of the inputs and show that orthogonal invariance   
+338 algorithm requires sample size of order $\dot { d } ^ { 2 }$ while there is a convnet requires only $O ( 1 )$ samples. Their   
+339 convnet essentially corresponds to the $d _ { 0 } = s = 1$ and $r = 2$ case of Theorem 4.1, in which the   
+340 dimension of this eigenspace (and indeed of all $r$ -eigenspace by treating $r$ as a finite constant as   
+341 $d \to \infty$ ) of $\mathsf { G A P } _ { \infty }$ is $O ( \bar { 1 } )$ while the dimension of the 2-eigenspace of $\mathsf { F C N } _ { \infty }$ is of order $d ^ { 2 }$ .
+
+# 342 6 Conclusion
+
+43 In this paper, we consider machine learning methods as an integrated system of data, models and   
+44 inference algorithms and study the basic symmetries of various machine learning systems. We surface   
+45 the importance of locality in modern machine learning systems through large scale empirical study   
+46 and through an eigendecomposition of one-layer infinite networks. However, we haven’t addressed   
+47 the two import questions (1) theoretical characterization of the effect of composing locality and (2)   
+48 the mathematical understanding of DIDE and how the prior is corrected by the coupled effect of data   
+49 and gradient descent. We leave them to future work.
+
+The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example:
+
+• Did you include the license to the code and datasets? [Yes] See Section ??.   
+• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   
+• Did you include the license to the code and datasets? [N/A]
+
+Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.
+
+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]   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
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+
+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)? [No]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [No]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(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)? [No]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
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+(b) Did you mention the license of the assets? [TODO]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [TODO]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [TODO]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [TODO]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [TODO]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [TODO]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [TODO]   
+References   
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+        "text": "What Breaks the Curse of Dimensionality in Deep Learning? ",
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+        "text": "Abstract ",
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+        "text": "1 Although learning in high dimensions is commonly believed to suffer from the   \n2 curse of dimensionality, modern machine learning methods often exhibit an as  \n3 tonishing power to tackle a wide range of challenging real-world learning prob  \n4 lems without using abundant amounts of data. How exactly these methods break   \n5 this curse remains a fundamental open question in the theory of deep learning.   \n6 While previous efforts have investigated this question by studying the data (D),   \n7 model (M), and inference algorithm (I) as independent modules, in this paper   \n8 we analyzes the triple (D, M, I) as an integrated system. We examine the basic   \n9 symmetries of such systems, focusing on four of the main architectures in deep   \n10 learning: fully-connected networks (FCN), locally-connected networks (LCN), and   \n11 convolutional networks with and without pooling (GAP/VEC). By computing an   \n12 eigen-decomposition of the infinite-width limits (aka Neural Kernels) of these   \n13 architectures, we characterize how inductive biases (locality, weight-sharing, pool  \n14 ing, etc) and the breaking of spurious symmetries can affect the performance of   \n15 these learning systems. Our theoretical analysis shows that for many real-world   \n16 tasks it is locality rather than symmetry that provides the first-order remedy to the   \n17 curse of dimensionality. Empirical results on state-of-the-art models on ImageNet   \n18 corroborate our results. ",
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+        "text": "19 1 Introduction ",
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+        "text": "20 Statistical problems with high-dimensional data are frequently plagued by the curse of dimensionality,   \n21 in which the number of samples required to solve the problem grows rapidly with the dimensionality   \n22 of the input. Classical theory explains this phenomenon as the consequence of basic geometric and   \n23 algebraic properties of high-dimensional spaces; for example, the number of $\\epsilon$ -cubes inside a unit   \n24 cube in $\\mathbb { R } ^ { \\hat { d } }$ grows exponentially like $\\epsilon ^ { - d }$ , and the number of degree $r$ polynomials in $\\mathbb { R } ^ { d }$ grows like a   \n25 power-law $d ^ { r }$ . Since for real-world problems $d$ is typically in the hundreds or thousands, classical   \n26 wisdom suggests that learning is likely to be infeasible. However, starting from the groundbreaking   \n27 work AlexNet [1], practitioners in deep learning have tackled a wide range of difficult real-world   \n28 learning problems ([2–6]) in high dimensions, once believed by many to be out-of-scope of current   \n29 techniques. The astonishing success of modern machine learning methods clearly contradicts the   \n30 curse of dimensinonality and therefore poses the fundamental question: mathematically, how do   \n31 modern machine learning methods break the curse of dimensionality?   \n32 To answer this question, we must trace back to the most fundamental ingredients of machine learning   \n33 methods. They are the data $( \\mathcal { D } )$ , the model $( \\mathcal { M } )$ , and the inference algorithm $( \\mathcal { T } )$ .   \n34 Data $( \\mathcal { D } )$ is of course central in machine learning. In the classical learning theory setting, the learning   \n35 objective usually has a power-law decay $m ^ { - \\bar { \\beta } }$ as the function of the number of training samples   \n36 $m$ . The theoretical bound on $\\beta$ is usually tiny, owing to the curse of dimensionality, and is of   \n37 limited practical utility for high-dimensional data. On the other hand, empirical measurements of   \n38 $\\beta$ in state-of-the-art deep learning models typically reveal values of $\\beta$ that are not at all small (e.g.   \n39 $\\beta = 0 . 4 3$ for ResNet in Fig.S2) even though $d$ is quite large (e.g. $\\dot { d } \\sim 1 0 ^ { 5 }$ for ImageNet). This   \n40 example suggests that the learning curve must have important functional dependence on $\\mathcal { M }$ and $\\mathcal { T }$ .   \n41 Indeed, as we will observe later, many of the best performing methods exhibit learning curves for   \n42 which $\\beta = \\beta ( m )$ actually increases as $m$ becomes larger, i.e. data makes the usage of data more   \n43 efficient. We call this phenomenon DIDE, for data improves data efficiency.   \n44 Designing machine learning models $( \\mathcal { M } )$ that maximize data-efficiency is critical to the success   \n45 of solving real-world tasks. Indeed, breakthroughs in machine learning are often driven by novel   \n46 architectures LeNet [7], AlexNet[1], Transformer [2], etc. While some of the inductive biases of these   \n47 methods are clear (e.g. translation symmetries of CNNs), others tend to build off of prior empirical   \n48 success and are less well-understood (e.g. the implicit bias of SGD). To build our understanding of   \n49 these biases and how they affect learning, we conduct a theoretical analysis of them in the infinite  \n50 width setting [8–12], which preserves most salient aspects of the architecture while enabling tractable   \n51 calculations. We classify all phenomena that could be explained by infinite networks alone as the   \n52 consequences of inductive biases.   \n53 The inference procedure $( \\mathcal { T } )$ is what enables learning in machine learning methods. It is widely   \n54 believed that modern inference methods, specifically gradient descent and variants, ‘implicitly‘ bias   \n55 the solutions of the networks towards those that generalize well and away from those that generalize   \n56 poorly [13–15]. The effects of the inference algorithm are intimately tied to the specifics of the model   \n57 (e.g. weight-sharing) and the data (e.g. augmentation), and might not be fully understood with a   \n58 fixed-data, fixed-model analysis. Indeed, good performance may derive from interactions between   \n59 $( { \\mathcal { M } } , { \\mathcal { T } } )$ , or $( \\mathcal { D } , \\mathcal { I } )$ , or even $( \\mathcal { D } , \\mathcal { M } , \\mathcal { T } )$ . In Sec. 3.1, we demonstrate the DIDE effect for a particular   \n60 choice of $( \\mathcal { D } , \\mathcal { M } , \\mathcal { T } )$ and show that this effect disappears if any one of $\\mathcal { D }$ , $\\mathcal { M }$ , or $\\mathcal { T }$ is altered. ",
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+        "text": "The above discussion highlights the insufficiency of treating $\\mathcal { D }$ , $\\mathcal { M }$ , and $\\mathcal { T }$ as separate non-interacting modules. They must be considered as an integrated system. Throughout this paper, we will refer to the triplet $( \\mathcal { D } , \\mathcal { M } , \\mathcal { T } )$ as a (machine) learning system and the tuple $( { \\mathcal { M } } , { \\mathcal { T } } )$ as the learning algorithm of the system that operates on $\\mathcal { D }$ . We summarize our contributions below. ",
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+        "text": "1. We surface the basic symmetries of various $( \\mathcal { D } , \\mathcal { M } , \\mathcal { T } )$ associated to four of the main architectures in deep learning $\\mathsf { F C N } _ { n }$ (fully-connected networks), $\\mathsf { L C N } _ { n }$ (locally-connected networks), ${ \\mathsf { V E C } } _ { n } / { \\mathsf { G A P } } _ { n }$ (convolution networks with a flattening /a global average pooling readout layer), their infinite width counterparts $\\mathsf { F C N } _ { \\infty } / \\mathsf { L C N } _ { \\infty } / \\mathsf { V E C } _ { \\infty } / \\mathsf { G A P } _ { \\infty }$ . Treating $\\mathsf { F C N } _ { n / \\infty }$ as the baseline model, we show that the locality from $\\mathsf { L C N } _ { n }$ and the weight-sharing from $\\mathsf { \\dot { V } E C } _ { n } / \\mathsf { G A P } _ { n }$ break spurious symmetries and lead to better systems. Empirically, we examine the relation between the symmetries and the performance of the systems in the infinite width setting and finite width setting with various of interventions. Surprisingly, we observe that state-of-the-art learning system (EfficientNet[16]) on ImageNet can learn almost equally well even the coordinate of the data are transformed by the symmetry group defined by $\\mathsf { L C N } _ { n }$ . ",
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+        "text": "2. We show that although the weight-sharing from ${ \\mathsf { V E C } } _ { n }$ provides coordinate information of the data to the system, as the width gets larger, it becomes harder for the learning algorithm to explore such information and at infinite width, the system restores the symmetry group that is identical to $\\mathsf { L C N } _ { n }$ , and is completely unaware of the coordinate information. As a consequence, the performance of the network, as a function of width, monotonically decays [12]. This is in stark contrast to recent finding that the performance of network is positively correlated to its width. We show that this phenomenon continues to hold even with various interventions (larger learning rate and l2 regularization) to the training procedures. However, with more data (e.g. data augmentation) ${ \\mathsf { V E C } } _ { n }$ can be on par with ${ \\mathsf { G A P } } _ { n }$ . ",
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+        "text": "3. The function space defined by $\\mathsf { L C N } _ { n }$ is a super set of that defined by ${ \\mathsf { V E C } } _ { n }$ . We prove the opposite is true. Therefore, ${ \\mathsf { V E C } } _ { n }$ is able to express functions in the space with a stronger inductive bias ${ \\mathsf { G A P } } _ { n }$ (translation invariance) and functions in a seemingly much larger class $\\mathsf { L C N } _ { n }$ . We hypothesize that as the dataset grows, the learned functions using ${ \\mathsf { V E C } } _ { n }$ is transitioned away from those learned using $\\mathsf { L C N } _ { n }$ and become closer to those learned using ${ \\mathsf { G A P } } _ { n }$ . This suggests, even though the prior (provided by human) is not $100 \\%$ correct, with the help of more data, gradient descent might be able to correct it, a possible explanation of DIDE. ",
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+        "text": "4. When the input space is the product of hyperspheres, we eigendecompose the kernels associated to one-hidden layer infinite width network, $\\mathsf { F C N } _ { \\infty }$ , $\\mathsf { V } \\bar { \\mathsf { E } } \\mathsf { C } _ { \\infty } = \\mathsf { L } \\bar { \\mathsf { C } } \\mathsf { N } _ { \\infty }$ and $\\mathsf { G A P } _ { \\infty }$ . We treat $\\mathsf { F C N } _ { \\infty }$ as the baseline, whose order $r$ eigenspace has dimension of order $d ^ { r }$ and eigenvalues of order $d ^ { - r }$ for $r \\geq 0$ [17]. We show that locality alone (i.e. $\\mathsf { V E C } _ { \\infty , \\mathsf { \\Lambda } }$ ) dramatically reduces the dimension of the $r$ -eigenspace for $r \\geq 2$ and the spectral gap between all $r$ -eigenspaces but $r = 0$ and $r = 1$ , making learning of higher order eigenspaces feasible with dramatically fewer samples and gradient steps. In addition, pooling (i.e. $\\mathsf { G A P } _ { \\infty } \\mathrm { \\Gamma } _ { \\infty } .$ ) reduces the dimension of $r$ -eigenspace for $r \\geq 1$ by a factor equal to the size of the pooling window, but it does not change the spectra in an essential way. ",
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+        "text": "02 Our empirical and theoretical results surface the importance of locality which, we believe, provides   \n03 the first-order remedy to the curse of dimensionality for many real-world tasks and which has been   \n04 largely overlooked. ",
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+        "type": "text",
+        "text": "2 Preliminary and Notation ",
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+        "text": "2.1 Neural Networks ",
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+        "text": "107 We focus our presentation on the supervised learning setting and more concretely, on image   \n108 recognition. Let $\\mathcal { D } \\subseteq ( \\mathbb { R } ^ { d } ) ^ { 3 } \\times \\mathbb { R } ^ { k } \\overset { \\cdot } { \\equiv } \\mathbb { R } ^ { 3 d } \\times \\mathbb { R } ^ { k }$ denote the data set (training and test) and   \n109 $\\mathcal { X } = \\{ x : ( x , y ) \\in \\mathcal { D } \\}$ and $\\mathcal { V } = \\{ y : ( x , y ) \\in \\mathcal { D } \\}$ denote the input space (images) and label space,   \n110 respectively. Here $d$ is the spatial dimension (e.g. $d = 3 2 \\times 3 2$ for CIFAR-10) of the images and 3 is   \n111 the total number of channels (i.e. RGB). We use $\\mathsf { F C N } _ { n }$ to denote a $L$ -hidden layer fully-connected   \n112 network with identical hidden widths $n _ { l } = n \\in \\mathbb { N }$ for $l = 1 , . . . , L$ and with readout width $n _ { L + 1 } = k$   \n113 (the number of logits). For each $x \\in \\mathbb { R } ^ { 3 d } = ( \\mathbb { R } ^ { d } ) ^ { 3 }$ , we use $h ^ { l } ( x ) , x ^ { l } ( x ) \\in \\mathbb R ^ { n _ { l } }$ to represent the pre  \n114 and post-activation functions at layer $l$ with input $x$ . The recurrence relation FCN is given by ",
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+        "text": "$$\n\\left\\{ \\begin{array} { l l } { h ^ { l + 1 } } & { = x ^ { l } W ^ { l + 1 } } \\\\ { x ^ { l + 1 } } & { = \\phi \\left( h ^ { l + 1 } \\right) } \\end{array} \\right. \\mathrm { a n d } \\ W _ { i , j } ^ { l } = \\frac { 1 } { \\sqrt { n _ { l } } } \\omega _ { i j } ^ { l } , \\quad \\omega _ { i j } ^ { l } \\sim \\mathcal { N } ( 0 , 1 )\n$$",
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+        "text": "115 where $\\phi$ is a point-wise activation function, $W ^ { l + 1 } \\in \\mathbb { R } ^ { n _ { l } \\times n _ { l + 1 } }$ are the weights and $\\omega _ { i j } ^ { l }$ are the   \n116 trainable parameters, drawn i.i.d. from a standard Gaussian $\\sim \\mathcal { N } ( 0 , 1 )$ at initialization. For simplicity   \n117 of the presentation, the bias terms and the hyperparameters (the variances of the weights) are omitted.   \n118 Adding them back won’t affect the conclusion of the paper.   \n119 For convolutional networks or locally-connected networks, the inputs are treated as tensors in $( \\mathbb { R } ^ { d } ) ^ { 3 }$ .   \n120 The recurrent relation of convolutional networks can be written as ",
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+        "img_path": "images/5d3bcbe7410cc1f2a4670226edc1659576022fac3b056fa89261ecfeb4e11184.jpg",
+        "text": "$$\nx _ { \\alpha , j } ^ { l + 1 } = \\phi ( h _ { \\alpha , j } ^ { l + 1 } ) \\quad \\mathrm { a n d } \\quad h _ { \\alpha , j } ^ { l + 1 } \\equiv { \\frac { 1 } { \\sqrt { ( 2 k + 1 ) n ^ { l } } } } \\sum _ { j = 1 } ^ { n ^ { l } } \\sum _ { \\beta = - k } ^ { k } x _ { \\alpha + \\beta , i } ^ { l } \\omega _ { i j , \\beta } ^ { l }\n$$",
+        "text_format": "latex",
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+        "text": "121 Here $\\alpha \\in [ d ]$ denote the spatial location, $i / j \\in [ n ]$ denotes the fanin/fanout channel indices. For   \n122 notational convenience, we assume circular padding and stride equal to 1 for all layers. The features   \n123 of the penultimate layer are 2D tensors and there are two commonly used approaches to map them   \n124 to the logit layer: stack a dense layer after either vectorizing the 2D tensor to a 1D vector or   \n125 applying a global average pooling layer to each channel. We use ${ \\mathsf { V E C } } _ { n } / { \\mathsf { G A P } } _ { n }$ to denote the network   \n126 obtain from the former/latter, which are known to be equipped with the inductive biases translation   \n127 equivariant/invariant. The readout layer of ${ \\mathsf { V E C } } _ { n } / { \\mathsf { G A P } } _ { n } ^ { - }$ could be written as ",
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+    {
+        "type": "equation",
+        "img_path": "images/867655db99805085c11dde037a3c5feaf5bdec1661d0163e550be9446a7e6ad1.jpg",
+        "text": "$$\nx _ { j } ^ { L + 1 } = \\frac { 1 } { \\sqrt { d n } } \\sum _ { \\alpha \\in [ d ] } x _ { \\alpha , i } ^ { L } w _ { \\alpha , i j } ^ { L + 1 } , x _ { j } ^ { L + 1 } = \\frac { 1 } { \\sqrt { n } } \\sum _ { i \\in [ n ] } \\left( \\frac { 1 } { d } \\sum _ { \\alpha \\in [ d ] } x _ { \\alpha , i } ^ { L } \\right) w _ { i j } ^ { L + 1 }\n$$",
+        "text_format": "latex",
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+        "text": "128 We briefly remark the the key difference between the two. In ${ \\mathsf { V E C } } _ { n }$ , each pixel in the penultimate   \n129 layer has its own (independent random) variable while pixels within the same channel shared the   \n130 same (random) variable in $\\mathsf { G A P } _ { n }$ . It is clear that the function space of ${ \\mathsf { V E C } } _ { n }$ contains that of ${ \\mathsf { G A P } } _ { n }$ .   \n131 Locally Connected Networks $\\mathsf { L C N } _ { n }$ [18, 19] are convolutional network without weight sharing   \n132 between spatial locations. $\\mathsf { L C N } _ { n }$ preserve the connectivity pattern, and thus topology, of a convnet.   \n133 Mathematically, the current formula is defind as in Equation 2 with all the shared parameters $\\omega _ { i j , \\beta } ^ { l }$   \n134 replaced by unshared $\\omega _ { i j , \\alpha , \\beta } ^ { l } \\sim \\mathcal { N } ( 0 , 1 )$   \n135 In this note, we assume that the $\\mathsf { L C N } _ { n }$ are always associated with a vectorization readout layer and it   \n136 is clear, as a function space, $\\mathsf { L C N } _ { n }$ is a super set of ${ \\mathsf { V E C } } _ { n }$ . Interestingly,the opposite is also true.   \n37 Theorem 2.1 (Sec. B). Let $\\mathsf { V E C } _ { n } / \\mathsf { L C N } _ { n } / \\mathsf { G A P } _ { n }$ denote the set of functions that can be represented   \n138 by $L$ -hidden layer $\\mathsf { V E C } _ { n } / \\mathsf { L C N } _ { n } / \\mathsf { G A P } _ { n }$ networks with hidden width $n$ . Then ",
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+        "bbox": [
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+        "img_path": "images/0559f45d6d241be4712fbd4c5cd0edca35fb8abaa7eeea61e7349867bbe24a01.jpg",
+        "text": "$$\n{ \\mathsf { G A P } } _ { n } \\subseteq { \\mathsf { V E C } } _ { n } \\subseteq { \\mathsf { L C N } } _ { n } \\subseteq { \\mathsf { V E C } } _ { d n }\n$$",
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+        "text": "139 The significance of this theorem is that if we consider the function space ${ \\mathsf { V E C } } _ { n }$ as a soft prior,   \n140 gradient descent could move it closer to a better prior ${ \\mathsf { G A P } } _ { n }$ (translation invariance) if the average   \n141 pooling is (approximately) learned in the readout layer or it might remain close to $\\mathsf { L C N } _ { n }$ . ",
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+    {
+        "type": "text",
+        "text": "2.2 Gradient Descent Training ",
+        "text_level": 1,
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+        "text": "143 We use $f$ to denote any functions defined by the architectures above and $\\theta$ to denote the collection   \n144 of all parameters. Denote by $\\theta _ { t }$ the time-dependence of the parameters and by $\\theta _ { 0 }$ their initial   \n145 values. We use $f _ { t } ( x ) \\equiv f ( \\dot { x } , \\theta _ { t } ) \\in \\mathbb R ^ { k }$ to denote the output (or logits) of the neural network at   \n146 time $t$ . Let $\\ell ( \\hat { y } , y ) : \\mathbb { R } ^ { k } \\times \\mathbb { R } ^ { k } \\to \\mathbb { R }$ denote the loss function where the first/second argument is   \n147 the prediction/true label. By applying continuous time gradient descent to minimize the objective   \n148 $\\begin{array} { r } { \\mathcal { L } = \\sum _ { ( x , y ) \\in \\mathcal { D } } \\ell ( f _ { t } ( x , \\theta ) , y ) } \\end{array}$ , the evolution of the parameters $\\theta$ and the logits $f$ can be written as ",
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+        "type": "equation",
+        "img_path": "images/12e219680be71a639c03a76355e11849d0b74d62552f2b7ba49f37c276d7cee7.jpg",
+        "text": "$$\n\\begin{array} { r } { \\dot { \\theta } _ { t } = - \\nabla _ { \\theta } f _ { t } ( \\mathcal { X } _ { T } ) ^ { T } \\nabla _ { f _ { t } ( \\mathcal { X } _ { T } ) } \\mathcal { L } , \\qquad \\dot { f } _ { t } ( \\mathcal { X } _ { T } ) = \\nabla _ { \\theta } f _ { t } ( \\mathcal { X } _ { T } ) \\dot { \\theta } _ { t } = - \\hat { \\Theta } _ { t } ( \\mathcal { X } _ { T } , \\mathcal { X } _ { T } ) \\nabla _ { f _ { t } ( \\mathcal { X } _ { T } ) } \\mathcal { L } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "149 where $f _ { t } ( \\mathcal { X } _ { T } ) = \\operatorname { v e c } \\left( [ f _ { t } \\left( x \\right) ] _ { x \\in \\mathcal { X } _ { T } } \\right)$ , the $k | \\mathcal { D } | \\times 1$ vector of concatenated logits for all examples, and   \n150 $\\nabla _ { f _ { t } ( \\mathcal { X } _ { T } ) } \\mathcal { L }$ is the gradient of the loss with respect to the model’s output, $f _ { t } ( \\mathcal { X } _ { T } )$ . $\\hat { \\Theta } _ { t } \\equiv \\hat { \\Theta } _ { t } ( \\mathcal { X } _ { T } , \\mathcal { X } _ { T } )$   \n151 is the tangent kernel at time $t$ , which is a $k | \\mathcal { D } | \\times k | \\mathcal { D } |$ kernel matrix ",
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+        "img_path": "images/8c6664f7d10902795be22c2281fa4cccd50503afe62f54cd19c792a42c323144.jpg",
+        "text": "$$\n\\hat { \\Theta } _ { t } = \\nabla _ { \\theta } f _ { t } ( \\mathcal { X } _ { T } ) \\nabla _ { \\theta } f _ { t } { ( \\mathcal { X } _ { T } ) } ^ { T }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "152 One can define the tangent kernel for general arguments, e.g. $\\hat { \\Theta } _ { t } ( x , \\mathcal { X } _ { T } )$ where $x$ is test input. At   \n153 finite-width, $\\hat { \\Theta }$ will depend on the specific random draw of the parameters and evolve with time. As   \n154 such, for a test point $x$ the prediction $f _ { t } ( x )$ depends on the random initalization and is also stochastic.   \n155 Note that the parameters are initialized randomly and the randomness will be carried out through the   \n156 training procedure. As a consequence, the prediction functions are stochastic. ",
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+        "type": "text",
+        "text": "57 2.3 Infinite Network: Gaussian Processes and the Neural Tangent Kernels ",
+        "text_level": 1,
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+        "text": "158 Neural Networks as Gaussian Processes (NNGP). As the width $n  \\infty$ , at initialization the   \n159 output $f _ { 0 } ( \\mathcal { X } )$ forms a Gaussian Process $f _ { 0 } ( \\mathcal { X } ) \\sim \\mathcal { G P } ( 0 , \\mathcal { K } ( \\mathcal { X } , \\mathcal { X } ) )$ , known as the NNGP [8, 20, 21].   \n160 Here $\\kappa$ is the GP kernel and can be computed in closed form for a variety of architectures. By treating   \n161 this infinite width network as a Bayesian model (aka Bayesian Neural Networks) and applying   \n162 Bayesian inference, the posterior is also a GP ",
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+        "img_path": "images/80442fb2d7bc1b42c25ae839d4e8d019700140631077ed59b7e2e5c6335eee64.jpg",
+        "text": "$$\n\\mathcal { N } \\left( \\mathcal { K } ( \\mathcal { X } _ { * } , \\mathcal { X } _ { T } ) \\mathcal { K } ^ { - 1 } ( \\mathcal { X } _ { T } , \\mathcal { X } _ { T } ) \\mathcal { Y } _ { T } , \\mathcal { K } ( \\mathcal { X } _ { * } , \\mathcal { X } _ { * } ) - \\mathcal { K } ( \\mathcal { X } _ { * } , \\mathcal { X } ) \\mathcal { K } ( \\mathcal { X } , \\mathcal { X } ) ^ { - 1 } \\mathcal { K } ( \\mathcal { X } _ { * } , \\mathcal { X } ) ^ { T } \\right)\n$$",
+        "text_format": "latex",
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+        "text": "163 Neural Tangent Kernelss(NTK). Recent advance in global convergence theory of over  \n164 parameterized networks [22–25, 12] has shown that under certain assumptions, the tangent kernels is   \n165 almost stationary over the course of training and is concentrated on its infinite width limit $\\Theta$ in the   \n166 sense there is a constant $C$ independent of $t$ and the network’s width $n$ such that ",
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+        "img_path": "images/bb42f17fd1d1f61b22881994484a29faea2e2fe0b6f55c11cb1e12d9ec5ed340.jpg",
+        "text": "$$\n\\operatorname* { s u p } _ { t \\geq 0 } \\| \\hat { \\Theta } _ { t } ( \\mathcal { X } _ { T } , \\mathcal { X } _ { T } ) - \\Theta ( \\mathcal { X } _ { T } , \\mathcal { X } _ { T } ) \\| _ { F } + \\| \\hat { \\Theta } _ { t } ( \\mathcal { X } _ { T } , \\mathcal { X } _ { * } ) - \\Theta ( \\mathcal { X } _ { T } , \\mathcal { X } _ { * } ) \\| _ { F } \\leq \\frac { C } { \\sqrt { n } } .\n$$",
+        "text_format": "latex",
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+        "text": "167 where is the infinite width limit of $\\Theta$ at initialization, whose existence has been proved in [22, 26].   \n168 As such, when the loss is the mean squared error (MSE), the mean prediction (marginarized over   \n169 random initialization) has the following closed form ",
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+        "text": "$$\n\\begin{array} { r } { f ( \\mathcal { X } _ { * } ) = \\Theta \\left( \\mathcal { X } _ { * } , \\mathcal { X } _ { T } \\right) \\Theta ^ { - 1 } ( \\mathcal { X } _ { T } , \\mathcal { X } _ { T } ) \\left( I - e ^ { - \\eta \\Theta ( \\mathcal { X } _ { T } , \\mathcal { X } _ { T } ) t } \\right) \\mathcal { V } , } \\end{array}\n$$",
+        "text_format": "latex",
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+        "text": "170 Letting $t \\to \\infty$ , the above solution is the same as that of the kernel ridgeless regression using the   \n171 infinite width tangent kernel $\\Theta$ . We use $\\mathsf { F C N } _ { \\infty } ( x ) , \\mathsf { L C N } _ { \\infty } ( x ) , \\mathsf { V E C } _ { \\infty } ( \\bar { x } )$ and ${ \\mathsf { G A P } } _ { \\infty } ( x )$ to denote   \n172 the infinite width solutions (either the GP inference or the NTK regression) for the corresponding   \n173 architectures, where we have suppressed the dependence on the training data $( \\mathcal { X } _ { T } , \\mathcal { Y } _ { T } )$ .   \n175 Symmetry is fundamental in physical systems. So is it in machine learning systems. We explore   \n176 symmetries of various machine learning systems in this section. Given $\\mathcal { D } = ( \\mathcal { X } , \\mathcal { Y } )$ and a transforma  \n177 tion on the input space $\\tau : \\mathbb { R } ^ { 3 d }  \\mathbb { R } ^ { 3 d }$ , we set $\\boldsymbol { \\tau } ( \\mathcal { D } ) = ( \\boldsymbol { \\tau } ( \\boldsymbol { \\mathcal { X } } ) , \\boldsymbol { \\mathcal { Y } } )$ . Let ${ \\mathrm { O } } ( 3 d )$ denote the orthogonal   \n178 group on the flatten input space $\\mathbb { R } ^ { 3 d }$ . The subgroup $0 ( 3 ) ^ { d } \\leq 0 ( 3 d )$ operates on the un-flattened   \n179 input $( \\mathbb { R } ^ { d } ) ^ { 3 }$ , whose element rotates each pixel $x _ { \\alpha } \\in \\mathbb { R } ^ { 3 }$ by an independent element $\\tau _ { \\alpha } \\in \\mathbf { O } ( 3 )$ . The   \n180 smaller subgroup ${ \\mathbf O } ( 3 ) \\otimes { \\mathbf I } _ { d } \\le { \\mathbf O } ( 3 ) ^ { d }$ applies the shared rotation (i.e. $\\tau _ { \\alpha } = \\tau$ to all $x _ { \\alpha }$ for $\\alpha \\in [ d ] )$ .   \n181 We use $ { \\mathbf { P } } ( 3 d )$ to denote the permutation group on $\\mathbb { R } ^ { 3 d }$ and $\\mathsf { P } ( 3 ) ^ { d }$ and $\\mathbf { P ( 3 ) } \\otimes \\mathbf { I } _ { d }$ are defined similarly.   \n182 Note that rotating $\\mathcal { X }$ by $\\tau$ is equivalent to transfer the original coordinate system by the adjoint   \n183 tranformation $\\tau ^ { * } = \\tau ^ { - 1 }$ .   \n184 For a deterministic (stochastic) learning algorithm $\\mathbf { \\mathcal { A } } = \\left( \\mathcal { M } , \\mathcal { T } \\right)$ , we use $\\boldsymbol { \\mathcal { A } } ( \\mathcal { D } _ { T } )$ to denote the learned   \n185 function (distribution of the learned functions) using training set $\\mathcal { D } _ { T }$ . We use $\\mathcal { A } ^ { \\tau } ( \\mathcal { D } _ { T } )$ to denote   \n186 the learned function(s) using $\\tau ( \\mathcal { D } _ { T } )$ and makes prediction on the transformed test point $\\tau ( \\mathcal { X } _ { * } )$ . In   \n187 another word, the learning algorithm is conducted in the input space whose coordinate system is   \n188 transformed by τ −1. ",
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+        "text": "Definition 1. Let $\\mathcal { G }$ be a group of transformations $\\mathbb { R } ^ { 3 d } \\to \\mathbb { R } ^ { 3 d }$ . We say a deterministic (stochastic) learning algorithm $\\mathbf { \\mathcal { A } } = \\left( \\mathcal { M } , \\mathcal { T } \\right)$ is $g$ -invariant if $\\mathcal { A } = \\mathcal { A } ^ { g }$ $\\boldsymbol { \\mathcal { A } } = ^ { d } \\_ { A } \\boldsymbol { g }$ ). In this case, we say the system $( \\mathcal { D } , \\mathcal { M } , \\mathcal { T } )$ is $g$ -invariant and use the notation $( \\mathcal { D } , \\mathcal { M } , \\mathcal { Z } ) = ( g \\mathcal { D } , \\mathcal { M } , \\mathcal { Z } )$ . If this holds for all $g \\in { \\mathcal { G } }$ , then we say the algorithm and the system are $\\mathcal { G }$ -invariant. ",
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+        "text": "193 If $( { \\mathcal { M } } , { \\mathcal { T } } )$ is the algorithm of minimum norm linear regressor, then $( \\mathcal { D } , \\mathcal { M } , \\mathcal { T } )$ is $\\mathrm { O } ( 3 ) ^ { d }$ -invariant;   \n194 see Sec.G for more details. Note that the symmetry (invariance) in our definition is a property of a   \n195 system and is different from the notion of symmetry that are commonly used in the machine learning   \n196 community, which is a property of a function (e.g. translation invariance). ",
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+        "text": "197 Theorem 3.1 (Sec.C). If the parameters of the networks are initialized with iid $\\mathcal { N } ( 0 , 1 )$ , then • $\\mathsf { F C N } _ { n / \\infty }$ are ${ \\bf O } ( 3 d )$ -invariant. 200 • ${ \\mathsf { V E C } } _ { n }$ is $\\mathbf { O ( 3 ) } \\otimes \\mathbf { I } _ { d }$ -invariant and ${ \\mathsf { V E C } } _ { \\infty }$ 201 is $O ( 3 ) ^ { d }$ -invariant. • $\\mathsf { L C N } _ { n / \\infty }$ are O(3)d-invariant. 202 • ${ \\mathsf { G A P } } _ { n / \\infty }$ are ${ \\mathbf { O } } ( 3 ) \\otimes { \\mathbf { I } } _ { d }$ -invariant. ",
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+        "text": "203 The ${ \\bf O } ( 3 d )$ -invariant of $\\mathsf { F C N } _ { \\infty }$ is because the NTK/NNGP kernel is an inner product kernel, namely,   \n204 there is a function $k$ such that the kernels have the form $k ( \\langle x , x ^ { \\prime } \\rangle )$ . The ${ \\bf O } ( 3 d )$ -invariant of finite   \n205 width $\\mathsf { F C N } _ { n }$ is due to the Gaussian initialization of the first layer which was first observed and   \n206 proved in [27]. Rotating the input by $\\tau \\in \\mathrm { O } ( 3 d )$ is equivalent to rotating the weight matrix $\\omega$ of   \n207 the first layer by $\\tau ^ { * }$ . Since for $\\omega \\in \\mathcal { N } ( 0 , 1 ) ^ { 3 d } \\tau ^ { * } \\omega = ^ { d } \\omega$ , at random initialization, the distribution   \n208 of the output functions (the prior) are unchanged if all inputs are rotated by the same element in   \n209 ${ \\mathrm { O } } ( 3 d )$ . This property continues to hold throughout the course of (continue/discrete) gradient descent   \n210 training with/without $L ^ { 2 }$ -regularization and Bayesian posterior inference. For the same reason, $\\mathsf { L C N } _ { n }$   \n211 is $O ( 3 ) ^ { d }$ -invariant because each patch of the image uses independent Gaussian random variables.   \n212 However, weight-sharing in ${ \\mathsf { V E C } } _ { n }$ and ${ \\mathsf { G A P } } _ { n }$ breaks the $\\mathrm { O } ( 3 ) ^ { d }$ symmetry, reducing it to ${ \\bf O ( 3 ) } \\otimes { \\bf I } _ { d }$ . ",
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+        "text": "13 For infinite networks, $\\mathsf { L C N } _ { \\infty } = \\mathsf { V E C } _ { \\infty }$ [28–31]. The kernels of ${ \\mathsf { V E C } } _ { \\infty }$ and $\\mathsf { G A P } _ { \\infty }$ are of the forms ",
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+        "type": "equation",
+        "img_path": "images/699ab7899146406a7828c24c348c706ce089f4e33af8474a8e6dd821712d4c07.jpg",
+        "text": "$$\n\\Theta _ { \\mathsf { V E C } } ( x , x ^ { \\prime } ) = k \\bigl ( \\{ \\langle x _ { \\alpha } , x _ { \\alpha } ^ { \\prime } \\rangle \\} _ { \\alpha \\in [ d ] } \\bigr ) \\quad \\mathrm { a n d } \\quad \\Theta _ { \\mathsf { G A P } } ( x , x ^ { \\prime } ) = k \\bigl ( \\{ \\langle x _ { \\alpha } , x _ { \\alpha ^ { \\prime } } ^ { \\prime } \\rangle \\} _ { \\alpha , \\alpha ^ { \\prime } \\in [ d ] } \\bigr ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "214 resp. The former depends only on the inner product between pixels in the same spatial location,   \n215 breaking the ${ \\mathrm { O } } ( 3 d )$ symmetry and reducing it to $O ( 3 ) ^ { d }$ . In addition, the latter depends also on the   \n216 inner products of pixels across different spatial locations due to pooling, which breaks the $O ( 3 ) ^ { d }$   \n217 symmetry and reduces it to ${ \\mathbf { O } } ( 3 ) \\otimes { \\mathbf { I } } _ { d }$ .   \n218 Note that $\\dim ( \\mathrm { O } ( 3 d ) ) = 3 d ( 3 d - 1 ) / 2$ , $\\dim ( \\mathbf { O } ( 3 ) ^ { d } ) = 3 d$ and $\\mathrm { d i m } ( \\mathrm { O } ( 3 ) \\otimes \\mathbf { I } _ { d } ) = 3$ . $\\mathsf { L C N } _ { n } / \\mathsf { V E C } _ { \\infty }$   \n219 dramatically reduces the dimension of the symmetry group. It is worth mentioning that while   \n220 $\\mathrm { d i m } ( \\mathrm { O } ( 3 d ) )$ many pairs of rotated and unrotated images are needed to recover the exact rotation in   \n221 ${ \\mathrm { O } } ( 3 d )$ , only 3 pairs are sufficient for $O ( 3 ) ^ { d }$ , same as that of ${ \\mathbf { O } } ( 3 ) \\otimes { \\mathbf { I } } _ { d }$ . The results of the paper   \n222 are presented in the most vanilla setting. Our methods can easily extend to more complicated   \n223 architectures like ResNet[32], MLP-Mixer[33] and etc. The symmetry groups of such systems   \n224 need to be computed in a case-by-case manner by identifying the invariant group of the random   \n225 initialization and training procedures. ",
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+            "Figure 1: Performance vs Symmetry. Machine learning systems are equipped with various kinds of symmetries. Transforming the system by the associated symmetry does not affect the performance of the system. However, injecting spurious symmetries beyond the associated symmetries could dramatically degrade their performance for both finite and infinite networks. "
+        ],
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+            "Figure 2: Even in the $\\mathrm { N N } +$ setting, ${ \\mathsf { V E C } } _ { n }$ is closer to ${ \\mathsf { G A P } } _ { n }$ for small $n$ and moves towards $\\mathsf { V E C } _ { \\infty }$ with more symmetries and/or larger $n$ and accuracy drops. "
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+        "type": "text",
+        "text": "226 3.1 Empirical Supports and Observations ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Performance under Rotations. We examinate the performance of: FCN, VEC, LCN, GAP and $\\mathsf { L A P ^ { 4 / 8 } }$ , when the coordinates of the data are transformed by six different groups ( $x$ -axis in Fig.1) using the standard dataset CIFAR-10. , Here $\\mathsf { L A P ^ { 4 / 8 } }$ is the same as GAP except the readout layer is replaced by the Local Average Pooling with window size $4 \\times 4 / 8 \\times 8$ . We consider 4 types of training methods: (1) NTK, i.e. infinite networks (2)NN, our baseline for finite width neural network which is trained with momemtum using a small learning rate and without $L ^ { 2 }$ regularizer and the network is centered $( + \\mathsf C )$ to reduce the variance from random initialization $( 3 ) \\mathrm { N N + = N N + L R + L 2 - C }$ , i.e. using a larger learning rate $( + \\mathsf { L R } )$ , adding $L ^ { 2 }$ regularization $( + \\mathsf { L } 2 ) ]$ and removing the centering $( - C )$ (4) $\\mathsf { N N + + = N N + + D A }$ , adding MixUp[34] data augmentation $( + \\mathsf { D A } )$ to $\\mathrm { N N } +$ . Overall, we observe that, for most of the cases in $\\mathrm { N T K / N N / N N } +$ , adding spurious symmetry to a system $( \\mathcal { D } , \\mathcal { M } , \\mathcal { Z } )$ degrades the performance towards that of the system invariant to that symmetry. Surprisingly, in the baseline NN, performance of $\\mathsf { V E C } _ { n } + \\mathsf { O } ( 3 ) \\otimes \\mathbf { I } _ { d }$ rotation is slightly worse than that of ${ \\mathsf { V E C } } _ { n } + \\mathbf { O } ( 3 ) ^ { d }$ and than that of $\\mathsf { L C N } _ { n }$ , indicating that the system with $\\mathcal { M } = \\mathsf { V E C } _ { n }$ is likely operating closely on the $\\mathrm { O } ( 3 ) ^ { d }$ symmetry. The interventions $- C + \\mathsf { L } 2 + \\mathsf { L } \\mathsf { R }$ in $\\mathrm { N N } +$ distinguishes the performance of $\\mathsf { V E C } _ { n } + \\mathsf { O } ( 3 ) \\otimes \\mathbf { I } _ { d }$ from ${ \\mathsf { V E C } } _ { n } + \\mathbf { O } ( 3 ) ^ { d }$ and $+ \\mathsf { D } \\mathsf { A }$ eventually closes the performance gap between $\\mathsf { V E C } _ { n } + \\mathsf { O } ( 3 ) \\otimes \\mathbf { I } _ { d }$ and ${ \\mathsf { G A P } } _ { n } + { \\mathsf { O } } ( 3 ) \\otimes \\mathbf { I } _ { d }$ , helping the system to be aware of the smaller symmetry ${ \\mathbf { O } } ( 3 ) \\otimes { \\mathbf { I } } _ { d }$ , escaping from the $\\mathrm { O } ( 3 ) ^ { d }$ symmetry. ",
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+        "type": "text",
+        "text": "244 Symmetry Breaking of ${ \\mathsf { V E C } } _ { n }$ . Assuming Equation 8, namely, the network is in the NTK regime, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/1f716e69428967e89b4cd8b29836cfb3eeac476439aba983c9dbfc535aff851f.jpg",
+        "text": "$$\n\\operatorname* { l i m } _ { n \\to \\infty } | \\mathbb { E } \\mathsf { V } \\mathsf { E } \\mathsf { C } _ { n } ( x ) - \\mathsf { V } \\mathsf { E } \\mathsf { C } _ { \\infty } ( x ) | + \\operatorname* { l i m } _ { n \\to \\infty } | \\mathbb { E } \\mathsf { V } \\mathsf { E } \\mathsf { C } _ { n } ( x ) - \\mathbb { E } \\mathsf { V } \\mathsf { E } \\mathsf { C } _ { n } ^ { \\tau } ( x ) | \\leq C n ^ { - \\frac { 1 } { 2 } }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "245 where the expectation $\\mathbb { E }$ is over random initialization and ${ \\mathsf { V E C } } _ { n } ( x )$ is the prediction of the test point   \n246 $x$ when $t = \\infty$ , i.e. training loss is $0 . \\mathsf { V E C } _ { n } ^ { \\tau }$ is the prediction of the $\\tau$ -rotated system, $\\tau \\in { \\mathrm { O } } ( 3 ) ^ { d }$ .   \n247 The $O ( 3 ) ^ { d }$ symmetry is restored as $n \\to \\infty$ . As such, for large $n$ , the system is approximately $\\mathrm { O } ( 3 ) ^ { d }$ -   \n248 invariant. We randomly sample $\\tau \\in { \\mathrm { O } } ( 3 ) ^ { d }$ and use the exponential map to construct a continuous   \n249 interpolation $\\tau _ { t } \\in \\mathbf { O } ( 3 ) ^ { d }$ with $\\tau _ { 0 } = \\mathbf { I d }$ and $\\tau _ { 1 } = \\tau$ . We train the network as in $\\Nu \\mathrm { N + + } ( + \\mathsf { L R } + \\mathsf { L } 2 - \\mathsf { C } )$   \n250 using different $n$ and $\\tau _ { t }$ and average the predictions over 10 random initialization as an approximation   \n251 of $\\bar { \\mathbb { E } } \\bar { \\mathsf { V } } \\bar { \\mathsf { E } } \\mathsf { C } _ { n } ^ { \\tau _ { t } } ( x )$ . Not surprisingly, as $n$ increases and/or $t$ increases, (1) test performance decays   \n252 monotonically (left panel in Fig.2), (2) the distance to $\\mathbb { E } G \\mathsf { A P } _ { n }$ increases monotonically (middle   \n253 panel) and (3) distance to ${ \\mathsf { V E C } } _ { \\infty }$ decrease monotonically (right panel). Clearly, the coordinate   \n254 information from the data is utilized by smaller width ${ \\mathsf { V E C } } _ { n }$ .   \n255 DIDE for ${ \\mathsf { V E C } } _ { n }$ . To understand the role of data, we vary the training set size of Cifar10 from about   \n256 $2 ^ { 6 }$ to $5 0 \\mathrm { k }$ (the whole un-augmented training set) and to $1 0 0 \\mathrm { k }$ (adding left-right flip augmentation) and   \n257 plot the learning curves in Fig.3. We observe dramatic speedup of learning for ${ \\mathsf { V E C } } _ { n }$ in the larger   \n258 data set regime, which isn’t the case for ${ \\mathsf { V E C } } _ { \\infty }$ (kernel), $\\mathsf { L C N } _ { n }$ , $\\mathsf { G A P } _ { \\infty }$ and even for ${ \\mathsf { G A P } } _ { n }$ after   \n259 $m = 2 ^ { 1 2 }$ . We argue that this is due to the prior (the function space defined by the model) is too large   \n260 (and not optimal) for the task and the coupled effect of more data together with inference procedures   \n261 corrects the prior, as it is suggested by Theorem 2.1. ",
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+        "type": "image",
+        "img_path": "images/619a9e725b2cd9e8a196f30b6c9e3ebd296f7f511e5e5af86bf84b2a003236a6.jpg",
+        "image_caption": [
+            "Figure 3: Data Bends Learning Curve of ${ \\mathsf { V E C } } _ { n }$ . We study the effect of training set size to the network’s performance for various models. In the small dataset regime, the slope of the learning curve (in the log-log plot) of ${ \\mathsf { V E C } } _ { n }$ is similar to that of $\\mathsf { V E C } _ { \\infty }$ and $\\mathsf { F C N } _ { n }$ . However, as the dataset gets larger, the slope increases significantly. This is hinted by Theorem 2.1. ",
+            "Figure 4: With coordinate of the input data rotated by $O ( 3 ) ^ { d }$ , state of the art models learn as good as without rotation. middle/right: slopes of the learning curves increases due to more data. DIDE "
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+        "type": "text",
+        "text": "DIDE for SOTA models. In the middle and right panels of Fig.S2, we provide additional evidence in a larger scale setting. We generate learning curves of ImageNet using ResNet50 and MLP-Mixer, a very recent architecture that contains no convolution layers except the first layer, which is a convolution with filter size and stride equal to (16, 16) (patches are disjoint). The symmetry group associated to ResNet is similar to that of ${ \\mathsf { G A P } } _ { n }$ which is relatively small. However, the symmetry group induced by the first layer of the Mixer is $\\mathbf { O } ( 3 \\times 1 6 ^ { 2 } ) \\otimes \\mathbf { I } _ { 1 4 ^ { 2 } }$ , where $\\mathrm { 3 \\times 1 6 ^ { 2 } }$ is number of entries in the $( 1 6 , 1 6 , 3 )$ patch (RGB channels) and $1 4 ^ { 2 } = 2 2 4 ^ { 2 } / 1 6 ^ { 2 }$ is the number of patches. Although the dimension of ${ \\mathrm { O } } ( 3 \\times 1 6 ^ { 2 } ) \\otimes { \\mathbf { I } } _ { 1 4 ^ { 2 } }$ is quite large (about $( 3 \\times 1 6 ^ { 2 } ) ^ { 2 } / 2 $ ), it is still dramatically smaller than that of applying a fully-connected layer to the flatten images, which ${ \\mathrm { O } } ( 3 \\times 2 2 4 ^ { 2 } )$ (about $( 3 \\times 2 2 4 ^ { 2 } ) ^ { 2 } / 2 )$ . In the middle panel of Fig.S2, we observe an almost perfect power-law scaling for the learning curve for the ResNet50 system with unrotated images. When the images are rotated by $O ( 3 ) ^ { d }$ $( d \\bar { = } 2 2 4 ^ { 2 }$ ), the learning curve is relatively flat in the smaller data regime (green dashed line). However, the data set grows, it eventually catches up (purple dashed line) as that of the unrotated setting; see Sec.E for ResNet34/101. In the third panel, we see the learning curves are much flatter (red) for the Mixer and even more so for the rotated images (green). Again, these curves are bent towards that of ResNet50 with unrorated images as data increases, indicating the prior was being corrected. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finally, in the left panel of Fig.S2, we compare the accuracy of state-of-the-art models trained on both unrotated and $O ( 3 { \\bar { ) } } ^ { d }$ rotated images. Surprisingly, the gap between the two are not large and becomes smaller for better performant models. For EfficientNet B7 1, the top-1 accuracy of the rotated system is only $1 . 2 \\%$ off from the unroated one. ",
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+    {
+        "type": "text",
+        "text": "4 Eigenecomposition of Neural Kernels ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To get insights into the inductive biases, we eigendecompose the kernels using spherical harmonics. We assume the input space ${ \\mathcal { X } } \\ = \\ \\{ \\xi \\ = \\ ( \\xi _ { 0 } , \\ldots , \\cdot \\xi _ { p - 1 } ) \\in \\ ( { \\sqrt { d _ { 0 } } } \\mathbb { S } ^ { ( d _ { 0 } - { \\bar { 1 } } ) } ) ^ { p } \\} \\ \\subseteq \\ \\mathbb { R } ^ { d _ { 0 } p } ,$ , i.e. ",
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+    {
+        "type": "text",
+        "text": "286 the $p$ -product of $( d _ { 0 } - 1 )$ -sphere with radius $\\sqrt { d _ { 0 } }$ . We call $\\xi _ { i } \\in \\sqrt { d _ { 0 } } \\mathbb { S } ^ { ( d _ { 0 } - 1 ) }$ a mini-patch and   \n287 $( \\xi _ { i } , \\xi _ { i + 1 } , \\ldots , \\xi _ { i + s - 1 } ) \\in ( \\sqrt { d _ { 0 } } \\mathbb { S } ^ { ( d _ { 0 } - 1 ) } ) ^ { s } \\}$ a patch for $i \\in [ p ]$ , where circular boundary condition is   \n288 assumed. We consider the asymptotic limit when $d _ { 0 } = \\bar { d ^ { \\alpha } } \\bar { , } p = d ^ { 1 - \\alpha }$ and $d = p d _ { 0 }  \\infty$ and treat   \n289 $0 \\textless \\alpha \\textless 1$ and $s$ as fixed constant. The input space $\\mathcal { X }$ is associated with the product measure√   \n290 $\\mu \\equiv \\sigma _ { d _ { 0 } } ^ { p }$ , where $\\sigma _ { d _ { 0 } }$ is the normalized uniform measure on $\\sqrt { d _ { 0 } } \\mathbb { S } ^ { ( d _ { 0 } - 1 ) }$ . The kernels associated to   \n291 the one-hidden layer infinite networks (either NNGP or NTK) has the following general forms ",
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+    {
+        "type": "equation",
+        "img_path": "images/1804821073042fe86fe806d309d801b286f8e0de100c088bdcd974499aa1add5.jpg",
+        "text": "$$\nk \\left( \\frac { 1 } { p } \\sum _ { i \\in [ p ] } \\xi _ { i } ^ { T } \\eta _ { i } / d _ { 0 } \\right) \\quad \\frac { 1 } { p } \\sum _ { i \\in [ p ] } k \\left( \\frac { 1 } { s } \\sum _ { b \\in [ s ] } \\xi _ { i + b } ^ { T } \\eta _ { i + b } / d _ { 0 } \\right) \\quad \\frac { 1 } { p ^ { 2 } } \\sum _ { i , j \\in [ p ] } k \\left( \\frac { 1 } { s } \\sum _ { b \\in [ s ] } \\xi _ { i + b } ^ { T } \\eta _ { j + b } / d _ { 0 } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "292 although that exact form of the (positive definite) kernel function $k : \\mathbb { R }  \\mathbb { R }$ depends on the kernel   \n293 types (NNGP vs NTK), activations, hyperparameters and etc. We assume the kernel is sufficiently   \n294 smooth in $( - 1 , 1 )$ and the Tayor expansion of $k ^ { ( r ) }$ converges uniformly in $[ - 1 , 1 ]$ for sufficiently   \n295 many $r \\in \\mathbb N$ . We use the notation that $A \\sim B$ if there are positive constants $c$ and $C$ such that   \n296 $c A \\leq B \\leq C A$ for $d$ sufficiently large. We use $\\kappa$ to represent any kernels above and consider it as a   \n297 Hilbert–Schmidt operator on $L ^ { \\tilde { 2 } } ( \\chi , \\breve { \\mu } )$ ",
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+        "img_path": "images/318ca75b8da6d9f146f2857a09ada0b63f69cfc47a75148c8ab2ffa428439704.jpg",
+        "text": "$$\n\\mathcal K f ( \\xi ) = \\int _ { \\mathcal K } { \\mathcal K } ( \\xi , \\eta ) f ( \\eta ) d \\mu , \\quad f \\in L ^ { 2 } ( \\mathcal { X } , \\mu ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "which is well-defined since 298 $\\mu$ is a probability measure and $k$ is bounded. Let $\\vec { r } = ( r _ { 0 } , \\ldots , r _ { p - 1 } ) \\in \\mathbb { N } ^ { p }$ , 299 $\\tau$ the shifting operator $\\tau \\vec { r } = \\left( r _ { p - 1 } , r _ { 0 } , \\ldots , r _ { p - 2 } \\right)$ . The $s$ -banded subset of $\\mathbb { N } ^ { p }$ is defined to be ",
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+        "text": "$$\nB ( \\mathbb { N } ^ { p } , s ) = \\{ { \\vec { r } } \\in \\mathbb { N } ^ { p } : \\operatorname { d i s t } ( \\operatorname { a r g m a x } _ { j } r _ { j } \\neq 0 , \\operatorname { a r g m i n } _ { j } r _ { j } \\neq 0 ) \\leq s - 1 \\}\n$$",
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+        "type": "text",
+        "text": "300 which is a quantifier used to restrict the support of a function on a patch. Here $\\mathrm { d i s t } ( i , j ) = \\operatorname* { m i n } \\{ | i -$   \n301 $j | , p - | i - \\bar { j } | \\}$ , a distance defined on the cyclic group $[ p ] = \\mathbb { Z } / p \\bar { \\mathbb { Z } }$ . The quotient space $B ( \\mathbb { N } ^ { p } , s ) / \\tau$   \n302 denotes a subset of $B ( \\mathbb { N } ^ { p } , s )$ by identifying $\\vec { v } = \\vec { v } ^ { \\prime }$ as the same element if ${ \\vec { v } } = \\tau ^ { a } { \\vec { r } } ^ { \\prime }$ for some $a \\in [ p ]$   \n303 Finally, $Y _ { r _ { j } , l _ { j } } ( \\xi _ { j } )$ is used to denote the $l _ { j }$ -th spherical harmonic of degree $r _ { j }$ in the unit sphere   \n304 $\\mathbb { S } ^ { ( d _ { 0 } - 1 ) }$ and has unit norm under the normalized measure on $\\mathbb { S } ^ { ( d _ { 0 } - 1 ) }$ . As such $Y _ { r _ { j } , l _ { j } } ( \\xi _ { j } / \\sqrt { d _ { 0 } } ) \\in$   \n305 $L ^ { 2 } ( \\sqrt { d _ { 0 } } \\mathbb { S } ^ { ( d _ { 0 } - 1 ) } , \\sigma _ { d _ { 0 } } )$ has unit norm. Recall that the total number of spherical harmonic of degree   \n306 $r _ { j }$ in $\\mathbb { S } ^ { ( d _ { 0 } - 1 ) }$ is $\\begin{array} { r } { N ( \\bar { d } _ { 0 } , r _ { j } ) = ( 2 r _ { j } + d _ { 0 } - 2 ) \\binom { d _ { 0 } + r _ { j } - 3 } { r _ { j } - 1 } \\sim d _ { 0 } ^ { r _ { j } } / r _ { j } ! } \\end{array}$ as $d _ { 0 } \\to \\infty$ . We use $N ( d _ { 0 } , \\vec { r } ) =$   \n307 $\\textstyle \\prod _ { j \\in [ p ] } N ( d _ { 0 } , r _ { j } )$ and $\\begin{array} { r } { [ N ( d _ { 0 } , \\vec { r } ) ] = \\prod _ { j \\in [ p ] } [ N ( \\vec { d _ { 0 } } , r _ { j } ) ] } \\end{array}$ , resp. Let ",
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+        "img_path": "images/8ca01ae6cc8976882c04fb7a82caf21a15d42c4d8df5d9f8664c114de296322d.jpg",
+        "text": "$$\n\\vec { Y } _ { \\vec { r } , \\vec { l } } ( \\xi ) = \\prod _ { j \\in [ p ] } Y _ { r _ { j } , l _ { j } } ( \\xi _ { j } )\n$$",
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+        "type": "text",
+        "text": "308 The following theorem shows that locality $\\mathrm { ( V E C _ { \\infty , } }$ ) dramatically reduces both the dimensions of   \n309 Eigendecomposition $r \\geq 1$ eigenspaces and the spectral gap between them. In addition, pooling   \n310 (i.e. translation symmetry of $\\mathsf { G A P } _ { n } \\mathrm { ~ . ~ }$ ) reduces their dimensions by a factor of $p$ . See Sec.E for the   \n311 implication of this theorem to learning. ",
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+        "type": "text",
+        "text": "312 Theorem 4.1. [Sec.D] We have the following eigendecomposition for the integral operator $\\kappa$ ",
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+        "img_path": "images/5346f5d54d0047c781071d54314fecfe2a34e9eb7408ab0174913831d096c797.jpg",
+        "text": "$$\n\\mathsf { H } = \\bigcup _ { r \\in \\mathbb { N } } \\mathsf { H } ^ { ( r ) } = \\bigcup _ { r \\in \\mathbb { N } } \\bigcup _ { \\vec { r } \\in Q ( \\mathcal { K } , r ) } \\mathsf { H } ^ { ( \\vec { r } ) } ,\n$$",
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+        "type": "text",
+        "text": "where 313 $Q ( K , r )$ is a quantifier defined below. If ${ \\mathfrak { r } } = 0 , \\mathsf { H } ^ { ( 0 ) }$ is the space of constant functions and the 314 eigenvalue is $\\sim k ( 0 )$ . For $r \\geq 1$ , ",
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+        "type": "text",
+        "text": "315 ",
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+        "type": "text",
+        "text": "1. $i f \\mathcal { K } = \\mathcal { K } _ { \\sf F C N }$ , then $Q ( \\mathcal { K } , r ) = \\{ \\vec { r } \\in \\mathbb { N } ^ { p } : | \\vec { r } | = r \\}$ and the unit eigenfunctions are ",
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+        "type": "equation",
+        "img_path": "images/506571688ba60a5f32c6dc232c42d93e3303d36816cc973cb3e5d6f7e3225f34.jpg",
+        "text": "$$\n\\{ \\mathsf { H } ^ { ( \\vec { r } ) } = \\mathrm { s p a n } \\{ Y _ { \\vec { r } , \\vec { l } } \\} _ { \\vec { l } \\in [ B ( d _ { 0 } , \\vec { r } ) ] }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "316 ",
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+        "type": "text",
+        "text": "2. $i f K = \\mathcal { K } _ { \\mathsf { V E C } }$ , ${ \\cal Q } ( { \\cal K } , r ) = \\{ \\vec { r } \\in { \\cal B } ( \\mathbb { N } ^ { p } , s ) : | \\vec { r } | = r \\}$ the unit eigenfunctions are ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/5f53190e452683130b08e6a18aeb73f7d5f4312ac938db3464e95a31a1440287.jpg",
+        "text": "$$\n\\left\\{ \\begin{array} { l l } { { \\displaystyle { \\sf H } _ { \\sf V E C } ^ { ( \\vec { r } ) } = \\mathrm { s p a n } \\left\\{ Y _ { \\vec { r } , \\vec { l } } \\right\\} _ { \\vec { l } \\in [ B ( d _ { 0 } , \\vec { r } ) ] } } _ { a \\mathrm { r } } } \\\\ { { \\mathrm { d i m } ( { \\sf H } _ { \\sf V E C } ^ { ( \\vec { r } ) } ) \\sim p ( s d _ { 0 } ) ^ { r } = s ^ { r } d ^ { 1 - \\alpha + r \\alpha } } } & { { a n d \\quad \\lambda ( { \\sf H } _ { \\sf V E C } ^ { ( \\vec { r } ) } ) \\sim p ^ { - 1 } ( s d _ { 0 } ) ^ { - r } \\delta ( k ^ { ( r ) } ( { 0 } ) ) } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+        "type": "image",
+        "img_path": "images/f333434682c669b1e856367975aa3b0a045c441b329742338d697f73dd0ac551.jpg",
+        "image_caption": [
+            "Figure 5: Eigenvalue Decay of Relu NTK of $\\mathsf { F C N } _ { \\infty }$ , ${ \\mathsf { V E C } } _ { \\infty }$ and $\\mathsf { G A P } _ { \\infty }$ . $d _ { 0 } = s = 3$ . The eigenvalues of $\\mathsf { G A P } _ { \\infty }$ decays faster because with $m = 1 5 k$ many samples, higher order eigenspace can be covered by $\\mathsf { G A P } _ { \\infty }$ but not $\\mathsf { F C N } _ { \\infty } / \\mathsf { V E C } _ { \\infty }$ due to Theorem 4.1. "
+        ],
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+    {
+        "type": "text",
+        "text": "317   \n318 ",
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+    },
+    {
+        "type": "text",
+        "text": "3. and finally, $i f K = \\mathcal { K } _ { \\mathsf { G A P } }$ , then $Q ( \\mathcal { K } , r ) = \\{ \\vec { r } \\in B ( \\mathbb { N } ^ { p } , s ) / \\tau : | \\vec { r } | = r \\}$ , the unit eigenfunctions are ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0e560d7c521bfd03ab1cd997e8d51fede895072042efc884a523caa0a9542373.jpg",
+        "text": "$$\n\\left\\{ \\begin{array} { l } { { \\sf H } _ { \\sf G A P } ^ { ( \\vec { r } ) } = \\mathrm { s p a n } \\left\\{ \\frac { 1 } { \\sqrt { p } } \\sum _ { \\tau \\in [ p ] } Y _ { \\vec { r } , \\vec { l } } ( \\tau \\xi ) \\right\\} _ { \\vec { l } \\in [ B ( d _ { 0 } , \\vec { r } ) ] } } \\\\ { \\dim ( { \\sf H } _ { \\sf G A P } ^ { ( r ) } ) \\sim ( s d _ { 0 } ) ^ { r } = s ^ { r } d ^ { r \\alpha } \\quad a n d \\quad \\lambda ( { \\sf H } _ { \\sf G A P } ^ { ( \\vec { r } ) } ) \\sim p ^ { - 1 } ( s d _ { 0 } ) ^ { - r } \\delta ( k ^ { ( r ) } ( 0 ) ) } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "319 5 Related Work ",
+        "text_level": 1,
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+        ],
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+    {
+        "type": "text",
+        "text": "The study of infinite networks dates back to seminal work by Neal [8] who showed the convergence of single hidden-layer networks to Gaussian Processes (GPs). Recently, there has been renewed interest in studying random, infinite, networks starting with concurrent work on “conjugate kernels” [10, 35] and “mean-field theory” [9, 36], taking a statistical learning and statistical physics view of points, resp. Since then this analysis has been extended to include a wide range for architectures [20, 21, 37, 29, 26, 38]. The inducing kernel is often referred to as the Neural Network Gaussian Process (NNGP) kernel. The neural tangent kernel (NTK), first introduced in Jacot et al. [22], along with followup work [12, 39] showed that the distribution of functions induced by gradient descent for infinite-width networks is a Gaussian Process with NTK as the kernel. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "329 The study of implicit bias (regularization) of gradient descent has received considerable interests.   \n330 The work [15, 40–43] demonstrate the convergence of SGD to the maximal margin solution for   \n331 logistic-type of losses during late time training. [44–50] study the early-time SGD dynamics, spectral   \n332 biases of neural networks. These results aim to explain the order of learning of neural networks:   \n333 functions of less complexity are usually learned before more complex functions.   \n334 [27] is the first to show that the prediction functions obtained from training FCN depend, in addition   \n335 on the labels, only on the covariance of the input data. This implies our result regarding the ${ \\mathrm { O } } ( 3 d )$   \n336 invariance of FCN. By utilizing this symmetry, recent work [51] constructs a particular task where   \n337 the label function is a second order polynomial of the inputs and show that orthogonal invariance   \n338 algorithm requires sample size of order $\\dot { d } ^ { 2 }$ while there is a convnet requires only $O ( 1 )$ samples. Their   \n339 convnet essentially corresponds to the $d _ { 0 } = s = 1$ and $r = 2$ case of Theorem 4.1, in which the   \n340 dimension of this eigenspace (and indeed of all $r$ -eigenspace by treating $r$ as a finite constant as   \n341 $d \\to \\infty$ ) of $\\mathsf { G A P } _ { \\infty }$ is $O ( \\bar { 1 } )$ while the dimension of the 2-eigenspace of $\\mathsf { F C N } _ { \\infty }$ is of order $d ^ { 2 }$ . ",
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+    {
+        "type": "text",
+        "text": "342 6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    },
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+        "type": "text",
+        "text": "43 In this paper, we consider machine learning methods as an integrated system of data, models and   \n44 inference algorithms and study the basic symmetries of various machine learning systems. We surface   \n45 the importance of locality in modern machine learning systems through large scale empirical study   \n46 and through an eigendecomposition of one-layer infinite networks. However, we haven’t addressed   \n47 the two import questions (1) theoretical characterization of the effect of composing locality and (2)   \n48 the mathematical understanding of DIDE and how the prior is corrected by the coupled effect of data   \n49 and gradient descent. We leave them to future work. ",
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+        "text": "The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example: ",
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+        "text": "• Did you include the license to the code and datasets? [Yes] See Section ??.   \n• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   \n• Did you include the license to the code and datasets? [N/A] ",
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+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No]   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [No]   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No] ",
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+        "text": "(a) If your work uses existing assets, did you cite the creators? [TODO]   \n(b) Did you mention the license of the assets? [TODO]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [TODO]   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [TODO]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [TODO] ",
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+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
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+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [TODO]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [TODO]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [TODO]   \nReferences   \n[1] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pages 1097–1105, 2012.   \n[2] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. arXiv preprint arXiv:1706.03762, 2017.   \n[3] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018.   \n[4] David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. Nature, 529(7587):484–489, 2016.   \n[5] Andrew W Senior, Richard Evans, John Jumper, James Kirkpatrick, Laurent Sifre, Tim Green, Chongli Qin, Augustin Žídek, Alexander WR Nelson, Alex Bridgland, et al. Improved protein structure prediction using potentials from deep learning. Nature, 577(7792):706–710, 2020.   \n[6] Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. Scaling laws for neural language models. arXiv preprint arXiv:2001.08361, 2020.   \n[7] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.   \n[8] Radford M. Neal. Priors for infinite networks (tech. rep. no. crg-tr-94-1). University of Toronto, 1994.   \n[9] Ben Poole, Subhaneil Lahiri, Maithra Raghu, Jascha Sohl-Dickstein, and Surya Ganguli. Exponential expressivity in deep neural networks through transient chaos. In Advances In Neural Information Processing Systems, 2016.   \n[10] Amit Daniely, Roy Frostig, and Yoram Singer. Toward deeper understanding of neural networks: The power of initialization and a dual view on expressivity. In Advances In Neural Information Processing Systems, 2016.   \n[11] Arthur Jacot, Franck Gabriel, and Clément Hongler. Neural tangent kernel: Convergence and generalization in neural networks. arXiv preprint arXiv:1806.07572, 2018.   \n[12] Jaehoon Lee, Lechao Xiao, Samuel S. Schoenholz, Yasaman Bahri, Roman Novak, Jascha Sohl-Dickstein, and Jeffrey Pennington. 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parse/train/aM7UsuOAzB3/aM7UsuOAzB3.md

@@ -0,0 +1,283 @@
+# Editing a Classifier by Rewriting Its Prediction Rules
+
+Shibani Santurkar∗ MIT shibani@mit.edu
+
+Dimitris Tsipras∗ MIT tsipras@mit.edu
+
+Mahalaxmi Elango MIT melango@mit.edu
+
+David Bau MIT davidbau@mit.edu
+
+Antonio Torralba MIT torralba@mit.edu
+
+Aleksander M ˛adry MIT madry@mit.edu
+
+# Abstract
+
+We present a methodology for modifying the behavior of a classifier by directly rewriting its prediction rules.1 Our approach requires virtually no additional data collection and can be applied to a variety of settings, including adapting a model to new environments, and modifying it to ignore spurious features.
+
+# 1 Introduction
+
+At the core of machine learning is the ability to automatically discover prediction rules from raw data. However, there is mounting evidence that not all of these rules are reliable [Torralba and Efros, 2011, Beery et al., 2018, Shetty et al., 2019, Agarwal et al., 2020, Xiao et al., 2020, Bissoto et al., 2020, Geirhos et al., 2020]. In particular, some rules could be based on biases in the training data: e.g., learning to associate cows with grass since they are typically depicted on pastures [Beery et al., 2018]. While such prediction rules may be useful in some scenarios, they will be irrelevant or misleading in others. This raises the question:
+
+How can we most effectively modify the way in which a given model makes its predictions?
+
+The canonical approach for performing such post hoc modifications is to intervene at the data level. For example, by gathering additional data that better reflects the real world (e.g., images of cows on the beach) and then using it to further train the model. Unfortunately, collecting such data can be challenging: how do we get cows to pose for us in a variety of environments? Furthermore, data collection is ultimately a very indirect way of specifying the intended model behavior. After all, even when data has been carefully curated to reflect a given real-world task, models still end up learning unintended prediction rules from it [Ponce et al., 2006, Torralba and Efros, 2011, Tsipras et al., 2020, Beyer et al., 2020].
+
+# Our contributions
+
+The goal of our work is to develop a toolkit that enables users to directly modify the prediction rules learned by an (image) classifier, as opposed to doing so implicitly via the data. Concretely:
+
+Editing prediction rules. We build on the recent work of Bau et al. [2020a] to develop a method for modifying a classifier’s prediction rules with essentially no additional data collection (Section 2). At a high level, our method enables the user to modify the weight of a layer so that the latent representations of a specific concept (e.g., snow) map to the representations of another (e.g., road). Crucially, this allows us to change the behavior of the classifier on all occurrences of that concept, beyond the specific examples (and the corresponding classes) used in the editing process.
+
+![](images/40a9b7c01376c758599bce5b69b31b05f386e5aa395df2fdce7fd4a688ea5ce8.jpg)  
+Figure 1: Editing prediction rules in pre-trained classifiers using a single exemplar. (a) We edit a VGG16 ImageNet classifier to map the representation of the concept “snow” to that of “asphalt road”. (b) This edit corrects classification errors on snowy scenes corresponding to various classes. (c) We edit a CLIP [Radford et al., 2021] model such that the text “iPod” maps to a blank area. (d) This change makes the model robust to the typographic attacks from Goh et al. [2021].
+
+Real-world scenarios. We demonstrate our approach in two scenarios motivated by real-world applications (Section 3). First, we focus on adapting an ImageNet classifier to a new environment: recognizing vehicles on snowy roads. Second, we consider the recent “typographic attack” of Goh et al. [2021] on a zero-shot CLIP [Radford et al., 2021] classifier: attaching a piece of paper with “iPod” written on it to various household items causes them to be incorrectly classified as “iPod.” In both settings, we find that our approach enables us to significantly improve model performance, using only a single, synthetic example to perform the edit—cf. Figure 1.
+
+Large-scale synthetic evaluation. To evaluate our method at scale, we develop an automated pipeline to generate a suite of varied test cases (Section 4). Our pipeline revolves around identifying specific concepts (e.g., “road" or “pasture”) in an existing dataset using pre-trained instance segmentation models and then modifying them using style transfer [Gatys et al., 2016] (e.g., to create “snowy road”). We find that our editing methodology is able to consistently correct a significant fraction of model failures induced by these transformations. In contrast, standard fine-tuning approaches are unable to do so given the same data, often causing more errors than they are fixing.
+
+Probing model behavior with counterfactuals. Beyond model editing, our concepttransformation pipeline can also be viewed as a scalable way of generating image counterfactuals. In Section 5, we demonstrate how such counterfactuals can be useful to gain insights into how a given model makes its predictions and pinpoint certain spurious correlations that it has picked up.
+
+# 2 A toolkit for editing prediction rules
+
+It has been widely observed that models pick up various context-specific correlations in the data— e.g., using the presence of “road” or a “wheel” to predict “car” (cf. Section 5). Such unreliable prediction rules (dependencies of predictions on specific input concepts) could hinder models when they encounter novel environments (e.g., snow-covered roads), and confusing or adversarial test conditions (e.g., cars with wooden wheels). Thus, a model designer might want to modify these rules before deploying their model.
+
+The canonical approach to modify a classifier post hoc is to collect additional data that captures the desired deployment scenario, and use it to retrain the model. However, even setting aside the challenges of data collection, it is not obvious a priori how much of an effect such retraining (e.g., via fine-tuning) will have. For instance, if we fine-tune our model on “cars” with wooden wheels, will it now recognize “scooters” or “trucks” with such wheels?
+
+The goal of this work is to instead develop a more direct way to modify a model’s behavior: rewriting its prediction rules in a targeted manner. For instance, in our previous example, we would ideally be able to modify the classifier to correctly recognize all vehicles with wooden wheels by simply teaching it to treat any wooden wheel as it would a standard one. Our approach is able to do exactly this. However, before describing this approach (Section 2.2), we first provide a brief overview of recent work by Bau et al. [2020a] which forms its basis.
+
+# 2.1 Background: Rewriting generative models
+
+Bau et al. [2020a] developed an approach for rewriting a deep generative model: specifically, enabling a user to replace all occurrences of one selected object (say, “dome”) in the generated images with another (say, “tree”), without changing the model’s behavior in other contexts. Their approach is motivated by the observation that, using a handful of example images, we can identify a vector in the model’s representation space that encodes a specific high-level concept [Kim et al., 2018, Bau et al., 2020a]. Building on this, Bau et al. [2020a] treat each layer of the model as an associative memory, which maps such a concept vector at each spatial location in its input (which we will refer to as the $k e y$ ) to another concept vector in its output (which we will call the value).
+
+In the simplest case, one can think of a linear layer with weights $W \in \mathbb { R } ^ { m x n }$ transforming the key $k \in \mathbb { R } ^ { n }$ to the value $v \in \mathbb { R } ^ { m }$ . In this setting, Bau et al. [2020a] formulate the rewrite operation as modifying the layer weights from $W$ to $W ^ { \prime }$ so that $v ^ { * } = W ^ { \prime } k ^ { * }$ , where $k ^ { * }$ corresponds to the old concept that we want to replace, and $v ^ { * }$ to the new concept. For instance, to replace “domes” with “trees” in the generated images, we would modify the layer so that the key $k ^ { * }$ for “dome” maps to the value $v ^ { * }$ for “tree”. Consequently, when this value is fed into the downstream layers of the network it would result in a tree in the final image. Crucially, this update should change the model’s behavior for every instance of the concept encoded in $k ^ { * }$ —i.e., all “domes” in the images should now be “trees”.
+
+To extend this approach to typical deep generative models, two challenges remain: (1) handling non-linear layers, and (2) ensuring that the edit doesn’t significantly hurt model behavior on other concepts. With these considerations in mind, Bau et al. [2020a] propose making the following rankone updates to the parameters $W$ of an arbitrary non-linear layer $f$ :
+
+$$
+\operatorname* { m i n } _ { \Lambda } \quad \sum _ { ( i , j ) \in { \cal S } } \big \| v _ { i j } ^ { * } - f ( k _ { i j } ^ { * } ; W ^ { \prime } ) \big \| \qquad \mathrm { s . t . } \quad W ^ { \prime } = W + \Lambda ( C ^ { - 1 } d ) ^ { \top } .
+$$
+
+Here, $S$ denotes the set of spatial locations in representation space for a single image corresponding to the concept of interest, $d$ is the top eigenvector of the keys $k _ { i j } ^ { * }$ at locations $( i , j ) \in S$ and $C =$ $\Sigma _ { d } k _ { d } k _ { d } ^ { \top }$ captures the second-order statistics for other keys $k _ { d }$ . Intuitively, the goal of this update is to modify the layer parameters to rewrite the desired key-value mapping in the most minimal way. We refer the reader to Bau et al. [2020a] for further details.
+
+# 2.2 Editing classifiers
+
+We now shift our attention to the focus of this work: editing classifiers. To describe our approach, we use the task of enabling classifiers to detect vehicles with “wooden wheels” as a running example. At a high level, we would like to apply the approach described in Section 2.1 to modify a chosen (potentially non-linear) layer $L$ of the network to rewrite the relevant key-value association. But, we need to first determine what these relevant keys and values are.
+
+Let us start with a single image $x$ , say, from class “car”, that contains the concept “wheel”. Let the location of the “wheel” in the image be denoted by a binary mask $m$ .2 Then, say we have access to a transformed version of $x$ —namely, $x ^ { \prime }$ —where the “car” has a “wooden wheel” (cf. Figure 2). This $x ^ { \prime }$ could be created by manually replacing the wheel, or by applying an automated procedure such as that in Section 4. In the rest of our study, we refer to a single $( x , x ^ { \prime } )$ pair as an exemplar.
+
+![](images/859c4c5ebcb8ee4c4952b8ac6842c0961b883f19fb9810e7ddc40c1bb4abc2a9.jpg)  
+Figure 2: Overview of our pipeline for directly editing the prediction-rules of a classifier. The edit in (a) seeks to modify the network to perceive wooden wheels as standard ones, using a small set of exemplar images (say from class “car”). To achieve this, we first obtain the keys $k _ { i j } ^ { * }$ corresponding to the new concept (here, “wooden wheel”), and the values $v _ { i j } ^ { * }$ corresponding to the original concept (here, “standard wheel”) in the input and output representation space of a layer $L$ respectively. We then update the weights $W$ of the layer to enforce this new key-value association (1). (b) To test our method, we measure the improvement in model performance on test instances containing the new concept—here, images of other vehicles with “wooden wheels”.
+
+Intuitively, we want the classifier to perceive the “wooden wheel” in the transformed image $x ^ { \prime }$ as it does the standard wheel in the original image $x$ . To achieve this, we must map the keys for wooden wheels to the value corresponding to their standard counterparts. In other words, the relevant keys $( k ^ { * } )$ correspond to the network’s representation of the concept in the transformed image $( x ^ { \prime } )$ directly before layer $L$ . Similarly, the relevant values $( v ^ { * } )$ that we want to map these keys to correspond to the network’s representation of the concept in the original images $( x )$ directly after layer $L$ . (The pertinent spatial regions in the representation space are simply determined by downsampling the mask to the appropriate dimensions.) Finally, the model edit is performed by feeding the resulting key-value pairs into the optimization problem (1) to determine the updated layer weights $W ^ { \prime }$ —cf. Figure 2 for an illustration of the overall process. Note that this approach can be easily extended to use multiple exemplars $( x _ { k } , x _ { k } ^ { \prime } )$ by simply expanding $S$ to include the union of relevant spatial locations (corresponding the concept of interest) across these exemplars.
+
+# 3 Does editing work in practice?
+
+To evaluate of our approach, we start by considering two scenarios motivated by real-world concerns: (i) adapting classifiers to handle novel weather conditions, and (ii) making models robust to typographic attacks Goh et al. [2021]. In both cases, we edit the model using a single exemplar, i.e., a single image that we manually annotate and modify. For comparison, we also consider two variants of fine-tuning using the same exemplar: (i) local fine-tuning, where we only train the weights of a single layer $L$ (similar to our editing approach); and (ii) global fine-tuning, where we also train all other layers between layer $L$ and the output of the model. It is worth noting that unlike fine-tuning, editing does not utilize class labels in any way. See Appendix A for experimental details.
+
+# 3.1 Tackling new environments: Vehicles on snow
+
+Our first use-case is adapting pre-trained classifiers to image subpopulations that are underrepresented in the training data. Specifically, we focus on the task of recognizing vehicles under heavy snow conditions—a setting that could be pertinent to self-driving cars—using a VGG16 classifier trained on ImageNet-1k. To study this problem, we collect a set of real photographs from road-related ImageNet classes using Flickr (details in Appendix A.5). We then rewrite the model’s prediction rules to map “snowy roads” to “road”. To do so, we create an exemplar by manually annotating the concept “road” in an ImageNet image from a different class (here, “police van”), and the manually replace it with snow texture obtained from Flickr. We then apply our editing methodology (cf. Section 2), using this single synthetic snow-to-road exemplar—see Figure 1.
+
+![](images/84f573b8d73f7e737b5dd5b6caace454b14d577f328e6d69b17308090244b78d.jpg)  
+Figure 3: (a) Adapting a pre-trained ImageNet VGG16 classifier to images of vehicles on snowy roads with a single exemplar. Fine-tuning (both local and global) does not improve accuracy, while editing to map “snowy road” “road” leads to a consistent improvement across multiple classes. (b) Improving the robustness of CLIP-ResNet-50 [Radford et al., 2021] models to typographic attacks [Goh et al., 2021]. Editing the model to map the text “iPod” $ ^ { \prime }$ “blank” using a single exemplar— either based on hand-written text on a physical teapot or from pasting typed text on an image of a “can opener”—completely corrects this vulnerability. While global fine-tuning can also improve model performance in this setting, it requires more careful hyperparameter tuning and typically hurts model performance in other contexts (Appendix Figure 9). Here, hyperparameters (cf. Appendix Table 2) are chosen based on the large-scale synthetic study in Section 4.1.
+
+In Figure 3a, we measure the error rate of the model on the new test set (vehicles in snow) before and after performing the rewrite. We find that our edits significantly improve the model’s error rate on these images, despite the fact that we only use a single synthetic exemplar (i.e., not a real “snowy road” photograph). Moreover, Figure 3a demonstrates that our method indeed changes the way that the model processes a concept (here “snow”) in a way that generalizes beyond the specific class used during editing (here, the exemplar was a “police van”) . In contrast, fine-tuning the model under the same setup does not improve its performance on these inputs.
+
+One potential concern is the impact of this process on the model’s accuracy on other ImageNet classes that contain snow (e.g., “ski”). On the 246 (of 50k) ImageNet test images that contain snow (identified using an MS-COCO-trained instance segmentation model [Chen et al., 2017]), the model’s accuracy pre-edit is $9 2 . 2 7 \%$ and post-edit is $9 1 . 0 5 \%$ —i.e., only 3/246 images are rendered incorrect by the edit. This indicates that the classifier is not disproportionately affected by the edit.
+
+# 3.2 Ignoring a spurious feature: Typographic attacks
+
+Our second use-case is modifying a model to ignore a spurious feature. We focus on the recentlydiscovered typographic attacks from Goh et al. [2021]: simply attaching a piece of paper with the text “iPod” on it is enough to make a zero-shot CLIP [Radford et al., 2021] classifier incorrectly classify an assortment of objects to be iPods. We reproduce these attacks on the ResNet-50 variant of the model—see Appendix Figure 7 for an illustration.
+
+To correct this behavior, we rewrite the model’s prediction rules to map the text “iPod” to “blank”. For the choice of our transformed input $x ^ { \prime }$ , we consider two variants: either a real photograph of a “teapot” with the typographic attack (Appendix Figure 7); or an ImageNet image of a “can opener” (randomly-chosen) with the typed text “iPod” pasted on it (Figure 1). The original image $x$ for our approach is obtained by replacing the handwritten/typed text with a white mask—cf. Figure 1. We then use this single training exemplar to perform the model edit.
+
+In both cases, we find that editing is able to fix all the errors caused by the typographic attacks, see Figure 3b. Interestingly, global fine-tuning also helps to correct many of these errors (potentially by adjusting class biases), albeit less reliably (for specific hyperparameters). However, unlike editing, fine-tuning also ends up damaging the model behavior in other scenarios—e.g., the model now spuriously associates the text “iPod” with the target class used for fine-tuning and/or has lower accuracy on normal “iPod” images from the test set (Appendix Figure 9).
+
+![](images/f8bd97b2557b2a74e1e52d0062bd969ee11c81341da24f89522fca48549f12e4.jpg)  
+Figure 4: Creating large-scale test sets for model rewriting. Given a standard dataset, we first identify salient concepts within the corresponding images using instance segmentation, and then apply a realistic transformation to each of these concepts using style transfer [Gatys et al., 2016]. We can then evaluate the effectiveness of a model rewriting technique based on the extent to which it can alleviate the model’s sensitivity (i.e., drop in accuracy) to such concept-level transformations.
+
+# 4 Large-scale synthetic evaluation
+
+The analysis of the previous section demonstrates that editing can improve model performance in realistic settings. However, due to the practical constraints of real-world data collection, this analysis was restricted to a relatively small test set. To corroborate the generality of our approach, we now develop a pipeline to automatically construct diverse rule-editing test cases. We then perform a large-scale evaluation and ablation of our editing method on this testbed.
+
+# 4.1 Synthesizing concept-level transformations
+
+At a high level, our goal is to automatically create a test set(s) in which a specific concept— possibly relevant to the detection of multiple dataset classes—undergoes a realistic transformation. To achieve this without additional data collection, we transform all instances of the concept of interest within existing datasets. For example, the “vehicles-on-snow” scenario of Section 3.1 can be synthetically reproduced by identifying the images within a standard dataset (say ImageNet [Deng et al., 2009, Russakovsky et al., 2015]) that contain segments of road and transforming these to make them resemble snow. Concretely, our pipeline (see Figure 4 for an illustration and Appendix A.6.1 for details), which takes as input an existing dataset, consists of the following two steps:
+
+1. Concept identification: In order to identify concepts within the dataset images in a scalable manner, we leverage pre-trained instance segmentation models. In particular, using state-of-the-art segmentation models—trained on MS-COCO [Lin et al., 2014] and LVIS [Gupta et al., 2019]—we are able to automatically generate concept segmentations for a range of high-level concepts (e.g., “grass”, “sea”, “tree”). 2. Concept transformation: We then transform the detected concept (within dataset images) in a consistent manner using existing methods for style transfer [Gatys et al., 2016, Ghiasi et al., 2017]. This allows us to preserve fine-grained image features and realism, while still exploring a range of potential transformations for a single concept. For our analysis, we manually curate a set of realistic transformations (e.g., “snow” and “graffiti”).
+
+Note that this concept-transformation pipeline does not require any additional training or data annotation. Thus it can be directly applied to new datasets, as long as we have access to a pre-trained segmentation model for the concepts of interest.
+
+# 4.2 Creating a suite of editing tasks
+
+We now utilize the concept transformations described above to create a benchmark for evaluating model rewriting methods. Intuitively, these transformations can capture invariances that the model should ideally have—e.g., recognizing vehicles correctly even when they have wooden wheels. In practice however, model accuracy on one or more classes (e.g., “car”, “scooter”) may degrade under these transformations. The goal of rewriting the model would thus be to fix these failure modes in a data-efficient manner. In this section, we evaluate our editing methodology—as well as the fine-tuning approaches discussed in Section 3—along this axis.
+
+Concretely, we focus on vision classifiers—specifically, VGG [Simonyan and Zisserman, 2015] and ResNet [He et al., 2015] models trained on the ImageNet [Deng et al., 2009, Russakovsky et al., 2015] and Places-365 [Zhou et al., 2017] datasets (cf. Appendix A.2). Each test set is constructed using the concept-transformation pipeline discussed above, based on a chosen concept-style pair (say “wheel”-“wooden”) 3. It consists of $N$ exemplars (pairs of original and transformed images, $( x , x ^ { \prime } ) )$ ) that belong to a single (randomly-chosen) target class in the dataset. All other transformed images containing the concept, including those from classes other than the target one, are used for validation and testing (30-70 split). We create two variants of the test set: one using the same style image as the exemplars (i.e., same wooden texture) for the transformation; and another using held-out style images (i.e., other wooden textures).
+
+To evaluate the impact of a method, we measure the change in model accuracy on the transformed examples (e.g., vehicles with “wooden wheel”s in Figure 2b). If the method is effective, then it should recover some of the incorrect predictions caused by the transformations. We only focus on the subset of examples $D$ that were correctly classified before the transformation, since we cannot expect to correct mistakes that do not stem from the transformation itself. Concretely, we measure the change in the number of mistakes made by the model on the transformed examples:
+
+$$
+\% \mathrm { e r r o r s ~ c o r r e c t e d } : = \frac { N _ { p r e } ( D ) - N _ { p o s t } ( D ) } { N _ { p r e } ( D ) }
+$$
+
+where $N _ { p r e / p o s t } ( D )$ denotes the number of transformed examples misclassified by the model before and after the rewrite, respectively. Note that this metric can range from $100 \%$ when rewriting leads to perfect classification on the transformed examples, to even a negative value when the rewriting process causes more mistakes that it fixes.
+
+In each case, we select the best hyperparameters—including the choice of the layer to modify— based on the validation set performance (cf. Appendix A.6.3). To quantify the effect of the modification on overall model behavior, we also measure the change in its (standard) test set performance. Since we are interested in rewrites that do not significantly hurt the overall model performance, we only consider hyperparameters that do not cause a large accuracy drop $( \leq 0 . 2 5 \% )$ . We found that the exact accuracy threshold did not have significant impact on the results—see Appendix Figures 15-18 for a demonstration of the full accuracy-effectiveness trade-off.
+
+# 4.3 The effectiveness of editing
+
+Recall that a key desideratum of our prediction-rule edits is that they should generalize. That is, if we modify the way that our model treats a specific concept, we want this modification to apply to every occurrence of that concept. For instance, if we edit a model to enforce that “wooden wheels” should be treated the same as regular “wheels” in the context of “car” images, we want the model to do the same when encountering other vehicles with “wooden wheels”. Thus, when analyzing performance in Figure 5, we consider inputs belonging to the class used to perform the edit separately.
+
+Editing. We find that editing is able to consistently correct mistakes in a manner that generalizes across classes. That is, editing is able to reduce errors in non-target classes, often by more than 20 percentage points, even when performed using only three exemplars from the target class. Moreover, this improvement extends to transformations using different variants of the style (e.g., textures of “wood”), other than those present in exemplars used to perform the modification.
+
+In Appendix B.1.3, we conduct ablation studies to get a better sense of the key algorithmic factors driving performance. Notably, we find that imposing the editing constraints (1) on the entirety of the image—as opposed to only focusing on key-value pairs that correspond to the concept of interest as proposed in Bau et al. [2020a]—leads to even better performance (cf. ‘-mask’ in Figure 5). We hypothesize that this has a regularizing effect as it constrains the weights to preserve the original mapping between keys and values in regions that do not contain the concept.
+
+![](images/954830d25274787fb89751f2439e475514c93ed0dd828113f0e3599c6ce2ece8.jpg)  
+(a) ImageNet-trained VGG16 (concepts from COCO)
+
+![](images/b4f310725c708c8ee76c97a86df04c821bc9ed04ef81945ca0c5c12f8ed6f004.jpg)  
+(b) Places-trained ResNet-18 (concepts from LVIS)   
+Figure 5: Editing vs. fine-tuning, averaged over concept-style pairs. We find that both methods (and their variants) are fairly successful at correcting errors on the target class (examples of which are used to perform the rewrite). This holds even when the transformation applied during testing is different from the one present in the train exemplars (e.g., a different texture of “wood”). However, crucially, only the improvements induced by editing generalize to other classes where the transformed concept is present. Fine-tuning fails in this setting—typically, causing more errors than it fixes. See Appendix Figures 10-13 for other experimental settings.
+
+Fine-tuning. Our baseline is the canonical fine-tuning approach, i.e., directly minimizing the model loss on the new data (here the transformed images) with respect to their label. Similar to Section 3, we consider both the local and global variants of fine-tuning. We find that while these approaches are able to correct model errors on transformed inputs from the class used to perform the modification, they typically decrease performance on other classes—i.e., they cause more errors than they fix. Moreover, even when we allow a larger drop in the model’s accuracy, or use more training exemplars, their performance often becomes worse such inputs (Appendix Figures 15-18).
+
+We present examples of errors corrected (or not) by editing and fine-tuning in Appendix Figure 14, and provide a per-concept/style break down in Appendix Figures 19 and 20.
+
+# 5 Beyond editing: Probing model behavior via counterfactuals
+
+In the previous section, we developed a scalable pipeline for creating concept-level transformations which we used to evaluate model rewriting methods. Here, we put forth another related use-case of that pipeline: debugging models to identify their (learned) prediction rules.
+
+In particular, observe that the resulting transformed inputs (cf. Figure 4) can be viewed as counterfactuals—a primitive commonly used in causal inference [Pearl, 2010] and interpretability [Goyal et al., 2019a,b, Bau et al., 2020b]. Counterfactuals can be used to identify the features that a model uses to make its prediction on a given input. For example, to understand whether the model relies on the presence of a “wheel” to recognize a “car” image, we can evaluate how the its prediction changes when just the wheel in the image is transformed. Based on this view, we now repurpose our concept transformations from Section 4 to discover a given classifier’s prediction rules.
+
+The effect of specific concepts. As a point of start, we study how sensitive the model’s prediction is to a given concept in terms of the accuracy drop caused by transformation of said concept. For instance, in Figure 6a, we find that the accuracy of a VGG16 ImageNet classifier drops by $2 5 \%$ on images of “croquet ball” when “grass” is transformed, whereas its accuracy on “collie” does not change. In line with previous studies [Zhang et al., 2007, Ribeiro et al., 2016, Rosenfeld et al., 2018, Barbu et al., 2019, Xiao et al., 2020], we also find that background concepts, such as “grass”, ���sea” and “sand”, have a large effect on model performance. We contrast this measure of influence across concepts for a single model (Appendix Figure 27), and across architectures for a single concept (Appendix B.2). Finally, we examine the effect of each concept stylization in Figure 6b.
+
+Per-class prediction rules. If instead we restrict our attention to a single class, we can pinpoint the set of concepts that the model relies on to detect this class. It turns out that aside from the main image object, ImageNet classifiers also heavily depend on co-occurring objects [Stock and Cisse,
+
+![](images/69b34d793490d5efbe31f255080c02c9e48df62634e74bbcd699266cfedb82b4.jpg)  
+(b) Style impact on accuracy   
+Figure 6: Model sensitivities diagnosed using our pipeline in a VGG16 ImageNet classifier. (a) Classes for which the model relies on “grass”: e.g., a “croquet ball” is not accurately recognized if “grass” is transformed, while “collie”s are not affected. (b) Applying different transformations to visual concepts reduces accuracy by varying amounts. (c) Transformations that hamper accuracy can highlight prediction rules: e.g., the class “groom” is sensitive to the presence of “dress.”
+
+2018, Tsipras et al., 2020, Beyer et al., 2020] in the image—e.g., “dress” for the class “groom”, “person” for the class “tench” (sic), and “road” for the class “race car” (cf. Figure 6c and Appendix Figure 29). We can also examine which styles hurt performance the most—e.g., we find that making “plants” “floral” hurts accuracy on the class “damselfly” $15 \%$ more than making them “snowy”.
+
+Overall, this pipeline provides a scalable path for model designers to analyze the invariances (and sensitivities) of their models with respect to various natural transformations of interest. We thus believe that this primitive holds promise for future interpretability and robustness studies.
+
+# 6 Related work
+
+High-level concepts in latent representations. There has been increasing interest in explaining the inner workings of deep classifiers through high-level concepts: e.g., by identifying individual neurons [Erhan et al., 2009, Zeiler and Fergus, 2014, Olah et al., 2017, Bau et al., 2017, Engstrom et al., 2019a] or activation vectors [Kim et al., 2018, Zhou et al., 2018, Chen et al., 2020] that correspond to human-understandable features. The effect of these features on model predictions can be analyzed by either inspecting the downstream weights of the model [Olah et al., 2018, Wong et al., 2021] or through counterfactual tests: e.g., via synthetic data [Goyal et al., 2019a]; by swapping features between individual images [Goyal et al., 2019b]; or by silencing sets of neurons [Bau et al., 2020b]. A parallel line of work aims to learn models that operate on data representations that explicitly encode high-level concepts: either by learning to predict a set of attributes [Lampert et al., 2009, Koh et al., 2020a] or by learning to segment inputs [Losch et al., 2019]. In our work, we identify concepts by manually selecting the relevant pixels in a handful of images and measure the impact of manipulating these features on model performance on a new test set.
+
+Model interventions. Direct manipulations of latent representations inside generative models have been used to create human-understandable changes in synthesized images [Bau et al., 2019, Jahanian et al., 2019, Goetschalckx et al., 2019, Shen et al., 2020, Härkönen et al., 2020, Wu et al., 2020]. Our work is inspired by that line of work as well as a recent finding that parameters of a generative model can be directly changed to alter generalized behavior [Bau et al., 2020a]. Unlike previous work, we edit classification models, changing rules that govern predictions rather than image synthesis. Concurrently with our work, there has been a series of methods proposed for editing factual knowledge in language models [Mitchell et al., 2021, De Cao et al., 2021, Dai et al., 2021].
+
+Ignoring spurious features. Prior work on preventing models from relying on spurious correlations is based on constraining model predictions to satisfy certain invariances. Examples include: training on counterfactuals (either by adding or removing objects from scenes [Shetty et al., 2018, 2019, Agarwal et al., 2020] or having human annotators edit text input [Kaushik et al., 2019]), learning representations that are simultaneously optimal across domains [Arjovsky et al., 2019], ensuring comparable performance across subpopulations [Sagawa et al., 2020], or enforcing consistency across inputs that depict the same entity [Heinze-Deml and Meinshausen, 2017]. In this work, we focus on a setting where the model designer is aware of undesirable correlations learned by the model and we provide the tools to rewrite them directly.
+
+Model robustness. A long line of work has been devoted to discovering and correcting failure modes of models. These studies focus on simulating variations in testing conditions that can arise during deployment, including: adversarial or natural input corruptions [Szegedy et al., 2014, Fawzi and Frossard, 2015, Fawzi et al., 2016, Engstrom et al., 2019b, Ford et al., 2019, Hendrycks and Dietterich, 2019, Kang et al., 2019], changes in the data collection process [Saenko et al., 2010, Torralba and Efros, 2011, Khosla et al., 2012, Tommasi and Tuytelaars, 2014, Recht et al., 2019], or variations in the data subpopulations present [Beery et al., 2018, Oren et al., 2019, Sagawa et al., 2020, Santurkar et al., 2021, Koh et al., 2020b]. Typical approaches for improving robustness in these contexts include robust optimization [Madry et al., 2018, Yin et al., 2019, Sagawa et al., 2020] and data augmentation schemes [Lopes et al., 2019, Hendrycks et al., 2019, Zhang et al., 2021]. Our rule-discovery and editing pipelines can be viewed as complementary to this work as they allows us to preemptively adjust the model’s prediction rules in anticipation of deployment conditions.
+
+Domain adaptation. The goal of domain adaptation is to adapt a model to a specific deployment environment using (potentially unlabeled) samples from it. This is typically achieved by either fine-tuning the model on the new domain [Donahue et al., 2014, Sharif Razavian et al., 2014, Kumar et al., 2020], learning the correspondence between the source and target domain, often in a latent representation space [Ben-David et al., 2007, Saenko et al., 2010, Ganin and Lempitsky, 2015, Courty et al., 2016, Gong et al., 2016], or updating the model’s batch normalization statistics [Li et al., 2016, Burns and Steinhardt, 2021]. These approaches all require a non-trivial amount of data from the target domain. The question of adaptation from a handful of samples has been explored [Motiian et al., 2017], but in a setting that requires samples across all target classes. In contrast, our method allows for generalization to new (potentially unknown) classes with even a single example.
+
+# 7 Conclusion
+
+We developed a general toolkit for performing targeted post hoc modifications to vision classifiers. Crucially, instead of specifying the desired behavior implicitly via the training data, our method allows users to directly edit the model’s prediction rules. By doing so, our approach makes it easier for users to encode their prior knowledge and preferences during the model debugging process. Additionally, a key benefit of this technique is that it fundamentally changes how the model processes a given concept—thus making it possible to edit its behavior beyond the specific class(es) used for editing. Finally, our edits do not require any additional data collection: they can be guided by as few as a single (synthetically-created) exemplar. We believe that this primitive opens up new avenues to interact with and correct our models before or during deployment.
+
+# Limitations and broader impact.
+
+Even though our methodology provides a general tool for model editing, performing such edits does require manual intervention and domain expertise. After all, the choice of what concept to edit— and its implications on the robustness of the model—lies with the model designer. For instance, in the vehicles-on-snow example, our objective was to have the model recognize any vehicle on snow the same way it would on a regular road—e.g., to adapt a system to different weather conditions. However, if our dataset contains classes for which the presence snow is absolutely essential for recognition, this might not be an appropriate edit to perform.
+
+Moreover, direct model editing is a departure from the standard way in which models are trained, and may have broader implications. While we have shown how it can be used to cause beneficial changes in pre-trained models, direct control of prediction rules could also make it easier for adversaries to introduce vulnerabilities into the model (e.g., by manipulating model behavior on a specific population demographic). Overall, direct model editing makes it clearer than ever that our models are a reflection of the goals and biases of we who create them—not only through the training tasks we choose, but now also through the rules that we rewrite.
+
+# Acknowledgments and Disclosure of Funding
+
+We thank the anonymous reviewers for their helpful comments and feedback.
+
+Work supported in part by the NSF grants CCF-1553428 and CNS-1815221, the DARPA SAILON HR0011-20-C-0022 grant, Open Philanthropy, a Google PhD fellowship, and a Facebook PhD fellowship. This material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) under Contract No. HR001120C0015.
+
+Research was sponsored by the United States Air Force Research Laboratory and the United States Air Force Artificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the United States Air Force or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
+
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+A. Sharif Razavian, H. Azizpour, J. Sullivan, and S. Carlsson. Cnn features off-the-shelf: an astounding baseline for recognition. In conference on computer vision and pattern recognition (CVPR) workshops, 2014.   
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parse/train/aM7UsuOAzB3/aM7UsuOAzB3_content_list.json

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+    {
+        "type": "text",
+        "text": "Editing a Classifier by Rewriting Its Prediction Rules ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Shibani Santurkar∗ MIT shibani@mit.edu ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Dimitris Tsipras∗ MIT tsipras@mit.edu ",
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+    {
+        "type": "text",
+        "text": "Mahalaxmi Elango MIT melango@mit.edu ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "David Bau MIT davidbau@mit.edu ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Antonio Torralba MIT torralba@mit.edu ",
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+    },
+    {
+        "type": "text",
+        "text": "Aleksander M ˛adry MIT madry@mit.edu ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We present a methodology for modifying the behavior of a classifier by directly rewriting its prediction rules.1 Our approach requires virtually no additional data collection and can be applied to a variety of settings, including adapting a model to new environments, and modifying it to ignore spurious features. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "At the core of machine learning is the ability to automatically discover prediction rules from raw data. However, there is mounting evidence that not all of these rules are reliable [Torralba and Efros, 2011, Beery et al., 2018, Shetty et al., 2019, Agarwal et al., 2020, Xiao et al., 2020, Bissoto et al., 2020, Geirhos et al., 2020]. In particular, some rules could be based on biases in the training data: e.g., learning to associate cows with grass since they are typically depicted on pastures [Beery et al., 2018]. While such prediction rules may be useful in some scenarios, they will be irrelevant or misleading in others. This raises the question: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "How can we most effectively modify the way in which a given model makes its predictions? ",
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+    {
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+        "text": "The canonical approach for performing such post hoc modifications is to intervene at the data level. For example, by gathering additional data that better reflects the real world (e.g., images of cows on the beach) and then using it to further train the model. Unfortunately, collecting such data can be challenging: how do we get cows to pose for us in a variety of environments? Furthermore, data collection is ultimately a very indirect way of specifying the intended model behavior. After all, even when data has been carefully curated to reflect a given real-world task, models still end up learning unintended prediction rules from it [Ponce et al., 2006, Torralba and Efros, 2011, Tsipras et al., 2020, Beyer et al., 2020]. ",
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+        "type": "text",
+        "text": "Our contributions ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The goal of our work is to develop a toolkit that enables users to directly modify the prediction rules learned by an (image) classifier, as opposed to doing so implicitly via the data. Concretely: ",
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+    {
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+        "text": "Editing prediction rules. We build on the recent work of Bau et al. [2020a] to develop a method for modifying a classifier’s prediction rules with essentially no additional data collection (Section 2). At a high level, our method enables the user to modify the weight of a layer so that the latent representations of a specific concept (e.g., snow) map to the representations of another (e.g., road). Crucially, this allows us to change the behavior of the classifier on all occurrences of that concept, beyond the specific examples (and the corresponding classes) used in the editing process. ",
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+        "image_caption": [
+            "Figure 1: Editing prediction rules in pre-trained classifiers using a single exemplar. (a) We edit a VGG16 ImageNet classifier to map the representation of the concept “snow” to that of “asphalt road”. (b) This edit corrects classification errors on snowy scenes corresponding to various classes. (c) We edit a CLIP [Radford et al., 2021] model such that the text “iPod” maps to a blank area. (d) This change makes the model robust to the typographic attacks from Goh et al. [2021]. "
+        ],
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+    {
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "Real-world scenarios. We demonstrate our approach in two scenarios motivated by real-world applications (Section 3). First, we focus on adapting an ImageNet classifier to a new environment: recognizing vehicles on snowy roads. Second, we consider the recent “typographic attack” of Goh et al. [2021] on a zero-shot CLIP [Radford et al., 2021] classifier: attaching a piece of paper with “iPod” written on it to various household items causes them to be incorrectly classified as “iPod.” In both settings, we find that our approach enables us to significantly improve model performance, using only a single, synthetic example to perform the edit—cf. Figure 1. ",
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+    {
+        "type": "text",
+        "text": "Large-scale synthetic evaluation. To evaluate our method at scale, we develop an automated pipeline to generate a suite of varied test cases (Section 4). Our pipeline revolves around identifying specific concepts (e.g., “road\" or “pasture”) in an existing dataset using pre-trained instance segmentation models and then modifying them using style transfer [Gatys et al., 2016] (e.g., to create “snowy road”). We find that our editing methodology is able to consistently correct a significant fraction of model failures induced by these transformations. In contrast, standard fine-tuning approaches are unable to do so given the same data, often causing more errors than they are fixing. ",
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+    {
+        "type": "text",
+        "text": "Probing model behavior with counterfactuals. Beyond model editing, our concepttransformation pipeline can also be viewed as a scalable way of generating image counterfactuals. In Section 5, we demonstrate how such counterfactuals can be useful to gain insights into how a given model makes its predictions and pinpoint certain spurious correlations that it has picked up. ",
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+    {
+        "type": "text",
+        "text": "2 A toolkit for editing prediction rules ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "It has been widely observed that models pick up various context-specific correlations in the data— e.g., using the presence of “road” or a “wheel” to predict “car” (cf. Section 5). Such unreliable prediction rules (dependencies of predictions on specific input concepts) could hinder models when they encounter novel environments (e.g., snow-covered roads), and confusing or adversarial test conditions (e.g., cars with wooden wheels). Thus, a model designer might want to modify these rules before deploying their model. ",
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+    {
+        "type": "text",
+        "text": "The canonical approach to modify a classifier post hoc is to collect additional data that captures the desired deployment scenario, and use it to retrain the model. However, even setting aside the challenges of data collection, it is not obvious a priori how much of an effect such retraining (e.g., via fine-tuning) will have. For instance, if we fine-tune our model on “cars” with wooden wheels, will it now recognize “scooters” or “trucks” with such wheels? ",
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+    {
+        "type": "text",
+        "text": "The goal of this work is to instead develop a more direct way to modify a model’s behavior: rewriting its prediction rules in a targeted manner. For instance, in our previous example, we would ideally be able to modify the classifier to correctly recognize all vehicles with wooden wheels by simply teaching it to treat any wooden wheel as it would a standard one. Our approach is able to do exactly this. However, before describing this approach (Section 2.2), we first provide a brief overview of recent work by Bau et al. [2020a] which forms its basis. ",
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+    {
+        "type": "text",
+        "text": "2.1 Background: Rewriting generative models ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Bau et al. [2020a] developed an approach for rewriting a deep generative model: specifically, enabling a user to replace all occurrences of one selected object (say, “dome”) in the generated images with another (say, “tree”), without changing the model’s behavior in other contexts. Their approach is motivated by the observation that, using a handful of example images, we can identify a vector in the model’s representation space that encodes a specific high-level concept [Kim et al., 2018, Bau et al., 2020a]. Building on this, Bau et al. [2020a] treat each layer of the model as an associative memory, which maps such a concept vector at each spatial location in its input (which we will refer to as the $k e y$ ) to another concept vector in its output (which we will call the value). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the simplest case, one can think of a linear layer with weights $W \\in \\mathbb { R } ^ { m x n }$ transforming the key $k \\in \\mathbb { R } ^ { n }$ to the value $v \\in \\mathbb { R } ^ { m }$ . In this setting, Bau et al. [2020a] formulate the rewrite operation as modifying the layer weights from $W$ to $W ^ { \\prime }$ so that $v ^ { * } = W ^ { \\prime } k ^ { * }$ , where $k ^ { * }$ corresponds to the old concept that we want to replace, and $v ^ { * }$ to the new concept. For instance, to replace “domes” with “trees” in the generated images, we would modify the layer so that the key $k ^ { * }$ for “dome” maps to the value $v ^ { * }$ for “tree”. Consequently, when this value is fed into the downstream layers of the network it would result in a tree in the final image. Crucially, this update should change the model’s behavior for every instance of the concept encoded in $k ^ { * }$ —i.e., all “domes” in the images should now be “trees”. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To extend this approach to typical deep generative models, two challenges remain: (1) handling non-linear layers, and (2) ensuring that the edit doesn’t significantly hurt model behavior on other concepts. With these considerations in mind, Bau et al. [2020a] propose making the following rankone updates to the parameters $W$ of an arbitrary non-linear layer $f$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/53a19541cfceaf172bedaeb9b4481504e3904ea3963f9ead4a8a671b9f5bcc20.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\Lambda } \\quad \\sum _ { ( i , j ) \\in { \\cal S } } \\big \\| v _ { i j } ^ { * } - f ( k _ { i j } ^ { * } ; W ^ { \\prime } ) \\big \\| \\qquad \\mathrm { s . t . } \\quad W ^ { \\prime } = W + \\Lambda ( C ^ { - 1 } d ) ^ { \\top } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Here, $S$ denotes the set of spatial locations in representation space for a single image corresponding to the concept of interest, $d$ is the top eigenvector of the keys $k _ { i j } ^ { * }$ at locations $( i , j ) \\in S$ and $C =$ $\\Sigma _ { d } k _ { d } k _ { d } ^ { \\top }$ captures the second-order statistics for other keys $k _ { d }$ . Intuitively, the goal of this update is to modify the layer parameters to rewrite the desired key-value mapping in the most minimal way. We refer the reader to Bau et al. [2020a] for further details. ",
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+        "type": "text",
+        "text": "2.2 Editing classifiers ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We now shift our attention to the focus of this work: editing classifiers. To describe our approach, we use the task of enabling classifiers to detect vehicles with “wooden wheels” as a running example. At a high level, we would like to apply the approach described in Section 2.1 to modify a chosen (potentially non-linear) layer $L$ of the network to rewrite the relevant key-value association. But, we need to first determine what these relevant keys and values are. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let us start with a single image $x$ , say, from class “car”, that contains the concept “wheel”. Let the location of the “wheel” in the image be denoted by a binary mask $m$ .2 Then, say we have access to a transformed version of $x$ —namely, $x ^ { \\prime }$ —where the “car” has a “wooden wheel” (cf. Figure 2). This $x ^ { \\prime }$ could be created by manually replacing the wheel, or by applying an automated procedure such as that in Section 4. In the rest of our study, we refer to a single $( x , x ^ { \\prime } )$ pair as an exemplar. ",
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+    {
+        "type": "image",
+        "img_path": "images/859c4c5ebcb8ee4c4952b8ac6842c0961b883f19fb9810e7ddc40c1bb4abc2a9.jpg",
+        "image_caption": [
+            "Figure 2: Overview of our pipeline for directly editing the prediction-rules of a classifier. The edit in (a) seeks to modify the network to perceive wooden wheels as standard ones, using a small set of exemplar images (say from class “car”). To achieve this, we first obtain the keys $k _ { i j } ^ { * }$ corresponding to the new concept (here, “wooden wheel”), and the values $v _ { i j } ^ { * }$ corresponding to the original concept (here, “standard wheel”) in the input and output representation space of a layer $L$ respectively. We then update the weights $W$ of the layer to enforce this new key-value association (1). (b) To test our method, we measure the improvement in model performance on test instances containing the new concept—here, images of other vehicles with “wooden wheels”. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            214,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, we want the classifier to perceive the “wooden wheel” in the transformed image $x ^ { \\prime }$ as it does the standard wheel in the original image $x$ . To achieve this, we must map the keys for wooden wheels to the value corresponding to their standard counterparts. In other words, the relevant keys $( k ^ { * } )$ correspond to the network’s representation of the concept in the transformed image $( x ^ { \\prime } )$ directly before layer $L$ . Similarly, the relevant values $( v ^ { * } )$ that we want to map these keys to correspond to the network’s representation of the concept in the original images $( x )$ directly after layer $L$ . (The pertinent spatial regions in the representation space are simply determined by downsampling the mask to the appropriate dimensions.) Finally, the model edit is performed by feeding the resulting key-value pairs into the optimization problem (1) to determine the updated layer weights $W ^ { \\prime }$ —cf. Figure 2 for an illustration of the overall process. Note that this approach can be easily extended to use multiple exemplars $( x _ { k } , x _ { k } ^ { \\prime } )$ by simply expanding $S$ to include the union of relevant spatial locations (corresponding the concept of interest) across these exemplars. ",
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+    {
+        "type": "text",
+        "text": "3 Does editing work in practice? ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To evaluate of our approach, we start by considering two scenarios motivated by real-world concerns: (i) adapting classifiers to handle novel weather conditions, and (ii) making models robust to typographic attacks Goh et al. [2021]. In both cases, we edit the model using a single exemplar, i.e., a single image that we manually annotate and modify. For comparison, we also consider two variants of fine-tuning using the same exemplar: (i) local fine-tuning, where we only train the weights of a single layer $L$ (similar to our editing approach); and (ii) global fine-tuning, where we also train all other layers between layer $L$ and the output of the model. It is worth noting that unlike fine-tuning, editing does not utilize class labels in any way. See Appendix A for experimental details. ",
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+    {
+        "type": "text",
+        "text": "3.1 Tackling new environments: Vehicles on snow ",
+        "text_level": 1,
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+        "text": "Our first use-case is adapting pre-trained classifiers to image subpopulations that are underrepresented in the training data. Specifically, we focus on the task of recognizing vehicles under heavy snow conditions—a setting that could be pertinent to self-driving cars—using a VGG16 classifier trained on ImageNet-1k. To study this problem, we collect a set of real photographs from road-related ImageNet classes using Flickr (details in Appendix A.5). We then rewrite the model’s prediction rules to map “snowy roads” to “road”. To do so, we create an exemplar by manually annotating the concept “road” in an ImageNet image from a different class (here, “police van”), and the manually replace it with snow texture obtained from Flickr. We then apply our editing methodology (cf. Section 2), using this single synthetic snow-to-road exemplar—see Figure 1. ",
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+            "Figure 3: (a) Adapting a pre-trained ImageNet VGG16 classifier to images of vehicles on snowy roads with a single exemplar. Fine-tuning (both local and global) does not improve accuracy, while editing to map “snowy road” “road” leads to a consistent improvement across multiple classes. (b) Improving the robustness of CLIP-ResNet-50 [Radford et al., 2021] models to typographic attacks [Goh et al., 2021]. Editing the model to map the text “iPod” $ ^ { \\prime }$ “blank” using a single exemplar— either based on hand-written text on a physical teapot or from pasting typed text on an image of a “can opener”—completely corrects this vulnerability. While global fine-tuning can also improve model performance in this setting, it requires more careful hyperparameter tuning and typically hurts model performance in other contexts (Appendix Figure 9). Here, hyperparameters (cf. Appendix Table 2) are chosen based on the large-scale synthetic study in Section 4.1. "
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+        "text": "In Figure 3a, we measure the error rate of the model on the new test set (vehicles in snow) before and after performing the rewrite. We find that our edits significantly improve the model’s error rate on these images, despite the fact that we only use a single synthetic exemplar (i.e., not a real “snowy road” photograph). Moreover, Figure 3a demonstrates that our method indeed changes the way that the model processes a concept (here “snow”) in a way that generalizes beyond the specific class used during editing (here, the exemplar was a “police van”) . In contrast, fine-tuning the model under the same setup does not improve its performance on these inputs. ",
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+        "text": "One potential concern is the impact of this process on the model’s accuracy on other ImageNet classes that contain snow (e.g., “ski”). On the 246 (of 50k) ImageNet test images that contain snow (identified using an MS-COCO-trained instance segmentation model [Chen et al., 2017]), the model’s accuracy pre-edit is $9 2 . 2 7 \\%$ and post-edit is $9 1 . 0 5 \\%$ —i.e., only 3/246 images are rendered incorrect by the edit. This indicates that the classifier is not disproportionately affected by the edit. ",
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+        "text": "3.2 Ignoring a spurious feature: Typographic attacks ",
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+        "text": "Our second use-case is modifying a model to ignore a spurious feature. We focus on the recentlydiscovered typographic attacks from Goh et al. [2021]: simply attaching a piece of paper with the text “iPod” on it is enough to make a zero-shot CLIP [Radford et al., 2021] classifier incorrectly classify an assortment of objects to be iPods. We reproduce these attacks on the ResNet-50 variant of the model—see Appendix Figure 7 for an illustration. ",
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+        "type": "text",
+        "text": "To correct this behavior, we rewrite the model’s prediction rules to map the text “iPod” to “blank”. For the choice of our transformed input $x ^ { \\prime }$ , we consider two variants: either a real photograph of a “teapot” with the typographic attack (Appendix Figure 7); or an ImageNet image of a “can opener” (randomly-chosen) with the typed text “iPod” pasted on it (Figure 1). The original image $x$ for our approach is obtained by replacing the handwritten/typed text with a white mask—cf. Figure 1. We then use this single training exemplar to perform the model edit. ",
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+        "text": "In both cases, we find that editing is able to fix all the errors caused by the typographic attacks, see Figure 3b. Interestingly, global fine-tuning also helps to correct many of these errors (potentially by adjusting class biases), albeit less reliably (for specific hyperparameters). However, unlike editing, fine-tuning also ends up damaging the model behavior in other scenarios—e.g., the model now spuriously associates the text “iPod” with the target class used for fine-tuning and/or has lower accuracy on normal “iPod” images from the test set (Appendix Figure 9). ",
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+            "Figure 4: Creating large-scale test sets for model rewriting. Given a standard dataset, we first identify salient concepts within the corresponding images using instance segmentation, and then apply a realistic transformation to each of these concepts using style transfer [Gatys et al., 2016]. We can then evaluate the effectiveness of a model rewriting technique based on the extent to which it can alleviate the model’s sensitivity (i.e., drop in accuracy) to such concept-level transformations. "
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+        "text": "4 Large-scale synthetic evaluation ",
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+        "text": "The analysis of the previous section demonstrates that editing can improve model performance in realistic settings. However, due to the practical constraints of real-world data collection, this analysis was restricted to a relatively small test set. To corroborate the generality of our approach, we now develop a pipeline to automatically construct diverse rule-editing test cases. We then perform a large-scale evaluation and ablation of our editing method on this testbed. ",
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+        "text": "4.1 Synthesizing concept-level transformations ",
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+        "text": "At a high level, our goal is to automatically create a test set(s) in which a specific concept— possibly relevant to the detection of multiple dataset classes—undergoes a realistic transformation. To achieve this without additional data collection, we transform all instances of the concept of interest within existing datasets. For example, the “vehicles-on-snow” scenario of Section 3.1 can be synthetically reproduced by identifying the images within a standard dataset (say ImageNet [Deng et al., 2009, Russakovsky et al., 2015]) that contain segments of road and transforming these to make them resemble snow. Concretely, our pipeline (see Figure 4 for an illustration and Appendix A.6.1 for details), which takes as input an existing dataset, consists of the following two steps: ",
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+        "text": "1. Concept identification: In order to identify concepts within the dataset images in a scalable manner, we leverage pre-trained instance segmentation models. In particular, using state-of-the-art segmentation models—trained on MS-COCO [Lin et al., 2014] and LVIS [Gupta et al., 2019]—we are able to automatically generate concept segmentations for a range of high-level concepts (e.g., “grass”, “sea”, “tree”). 2. Concept transformation: We then transform the detected concept (within dataset images) in a consistent manner using existing methods for style transfer [Gatys et al., 2016, Ghiasi et al., 2017]. This allows us to preserve fine-grained image features and realism, while still exploring a range of potential transformations for a single concept. For our analysis, we manually curate a set of realistic transformations (e.g., “snow” and “graffiti”). ",
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+        "text": "Note that this concept-transformation pipeline does not require any additional training or data annotation. Thus it can be directly applied to new datasets, as long as we have access to a pre-trained segmentation model for the concepts of interest. ",
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+        "text": "4.2 Creating a suite of editing tasks ",
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+        "text": "We now utilize the concept transformations described above to create a benchmark for evaluating model rewriting methods. Intuitively, these transformations can capture invariances that the model should ideally have—e.g., recognizing vehicles correctly even when they have wooden wheels. In practice however, model accuracy on one or more classes (e.g., “car”, “scooter”) may degrade under these transformations. The goal of rewriting the model would thus be to fix these failure modes in a data-efficient manner. In this section, we evaluate our editing methodology—as well as the fine-tuning approaches discussed in Section 3—along this axis. ",
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+        "text": "Concretely, we focus on vision classifiers—specifically, VGG [Simonyan and Zisserman, 2015] and ResNet [He et al., 2015] models trained on the ImageNet [Deng et al., 2009, Russakovsky et al., 2015] and Places-365 [Zhou et al., 2017] datasets (cf. Appendix A.2). Each test set is constructed using the concept-transformation pipeline discussed above, based on a chosen concept-style pair (say “wheel”-“wooden”) 3. It consists of $N$ exemplars (pairs of original and transformed images, $( x , x ^ { \\prime } ) )$ ) that belong to a single (randomly-chosen) target class in the dataset. All other transformed images containing the concept, including those from classes other than the target one, are used for validation and testing (30-70 split). We create two variants of the test set: one using the same style image as the exemplars (i.e., same wooden texture) for the transformation; and another using held-out style images (i.e., other wooden textures). ",
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+        "text": "To evaluate the impact of a method, we measure the change in model accuracy on the transformed examples (e.g., vehicles with “wooden wheel”s in Figure 2b). If the method is effective, then it should recover some of the incorrect predictions caused by the transformations. We only focus on the subset of examples $D$ that were correctly classified before the transformation, since we cannot expect to correct mistakes that do not stem from the transformation itself. Concretely, we measure the change in the number of mistakes made by the model on the transformed examples: ",
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+        "img_path": "images/d73f4ace6842218d076e777e3818db56624001d7fccf50438541b66901c79944.jpg",
+        "text": "$$\n\\% \\mathrm { e r r o r s ~ c o r r e c t e d } : = \\frac { N _ { p r e } ( D ) - N _ { p o s t } ( D ) } { N _ { p r e } ( D ) }\n$$",
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+        "text": "where $N _ { p r e / p o s t } ( D )$ denotes the number of transformed examples misclassified by the model before and after the rewrite, respectively. Note that this metric can range from $100 \\%$ when rewriting leads to perfect classification on the transformed examples, to even a negative value when the rewriting process causes more mistakes that it fixes. ",
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+        "text": "In each case, we select the best hyperparameters—including the choice of the layer to modify— based on the validation set performance (cf. Appendix A.6.3). To quantify the effect of the modification on overall model behavior, we also measure the change in its (standard) test set performance. Since we are interested in rewrites that do not significantly hurt the overall model performance, we only consider hyperparameters that do not cause a large accuracy drop $( \\leq 0 . 2 5 \\% )$ . We found that the exact accuracy threshold did not have significant impact on the results—see Appendix Figures 15-18 for a demonstration of the full accuracy-effectiveness trade-off. ",
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+        "text": "4.3 The effectiveness of editing ",
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+        "text": "Recall that a key desideratum of our prediction-rule edits is that they should generalize. That is, if we modify the way that our model treats a specific concept, we want this modification to apply to every occurrence of that concept. For instance, if we edit a model to enforce that “wooden wheels” should be treated the same as regular “wheels” in the context of “car” images, we want the model to do the same when encountering other vehicles with “wooden wheels”. Thus, when analyzing performance in Figure 5, we consider inputs belonging to the class used to perform the edit separately. ",
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+        "type": "text",
+        "text": "Editing. We find that editing is able to consistently correct mistakes in a manner that generalizes across classes. That is, editing is able to reduce errors in non-target classes, often by more than 20 percentage points, even when performed using only three exemplars from the target class. Moreover, this improvement extends to transformations using different variants of the style (e.g., textures of “wood”), other than those present in exemplars used to perform the modification. ",
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+        "type": "text",
+        "text": "In Appendix B.1.3, we conduct ablation studies to get a better sense of the key algorithmic factors driving performance. Notably, we find that imposing the editing constraints (1) on the entirety of the image—as opposed to only focusing on key-value pairs that correspond to the concept of interest as proposed in Bau et al. [2020a]—leads to even better performance (cf. ‘-mask’ in Figure 5). We hypothesize that this has a regularizing effect as it constrains the weights to preserve the original mapping between keys and values in regions that do not contain the concept. ",
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+        "image_caption": [
+            "(a) ImageNet-trained VGG16 (concepts from COCO) "
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+        "image_caption": [
+            "(b) Places-trained ResNet-18 (concepts from LVIS) ",
+            "Figure 5: Editing vs. fine-tuning, averaged over concept-style pairs. We find that both methods (and their variants) are fairly successful at correcting errors on the target class (examples of which are used to perform the rewrite). This holds even when the transformation applied during testing is different from the one present in the train exemplars (e.g., a different texture of “wood”). However, crucially, only the improvements induced by editing generalize to other classes where the transformed concept is present. Fine-tuning fails in this setting—typically, causing more errors than it fixes. See Appendix Figures 10-13 for other experimental settings. "
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+        "type": "text",
+        "text": "Fine-tuning. Our baseline is the canonical fine-tuning approach, i.e., directly minimizing the model loss on the new data (here the transformed images) with respect to their label. Similar to Section 3, we consider both the local and global variants of fine-tuning. We find that while these approaches are able to correct model errors on transformed inputs from the class used to perform the modification, they typically decrease performance on other classes—i.e., they cause more errors than they fix. Moreover, even when we allow a larger drop in the model’s accuracy, or use more training exemplars, their performance often becomes worse such inputs (Appendix Figures 15-18). ",
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+        "type": "text",
+        "text": "We present examples of errors corrected (or not) by editing and fine-tuning in Appendix Figure 14, and provide a per-concept/style break down in Appendix Figures 19 and 20. ",
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+        "type": "text",
+        "text": "5 Beyond editing: Probing model behavior via counterfactuals ",
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+        "type": "text",
+        "text": "In the previous section, we developed a scalable pipeline for creating concept-level transformations which we used to evaluate model rewriting methods. Here, we put forth another related use-case of that pipeline: debugging models to identify their (learned) prediction rules. ",
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+        "type": "text",
+        "text": "In particular, observe that the resulting transformed inputs (cf. Figure 4) can be viewed as counterfactuals—a primitive commonly used in causal inference [Pearl, 2010] and interpretability [Goyal et al., 2019a,b, Bau et al., 2020b]. Counterfactuals can be used to identify the features that a model uses to make its prediction on a given input. For example, to understand whether the model relies on the presence of a “wheel” to recognize a “car” image, we can evaluate how the its prediction changes when just the wheel in the image is transformed. Based on this view, we now repurpose our concept transformations from Section 4 to discover a given classifier’s prediction rules. ",
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+        "type": "text",
+        "text": "The effect of specific concepts. As a point of start, we study how sensitive the model’s prediction is to a given concept in terms of the accuracy drop caused by transformation of said concept. For instance, in Figure 6a, we find that the accuracy of a VGG16 ImageNet classifier drops by $2 5 \\%$ on images of “croquet ball” when “grass” is transformed, whereas its accuracy on “collie” does not change. In line with previous studies [Zhang et al., 2007, Ribeiro et al., 2016, Rosenfeld et al., 2018, Barbu et al., 2019, Xiao et al., 2020], we also find that background concepts, such as “grass”, “sea” and “sand”, have a large effect on model performance. We contrast this measure of influence across concepts for a single model (Appendix Figure 27), and across architectures for a single concept (Appendix B.2). Finally, we examine the effect of each concept stylization in Figure 6b. ",
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+    {
+        "type": "text",
+        "text": "Per-class prediction rules. If instead we restrict our attention to a single class, we can pinpoint the set of concepts that the model relies on to detect this class. It turns out that aside from the main image object, ImageNet classifiers also heavily depend on co-occurring objects [Stock and Cisse, ",
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+    {
+        "type": "image",
+        "img_path": "images/69b34d793490d5efbe31f255080c02c9e48df62634e74bbcd699266cfedb82b4.jpg",
+        "image_caption": [
+            "(b) Style impact on accuracy ",
+            "Figure 6: Model sensitivities diagnosed using our pipeline in a VGG16 ImageNet classifier. (a) Classes for which the model relies on “grass”: e.g., a “croquet ball” is not accurately recognized if “grass” is transformed, while “collie”s are not affected. (b) Applying different transformations to visual concepts reduces accuracy by varying amounts. (c) Transformations that hamper accuracy can highlight prediction rules: e.g., the class “groom” is sensitive to the presence of “dress.” "
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+    {
+        "type": "text",
+        "text": "2018, Tsipras et al., 2020, Beyer et al., 2020] in the image—e.g., “dress” for the class “groom”, “person” for the class “tench” (sic), and “road” for the class “race car” (cf. Figure 6c and Appendix Figure 29). We can also examine which styles hurt performance the most—e.g., we find that making “plants” “floral” hurts accuracy on the class “damselfly” $15 \\%$ more than making them “snowy”. ",
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+    {
+        "type": "text",
+        "text": "Overall, this pipeline provides a scalable path for model designers to analyze the invariances (and sensitivities) of their models with respect to various natural transformations of interest. We thus believe that this primitive holds promise for future interpretability and robustness studies. ",
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+    {
+        "type": "text",
+        "text": "6 Related work ",
+        "text_level": 1,
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+        "text": "High-level concepts in latent representations. There has been increasing interest in explaining the inner workings of deep classifiers through high-level concepts: e.g., by identifying individual neurons [Erhan et al., 2009, Zeiler and Fergus, 2014, Olah et al., 2017, Bau et al., 2017, Engstrom et al., 2019a] or activation vectors [Kim et al., 2018, Zhou et al., 2018, Chen et al., 2020] that correspond to human-understandable features. The effect of these features on model predictions can be analyzed by either inspecting the downstream weights of the model [Olah et al., 2018, Wong et al., 2021] or through counterfactual tests: e.g., via synthetic data [Goyal et al., 2019a]; by swapping features between individual images [Goyal et al., 2019b]; or by silencing sets of neurons [Bau et al., 2020b]. A parallel line of work aims to learn models that operate on data representations that explicitly encode high-level concepts: either by learning to predict a set of attributes [Lampert et al., 2009, Koh et al., 2020a] or by learning to segment inputs [Losch et al., 2019]. In our work, we identify concepts by manually selecting the relevant pixels in a handful of images and measure the impact of manipulating these features on model performance on a new test set. ",
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+    {
+        "type": "text",
+        "text": "Model interventions. Direct manipulations of latent representations inside generative models have been used to create human-understandable changes in synthesized images [Bau et al., 2019, Jahanian et al., 2019, Goetschalckx et al., 2019, Shen et al., 2020, Härkönen et al., 2020, Wu et al., 2020]. Our work is inspired by that line of work as well as a recent finding that parameters of a generative model can be directly changed to alter generalized behavior [Bau et al., 2020a]. Unlike previous work, we edit classification models, changing rules that govern predictions rather than image synthesis. Concurrently with our work, there has been a series of methods proposed for editing factual knowledge in language models [Mitchell et al., 2021, De Cao et al., 2021, Dai et al., 2021]. ",
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+    {
+        "type": "text",
+        "text": "Ignoring spurious features. Prior work on preventing models from relying on spurious correlations is based on constraining model predictions to satisfy certain invariances. Examples include: training on counterfactuals (either by adding or removing objects from scenes [Shetty et al., 2018, 2019, Agarwal et al., 2020] or having human annotators edit text input [Kaushik et al., 2019]), learning representations that are simultaneously optimal across domains [Arjovsky et al., 2019], ensuring comparable performance across subpopulations [Sagawa et al., 2020], or enforcing consistency across inputs that depict the same entity [Heinze-Deml and Meinshausen, 2017]. In this work, we focus on a setting where the model designer is aware of undesirable correlations learned by the model and we provide the tools to rewrite them directly. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Model robustness. A long line of work has been devoted to discovering and correcting failure modes of models. These studies focus on simulating variations in testing conditions that can arise during deployment, including: adversarial or natural input corruptions [Szegedy et al., 2014, Fawzi and Frossard, 2015, Fawzi et al., 2016, Engstrom et al., 2019b, Ford et al., 2019, Hendrycks and Dietterich, 2019, Kang et al., 2019], changes in the data collection process [Saenko et al., 2010, Torralba and Efros, 2011, Khosla et al., 2012, Tommasi and Tuytelaars, 2014, Recht et al., 2019], or variations in the data subpopulations present [Beery et al., 2018, Oren et al., 2019, Sagawa et al., 2020, Santurkar et al., 2021, Koh et al., 2020b]. Typical approaches for improving robustness in these contexts include robust optimization [Madry et al., 2018, Yin et al., 2019, Sagawa et al., 2020] and data augmentation schemes [Lopes et al., 2019, Hendrycks et al., 2019, Zhang et al., 2021]. Our rule-discovery and editing pipelines can be viewed as complementary to this work as they allows us to preemptively adjust the model’s prediction rules in anticipation of deployment conditions. ",
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+    {
+        "type": "text",
+        "text": "Domain adaptation. The goal of domain adaptation is to adapt a model to a specific deployment environment using (potentially unlabeled) samples from it. This is typically achieved by either fine-tuning the model on the new domain [Donahue et al., 2014, Sharif Razavian et al., 2014, Kumar et al., 2020], learning the correspondence between the source and target domain, often in a latent representation space [Ben-David et al., 2007, Saenko et al., 2010, Ganin and Lempitsky, 2015, Courty et al., 2016, Gong et al., 2016], or updating the model’s batch normalization statistics [Li et al., 2016, Burns and Steinhardt, 2021]. These approaches all require a non-trivial amount of data from the target domain. The question of adaptation from a handful of samples has been explored [Motiian et al., 2017], but in a setting that requires samples across all target classes. In contrast, our method allows for generalization to new (potentially unknown) classes with even a single example. ",
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+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We developed a general toolkit for performing targeted post hoc modifications to vision classifiers. Crucially, instead of specifying the desired behavior implicitly via the training data, our method allows users to directly edit the model’s prediction rules. By doing so, our approach makes it easier for users to encode their prior knowledge and preferences during the model debugging process. Additionally, a key benefit of this technique is that it fundamentally changes how the model processes a given concept—thus making it possible to edit its behavior beyond the specific class(es) used for editing. Finally, our edits do not require any additional data collection: they can be guided by as few as a single (synthetically-created) exemplar. We believe that this primitive opens up new avenues to interact with and correct our models before or during deployment. ",
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+    {
+        "type": "text",
+        "text": "Limitations and broader impact. ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Even though our methodology provides a general tool for model editing, performing such edits does require manual intervention and domain expertise. After all, the choice of what concept to edit— and its implications on the robustness of the model—lies with the model designer. For instance, in the vehicles-on-snow example, our objective was to have the model recognize any vehicle on snow the same way it would on a regular road—e.g., to adapt a system to different weather conditions. However, if our dataset contains classes for which the presence snow is absolutely essential for recognition, this might not be an appropriate edit to perform. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Moreover, direct model editing is a departure from the standard way in which models are trained, and may have broader implications. While we have shown how it can be used to cause beneficial changes in pre-trained models, direct control of prediction rules could also make it easier for adversaries to introduce vulnerabilities into the model (e.g., by manipulating model behavior on a specific population demographic). Overall, direct model editing makes it clearer than ever that our models are a reflection of the goals and biases of we who create them—not only through the training tasks we choose, but now also through the rules that we rewrite. ",
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We thank the anonymous reviewers for their helpful comments and feedback. ",
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+    },
+    {
+        "type": "text",
+        "text": "Work supported in part by the NSF grants CCF-1553428 and CNS-1815221, the DARPA SAILON HR0011-20-C-0022 grant, Open Philanthropy, a Google PhD fellowship, and a Facebook PhD fellowship. This material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) under Contract No. HR001120C0015. ",
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+    },
+    {
+        "type": "text",
+        "text": "Research was sponsored by the United States Air Force Research Laboratory and the United States Air Force Artificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the United States Air Force or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. ",
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+    {
+        "type": "text",
+        "text": "References ",
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+    },
+    {
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parse/train/b7ZRqEFXdQ/b7ZRqEFXdQ.md

@@ -0,0 +1,361 @@
+# IMPROVING SEQUENCE GENERATIVE ADVERSARIAL NETWORKS WITH FEATURE STATISTICS ALIGNMENT
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Generative Adversarial Networks (GAN) are facing great challenges in synthesizing sequences of discrete elements, such as mode dropping and unstable training. The binary classifier in the discriminator may limit the capacity of learning signals and thus hinder the advance of adversarial training. To address such issues, apart from the binary classification feedback, we harness a Feature Statistics Alignment (FSA) paradigm to deliver fine-grained signals in the latent high-dimensional representation space. Specifically, FSA forces the mean statistics of the fake data distribution to approach that of real data as close as possible in a finite-dimensional feature space. Experiments on synthetic and real benchmark datasets show the superior performance in quantitative evaluation and demonstrate the effectiveness of our approach to discrete sequence generation. To the best of our knowledge, the proposed architecture is the first that employs feature alignment regularization in the Gumbel-Softmax based GAN framework for sequence generation.
+
+# 1 INTRODUCTION
+
+Unsupervised sequence generation is the cornerstone for a plethora of applications, such as machine translation (Wu et al., 2016), image captioning (Anderson et al., 2018), and dialogue generation (Li et al., 2017). The most common approach to autoregressive sequence modeling is maximizing the likelihood of each token in the sequence given the previous partial observation. However, using maximum likelihood estimation (MLE) for sequence modeling is inherently prone to the exposure bias problem (Bengio et al., 2015), which results from the discrepancy between the training and inference stage: the generator predicts the next token conditioned on its previously generated ones during inference but conditioned on its prefix ground-truth tokens during training, yielding accumulative mismatch along with the increment of generated sequence length.
+
+Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) can serve as an alternative to models trained by MLE, which have achieved promising results in generating sequences of discrete elements, in particular, language sequences (Kusner & Hernandez-Lobato, 2016; Yu et al., 2017; Lin ´ et al., 2017; Guo et al., 2018; Fedus et al., 2018; Nie et al., 2019; de Masson d’Autume et al., 2019; Zhou et al., 2020; Scialom et al., 2020). GANs consist of two competing networks: a discriminator that is trained to distinguish the generated samples from real data, and a generator that aims to generate high-quality samples to fool the discriminator.
+
+Although having succeeded in avoiding exposure bias issues, GANs still suffer from some intrinsic problems, such as mode dropping, reward sparsity, and training instability. To enrich the informativeness of the discriminator’s training signal, several approaches have been proposed by measuring the latent features, such as feature distribution matching (Zhang et al., 2017; Chen et al., 2018) and comparative discriminators (Lin et al., 2017; Zhou et al., 2020). Zhang et al. (2017) and Chen et al. (2018) leveraged feature matching mechanism by minimizing the kernel-based moment-matching metric, such as Maximum Mean Discrepancy and Earth-Mover’s Distance, between encoded features. However, merely adopting feature matching in lieu of the original learning signal may lack some guiding feedback at the initial stage of training.
+
+Another approach is to compare the finite latent features with comparative discriminators like ranker and relativistic discriminator. Lin et al. (2017) maintained that the binary classification in the discriminator network limits the learning capacity of tasks because the diversity and richness are circumscribed by the degenerated distribution. RankGAN (Lin et al., 2017) replaced the binary classifier with a pairwise feature ranker by comparing the similarities between sample features in the latent space. SAL (Zhou et al., 2020) classified the encoded features of constructed pairwise training examples into three categories, i.e., better / worse / indistinguishable. Nevertheless, adopting a comparative discriminator could provide the fine-grained signals for updating the generator network and may require some coarse credits for further improvements.
+
+![](images/d5c43bed8bb6e354f0c6608a642db50fbc6ddae793dfb98329f54419432a6ef2.jpg)  
+Figure 1: (a) Standard GANs using a binary classifier as its discriminator; (b) GANs with Feature Statistics Alignment and relativistic discriminator that provide more instructive signals for updating the generator.
+
+In this work, we propose to improve the GANs for sequence generation by jointly considering both the feature statistics matching and relativistic discriminator to serve as fine-grained and coarse learning signals respectively. We leverage the Feature Statistics Alignment (FSA) paradigm to embed the latent feature representations in a finite feature space and force the distribution of generated samples to approach the real data distribution by minimizing the distance between their respective feature representation centroids. Intuitively, matching the mean feature representations of fake and real samples could make the two data distributions closer. Besides, the relativistic discriminator (Jolicoeur-Martineau, 2019) is employed to measure the comparative information between generated and real sequences and empirically to show the effectiveness during the model training.
+
+Our experimental results illustrate the effectiveness of FSA techniques and large batch size to alleviate the gradient vanishing problem and stabilize the training process in comparison with the vanilla Gumbel-Softmax GANs. Besides, our models could generate discrete text sequences with high quality in terms of the semantic coherence and grammatical correctness of language, as evaluated with crowdsourcing. Furthermore, we empirically demonstrate that the proposed architecture overshadows most existing models in terms of quantitative and qualitative evaluation. To the best of our knowledge, the proposed framework is the first to adopt the statistics feature alignment paradigm in the Gumbel-Softmax based GAN framework for discrete sequence generation.
+
+# 2 ADVERSARIAL SEQUENCE GENERATION
+
+Adversarial sequence generation has attracted broad attention for its properties to solve the exposure bias issue suffered with maximum likelihood estimation (MLE) for generating language sequences. Based on the game theory, its goal is to train a generator network $G \big ( z ; \pmb { \theta } ^ { ( G ) } \big )$ that produces samples from the data distribution $p _ { \mathrm { d a t a } } ( \pmb { x } )$ by decoding the randomly initialized noise $_ { z }$ (i.e., standard normal distribution) into the sequence $\pmb { x } = G ( \pmb { z } ; \pmb { \theta } ^ { ( G ) } )$ , where the training signal is provided by the discriminator network $D ( \pmb { x } ; \pmb { \phi } ^ { ( D ) } )$ that is trained to distinguish between the samples drawn from the real data distribution $p _ { \mathrm { d a t a } }$ and those produced by the generator. The minimax objective of adversarial training is formulated as:
+
+$$
+\operatorname* { m i n } _ { \theta ^ { ( G ) } } \operatorname* { m a x } _ { \phi ^ { ( D ) } } \mathbb { E } _ { \mathbf { x } \sim p _ { \mathrm { d a t a } } } \big [ \log D ( \mathbf { \boldsymbol { x } } ; \phi ^ { ( D ) } ) \big ] + \mathbb { E } _ { \mathbf { \boldsymbol { z } } \sim p _ { \mathbf { z } } } \big [ \log \big ( 1 - D _ { \phi ^ { ( D ) } } \big ( G ( \boldsymbol { z } ; \boldsymbol { \theta } ^ { ( G ) } ) \big ) \big ) \big ] .
+$$
+
+Despite the impressive results of GANs in the sequence generation (Yu et al., 2017; Gulrajani et al., 2017; Scialom et al., 2020), there are still several fundamental issues in the GAN training: (a) Training instability, which arises from the intrinsic nature of minimax games in GANs; (b) Mode dropping, which is the fact that GANs only generate samples with limited patterns in the real data distribution instead of attending to diverse patterns (Chen et al., 2018); (c) Reward sparsity, which is because that it is easier to train the discriminator than the generator, making it difficult to acquire the instructive feedback (Zhou et al., 2020).
+
+Due to the non-differentiability of gradients caused by sampling operations between the generator and discriminator for sequence generation, the majority of previous works have resorted to reinforcement learning (RL) heuristics with Monte Carlo search to collect the credits from the discriminator. The usage of RL may further deteriorate the instability of model training and exacerbate the reward sparsity problem. Gumbel-Softmax relaxation has proven to be an alternative to RL techniques (Kusner & Hernandez-Lobato, 2016; Nie et al., 2019). How to efficiently train GANs with ´ the Gumbel-Softmax trick still remains under-explored. Therefore, we utilize the Gumbel-Softmax reparameterization instead of conventional policy gradients in our framework.
+
+# 3 METHODOLOGY
+
+As illustrated in Fig. 1, standard GANs employ the real/fake binary classifier as the discriminator, which is prone to be overtrained in comparison with the generator (Salimans et al., 2016). To prevent the discriminator from overfitting and further stabilize the training process, we propose to leverage the FSA techniques and relativistic discriminator by comparing the fake and real distributions from two different aspects.
+
+Compared with conventional GANs, the proposed framework enjoys the following advantages: (a) In the earlier training stage, the relativistic discriminator could estimate the probability that how much better the real data is in comparison with the generated samples. This could provide the relatively “coarse” credits to the generator. (b) When it comes to the later training stage and the quality of generated samples becomes high, FSA measures the “fine-grained” difference between latent features of fake and real data batches, precluding the discriminator from being overly confident. In other words, the FSA can be regarded as a kind of regularization for adversarial training.
+
+# 3.1 FEATURE STATISTICS ALIGNMENT
+
+Intuitively, aligning the statistics of embedded feature representations increases the opportunities to capture the various modes of the data distribution. For brevity, we utilize the first-order mean statistics in our framework and leave the higher-order statistics for future work.
+
+Denoting the feature extractor as $F _ { \omega }$ parameterized by $\omega$ , a minibatch of real data samples as $_ { \textbf { \em x } }$ with the batch size of $N$ , we propose two variants of FSA formulations, which calculate the mean squared difference and Euclidean distance between the minibatch centroids of real and fake feature representations. To reduce the parameter amount, we share the weights between the discriminator and feature extractor of FSA.
+
+Mean Squared Alignment (MSA) We take the mean squared difference between the centroids of fake and generated distributions as the feature alignment metric:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { M S A } } = \left\| \mathbb { E } _ { { \pmb { x } } \sim p _ { \mathrm { d a t a } } } \big [ F _ { \omega } ( { \pmb x } ) \big ] - \mathbb { E } _ { { \pmb z } \sim p _ { \pmb z } } \big [ F _ { \omega } \big ( G ( { \pmb z } ; { \pmb \theta } ^ { ( G ) } ) \big ) \big ] \right\| _ { 2 } ^ { 2 } } \\ { = \| \displaystyle \frac { 1 } { N } \sum _ { i = 1 } ^ { N } F _ { \omega } ( \pmb x _ { i } ) - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } F _ { \omega } ( G ( { \pmb z } _ { i } ; { \pmb \theta } ^ { ( G ) } ) ) \big \| _ { 2 } ^ { 2 } . } \end{array}
+$$
+
+Mean Distance Alignment (MDA) Another intuitive approach is to calculate the distance between two sample centroids, which is equivalent to the square root of MSA mathematically:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \mathrm { M D A } } = \left\| \mathbb { E } _ { { \pmb { x } } \sim p _ { \mathrm { d a t a } } } \big [ F _ { \omega } ( { \pmb x } ) \big ] - \mathbb { E } _ { { \pmb z } \sim p _ { \pmb z } } \big [ F _ { \omega } \big ( { \pmb G } ( { \pmb z } ; { \pmb \theta } ^ { ( G ) } ) \big ) \big ] \right\| _ { 2 } } \\ & { \qquad = \sqrt { L _ { \mathrm { M S A } } } . } \end{array}
+$$
+
+By forcing the mean statistics of fake data to be close to the real samples, the generator could receive more informative signals in the training process. It is worth noting that the large batch size is helpful to reduce the variance of small mini-batches.
+
+# 3.2 RELATIVISTIC DISCRIMINATOR
+
+We consider the Relativistic Discriminator (Jolicoeur-Martineau, 2019) to take into account the relative confidence that the given real data is more realistic than the randomly sampled fake data. In the standard GAN, defining the discriminator as $D ( \pmb { x } ) = \mathrm { s i g m o i d } ( H ( \pmb { x } ) )$ , where $H ( \cdot )$ represents the non-transformed layer before the final non-linearity. The objectives for the discriminator and generator in terms of the Relativistic Discriminator are defined as:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \mathrm { R D } } = - \mathbb { E } _ { { \pmb x } \sim p _ { \mathrm { d a t a } } , { \pmb z } \sim p _ { z } } [ \log a \big ( H ( { \pmb x } ) - H ( G ( { \pmb z } ; { \pmb \theta } ^ { ( G ) } ) ) \big ) ] , } \\ & { \mathcal { L } _ { \mathrm { R G } } = - \mathbb { E } _ { { \pmb x } \sim p _ { \mathrm { d a t a } } , { \pmb z } \sim p _ { z } } [ \log a \big ( H ( G ( { \pmb z } ; { \pmb \theta } ^ { ( G ) } ) ) - H ( { \pmb x } ) \big ) ] , } \end{array}
+$$
+
+where $a$ represents the activation function to be relativistic (we use sigmoid function in our experiments), RD and RG denote the loss terms for the discriminator and generator respectively.
+
+# 3.3 OVERALL TRAINING OBJECTIVES
+
+Previous works using moment matching techniques to support the training on both the discriminator and generator (Zhang et al., 2017; Chen et al., 2018). However, the generator is always more difficult to train than the discriminator, resulting in the training instability and reward sparsity. To relieve these issues, we only adopt the FSA techniques to enhance the generator but keep the objective of the discriminator unchanged. This could pass more informative signals only to the generator, and also prevent the discriminator to be overtrained.
+
+Therefore, the overall training objectives for the proposed framework are defined as:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \mathrm { D } } = \mathcal { L } _ { \mathrm { R D } } , } \\ & { \mathcal { L } _ { \mathrm { G } } = \mathcal { L } _ { \mathrm { R G } } + \mathcal { L } _ { \mathrm { F S A } } , } \end{array}
+$$
+
+where $\mathcal { L } _ { \mathrm { F S A } }$ takes the form of $\mathcal { L } _ { \mathrm { M S A } }$ and ${ \mathcal { L } } _ { \mathrm { M D A } }$ as Eq. 2 and Eq. 4.
+
+The goal of the discriminator is to maximize the gap between the generated and real data, whereas the generator jointly considers two different aspects simultaneously: it not only competes with the discriminator by maximizing the gap in terms of the relativistic signals but takes into account the additional leaked feature information from the discriminator. The idea of leaked features from the discriminator is similar to LeakGAN (Guo et al., 2018). The FSA term on the RHS can also be regarded as a dynamic regularizer for the sequence generator.
+
+# 3.4 TRAINING WITH DISCRETE SEQUENCE
+
+# 3.4.1 GUMBEL-SOFTMAX DISTRIBUTION
+
+Conventional GANs for generating discrete sequences are inherently unable to backpropagate the gradient through samples due to the non-differentiable sampling from a categorical distribution. Gumbel-Softmax distribution (Jang et al., 2017; Maddison et al., 2017) was proposed to deal with this issue by smoothly annealed to approximate the categorical distribution.
+
+Denoting the output probabilities $\pi _ { 1 } , \pi _ { 2 } , \cdots , \pi _ { | V | }$ , where $| V |$ represents the output vocabulary size in the generator, the Gumbel-Max trick (Maddison et al., 2014) can be parameterized as:
+
+$$
+y _ { i } = \mathrm { o n e } \mathrm { . h o t } \big ( \arg \operatorname* { m a x } _ { i } [ g _ { i } + \log \pi _ { i } ] \big ) \quad \mathrm { ~ f o r ~ } i = 1 , \cdots , | V | ,
+$$
+
+where $\{ g _ { i } | i = 1 , \cdots , | V | \}$ are i.i.d from the Gumbel(0,1) distribution, that is, $g _ { i } = - \log ( - \log u _ { i } )$ with $u _ { i }$ is drawn from a standard uniform distribution Uniform(0,1). one hot represents the $| V |$ - dimensional one hot encoding.
+
+Gumbel-Softmax approximates the non-differential arg max operation using the softmax function:
+
+$$
+\hat { y } _ { i } = \frac { \exp \left( ( \log ( \pi _ { i } ) + g _ { i } ) / \tau \right) } { \sum _ { j = 1 } ^ { | V | } \exp \left( ( \log ( \pi _ { j } ) + g _ { j } ) / \tau \right) } , \quad \mathrm { ~ f o r ~ } i = 1 , \cdots , | V | ,
+$$
+
+where $\tau$ denotes the softmax temperature to modulate the exploitation and exploration during training. When $\tau$ approaches too high, the approximation is nearly equiprobable, encouraging the generator to explore different options. In contrast, the lower $\tau$ could discourage the exploration and tend to exploit during training. In particular, when $\tau  0$ , $\hat { y } _ { i }$ approaches the result of one hot operator as in Eq. 10, whereas $\hat { y } _ { i }$ will degenerated into a uniform distribution when $\tau  \infty$ .
+
+# 3.4.2 ARCHITECTURE AND ADVERSARIAL TRAINING
+
+RNN Generator Due to the free-running mode of GAN’s generator, it is unsuitable to adopt the transformer-based models due to the non-recurrence nature. Thus, we investigate the Recurrent Neural Network (RNN) based models, such as Long Short-Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997), and Relational Memory Core (RMC) (Santoro et al., 2018).
+
+CNN Discriminator & Feature Extractor We use the convolutional neural networks (CNN) architecture (Kim, 2014) as the discriminator and feature extractor for input sequences. We employ the multi-channel convolution using multiple filters with various window sizes to extract the distinct n-gram features, followed by a max-over-time pooling operation to gather the most salient features, i.e., features with the highest value for each feature map.
+
+Adversarial Training Algorithm Alg. 1 illustrates the overall training process of the proposed framework. The Relativistic Discriminator and the generator could reach the Nash Equilibrium when the generator could fool the discriminator into accepting its output as being true. Since the discriminator is easy to be overtrained, we do not pretrain the discriminator but only pretrain the generator using MLE for few epochs.
+
+1: Require: generator $G _ { \theta }$ ; discriminator $D _ { \phi }$ ; samples of real data $\mathbb { S }$ ; generator training step $g$ ;   
+discriminator training step $k$ ; the generator pretraining epochs $m$ .   
+2: Pretrain $G _ { \theta }$ using MLE on $\mathbb { S }$ for $m$ epochs   
+3: repeat   
+4: for $g$ steps do   
+5: Sample a minibatch from real data $\mathbb { S }$   
+6: Generate a minibatch of samples $\mathbf { \boldsymbol { x } } _ { g } \sim G _ { \theta }$   
+7: Update $G _ { \theta }$ via Eq.(9)   
+8: end for   
+9: for $k$ steps do   
+10: Sample a minibatch from real data $\mathbb { S }$   
+11: Sample a minibatch from the generated data   
+12: Train the discriminator $D _ { \phi }$ by Eq.(8)   
+13: end for   
+14: until convergence
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETTING
+
+Similar to (Lin et al., 2017; Guo et al., 2018; Nie et al., 2019; Zhou et al., 2020), we evaluate the proposed framework based on the Texygen benchmark platform (Zhu et al., 2018) for adversarial text generation. Experiments were conducted on the synthetic and real datasets: (a) synthetic data, which is generated by an oracle single-layer LSTM as in (Yu et al., 2017); (b) MS COCO Image Caption dataset (Chen et al., 2015); (c) EMNLP WMT 2017 News dataset (Guo et al., 2018). Table 1 summarizes the statistics of benchmark datasets for evaluation.
+
+Algorithm 1 Adversarial Training with Feature Statisitcs Alignment   
+Table 1: Summary of experimental datasets.   
+
+<table><tr><td>dataset</td><td>vocabulary size</td><td> sequence length</td><td>training set</td><td>test set</td></tr><tr><td>synthetic data</td><td>5,000</td><td>20/40</td><td>10,000</td><td>10,000</td></tr><tr><td>MS COCO</td><td>4,657</td><td>37</td><td>10,000</td><td>10,000</td></tr><tr><td>EMNLP2017 WMT News</td><td>5,255</td><td>51</td><td>27,8586</td><td>10,000</td></tr></table>
+
+For the synthetic data experiments, we utilize a single-layer LSTM initialized by the standard normal distribution as the oracle model, which is used to generate 10,000 samples of length 20 and 40 respectively as the real samples. We use the negative log-likelihood (NLL) under the oracle data distribution for evaluation, termed $\mathrm { N L L } _ { \mathrm { o r a c l e } }$ . For the real data experiments, the BLEU score (Papineni et al., 2002) serves as a metric to evaluate the n-gram statistics overlapping on the whole dataset.
+
+To measure the diversity of generated samples, the NLL of the generator (denoted as ${ \mathrm { N L L } } _ { \mathrm { g e n . } }$ ) is used by computing the NLL of reference samples in the test set by the generator. Considering that BLEU scores always focus on the local text statistics and may be insufficient for evaluating the overall quality of texts, we conducted additional human evaluation via crowdsourcing on the generated samples of all comparison models. See Appendix A for more experimental details.
+
+We compare the proposed framework with the MLE baseline and other state-of-the-art models, involving SeqGAN (Yu et al., 2017), RankGAN (Lin et al., 2017), LeakGAN (Guo et al., 2018), RelGAN (Nie et al., 2019), and Self-Adversarial Learning (SAL) (Zhou et al., 2020).
+
+# 4.2 EXPERIMENTAL RESULTS
+
+# 4.2.1 SYNTHETIC DATA
+
+Table 2 illustrates the performance of different models on NLLoracle. Since our experiments achieved better results using MSA instead of MDA on synthetic data, we only report the optimal results with MSA in Table 2. Our models outperform other prevalent adversarial models in terms of the generated sample quality, demonstrating the effectiveness of our proposed method.
+
+In practice, we found that the LSTM generator could outperform the RMC generator for producing the sequence with the length of 20, whereas RMC generators exceed LSTMs for long sequence generation. This may be due to the fact that LSTMs may forget the long-term dependencies with the sequence length increases, but the self-attention based relational memory cell in RMC could mitigate the issues using interactive memory slots. This demonstrates that the proposed model could produce samples with high quality. As to the diversity, our method achieves competitive ${ \mathrm { N L L } } _ { \mathrm { g e n } }$ score compared with baselines models (see Appendix C.1 for the detail).
+
+Table 2: The $\mathrm { N L L } _ { \mathrm { o r a c l e } }$ performance of different models $\tau = 1$ ) on the synthetic dataset with the sequence length of 20 and 40 respectively. For NLL, the lower, the better.   
+
+<table><tr><td>Length</td><td>MLE</td><td>SeqGAN</td><td>RankGAN</td><td>LeakGAN</td><td>RelGAN</td><td>SAL</td><td>Ours (LSTM)</td><td>Ours (RMC)</td><td>Real</td></tr><tr><td>20</td><td>9.038</td><td>8.736</td><td>8.247</td><td>7.038</td><td>6.680</td><td>7.71</td><td>5.047</td><td>5.819</td><td>5.750</td></tr><tr><td>40</td><td>10.411</td><td>10.310</td><td>9.958</td><td>7.191</td><td>6.765</td><td>9.31</td><td>5.909</td><td>5.087</td><td>4.071</td></tr></table>
+
+# 4.2.2 MS COCO DATASET
+
+To further test the performance on the real data, we run and evaluate our model on MS COCO image caption datasets. The data and preprocessing remain the same as in Texygen (Zhu et al., 2018). Empirically, we found that RMC achieves better results in terms of the long sequences, and thus reports the results using the RMC generator if not otherwise specified.
+
+Table 3: BLEU and ${ \mathrm { N L L } } _ { \mathrm { g e n } }$ performance on MS COCO image captions with $\tau \ : = \ : 0 . 1$ for the proposed models with MSA and MDA. For BLEU scores, the higher, the better.   
+
+<table><tr><td>Model</td><td>BLEU-2</td><td>BLEU-3</td><td>BLEU-4</td><td>BLEU-5</td><td>NLLgen</td></tr><tr><td>MLE</td><td>0.731</td><td>0.497</td><td>0.305</td><td>0.189</td><td>0.718</td></tr><tr><td>SeqGAN</td><td>0.745</td><td>0.498</td><td>0.294</td><td>0.180</td><td>1.082</td></tr><tr><td>RankGAN</td><td>0.743</td><td>0.467</td><td>0.264</td><td>0.156</td><td>1.344</td></tr><tr><td>LeakGAN</td><td>0.746</td><td>0.528</td><td>0.355</td><td>0.230</td><td>0.679</td></tr><tr><td>RelGAN</td><td>0.849</td><td>0.687</td><td>0.502</td><td>0.331</td><td>0.756</td></tr><tr><td>SAL</td><td>0.785</td><td>0.581</td><td>0.362</td><td>0.227</td><td>0.873</td></tr><tr><td>Ours (MSA)</td><td>0.959</td><td>0.866</td><td>0.759</td><td>0.630</td><td>0.760</td></tr><tr><td>Ours (MDA)</td><td>0.938</td><td>0.863</td><td>0.731</td><td>0.582</td><td>0.717</td></tr></table>
+
+Table 3 exhibits the final results of the BLEU and ${ \mathrm { N L L } } _ { \mathrm { g e n } }$ scores on different comparison models. Our models reported in Table 3 set the softmax temperature as 0.1 for the model with MSA and that with MDA. Notably, our model shows the significant improvement on previous methods, consistently overshadowing the state-of-the-art models in terms of the sample quality (indicated by BLEU scores) while maintaining the diversity (indicated by $\mathrm { N L L _ { g e n } } ,$ ).
+
+# 4.2.3 EMNLP2017 WMT NEWS DATASET
+
+Table 4 presents the considerable improvements of our model (w.r.t. quality) on EMNLP2017 WMT News dataset, with the temperature of 1 for models with MSA and MDA. The maximum length in the EMNLP2017 dataset is 51, greatly challenging the generation task.
+
+It can be observed that only RankGAN and LeakGAN slightly outrank the MLE in terms of the ${ \mathrm { N L L } } _ { \mathrm { g e n } }$ , which may be due to the fact that feature ranking or leakage information could pass more internal information from the discriminator to the generator, and thereby smoothly assist the training process of the generator. By leveraging the FSA methods, our model greatly outperforms these models and yielding the long sequences with promising qualities.
+
+Table 4: The BLEU and ${ \mathrm { N L L } } _ { \mathrm { g e n } }$ performance on EMNLP2017 WMT News dataset with temperatures of 1 for the proposed models with MSA and MDA.   
+
+<table><tr><td>Model</td><td>BLEU-2</td><td>BLEU-3</td><td>BLEU-4</td><td>BLEU-5</td><td>NLLgen</td></tr><tr><td>MLE</td><td>0.768</td><td>0.473</td><td>0.240</td><td>0.126</td><td>2.382</td></tr><tr><td>SeqGAN</td><td>0.777</td><td>0.491</td><td>0.261</td><td>0.138</td><td>2.773</td></tr><tr><td>RankGAN</td><td>0.727</td><td>0.435</td><td>0.209</td><td>0.101</td><td>3.345</td></tr><tr><td>LeakGAN</td><td>0.826</td><td>0.645</td><td>0.437</td><td>0.272</td><td>2.356</td></tr><tr><td>RelGAN</td><td>0.881</td><td>0.705</td><td>0.501</td><td>0.319</td><td>2.482</td></tr><tr><td>SAL</td><td>0.788</td><td>0.523</td><td>0.281</td><td>0.149</td><td>2.578</td></tr><tr><td>Ours (MSA)</td><td>0.932</td><td>0.798</td><td>0.585</td><td>0.404</td><td>3.999</td></tr><tr><td>Ours (MDA)</td><td>0.916</td><td>0.784</td><td>0.592</td><td>0.386</td><td>2.732</td></tr></table>
+
+Apart from the automatic quantitative evaluation, we also conducted human evaluation on the MS COCO Image Captioning dataset. We randomly sampled 100 sentences for each model and the real data, then asked 12 different people to score them on the scale of 1-5 with anonymizing the model’s identity. Please see Appendix. C.2 for more details of human evaluation.
+
+Our models received the highest score in comparison with other models, where the model with MSA receives higher credits than that with MDA. By manually going through the generated samples, we found that models with MSA tend to generate long sentences with higher quality but run into the mode dropping issue. In contrast, models with MDA posses a better trade-off between the diversity and quality of samples, yielding promising results in comparison with previous models.
+
+Table 5: Mean and standard deviation results of human evaluation w.r.t different models on MS COCO Image Caption dataset. Note that “Real” indicates the real data samples.   
+
+<table><tr><td>Model</td><td>MLE</td><td>SeqGAN</td><td>RankGAN</td><td>LeakGAN</td><td>MaliGAN</td></tr><tr><td>Human score</td><td>3.127 ± 0.124</td><td>3.062 ± 0.100</td><td>3.048 ± 0.127</td><td>3.018 ± 0.104</td><td>3.031 ± 0.123</td></tr><tr><td>Model</td><td>TextGAN</td><td>RelGAN</td><td>Ours (MSA)</td><td>Ours (MDA)</td><td>Real</td></tr><tr><td>Human score</td><td>1.973 ± 0.168</td><td>3.687 ± 0.181</td><td>4.209 ± 0.245</td><td>3.878 ± 0.276</td><td>3.444 ± 0.122</td></tr></table>
+
+# 4.3 DISCUSSION
+
+Fig. 2 (Left) illustrates the ablation test of FSA in terms of the BLEU-4 score (See Appendix D.2 for all the results). It can be observed that the increasing trend for models with FSA outreached that w/o feature alignment mechanism and our models achieve superior performance in the later training stage. The utilization of FSA leads to significant performance gains, as it could provide the consecutive “fined-grained” smoother learning signals to update the generator.
+
+Fig. 3 demonstrates the relative importance of each component of our models with an ablation test. The usage of FSA results in the most significant performance gain, followed by Gumbel-Softmax, and large batch size. A large batch size can boost the performance due to the variance reduction, accurate estimates of feature statistics, and stabilizing effect during the adversarial training (see Fig. 2 (Right)).
+
+As to the Gumbel-Softmax trick, we conduct experiments on various temperature values and find that a fined-tuned temperature hyperparameter could account for the second important factor to contribute to the overall performance boost. Please check Appendix D.3 for the detailed results w.r.t the Gumbel-Softmax temperature.
+
+![](images/2b18d2132e606bf0e13a2708dbda85111aef2e0ce4f1ba72800a36a18cbba091.jpg)  
+Figure 2: (Left) BLEU-4 scores of the proposed model w/ and w/o FSA mechanism. (Right) BLEU4 scores of the proposed models with various batch size (64 v.s. 128). The vertical dash lines indicate the end of generator pretraining. MS COCO results unless otherwise specified.
+
+# 5 RELATED WORK
+
+There has been a large category of GANs for sequence generation, which heavily rely on the RL paradigm. SeqGAN (Yu et al., 2017) regards the sequence generation as a Markov decision making process, estimates the rewards via Monte Carlo search, and trains the generator with policy gradient. RankGAN (Lin et al., 2017) and SAL (Zhou et al., 2020) replace the binary classifier in the discriminator as comparative discriminators to take into account the re
+
+![](images/a9b4402d9359f78aa8566845566a8ea70af6b5177cb955b27152c7882c60c327.jpg)  
+Figure 3: Ablation Study of the proposed model.
+
+lation between constructed pair samples. MaliGAN (Che et al., 2017) utilizes the information in the discriminator as an additional source of training signals on the MLE objective to reduce the variance of gradients. LeakGAN (Guo et al., 2018) leaks the intermediate feature information via a manager to guide the generator, which is inspired by hierarchical RL. ColdGAN (Scialom et al., 2020) integrates the advance of importance sampling, Proximal Policy Optimization (PPO) algorithm (Schulman et al., 2017), and nucleus sampling (Holtzman et al., 2019) for finetuning the pretrained T5 (Raffel et al., 2019) and BART (Lewis et al., 2019).
+
+Another approach applies non-RL methods for adversarial sequence generation by either approximating the categorical sampling or directly using the continuous latent representation. TextGAN (Zhang et al., 2017) uses feature matching techniques via a kernelized discrepancy in the Reproducing Kernel Hilbert Space. FMGAN (Chen et al., 2018) proposes to match the feature distributions using a Feature-Mover’s Distance. Similar to our proposed model, both of them apply annealed soft-argmax for approximation. ARAML Ke et al. (2019) utilizes Reward Augmented Maximum Likelihood by sampling from the stationary distribution to acquire rewards. However, none of them adopt the Gumbel-Max trick to reparameterize the categorical sampling. Besides, they applied feature matching as the training objectives of both the discriminator and generator, whereas we only apply the feature statistics matching to modulate the generator. Gumbel-Softmax (GS) GAN (Kusner & Hernandez-Lobato, 2016) and RelGAN (Nie et al., 2019) prove the effectiveness ´ of Gumbel-Softmax on unsupervised sequence generation. DialogeWAR Gu et al. (2018) employs a GS GAN within the latent variable space for dialogue generation. However, improving the training of GS GANs still remains an open problem. Our model aims to promote the GS GAN with the proposed FSA and other techniques to boost the training of language GANs. We also report a list of other techniques we tried but proved to be unsuccessful or unnecessary in Appendix B.
+
+# 6 CONCLUSION
+
+We propose an adversarial training framework for discrete sequence generation, by leveraging the advance of Feature Statistics Alignment and Gumbel-Softmax relaxation. Our model empirically shows superior performance in terms of the quantitative and human evaluation. In the future, it would be a promising direction to extend the proposed model to conditional text generation, such as text style transfer.
+
+# REFERENCES
+
+Peter Anderson, Xiaodong He, Chris Buehler, Damien Teney, Mark Johnson, Stephen Gould, and Lei Zhang. Bottom-up and top-down attention for image captioning and visual question answering. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 6077–6086, 2018.
+
+Samy Bengio, Oriol Vinyals, Navdeep Jaitly, and Noam Shazeer. Scheduled sampling for sequence prediction with recurrent neural networks. In Advances in Neural Information Processing Systems, pp. 1171–1179, 2015.
+
+Tong Che, Yanran Li, R. Zhang, R. Devon Hjelm, W. Li, Y. Song, and Yoshua Bengio. Maximumlikelihood augmented discrete generative adversarial networks. ArXiv, abs/1702.07983, 2017.
+
+Liqun Chen, Shuyang Dai, Chenyang Tao, Dinghan Shen, Zhe Gan, H. Zhang, Yizhe Zhang, and L. Carin. Adversarial text generation via feature-mover’s distance. ArXiv, abs/1809.06297, 2018.
+
+Xinlei Chen, Hao Fang, Tsung-Yi Lin, Ramakrishna Vedantam, Saurabh Gupta, Piotr Dollar, and ´ C Lawrence Zitnick. Microsoft coco captions: Data collection and evaluation server. arXiv preprint arXiv:1504.00325, 2015.
+
+Cyprien de Masson d’Autume, Mihaela Rosca, Jack W. Rae, and Shakir Mohamed. Training language gans from scratch. ArXiv, abs/1905.09922, 2019.
+
+William Fedus, Ian Goodfellow, and Andrew M Dai. Maskgan: Better text generation via filling in the . arXiv preprint arXiv:1801.07736, 2018.
+
+Ian J. Goodfellow, Jean Pouget-Abadie, M. Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron C. Courville, and Yoshua Bengio. Generative adversarial nets. In NIPS, 2014.
+
+Xiaodong Gu, Kyunghyun Cho, Jung-Woo Ha, and Sunghun Kim. Dialogwae: Multimodal response generation with conditional wasserstein auto-encoder. arXiv preprint arXiv:1805.12352, 2018.
+
+Ishaan Gulrajani, F. Ahmed, Mart´ın Arjovsky, Vincent Dumoulin, and Aaron C. Courville. Improved training of wasserstein gans. In NIPS, 2017.
+
+Jiaxian Guo, S. Lu, Han Cai, W. Zhang, Y. Yu, and J. Wang. Long text generation via adversarial training with leaked information. ArXiv, abs/1709.08624, 2018.
+
+Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 9(8): 1735–1780, 1997.
+
+Ari Holtzman, Jan Buys, Li Du, Maxwell Forbes, and Yejin Choi. The curious case of neural text degeneration. arXiv preprint arXiv:1904.09751, 2019.
+
+Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterization with gumbel-softmax. ArXiv, abs/1611.01144, 2017.
+
+Alexia Jolicoeur-Martineau. The relativistic discriminator: a key element missing from standard gan. ArXiv, abs/1807.00734, 2019.
+
+Pei Ke, Fei Huang, Minlie Huang, and Xiaoyan Zhu. Araml: A stable adversarial training framework for text generation. arXiv preprint arXiv:1908.07195, 2019.
+
+Yoon Kim. Convolutional neural networks for sentence classification. arXiv preprint arXiv:1408.5882, 2014.
+
+Matt J. Kusner and Jose Miguel Hern ´ andez-Lobato. Gans for sequences of discrete elements with ´ the gumbel-softmax distribution. ArXiv, abs/1611.04051, 2016.
+
+Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Ves Stoyanov, and Luke Zettlemoyer. Bart: Denoising sequence-to-sequence pretraining for natural language generation, translation, and comprehension. arXiv preprint arXiv:1910.13461, 2019.
+
+Jiwei Li, Will Monroe, Tianlin Shi, Sebastien Jean, Alan Ritter, and Dan Jurafsky. Adversarial´ learning for neural dialogue generation. arXiv preprint arXiv:1701.06547, 2017.   
+Kevin Lin, Dianqi Li, Xiaodong He, M. Sun, and Zhengyou Zhang. Adversarial ranking for language generation. ArXiv, abs/1705.11001, 2017.   
+Chris J Maddison, Daniel Tarlow, and Tom Minka. A\* sampling. In Advances in Neural Information Processing Systems, pp. 3086–3094, 2014.   
+Chris J. Maddison, A. Mnih, and Y. Teh. The concrete distribution: A continuous relaxation of discrete random variables. ArXiv, abs/1611.00712, 2017.   
+Weili Nie, Nina Narodytska, and Ankit B. Patel. Relgan: Relational generative adversarial networks for text generation. In ICLR, 2019.   
+Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. Bleu: a method for automatic evaluation of machine translation. In Proceedings of the 40th annual meeting of the Association for Computational Linguistics, pp. 311–318, 2002.   
+Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. Exploring the limits of transfer learning with a unified text-to-text transformer. arXiv preprint arXiv:1910.10683, 2019.   
+Tim Salimans, Ian J. Goodfellow, W. Zaremba, Vicki Cheung, A. Radford, and Xi Chen. Improved techniques for training gans. ArXiv, abs/1606.03498, 2016.   
+Adam Santoro, Ryan Faulkner, David Raposo, Jack Rae, Mike Chrzanowski, Theophane Weber, Daan Wierstra, Oriol Vinyals, Razvan Pascanu, and Timothy Lillicrap. Relational recurrent neural networks. In Advances in neural information processing systems, pp. 7299–7310, 2018.   
+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.   
+Thomas Scialom, Paul-Alexis Dray, Sylvain Lamprier, Benjamin Piwowarski, and Jacopo Staiano. Coldgans: Taming language gans with cautious sampling strategies. ArXiv, abs/2006.04643, 2020.   
+Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, et al. Google’s neural machine translation system: Bridging the gap between human and machine translation. arXiv preprint arXiv:1609.08144, 2016.   
+Lantao Yu, W. Zhang, J. Wang, and Y. Yu. Seqgan: Sequence generative adversarial nets with policy gradient. In AAAI, 2017.   
+Yizhe Zhang, Zhe Gan, K. Fan, Z. Chen, Ricardo Henao, Dinghan Shen, and L. Carin. Adversarial feature matching for text generation. In ICML, 2017.   
+Wangchunshu Zhou, Tao Ge, Ke Xu, Furu Wei, and M. Zhou. Self-adversarial learning with comparative discrimination for text generation. ArXiv, abs/2001.11691, 2020.   
+Yaoming Zhu, S. Lu, L. Zheng, Jiaxian Guo, W. Zhang, J. Wang, and Y. Yu. Texygen: A benchmarking platform for text generation models. The 41st International ACM SIGIR Conference on Research Development in Information Retrieval, 2018.
+
+# A EXPERIMENTAL DETAILS
+
+A.1 TRAINING DETAILS
+
+Generator The input embedding dimension for the generator is set to 32. As for the LSTM generator, we use the single-layer LSTM with the hidden dimension of 32. Then adopt a linear transformation to get the logits at each time step, and iteratively feed the sampled output tokens into the generator at the next time step. As for the RMC generator, we follow the experimental settings as (Nie et al., 2019), setting the memory size as 256, memory slots as 1, attention head number as 2. After the one-layer RMC, a linear projection is applied to get the output logits at each step.
+
+CNN Discriminator The input embedding dimension for the discriminator is set to 64. We adopt the filter size of $\{ 2 , 3 , 4 , 5 \}$ with the number of 300 channels for each. A max-over-time pooling is adopted after the convolution layer. Afterward, a highway layer that is identical to SeqGAN (Yu et al., 2017) is used followed by a linear transformation with the dimension of 100. Finally, apply a linear transformation to get the final logits. The feature extractor for Feature Statistics Alignment shares the identical architecture and weights with the CNN discriminator, and the leaked feature dimension is set to 100.
+
+Optimization We use Adam optimizer with $\beta _ { 1 } = 0 . 9$ and $\beta _ { 2 } = 0 . 9 9 9$ . The initial learning rate for the generator was set to 1e-2 and 1e-4 for pretraining and adversarial training. We set the initial learning rate as 1e-4 for the discriminator during the adversarial training. To prevent overfitting, we clip the gradients of parameters whose $\mathrm { L _ { 2 } }$ norm exceeds 5.
+
+Training Procedure We conduct experiments to finetune the following experiments: the batch size of $\{ 3 2 , 6 4 , 1 2 8 \}$ , the Gumbel-Softmax temperature $\tau \in \{ 1 , 0 . 5 , 0 . 1 , 0 . 0 1 . 0 . 0 0 1 \}$ . The training steps of generator and discriminators are set to $g = 1$ and $d = 5$ , respectively. The generator is pretrained for 150 epochs before adversarial training. Finally, the optimal batch size is set to 128 for both synthetic and real datasets. It is worth noting that we also test the batch size to 256, which requires too much GPU resource but do not show obvious improvement.
+
+# B NEGATIVE RESULTS
+
+Here we list some approaches that we tried but proved unsuccessful:
+
+• Using Mogrifier LSTM as the generator, which achieves similar results as vanilla LSTMs on the synthetic data. Using a Wasserstein loss instead of current Relativistic Discriminator. Not as stable as current solutions. Using the Transformer model as the discriminator. It achieves unsatisfied results with the current experimental settings. Using interleaved training instead of two-stage training, i.e., adversarial training after pretraining. It is unsuccessful to train the generator for 15 iterations after one iteration using MLE.   
+• Using top- $\mathbf { \nabla } \cdot \mathbf { k }$ sampling and nucleus sampling, instead of the argmax in the Gumbel-Max trick. This does not always boost the final performance.   
+• Using a hinge loss on the discriminator. This did not improve over the current relativistic loss.
+
+# C EVALUATION DETAILS
+
+# C.1 $\mathrm { N L L } _ { \mathrm { G E N } }$ ON SYNTHETIC DATA
+
+Table 6 reports the ${ \mathrm { N L L } } _ { \mathrm { g e n } }$ metric of comparison models. It can be seen that the proposed model with RMC generator achieves comparative results in comparison with baselines in terms of both short and long texts.
+
+Table 6: The ${ \mathrm { N L L } } _ { \mathrm { g e n } }$ performance of different models $\mathit { \Pi } ( \tau = 1 )$ ) on the synthetic dataset with the sequence length of 20 and 40 respectively. For the NLL score, the lower, the better.   
+
+<table><tr><td>Length</td><td>MLE</td><td>SeqGAN</td><td>RankGAN</td><td>SAL</td><td>Ours (LSTM)</td><td>Ours (RMC)</td></tr><tr><td>20</td><td>5.96</td><td>6.61</td><td>7.14</td><td>6.58</td><td>7.73</td><td>5.12</td></tr><tr><td>40</td><td>6.55</td><td>6.98</td><td>7.05</td><td>6.97</td><td>7.59</td><td>6.89</td></tr></table>
+
+# C.2 HUMAN EVALUATION
+
+Acceptance (i.e. if a sentence is acceptable), grammaticality (i.e., if a sentence is grammatically correct), and meaningfulness (i.e., if a sentence makes sense) are three main standards for the text quality evaluation. Please note that any minor text formatting issues which will not negatively influence the understanding and correctness of the sentences (e.g., punctuation, capitalization, spelling errors, extra spaces) can be ignored. Please also note: a sentence consists of less than 10 words should get one point deducted. Table 7 below gives more detailed criteria.
+
+It is worth to mention that the human evaluation is used to measure the quality of generated sentences rather than the diversity.
+
+Table 7: The human evaluation scale from 1 to 5 with corresponding criteria and example sentences.   
+
+<table><tr><td>Scale</td><td>Criterion&amp;Example</td></tr><tr><td>5 -Excellent</td><td>Grammatical,acceptable,and meaningful. For example,“a man is carving under yellow planes .&quot;</td></tr><tr><td>4-Good</td><td>Include 1 to 2 tiny grammatical errors,and the whole sentence is mostly acceptable and meaningful.For example,“two giraffe standing in front of them.&quot;</td></tr><tr><td>3-Fair</td><td>Include major grammatical errors, but the whole sentence is still acceptable and making sense.For example,“a kitchen with a grill roll from him .”</td></tr><tr><td>2-Poor</td><td>Include severe grammatical errors,and the whole sentence does not make sense,but some parts are still acceptable.For example,“a motorcycle on a paved road on the freeway &quot;</td></tr><tr><td>1 - Unacceptable</td><td>It is basically a string of words with random order and totally ungrammatical.The entire sentence does not make any sense.For example,“a city .&quot;</td></tr></table>
+
+# C.3 HUMAN EVALUATION ANALYSIS
+
+The model with MSA tends to generate grammatically correct sentences, and the sentences tend to be longer. For example, “A man is sitting on a motorcycle on a busy street, in a city.” Though it produces the samples with high quality by human evaluation, however, it does not solve the mode dropping collapse.
+
+In contrast, models with MDA tend to generate a more variety of sentences rather than repeated ones. Most sentences are grammatically correct and meaningful. They follow the SVO sentence structure with Preposition Phrase (PP) placed at the acceptable position in a sentence. Even though some of the auxiliary or main action verbs are missing, the meaning of each sentence can still be understandable and making sense. There is no obvious mode dropping issues according to the generated samples of MDA.
+
+# D DETAILED RESULTS OF ABLATION STUDY
+
+D.1 IMPACT OF FEATURE STATISTICS ALIGNMENT
+
+See Fig. 4 for the results of the ablation study on FSA techniques.
+
+# D.2 IMPACT OF LARGE BATCH SIZE
+
+Fig. 5 shows the training curve of our model on various batch sizes. It is observed that the increase of batch size could provide the performance boost, due to the variance reduction of gradients and the stability of adversarial dynamics.
+
+![](images/8b052c00cb7a44407c818c647687efa2d7394abb0d6cb755ce51ac76f547d1ea.jpg)  
+Figure 4: Training curves of BLEU scores on MS COCO Image Caption dataset w/ and w/o FSA mechanism.
+
+D.3 IMPACT OF GUMBEL-SOFTMAX TEMPERATURE
+
+Fig. 6 reports the BLEU scores with different temperatures on MS COCO dataset. It can be seen that a suitable $\tau$ could greatly advance the automatic evaluation scores.
+
+E GENERATED SAMPLES ON REAL DATASET
+
+E.1 GENERATED SAMPLES ON MS COCO DATASET
+
+Table 8 displays samples of generated samples from all baseline models and references on MS COCO dataset.
+
+Table 9 shows the randomly sampled sentences from the proposed models generated on MS COCO Dataset.
+
+# E.2 GENERATED SAMPLES ON EMNLP2017 WMT NEWS DATASET
+
+Table 10 presents the random sampled sentences from our models generated on EMNLP2017 WMT News Dataset.
+
+![](images/fcac4f799553fb88a4058f3d6856f45574e6e2183dd5cd1376d8decb6049e58d.jpg)  
+Figure 5: Training curves of BLEU scores on MS COCO Image Caption dataset with various batch sizes.
+
+Table 8: Samples of baseline models and real dataset on MS COCO Image Captioning dataset.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Samples</td></tr><tr><td rowspan=1 colspan=1>Real</td><td rowspan=1 colspan=1>a single kite flies high above a body of water as a person stands on the edge of the water .a man wearing an apron in an industrial kitchen reaching for a pot .</td></tr><tr><td rowspan=1 colspan=1>MLE</td><td rowspan=1 colspan=1>a man watches on his bike,ina lake on a feld .a women is standing behind an orange table in helmet on a child in the background .</td></tr><tr><td rowspan=1 colspan=1>SeqGAN</td><td rowspan=1 colspan=1>some people sitting on top of luggage near a truck .a man sitting in a bath tub on tops.</td></tr><tr><td rowspan=1 colspan=1>TextGAN</td><td rowspan=1 colspan=1>a man riding a motorcycle .a bathroom with a sink,and a table.</td></tr><tr><td rowspan=1 colspan=1>LeakGAN</td><td rowspan=1 colspan=1>a man standing next to her cellphone on a street sign .a woman is holding a child in the air.</td></tr><tr><td rowspan=1 colspan=1>MaliGAN</td><td rowspan=1 colspan=1>a woman is standing and another oak cake on a drain .a man standing in a kitchen with her laptop and two tables</td></tr><tr><td rowspan=1 colspan=1>RankGAN</td><td rowspan=1 colspan=1>a colorful bike is is down next to a large mirror .a man is riding a bike down a track.</td></tr><tr><td rowspan=1 colspan=1>RelGAN</td><td rowspan=1 colspan=1>a woman walking with a dog in the city in front of a city bus .a man sitting on a bed in a room with a chair on the couch.</td></tr><tr><td rowspan=1 colspan=1>Ours (MSA)</td><td rowspan=1 colspan=1>a man is sitting on a motorcycle on a busy street,in a city .a man siting on a motorcycle on a crowded street near a building ,with a bicycle in a parking lot .</td></tr><tr><td rowspan=1 colspan=1>Ours (MDA)</td><td rowspan=1 colspan=1>a person is riding a motorcycle on a city street with a woman standing on the back of it .a man with a woman standing next to a fire hydrant wearing a backpack .</td></tr></table>
+
+![](images/e5732d522c487e9e96adea2010fc54ecc8a794c31cc9773a6d7f41db2dc71c02.jpg)  
+Figure 6: Training curves of BLEU scores on MS COCO Image Caption dataset with various Gumbel-Softmax temperature values.
+
+Table 10: Randomly sampled 10 samples trained on EMNLP2017 WMT News dataset, with MDA (top row) and MSA (bottom row).   
+
+<table><tr><td>the british people would have tocommit to traveling toeurope and has beena priority for thefrst time ina decade. his priority finally becomes a hope for the police to be informed by the attck and is not on the scene. there is more thanayear before the startof thedayafter the european’s first,but it was not in the lasttwo years. it is notsomething thatisabouttobea15.6percentinthefourthquarter,accordingtoareportfromthethirdofthe week. “i’ve been a part of our last two years,”he said in a statement from the bbc’s today . now that’swhy we have to be a part of theUK. i&#x27;mnotsayingthat wasthe firstofthe Kindofpeople who werein the wrong butitistruethatit isyettobe determined. he willbeakey for the firsttime inadecade,and has helpedto stopthespreadofthe decade-overthe past year. so what if that’s the reason that they can be within the last two years. it was one ofthe mostinthefirstquarter,butitwas thefirstof thenearlytwo monthssince thestartofthefirstofthe day.</td></tr><tr><td>if you’re a new,and you have to be a part of the team. in a fox news,she has already been a major despite a conflict in the world . when you’re in the world,when they are growing. buti’ve been a part of the group for christmas . instead,there is no evidence to suggest that the united kingdom . the department of health and the enforcement and defense agencies last Thursday . if you’re the only in the world,and i’m sure.</td></tr></table>

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+        "text": "IMPROVING SEQUENCE GENERATIVE ADVERSARIAL NETWORKS WITH FEATURE STATISTICS ALIGNMENT ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+        "text": "ABSTRACT ",
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+        "text": "Generative Adversarial Networks (GAN) are facing great challenges in synthesizing sequences of discrete elements, such as mode dropping and unstable training. The binary classifier in the discriminator may limit the capacity of learning signals and thus hinder the advance of adversarial training. To address such issues, apart from the binary classification feedback, we harness a Feature Statistics Alignment (FSA) paradigm to deliver fine-grained signals in the latent high-dimensional representation space. Specifically, FSA forces the mean statistics of the fake data distribution to approach that of real data as close as possible in a finite-dimensional feature space. Experiments on synthetic and real benchmark datasets show the superior performance in quantitative evaluation and demonstrate the effectiveness of our approach to discrete sequence generation. To the best of our knowledge, the proposed architecture is the first that employs feature alignment regularization in the Gumbel-Softmax based GAN framework for sequence generation. ",
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+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Unsupervised sequence generation is the cornerstone for a plethora of applications, such as machine translation (Wu et al., 2016), image captioning (Anderson et al., 2018), and dialogue generation (Li et al., 2017). The most common approach to autoregressive sequence modeling is maximizing the likelihood of each token in the sequence given the previous partial observation. However, using maximum likelihood estimation (MLE) for sequence modeling is inherently prone to the exposure bias problem (Bengio et al., 2015), which results from the discrepancy between the training and inference stage: the generator predicts the next token conditioned on its previously generated ones during inference but conditioned on its prefix ground-truth tokens during training, yielding accumulative mismatch along with the increment of generated sequence length. ",
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+        "text": "Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) can serve as an alternative to models trained by MLE, which have achieved promising results in generating sequences of discrete elements, in particular, language sequences (Kusner & Hernandez-Lobato, 2016; Yu et al., 2017; Lin ´ et al., 2017; Guo et al., 2018; Fedus et al., 2018; Nie et al., 2019; de Masson d’Autume et al., 2019; Zhou et al., 2020; Scialom et al., 2020). GANs consist of two competing networks: a discriminator that is trained to distinguish the generated samples from real data, and a generator that aims to generate high-quality samples to fool the discriminator. ",
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+        "text": "Although having succeeded in avoiding exposure bias issues, GANs still suffer from some intrinsic problems, such as mode dropping, reward sparsity, and training instability. To enrich the informativeness of the discriminator’s training signal, several approaches have been proposed by measuring the latent features, such as feature distribution matching (Zhang et al., 2017; Chen et al., 2018) and comparative discriminators (Lin et al., 2017; Zhou et al., 2020). Zhang et al. (2017) and Chen et al. (2018) leveraged feature matching mechanism by minimizing the kernel-based moment-matching metric, such as Maximum Mean Discrepancy and Earth-Mover’s Distance, between encoded features. However, merely adopting feature matching in lieu of the original learning signal may lack some guiding feedback at the initial stage of training. ",
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+        "text": "Another approach is to compare the finite latent features with comparative discriminators like ranker and relativistic discriminator. Lin et al. (2017) maintained that the binary classification in the discriminator network limits the learning capacity of tasks because the diversity and richness are circumscribed by the degenerated distribution. RankGAN (Lin et al., 2017) replaced the binary classifier with a pairwise feature ranker by comparing the similarities between sample features in the latent space. SAL (Zhou et al., 2020) classified the encoded features of constructed pairwise training examples into three categories, i.e., better / worse / indistinguishable. Nevertheless, adopting a comparative discriminator could provide the fine-grained signals for updating the generator network and may require some coarse credits for further improvements. ",
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+            "Figure 1: (a) Standard GANs using a binary classifier as its discriminator; (b) GANs with Feature Statistics Alignment and relativistic discriminator that provide more instructive signals for updating the generator. "
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+        "type": "text",
+        "text": "In this work, we propose to improve the GANs for sequence generation by jointly considering both the feature statistics matching and relativistic discriminator to serve as fine-grained and coarse learning signals respectively. We leverage the Feature Statistics Alignment (FSA) paradigm to embed the latent feature representations in a finite feature space and force the distribution of generated samples to approach the real data distribution by minimizing the distance between their respective feature representation centroids. Intuitively, matching the mean feature representations of fake and real samples could make the two data distributions closer. Besides, the relativistic discriminator (Jolicoeur-Martineau, 2019) is employed to measure the comparative information between generated and real sequences and empirically to show the effectiveness during the model training. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our experimental results illustrate the effectiveness of FSA techniques and large batch size to alleviate the gradient vanishing problem and stabilize the training process in comparison with the vanilla Gumbel-Softmax GANs. Besides, our models could generate discrete text sequences with high quality in terms of the semantic coherence and grammatical correctness of language, as evaluated with crowdsourcing. Furthermore, we empirically demonstrate that the proposed architecture overshadows most existing models in terms of quantitative and qualitative evaluation. To the best of our knowledge, the proposed framework is the first to adopt the statistics feature alignment paradigm in the Gumbel-Softmax based GAN framework for discrete sequence generation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 ADVERSARIAL SEQUENCE GENERATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Adversarial sequence generation has attracted broad attention for its properties to solve the exposure bias issue suffered with maximum likelihood estimation (MLE) for generating language sequences. Based on the game theory, its goal is to train a generator network $G \\big ( z ; \\pmb { \\theta } ^ { ( G ) } \\big )$ that produces samples from the data distribution $p _ { \\mathrm { d a t a } } ( \\pmb { x } )$ by decoding the randomly initialized noise $_ { z }$ (i.e., standard normal distribution) into the sequence $\\pmb { x } = G ( \\pmb { z } ; \\pmb { \\theta } ^ { ( G ) } )$ , where the training signal is provided by the discriminator network $D ( \\pmb { x } ; \\pmb { \\phi } ^ { ( D ) } )$ that is trained to distinguish between the samples drawn from the real data distribution $p _ { \\mathrm { d a t a } }$ and those produced by the generator. The minimax objective of adversarial training is formulated as: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4fb6383f5187dc80d7abc00f714b33f7139996af56f0a63c203c9bf33330d360.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta ^ { ( G ) } } \\operatorname* { m a x } _ { \\phi ^ { ( D ) } } \\mathbb { E } _ { \\mathbf { x } \\sim p _ { \\mathrm { d a t a } } } \\big [ \\log D ( \\mathbf { \\boldsymbol { x } } ; \\phi ^ { ( D ) } ) \\big ] + \\mathbb { E } _ { \\mathbf { \\boldsymbol { z } } \\sim p _ { \\mathbf { z } } } \\big [ \\log \\big ( 1 - D _ { \\phi ^ { ( D ) } } \\big ( G ( \\boldsymbol { z } ; \\boldsymbol { \\theta } ^ { ( G ) } ) \\big ) \\big ) \\big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Despite the impressive results of GANs in the sequence generation (Yu et al., 2017; Gulrajani et al., 2017; Scialom et al., 2020), there are still several fundamental issues in the GAN training: (a) Training instability, which arises from the intrinsic nature of minimax games in GANs; (b) Mode dropping, which is the fact that GANs only generate samples with limited patterns in the real data distribution instead of attending to diverse patterns (Chen et al., 2018); (c) Reward sparsity, which is because that it is easier to train the discriminator than the generator, making it difficult to acquire the instructive feedback (Zhou et al., 2020). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Due to the non-differentiability of gradients caused by sampling operations between the generator and discriminator for sequence generation, the majority of previous works have resorted to reinforcement learning (RL) heuristics with Monte Carlo search to collect the credits from the discriminator. The usage of RL may further deteriorate the instability of model training and exacerbate the reward sparsity problem. Gumbel-Softmax relaxation has proven to be an alternative to RL techniques (Kusner & Hernandez-Lobato, 2016; Nie et al., 2019). How to efficiently train GANs with ´ the Gumbel-Softmax trick still remains under-explored. Therefore, we utilize the Gumbel-Softmax reparameterization instead of conventional policy gradients in our framework. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As illustrated in Fig. 1, standard GANs employ the real/fake binary classifier as the discriminator, which is prone to be overtrained in comparison with the generator (Salimans et al., 2016). To prevent the discriminator from overfitting and further stabilize the training process, we propose to leverage the FSA techniques and relativistic discriminator by comparing the fake and real distributions from two different aspects. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Compared with conventional GANs, the proposed framework enjoys the following advantages: (a) In the earlier training stage, the relativistic discriminator could estimate the probability that how much better the real data is in comparison with the generated samples. This could provide the relatively “coarse” credits to the generator. (b) When it comes to the later training stage and the quality of generated samples becomes high, FSA measures the “fine-grained” difference between latent features of fake and real data batches, precluding the discriminator from being overly confident. In other words, the FSA can be regarded as a kind of regularization for adversarial training. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.1 FEATURE STATISTICS ALIGNMENT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Intuitively, aligning the statistics of embedded feature representations increases the opportunities to capture the various modes of the data distribution. For brevity, we utilize the first-order mean statistics in our framework and leave the higher-order statistics for future work. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Denoting the feature extractor as $F _ { \\omega }$ parameterized by $\\omega$ , a minibatch of real data samples as $_ { \\textbf { \\em x } }$ with the batch size of $N$ , we propose two variants of FSA formulations, which calculate the mean squared difference and Euclidean distance between the minibatch centroids of real and fake feature representations. To reduce the parameter amount, we share the weights between the discriminator and feature extractor of FSA. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Mean Squared Alignment (MSA) We take the mean squared difference between the centroids of fake and generated distributions as the feature alignment metric: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/62db8fa5f13fb9cc5ebcf50f3027058aaf4265664763853ade807d97fe7d4845.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { M S A } } = \\left\\| \\mathbb { E } _ { { \\pmb { x } } \\sim p _ { \\mathrm { d a t a } } } \\big [ F _ { \\omega } ( { \\pmb x } ) \\big ] - \\mathbb { E } _ { { \\pmb z } \\sim p _ { \\pmb z } } \\big [ F _ { \\omega } \\big ( G ( { \\pmb z } ; { \\pmb \\theta } ^ { ( G ) } ) \\big ) \\big ] \\right\\| _ { 2 } ^ { 2 } } \\\\ { = \\| \\displaystyle \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } F _ { \\omega } ( \\pmb x _ { i } ) - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } F _ { \\omega } ( G ( { \\pmb z } _ { i } ; { \\pmb \\theta } ^ { ( G ) } ) ) \\big \\| _ { 2 } ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Mean Distance Alignment (MDA) Another intuitive approach is to calculate the distance between two sample centroids, which is equivalent to the square root of MSA mathematically: ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ddcb9a57116bf4143aa669f5e948fb471f94e8c801f68e864353139e5aaf09de.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\mathrm { M D A } } = \\left\\| \\mathbb { E } _ { { \\pmb { x } } \\sim p _ { \\mathrm { d a t a } } } \\big [ F _ { \\omega } ( { \\pmb x } ) \\big ] - \\mathbb { E } _ { { \\pmb z } \\sim p _ { \\pmb z } } \\big [ F _ { \\omega } \\big ( { \\pmb G } ( { \\pmb z } ; { \\pmb \\theta } ^ { ( G ) } ) \\big ) \\big ] \\right\\| _ { 2 } } \\\\ & { \\qquad = \\sqrt { L _ { \\mathrm { M S A } } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "By forcing the mean statistics of fake data to be close to the real samples, the generator could receive more informative signals in the training process. It is worth noting that the large batch size is helpful to reduce the variance of small mini-batches. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 RELATIVISTIC DISCRIMINATOR ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We consider the Relativistic Discriminator (Jolicoeur-Martineau, 2019) to take into account the relative confidence that the given real data is more realistic than the randomly sampled fake data. In the standard GAN, defining the discriminator as $D ( \\pmb { x } ) = \\mathrm { s i g m o i d } ( H ( \\pmb { x } ) )$ , where $H ( \\cdot )$ represents the non-transformed layer before the final non-linearity. The objectives for the discriminator and generator in terms of the Relativistic Discriminator are defined as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5cbb36e20a6dcc228a2f64d99d5da2aabcaf92c1ccdfdec162b6a0aff50e7fa5.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\mathrm { R D } } = - \\mathbb { E } _ { { \\pmb x } \\sim p _ { \\mathrm { d a t a } } , { \\pmb z } \\sim p _ { z } } [ \\log a \\big ( H ( { \\pmb x } ) - H ( G ( { \\pmb z } ; { \\pmb \\theta } ^ { ( G ) } ) ) \\big ) ] , } \\\\ & { \\mathcal { L } _ { \\mathrm { R G } } = - \\mathbb { E } _ { { \\pmb x } \\sim p _ { \\mathrm { d a t a } } , { \\pmb z } \\sim p _ { z } } [ \\log a \\big ( H ( G ( { \\pmb z } ; { \\pmb \\theta } ^ { ( G ) } ) ) - H ( { \\pmb x } ) \\big ) ] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $a$ represents the activation function to be relativistic (we use sigmoid function in our experiments), RD and RG denote the loss terms for the discriminator and generator respectively. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 OVERALL TRAINING OBJECTIVES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Previous works using moment matching techniques to support the training on both the discriminator and generator (Zhang et al., 2017; Chen et al., 2018). However, the generator is always more difficult to train than the discriminator, resulting in the training instability and reward sparsity. To relieve these issues, we only adopt the FSA techniques to enhance the generator but keep the objective of the discriminator unchanged. This could pass more informative signals only to the generator, and also prevent the discriminator to be overtrained. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Therefore, the overall training objectives for the proposed framework are defined as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/66ae6b23eb6729e9121f4f517f429a453f7302f51445f98dcef575c6cf30b26c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\mathrm { D } } = \\mathcal { L } _ { \\mathrm { R D } } , } \\\\ & { \\mathcal { L } _ { \\mathrm { G } } = \\mathcal { L } _ { \\mathrm { R G } } + \\mathcal { L } _ { \\mathrm { F S A } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            431,
+            420,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } _ { \\mathrm { F S A } }$ takes the form of $\\mathcal { L } _ { \\mathrm { M S A } }$ and ${ \\mathcal { L } } _ { \\mathrm { M D A } }$ as Eq. 2 and Eq. 4. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The goal of the discriminator is to maximize the gap between the generated and real data, whereas the generator jointly considers two different aspects simultaneously: it not only competes with the discriminator by maximizing the gap in terms of the relativistic signals but takes into account the additional leaked feature information from the discriminator. The idea of leaked features from the discriminator is similar to LeakGAN (Guo et al., 2018). The FSA term on the RHS can also be regarded as a dynamic regularizer for the sequence generator. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.4 TRAINING WITH DISCRETE SEQUENCE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.4.1 GUMBEL-SOFTMAX DISTRIBUTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Conventional GANs for generating discrete sequences are inherently unable to backpropagate the gradient through samples due to the non-differentiable sampling from a categorical distribution. Gumbel-Softmax distribution (Jang et al., 2017; Maddison et al., 2017) was proposed to deal with this issue by smoothly annealed to approximate the categorical distribution. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Denoting the output probabilities $\\pi _ { 1 } , \\pi _ { 2 } , \\cdots , \\pi _ { | V | }$ , where $| V |$ represents the output vocabulary size in the generator, the Gumbel-Max trick (Maddison et al., 2014) can be parameterized as: ",
+        "bbox": [
+            174,
+            690,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/229cdbc4088a91f473c5cbe8c3cc562cb62f979e1bbeb74260850086ffda37d0.jpg",
+        "text": "$$\ny _ { i } = \\mathrm { o n e } \\mathrm { . h o t } \\big ( \\arg \\operatorname* { m a x } _ { i } [ g _ { i } + \\log \\pi _ { i } ] \\big ) \\quad \\mathrm { ~ f o r ~ } i = 1 , \\cdots , | V | ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            720,
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+            743
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\{ g _ { i } | i = 1 , \\cdots , | V | \\}$ are i.i.d from the Gumbel(0,1) distribution, that is, $g _ { i } = - \\log ( - \\log u _ { i } )$ with $u _ { i }$ is drawn from a standard uniform distribution Uniform(0,1). one hot represents the $| V |$ - dimensional one hot encoding. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Gumbel-Softmax approximates the non-differential arg max operation using the softmax function: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0e114126b3b378aa32c44d5fbb4df65eafc4e01e64c43ad7faec5a1050a9a40e.jpg",
+        "text": "$$\n\\hat { y } _ { i } = \\frac { \\exp \\left( ( \\log ( \\pi _ { i } ) + g _ { i } ) / \\tau \\right) } { \\sum _ { j = 1 } ^ { | V | } \\exp \\left( ( \\log ( \\pi _ { j } ) + g _ { j } ) / \\tau \\right) } , \\quad \\mathrm { ~ f o r ~ } i = 1 , \\cdots , | V | ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            811,
+            696,
+            853
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau$ denotes the softmax temperature to modulate the exploitation and exploration during training. When $\\tau$ approaches too high, the approximation is nearly equiprobable, encouraging the generator to explore different options. In contrast, the lower $\\tau$ could discourage the exploration and tend to exploit during training. In particular, when $\\tau  0$ , $\\hat { y } _ { i }$ approaches the result of one hot operator as in Eq. 10, whereas $\\hat { y } _ { i }$ will degenerated into a uniform distribution when $\\tau  \\infty$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.4.2 ARCHITECTURE AND ADVERSARIAL TRAINING",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "RNN Generator Due to the free-running mode of GAN’s generator, it is unsuitable to adopt the transformer-based models due to the non-recurrence nature. Thus, we investigate the Recurrent Neural Network (RNN) based models, such as Long Short-Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997), and Relational Memory Core (RMC) (Santoro et al., 2018). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "CNN Discriminator & Feature Extractor We use the convolutional neural networks (CNN) architecture (Kim, 2014) as the discriminator and feature extractor for input sequences. We employ the multi-channel convolution using multiple filters with various window sizes to extract the distinct n-gram features, followed by a max-over-time pooling operation to gather the most salient features, i.e., features with the highest value for each feature map. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Adversarial Training Algorithm Alg. 1 illustrates the overall training process of the proposed framework. The Relativistic Discriminator and the generator could reach the Nash Equilibrium when the generator could fool the discriminator into accepting its output as being true. Since the discriminator is easy to be overtrained, we do not pretrain the discriminator but only pretrain the generator using MLE for few epochs. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1: Require: generator $G _ { \\theta }$ ; discriminator $D _ { \\phi }$ ; samples of real data $\\mathbb { S }$ ; generator training step $g$ ;   \ndiscriminator training step $k$ ; the generator pretraining epochs $m$ .   \n2: Pretrain $G _ { \\theta }$ using MLE on $\\mathbb { S }$ for $m$ epochs   \n3: repeat   \n4: for $g$ steps do   \n5: Sample a minibatch from real data $\\mathbb { S }$   \n6: Generate a minibatch of samples $\\mathbf { \\boldsymbol { x } } _ { g } \\sim G _ { \\theta }$   \n7: Update $G _ { \\theta }$ via Eq.(9)   \n8: end for   \n9: for $k$ steps do   \n10: Sample a minibatch from real data $\\mathbb { S }$   \n11: Sample a minibatch from the generated data   \n12: Train the discriminator $D _ { \\phi }$ by Eq.(8)   \n13: end for   \n14: until convergence ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETTING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Similar to (Lin et al., 2017; Guo et al., 2018; Nie et al., 2019; Zhou et al., 2020), we evaluate the proposed framework based on the Texygen benchmark platform (Zhu et al., 2018) for adversarial text generation. Experiments were conducted on the synthetic and real datasets: (a) synthetic data, which is generated by an oracle single-layer LSTM as in (Yu et al., 2017); (b) MS COCO Image Caption dataset (Chen et al., 2015); (c) EMNLP WMT 2017 News dataset (Guo et al., 2018). Table 1 summarizes the statistics of benchmark datasets for evaluation. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/079b0282959a5ad5340cbf7574da899fc1f1201ce23089e6e055e047f9737dd3.jpg",
+        "table_caption": [
+            "Algorithm 1 Adversarial Training with Feature Statisitcs Alignment ",
+            "Table 1: Summary of experimental datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>dataset</td><td>vocabulary size</td><td> sequence length</td><td>training set</td><td>test set</td></tr><tr><td>synthetic data</td><td>5,000</td><td>20/40</td><td>10,000</td><td>10,000</td></tr><tr><td>MS COCO</td><td>4,657</td><td>37</td><td>10,000</td><td>10,000</td></tr><tr><td>EMNLP2017 WMT News</td><td>5,255</td><td>51</td><td>27,8586</td><td>10,000</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For the synthetic data experiments, we utilize a single-layer LSTM initialized by the standard normal distribution as the oracle model, which is used to generate 10,000 samples of length 20 and 40 respectively as the real samples. We use the negative log-likelihood (NLL) under the oracle data distribution for evaluation, termed $\\mathrm { N L L } _ { \\mathrm { o r a c l e } }$ . For the real data experiments, the BLEU score (Papineni et al., 2002) serves as a metric to evaluate the n-gram statistics overlapping on the whole dataset. ",
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "To measure the diversity of generated samples, the NLL of the generator (denoted as ${ \\mathrm { N L L } } _ { \\mathrm { g e n . } }$ ) is used by computing the NLL of reference samples in the test set by the generator. Considering that BLEU scores always focus on the local text statistics and may be insufficient for evaluating the overall quality of texts, we conducted additional human evaluation via crowdsourcing on the generated samples of all comparison models. See Appendix A for more experimental details. ",
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+    {
+        "type": "text",
+        "text": "We compare the proposed framework with the MLE baseline and other state-of-the-art models, involving SeqGAN (Yu et al., 2017), RankGAN (Lin et al., 2017), LeakGAN (Guo et al., 2018), RelGAN (Nie et al., 2019), and Self-Adversarial Learning (SAL) (Zhou et al., 2020). ",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.2.1 SYNTHETIC DATA ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Table 2 illustrates the performance of different models on NLLoracle. Since our experiments achieved better results using MSA instead of MDA on synthetic data, we only report the optimal results with MSA in Table 2. Our models outperform other prevalent adversarial models in terms of the generated sample quality, demonstrating the effectiveness of our proposed method. ",
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+    {
+        "type": "text",
+        "text": "In practice, we found that the LSTM generator could outperform the RMC generator for producing the sequence with the length of 20, whereas RMC generators exceed LSTMs for long sequence generation. This may be due to the fact that LSTMs may forget the long-term dependencies with the sequence length increases, but the self-attention based relational memory cell in RMC could mitigate the issues using interactive memory slots. This demonstrates that the proposed model could produce samples with high quality. As to the diversity, our method achieves competitive ${ \\mathrm { N L L } } _ { \\mathrm { g e n } }$ score compared with baselines models (see Appendix C.1 for the detail). ",
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+    {
+        "type": "table",
+        "img_path": "images/58837c43d39b61154bb3fe99d903150ad8bf548b0a8fab2d2cf97f1a46c20201.jpg",
+        "table_caption": [
+            "Table 2: The $\\mathrm { N L L } _ { \\mathrm { o r a c l e } }$ performance of different models $\\tau = 1$ ) on the synthetic dataset with the sequence length of 20 and 40 respectively. For NLL, the lower, the better. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Length</td><td>MLE</td><td>SeqGAN</td><td>RankGAN</td><td>LeakGAN</td><td>RelGAN</td><td>SAL</td><td>Ours (LSTM)</td><td>Ours (RMC)</td><td>Real</td></tr><tr><td>20</td><td>9.038</td><td>8.736</td><td>8.247</td><td>7.038</td><td>6.680</td><td>7.71</td><td>5.047</td><td>5.819</td><td>5.750</td></tr><tr><td>40</td><td>10.411</td><td>10.310</td><td>9.958</td><td>7.191</td><td>6.765</td><td>9.31</td><td>5.909</td><td>5.087</td><td>4.071</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.2.2 MS COCO DATASET ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To further test the performance on the real data, we run and evaluate our model on MS COCO image caption datasets. The data and preprocessing remain the same as in Texygen (Zhu et al., 2018). Empirically, we found that RMC achieves better results in terms of the long sequences, and thus reports the results using the RMC generator if not otherwise specified. ",
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+        "type": "table",
+        "img_path": "images/d1c0c24042a664f10b54e69b06e0c0bcd811dcadb83dea3ad747d1235a4f927a.jpg",
+        "table_caption": [
+            "Table 3: BLEU and ${ \\mathrm { N L L } } _ { \\mathrm { g e n } }$ performance on MS COCO image captions with $\\tau \\ : = \\ : 0 . 1$ for the proposed models with MSA and MDA. For BLEU scores, the higher, the better. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>BLEU-2</td><td>BLEU-3</td><td>BLEU-4</td><td>BLEU-5</td><td>NLLgen</td></tr><tr><td>MLE</td><td>0.731</td><td>0.497</td><td>0.305</td><td>0.189</td><td>0.718</td></tr><tr><td>SeqGAN</td><td>0.745</td><td>0.498</td><td>0.294</td><td>0.180</td><td>1.082</td></tr><tr><td>RankGAN</td><td>0.743</td><td>0.467</td><td>0.264</td><td>0.156</td><td>1.344</td></tr><tr><td>LeakGAN</td><td>0.746</td><td>0.528</td><td>0.355</td><td>0.230</td><td>0.679</td></tr><tr><td>RelGAN</td><td>0.849</td><td>0.687</td><td>0.502</td><td>0.331</td><td>0.756</td></tr><tr><td>SAL</td><td>0.785</td><td>0.581</td><td>0.362</td><td>0.227</td><td>0.873</td></tr><tr><td>Ours (MSA)</td><td>0.959</td><td>0.866</td><td>0.759</td><td>0.630</td><td>0.760</td></tr><tr><td>Ours (MDA)</td><td>0.938</td><td>0.863</td><td>0.731</td><td>0.582</td><td>0.717</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 3 exhibits the final results of the BLEU and ${ \\mathrm { N L L } } _ { \\mathrm { g e n } }$ scores on different comparison models. Our models reported in Table 3 set the softmax temperature as 0.1 for the model with MSA and that with MDA. Notably, our model shows the significant improvement on previous methods, consistently overshadowing the state-of-the-art models in terms of the sample quality (indicated by BLEU scores) while maintaining the diversity (indicated by $\\mathrm { N L L _ { g e n } } ,$ ). ",
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+    {
+        "type": "text",
+        "text": "4.2.3 EMNLP2017 WMT NEWS DATASET ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 4 presents the considerable improvements of our model (w.r.t. quality) on EMNLP2017 WMT News dataset, with the temperature of 1 for models with MSA and MDA. The maximum length in the EMNLP2017 dataset is 51, greatly challenging the generation task. ",
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+    {
+        "type": "text",
+        "text": "It can be observed that only RankGAN and LeakGAN slightly outrank the MLE in terms of the ${ \\mathrm { N L L } } _ { \\mathrm { g e n } }$ , which may be due to the fact that feature ranking or leakage information could pass more internal information from the discriminator to the generator, and thereby smoothly assist the training process of the generator. By leveraging the FSA methods, our model greatly outperforms these models and yielding the long sequences with promising qualities. ",
+        "bbox": [
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 4: The BLEU and ${ \\mathrm { N L L } } _ { \\mathrm { g e n } }$ performance on EMNLP2017 WMT News dataset with temperatures of 1 for the proposed models with MSA and MDA. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>BLEU-2</td><td>BLEU-3</td><td>BLEU-4</td><td>BLEU-5</td><td>NLLgen</td></tr><tr><td>MLE</td><td>0.768</td><td>0.473</td><td>0.240</td><td>0.126</td><td>2.382</td></tr><tr><td>SeqGAN</td><td>0.777</td><td>0.491</td><td>0.261</td><td>0.138</td><td>2.773</td></tr><tr><td>RankGAN</td><td>0.727</td><td>0.435</td><td>0.209</td><td>0.101</td><td>3.345</td></tr><tr><td>LeakGAN</td><td>0.826</td><td>0.645</td><td>0.437</td><td>0.272</td><td>2.356</td></tr><tr><td>RelGAN</td><td>0.881</td><td>0.705</td><td>0.501</td><td>0.319</td><td>2.482</td></tr><tr><td>SAL</td><td>0.788</td><td>0.523</td><td>0.281</td><td>0.149</td><td>2.578</td></tr><tr><td>Ours (MSA)</td><td>0.932</td><td>0.798</td><td>0.585</td><td>0.404</td><td>3.999</td></tr><tr><td>Ours (MDA)</td><td>0.916</td><td>0.784</td><td>0.592</td><td>0.386</td><td>2.732</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Apart from the automatic quantitative evaluation, we also conducted human evaluation on the MS COCO Image Captioning dataset. We randomly sampled 100 sentences for each model and the real data, then asked 12 different people to score them on the scale of 1-5 with anonymizing the model’s identity. Please see Appendix. C.2 for more details of human evaluation. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our models received the highest score in comparison with other models, where the model with MSA receives higher credits than that with MDA. By manually going through the generated samples, we found that models with MSA tend to generate long sentences with higher quality but run into the mode dropping issue. In contrast, models with MDA posses a better trade-off between the diversity and quality of samples, yielding promising results in comparison with previous models. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/fb2434071e1f6e51cf22dc484c28d94a8310a7c16307cd9f3eb3647596801ebd.jpg",
+        "table_caption": [
+            "Table 5: Mean and standard deviation results of human evaluation w.r.t different models on MS COCO Image Caption dataset. Note that “Real” indicates the real data samples. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>MLE</td><td>SeqGAN</td><td>RankGAN</td><td>LeakGAN</td><td>MaliGAN</td></tr><tr><td>Human score</td><td>3.127 ± 0.124</td><td>3.062 ± 0.100</td><td>3.048 ± 0.127</td><td>3.018 ± 0.104</td><td>3.031 ± 0.123</td></tr><tr><td>Model</td><td>TextGAN</td><td>RelGAN</td><td>Ours (MSA)</td><td>Ours (MDA)</td><td>Real</td></tr><tr><td>Human score</td><td>1.973 ± 0.168</td><td>3.687 ± 0.181</td><td>4.209 ± 0.245</td><td>3.878 ± 0.276</td><td>3.444 ± 0.122</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.3 DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Fig. 2 (Left) illustrates the ablation test of FSA in terms of the BLEU-4 score (See Appendix D.2 for all the results). It can be observed that the increasing trend for models with FSA outreached that w/o feature alignment mechanism and our models achieve superior performance in the later training stage. The utilization of FSA leads to significant performance gains, as it could provide the consecutive “fined-grained” smoother learning signals to update the generator. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Fig. 3 demonstrates the relative importance of each component of our models with an ablation test. The usage of FSA results in the most significant performance gain, followed by Gumbel-Softmax, and large batch size. A large batch size can boost the performance due to the variance reduction, accurate estimates of feature statistics, and stabilizing effect during the adversarial training (see Fig. 2 (Right)). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As to the Gumbel-Softmax trick, we conduct experiments on various temperature values and find that a fined-tuned temperature hyperparameter could account for the second important factor to contribute to the overall performance boost. Please check Appendix D.3 for the detailed results w.r.t the Gumbel-Softmax temperature. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/2b18d2132e606bf0e13a2708dbda85111aef2e0ce4f1ba72800a36a18cbba091.jpg",
+        "image_caption": [
+            "Figure 2: (Left) BLEU-4 scores of the proposed model w/ and w/o FSA mechanism. (Right) BLEU4 scores of the proposed models with various batch size (64 v.s. 128). The vertical dash lines indicate the end of generator pretraining. MS COCO results unless otherwise specified. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "There has been a large category of GANs for sequence generation, which heavily rely on the RL paradigm. SeqGAN (Yu et al., 2017) regards the sequence generation as a Markov decision making process, estimates the rewards via Monte Carlo search, and trains the generator with policy gradient. RankGAN (Lin et al., 2017) and SAL (Zhou et al., 2020) replace the binary classifier in the discriminator as comparative discriminators to take into account the re",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/a9b4402d9359f78aa8566845566a8ea70af6b5177cb955b27152c7882c60c327.jpg",
+        "image_caption": [
+            "Figure 3: Ablation Study of the proposed model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "lation between constructed pair samples. MaliGAN (Che et al., 2017) utilizes the information in the discriminator as an additional source of training signals on the MLE objective to reduce the variance of gradients. LeakGAN (Guo et al., 2018) leaks the intermediate feature information via a manager to guide the generator, which is inspired by hierarchical RL. ColdGAN (Scialom et al., 2020) integrates the advance of importance sampling, Proximal Policy Optimization (PPO) algorithm (Schulman et al., 2017), and nucleus sampling (Holtzman et al., 2019) for finetuning the pretrained T5 (Raffel et al., 2019) and BART (Lewis et al., 2019). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Another approach applies non-RL methods for adversarial sequence generation by either approximating the categorical sampling or directly using the continuous latent representation. TextGAN (Zhang et al., 2017) uses feature matching techniques via a kernelized discrepancy in the Reproducing Kernel Hilbert Space. FMGAN (Chen et al., 2018) proposes to match the feature distributions using a Feature-Mover’s Distance. Similar to our proposed model, both of them apply annealed soft-argmax for approximation. ARAML Ke et al. (2019) utilizes Reward Augmented Maximum Likelihood by sampling from the stationary distribution to acquire rewards. However, none of them adopt the Gumbel-Max trick to reparameterize the categorical sampling. Besides, they applied feature matching as the training objectives of both the discriminator and generator, whereas we only apply the feature statistics matching to modulate the generator. Gumbel-Softmax (GS) GAN (Kusner & Hernandez-Lobato, 2016) and RelGAN (Nie et al., 2019) prove the effectiveness ´ of Gumbel-Softmax on unsupervised sequence generation. DialogeWAR Gu et al. (2018) employs a GS GAN within the latent variable space for dialogue generation. However, improving the training of GS GANs still remains an open problem. Our model aims to promote the GS GAN with the proposed FSA and other techniques to boost the training of language GANs. We also report a list of other techniques we tried but proved to be unsuccessful or unnecessary in Appendix B. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            813,
+            318,
+            829
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We propose an adversarial training framework for discrete sequence generation, by leveraging the advance of Feature Statistics Alignment and Gumbel-Softmax relaxation. Our model empirically shows superior performance in terms of the quantitative and human evaluation. In the future, it would be a promising direction to extend the proposed model to conditional text generation, such as text style transfer. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "text": "Jiwei Li, Will Monroe, Tianlin Shi, Sebastien Jean, Alan Ritter, and Dan Jurafsky. Adversarial´ learning for neural dialogue generation. arXiv preprint arXiv:1701.06547, 2017.   \nKevin Lin, Dianqi Li, Xiaodong He, M. Sun, and Zhengyou Zhang. Adversarial ranking for language generation. ArXiv, abs/1705.11001, 2017.   \nChris J Maddison, Daniel Tarlow, and Tom Minka. A\\* sampling. In Advances in Neural Information Processing Systems, pp. 3086–3094, 2014.   \nChris J. Maddison, A. Mnih, and Y. Teh. The concrete distribution: A continuous relaxation of discrete random variables. ArXiv, abs/1611.00712, 2017.   \nWeili Nie, Nina Narodytska, and Ankit B. Patel. Relgan: Relational generative adversarial networks for text generation. In ICLR, 2019.   \nKishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. Bleu: a method for automatic evaluation of machine translation. In Proceedings of the 40th annual meeting of the Association for Computational Linguistics, pp. 311–318, 2002.   \nColin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. Exploring the limits of transfer learning with a unified text-to-text transformer. arXiv preprint arXiv:1910.10683, 2019.   \nTim Salimans, Ian J. Goodfellow, W. Zaremba, Vicki Cheung, A. Radford, and Xi Chen. Improved techniques for training gans. ArXiv, abs/1606.03498, 2016.   \nAdam Santoro, Ryan Faulkner, David Raposo, Jack Rae, Mike Chrzanowski, Theophane Weber, Daan Wierstra, Oriol Vinyals, Razvan Pascanu, and Timothy Lillicrap. Relational recurrent neural networks. In Advances in neural information processing systems, pp. 7299–7310, 2018.   \nJohn Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.   \nThomas Scialom, Paul-Alexis Dray, Sylvain Lamprier, Benjamin Piwowarski, and Jacopo Staiano. Coldgans: Taming language gans with cautious sampling strategies. ArXiv, abs/2006.04643, 2020.   \nYonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, et al. Google’s neural machine translation system: Bridging the gap between human and machine translation. arXiv preprint arXiv:1609.08144, 2016.   \nLantao Yu, W. Zhang, J. Wang, and Y. Yu. Seqgan: Sequence generative adversarial nets with policy gradient. In AAAI, 2017.   \nYizhe Zhang, Zhe Gan, K. Fan, Z. Chen, Ricardo Henao, Dinghan Shen, and L. Carin. Adversarial feature matching for text generation. In ICML, 2017.   \nWangchunshu Zhou, Tao Ge, Ke Xu, Furu Wei, and M. Zhou. Self-adversarial learning with comparative discrimination for text generation. ArXiv, abs/2001.11691, 2020.   \nYaoming Zhu, S. Lu, L. Zheng, Jiaxian Guo, W. Zhang, J. Wang, and Y. Yu. Texygen: A benchmarking platform for text generation models. The 41st International ACM SIGIR Conference on Research Development in Information Retrieval, 2018. ",
+        "bbox": [
+            171,
+            98,
+            828,
+            799
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            102,
+            421,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 TRAINING DETAILS ",
+        "bbox": [
+            176,
+            132,
+            354,
+            147
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Generator The input embedding dimension for the generator is set to 32. As for the LSTM generator, we use the single-layer LSTM with the hidden dimension of 32. Then adopt a linear transformation to get the logits at each time step, and iteratively feed the sampled output tokens into the generator at the next time step. As for the RMC generator, we follow the experimental settings as (Nie et al., 2019), setting the memory size as 256, memory slots as 1, attention head number as 2. After the one-layer RMC, a linear projection is applied to get the output logits at each step. ",
+        "bbox": [
+            174,
+            159,
+            825,
+            242
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "CNN Discriminator The input embedding dimension for the discriminator is set to 64. We adopt the filter size of $\\{ 2 , 3 , 4 , 5 \\}$ with the number of 300 channels for each. A max-over-time pooling is adopted after the convolution layer. Afterward, a highway layer that is identical to SeqGAN (Yu et al., 2017) is used followed by a linear transformation with the dimension of 100. Finally, apply a linear transformation to get the final logits. The feature extractor for Feature Statistics Alignment shares the identical architecture and weights with the CNN discriminator, and the leaked feature dimension is set to 100. ",
+        "bbox": [
+            173,
+            257,
+            825,
+            354
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Optimization We use Adam optimizer with $\\beta _ { 1 } = 0 . 9$ and $\\beta _ { 2 } = 0 . 9 9 9$ . The initial learning rate for the generator was set to 1e-2 and 1e-4 for pretraining and adversarial training. We set the initial learning rate as 1e-4 for the discriminator during the adversarial training. To prevent overfitting, we clip the gradients of parameters whose $\\mathrm { L _ { 2 } }$ norm exceeds 5. ",
+        "bbox": [
+            174,
+            369,
+            825,
+            426
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Training Procedure We conduct experiments to finetune the following experiments: the batch size of $\\{ 3 2 , 6 4 , 1 2 8 \\}$ , the Gumbel-Softmax temperature $\\tau \\in \\{ 1 , 0 . 5 , 0 . 1 , 0 . 0 1 . 0 . 0 0 1 \\}$ . The training steps of generator and discriminators are set to $g = 1$ and $d = 5$ , respectively. The generator is pretrained for 150 epochs before adversarial training. Finally, the optimal batch size is set to 128 for both synthetic and real datasets. It is worth noting that we also test the batch size to 256, which requires too much GPU resource but do not show obvious improvement. ",
+        "bbox": [
+            174,
+            440,
+            825,
+            525
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B NEGATIVE RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            544,
+            379,
+            560
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Here we list some approaches that we tried but proved unsuccessful: ",
+        "bbox": [
+            173,
+            575,
+            620,
+            590
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "• Using Mogrifier LSTM as the generator, which achieves similar results as vanilla LSTMs on the synthetic data. Using a Wasserstein loss instead of current Relativistic Discriminator. Not as stable as current solutions. Using the Transformer model as the discriminator. It achieves unsatisfied results with the current experimental settings. Using interleaved training instead of two-stage training, i.e., adversarial training after pretraining. It is unsuccessful to train the generator for 15 iterations after one iteration using MLE.   \n• Using top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ sampling and nucleus sampling, instead of the argmax in the Gumbel-Max trick. This does not always boost the final performance.   \n• Using a hinge loss on the discriminator. This did not improve over the current relativistic loss. ",
+        "bbox": [
+            215,
+            601,
+            825,
+            805
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C EVALUATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            824,
+            397,
+            840
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C.1 $\\mathrm { N L L } _ { \\mathrm { G E N } }$ ON SYNTHETIC DATA ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            854,
+            426,
+            871
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Table 6 reports the ${ \\mathrm { N L L } } _ { \\mathrm { g e n } }$ metric of comparison models. It can be seen that the proposed model with RMC generator achieves comparative results in comparison with baselines in terms of both short and long texts. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/ac842b9b2dac547e2fd0199716b67c05041711ea4ba0e8de5d3744507493634c.jpg",
+        "table_caption": [
+            "Table 6: The ${ \\mathrm { N L L } } _ { \\mathrm { g e n } }$ performance of different models $\\mathit { \\Pi } ( \\tau = 1 )$ ) on the synthetic dataset with the sequence length of 20 and 40 respectively. For the NLL score, the lower, the better. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Length</td><td>MLE</td><td>SeqGAN</td><td>RankGAN</td><td>SAL</td><td>Ours (LSTM)</td><td>Ours (RMC)</td></tr><tr><td>20</td><td>5.96</td><td>6.61</td><td>7.14</td><td>6.58</td><td>7.73</td><td>5.12</td></tr><tr><td>40</td><td>6.55</td><td>6.98</td><td>7.05</td><td>6.97</td><td>7.59</td><td>6.89</td></tr></table>",
+        "bbox": [
+            236,
+            133,
+            759,
+            190
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C.2 HUMAN EVALUATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            207,
+            369,
+            222
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Acceptance (i.e. if a sentence is acceptable), grammaticality (i.e., if a sentence is grammatically correct), and meaningfulness (i.e., if a sentence makes sense) are three main standards for the text quality evaluation. Please note that any minor text formatting issues which will not negatively influence the understanding and correctness of the sentences (e.g., punctuation, capitalization, spelling errors, extra spaces) can be ignored. Please also note: a sentence consists of less than 10 words should get one point deducted. Table 7 below gives more detailed criteria. ",
+        "bbox": [
+            174,
+            233,
+            825,
+            316
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "It is worth to mention that the human evaluation is used to measure the quality of generated sentences rather than the diversity. ",
+        "bbox": [
+            176,
+            324,
+            821,
+            352
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/53f94dce573bac8129f4a6901bbbf205f91d6bf9489eb1b2f499b3fc8f5a8c30.jpg",
+        "table_caption": [
+            "Table 7: The human evaluation scale from 1 to 5 with corresponding criteria and example sentences. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Scale</td><td>Criterion&amp;Example</td></tr><tr><td>5 -Excellent</td><td>Grammatical,acceptable,and meaningful. For example,“a man is carving under yellow planes .&quot;</td></tr><tr><td>4-Good</td><td>Include 1 to 2 tiny grammatical errors,and the whole sentence is mostly acceptable and meaningful.For example,“two giraffe standing in front of them.&quot;</td></tr><tr><td>3-Fair</td><td>Include major grammatical errors, but the whole sentence is still acceptable and making sense.For example,“a kitchen with a grill roll from him .”</td></tr><tr><td>2-Poor</td><td>Include severe grammatical errors,and the whole sentence does not make sense,but some parts are still acceptable.For example,“a motorcycle on a paved road on the freeway &quot;</td></tr><tr><td>1 - Unacceptable</td><td>It is basically a string of words with random order and totally ungrammatical.The entire sentence does not make any sense.For example,“a city .&quot;</td></tr></table>",
+        "bbox": [
+            173,
+            392,
+            825,
+            541
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C.3 HUMAN EVALUATION ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            568,
+            444,
+            582
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The model with MSA tends to generate grammatically correct sentences, and the sentences tend to be longer. For example, “A man is sitting on a motorcycle on a busy street, in a city.” Though it produces the samples with high quality by human evaluation, however, it does not solve the mode dropping collapse. ",
+        "bbox": [
+            174,
+            594,
+            825,
+            651
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In contrast, models with MDA tend to generate a more variety of sentences rather than repeated ones. Most sentences are grammatically correct and meaningful. They follow the SVO sentence structure with Preposition Phrase (PP) placed at the acceptable position in a sentence. Even though some of the auxiliary or main action verbs are missing, the meaning of each sentence can still be understandable and making sense. There is no obvious mode dropping issues according to the generated samples of MDA. ",
+        "bbox": [
+            173,
+            657,
+            825,
+            741
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D DETAILED RESULTS OF ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            762,
+            560,
+            780
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D.1 IMPACT OF FEATURE STATISTICS ALIGNMENT ",
+        "bbox": [
+            174,
+            795,
+            539,
+            809
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "See Fig. 4 for the results of the ablation study on FSA techniques. ",
+        "bbox": [
+            174,
+            821,
+            604,
+            837
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D.2 IMPACT OF LARGE BATCH SIZE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            854,
+            437,
+            869
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Fig. 5 shows the training curve of our model on various batch sizes. It is observed that the increase of batch size could provide the performance boost, due to the variance reduction of gradients and the stability of adversarial dynamics. ",
+        "bbox": [
+            174,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/8b052c00cb7a44407c818c647687efa2d7394abb0d6cb755ce51ac76f547d1ea.jpg",
+        "image_caption": [
+            "Figure 4: Training curves of BLEU scores on MS COCO Image Caption dataset w/ and w/o FSA mechanism. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            104,
+            820,
+            462
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D.3 IMPACT OF GUMBEL-SOFTMAX TEMPERATURE",
+        "bbox": [
+            173,
+            518,
+            545,
+            534
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Fig. 6 reports the BLEU scores with different temperatures on MS COCO dataset. It can be seen that a suitable $\\tau$ could greatly advance the automatic evaluation scores. ",
+        "bbox": [
+            173,
+            544,
+            823,
+            573
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "E GENERATED SAMPLES ON REAL DATASET ",
+        "bbox": [
+            173,
+            593,
+            560,
+            609
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "E.1 GENERATED SAMPLES ON MS COCO DATASET ",
+        "bbox": [
+            174,
+            623,
+            549,
+            638
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 8 displays samples of generated samples from all baseline models and references on MS COCO dataset. ",
+        "bbox": [
+            176,
+            650,
+            823,
+            679
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 9 shows the randomly sampled sentences from the proposed models generated on MS COCO Dataset. ",
+        "bbox": [
+            174,
+            685,
+            821,
+            713
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "E.2 GENERATED SAMPLES ON EMNLP2017 WMT NEWS DATASET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            731,
+            658,
+            746
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 10 presents the random sampled sentences from our models generated on EMNLP2017 WMT News Dataset. ",
+        "bbox": [
+            173,
+            757,
+            820,
+            785
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/fcac4f799553fb88a4058f3d6856f45574e6e2183dd5cd1376d8decb6049e58d.jpg",
+        "image_caption": [
+            "Figure 5: Training curves of BLEU scores on MS COCO Image Caption dataset with various batch sizes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            123,
+            820,
+            483
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/87e9ca44ecc48a533cb089ec8093760b86e394079a6a91f3a33dec7e80491e97.jpg",
+        "table_caption": [
+            "Table 8: Samples of baseline models and real dataset on MS COCO Image Captioning dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Samples</td></tr><tr><td rowspan=1 colspan=1>Real</td><td rowspan=1 colspan=1>a single kite flies high above a body of water as a person stands on the edge of the water .a man wearing an apron in an industrial kitchen reaching for a pot .</td></tr><tr><td rowspan=1 colspan=1>MLE</td><td rowspan=1 colspan=1>a man watches on his bike,ina lake on a feld .a women is standing behind an orange table in helmet on a child in the background .</td></tr><tr><td rowspan=1 colspan=1>SeqGAN</td><td rowspan=1 colspan=1>some people sitting on top of luggage near a truck .a man sitting in a bath tub on tops.</td></tr><tr><td rowspan=1 colspan=1>TextGAN</td><td rowspan=1 colspan=1>a man riding a motorcycle .a bathroom with a sink,and a table.</td></tr><tr><td rowspan=1 colspan=1>LeakGAN</td><td rowspan=1 colspan=1>a man standing next to her cellphone on a street sign .a woman is holding a child in the air.</td></tr><tr><td rowspan=1 colspan=1>MaliGAN</td><td rowspan=1 colspan=1>a woman is standing and another oak cake on a drain .a man standing in a kitchen with her laptop and two tables</td></tr><tr><td rowspan=1 colspan=1>RankGAN</td><td rowspan=1 colspan=1>a colorful bike is is down next to a large mirror .a man is riding a bike down a track.</td></tr><tr><td rowspan=1 colspan=1>RelGAN</td><td rowspan=1 colspan=1>a woman walking with a dog in the city in front of a city bus .a man sitting on a bed in a room with a chair on the couch.</td></tr><tr><td rowspan=1 colspan=1>Ours (MSA)</td><td rowspan=1 colspan=1>a man is sitting on a motorcycle on a busy street,in a city .a man siting on a motorcycle on a crowded street near a building ,with a bicycle in a parking lot .</td></tr><tr><td rowspan=1 colspan=1>Ours (MDA)</td><td rowspan=1 colspan=1>a person is riding a motorcycle on a city street with a woman standing on the back of it .a man with a woman standing next to a fire hydrant wearing a backpack .</td></tr></table>",
+        "bbox": [
+            171,
+            584,
+            826,
+            902
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/e5732d522c487e9e96adea2010fc54ecc8a794c31cc9773a6d7f41db2dc71c02.jpg",
+        "image_caption": [
+            "Figure 6: Training curves of BLEU scores on MS COCO Image Caption dataset with various Gumbel-Softmax temperature values. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            136,
+            818,
+            497
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/72df0b3d214134fa5361983563e0be06451be26ef24db586d1fe1ce943d119ef.jpg",
+        "table_caption": [
+            "Table 10: Randomly sampled 10 samples trained on EMNLP2017 WMT News dataset, with MDA (top row) and MSA (bottom row). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>the british people would have tocommit to traveling toeurope and has beena priority for thefrst time ina decade. his priority finally becomes a hope for the police to be informed by the attck and is not on the scene. there is more thanayear before the startof thedayafter the european’s first,but it was not in the lasttwo years. it is notsomething thatisabouttobea15.6percentinthefourthquarter,accordingtoareportfromthethirdofthe week. “i’ve been a part of our last two years,”he said in a statement from the bbc’s today . now that’swhy we have to be a part of theUK. i&#x27;mnotsayingthat wasthe firstofthe Kindofpeople who werein the wrong butitistruethatit isyettobe determined. he willbeakey for the firsttime inadecade,and has helpedto stopthespreadofthe decade-overthe past year. so what if that’s the reason that they can be within the last two years. it was one ofthe mostinthefirstquarter,butitwas thefirstof thenearlytwo monthssince thestartofthefirstofthe day.</td></tr><tr><td>if you’re a new,and you have to be a part of the team. in a fox news,she has already been a major despite a conflict in the world . when you’re in the world,when they are growing. buti’ve been a part of the group for christmas . instead,there is no evidence to suggest that the united kingdom . the department of health and the enforcement and defense agencies last Thursday . if you’re the only in the world,and i’m sure.</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    }
+]

parse/train/dmCL033_YwO/dmCL033_YwO.md

@@ -0,0 +1,365 @@
+# DEEPERGCN: TRAINING DEEPER GCNS WITH GENERALIZED AGGREGATION FUNCTIONS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Graph Convolutional Networks (GCNs) have been drawing significant attention with the power of representation learning on graphs. Recent works developed frameworks to train deep GCNs. Such works show impressive results in tasks like point cloud classification and segmentation, and protein interaction prediction. In this work, we study the performance of such deep models in large scale graph datasets from the Open Graph Benchmark (OGB). In particular, we look at the effect of adequately choosing an aggregation function, and its effect on final performance. Common choices of aggregation are mean, max, and sum. It has shown that GCNs are sensitive to such aggregations when applied to different datasets. We further validate this point and propose to alleviate it by introducing a novel Generalized Aggregation Function. Our new aggregation not only covers all commonly used ones, but also can be tuned to learn customized functions for different tasks. Our generalized aggregation is fully differentiable, and thus its parameters can be learned in an end-to-end fashion. We add our generalized aggregation into a deep GCN framework and show it achieves state-of-the-art results in six benchmarks from OGB.
+
+# 1 INTRODUCTION
+
+The rise of availability of non-Euclidean data (Bronstein et al., 2017) has recently shed interest into the topic of Graph Convolutional Networks (GCNs). GCNs provide powerful deep learning architectures for irregular data, like point clouds and graphs. GCNs have proven valuable for applications in social networks (Tang & Liu, 2009), drug discovery (Zitnik & Leskovec, 2017; Wale et al., 2008), recommendation engines (Monti et al., 2017b; Ying et al., 2018), and point clouds (Wang et al., 2018; Li et al., 2019b). Recent works looked at frameworks to train deeper GCN architectures (Li et al., 2019b;a). These works demonstrate how increased depth leads to state-of-the-art performance on tasks like point cloud classification and segmentation, and protein interaction prediction. The power of deep models become more evident with the introduction of more challenging and largescale graph datasets. Such datasets were recently introduced in the Open Graph Benchmark (OGB) (Hu et al., 2020), for tasks of node classification, link prediction, and graph classification.
+
+Graph convolutions in GCNs are based on the notion of message passing (Gilmer et al., 2017). To compute a new node feature at each GCN layer, information is aggregated from the node and its connected neighbors. Given the nature of graphs, aggregation functions must be permutation invariant. This property guarantees invariance/equivariance to isomorphic graphs (Battaglia et al., 2018; Xu et al., 2019b; Maron et al., 2019a). Popular choices for aggregation functions are mean (Kipf & Welling, 2016), max (Hamilton et al., 2017), and sum (Xu et al., 2019b). Recent works suggest different aggregations have different performance impact depending on the task. For example, mean and sum perform best in node classification (Kipf & Welling, 2016), while max is favorable for dealing with 3D point clouds (Qi et al., 2017; Wang et al., 2019). Currently, all works rely on empirical analysis to choose aggregation functions.
+
+In DeepGCNs (Li et al. (2019b)), the authors complement aggregation functions with residual and dense connections, and dilated convolutions, in order to train very deep GCNs. Equipped with these new modules, GCNs with more than 100 layers can be reliably trained. Despite the potential of these new modules (Kipf & Welling, 2016; Hamilton et al., 2017; Velickovi ˇ c et al., 2018; Xu et al., 2019a), ´ it is still unclear if they are the ideal choice for DeepGCNs when handling large-scale graphs.
+
+![](images/55f48c1d200be91b6486c48d52bb04aec12d8c03ac28554d72669f52f9e2b12b.jpg)  
+Figure 1: Illustration of Generalized Message Aggregation Functions
+
+In this work, we analyze the performance of GCNs on large-scale graphs. In particular, we look at the effect of aggregation functions in performance. We unify aggregation functions by proposing a novel Generalized Aggregation Function (Figure 1) suited for graph convolutions. We show how our function covers all commonly used aggregations (mean, max, and sum), and its parameters can be tuned to learn customized functions for different tasks. Our novel aggregation is fully differentiable and can be learned in an end-to-end fashion in a deep GCN framework. In our experiments, we show the performance of baseline aggregations in various large-scale graph datasets. We then introduce our generalized aggregation and observe improved performance with the correct choice of aggregation parameters. Finally, we demonstrate how learning the parameters of our generalized aggregation, in an end-to-end fashion, leads to state-of-the-art performance in several OGB benchmarks. Our analysis indicates the choice of suitable aggregations is imperative to the performance of different tasks. A differentiable generalized aggregation function ensures the correct aggregation is used for each learning scenario.
+
+We summarize our contributions as two-fold: (1) We propose a novel Generalized Aggregation Function. This new function is suitable for GCNs, as it enjoys a permutation invariant property. We show how our generalized aggregation covers commonly used functions such as mean, max, and sum in graph convolutions. Additionally, we show how its parameters can be tuned to improve performance on diverse GCN tasks. Since this new function is fully differentiable, we show how its parameters can be learned in an end-to-end fashion. (2) We run extensive experiments on seven datasets from the Open Graph Benchmark (OGB). Our results show that combining depth with our generalized aggregation function achieves state-of-the-art in several of these benchmarks.
+
+# 2 RELATED WORK
+
+Graph Convolutional Networks (GCNs). Current GCN algorithms can be divided into two categories: spectral-based and spatial-based. Based on spectral graph theory, Bruna et al. (2013) firstly developed graph convolutions using the Fourier basis of a given graph in the spectral domain. Later, many methods proposed to apply improvements, extensions, and approximations on spectral-based GCNs (Kipf & Welling, 2016; Defferrard et al., 2016; Henaff et al., 2015; Levie et al., 2018; Li et al., 2018; Wu et al., 2019). Spatial-based GCNs (Scarselli et al., 2008; Hamilton et al., 2017; Monti et al., 2017a; Niepert et al., 2016; Gao et al., 2018; Xu et al., 2019b; Velickovi ˇ c et al., 2018) ´ define graph convolution operations directly on the graph by aggregating information from neighbor nodes. To address the scalability issue of GCNs on large-scale graphs, two main categories of algorithms exist: sampling-based (Hamilton et al., 2017; Chen et al., 2018b; Li et al., 2018; Chen et al., 2018a; Zeng et al., 2020) and clustering-based (Chiang et al., 2019).
+
+Training Deep GCNs. Despite the rapid and fruitful progress of GCNs, most prior work employs shallow GCNs. Several works attempt different ways of training deeper GCNs (Hamilton et al., 2017; Armeni et al., 2017; Rahimi et al., 2018; Xu et al., 2018). However, all these approaches are limited to 10 layers of depth, after which GCN performance would degrade because of vanishing gradient and over-smoothingLi et al. (2018). Inspired by the merits of training deep CNN-based networks (He et al., 2016a; Huang et al., 2017; Yu & Koltun, 2016), DeepGCNs (Li et al., 2019b) propose to train very deep GCNs (56 layers) by adapting residual/dense connections (ResGCN/DenseGCN) and dilated convolutions to GCNs. DeepGCN variants achieve state-of-the art results on S3DIS point cloud semantic segmentation (Armeni et al., 2017) and the PPI dataset. Many recent works focus on further addressing this phenomenon (Klicpera et al., 2019; Rong et al., 2020; Zhao & Akoglu, 2020; Chen et al., 2020; Gong et al., 2020; Rossi et al., 2020). In particular, Klicpera et al. (2019) propose a PageRank-based message passing mechanism involving the root node in the loop. Alternatively, DropEdge (Rong et al., 2020) randomly removes edges from the graph, and PairNorm (Zhao & Akoglu, 2020) develops a novel normalization layer. We find that the choice of aggregation may also limit the power of deep GCNs. In this work, we thoroughly study the important relation between aggregation functions and deep GCN architectures.
+
+Aggregation Functions for GCNs. GCNs update a node’s feature vector by aggregating feature information from its neighbors in the graph. Many different neighborhood aggregation functions that possess a permutation invariant property have been proposed (Hamilton et al., 2017; Velickovi ˇ c´ et al., 2018; Xu et al., 2019b). Specifically, Hamilton et al. (2017) examine mean, max, and LSTM aggregators, and they empirically find that max and LSTM achieve the best performance. Graph attention networks (GATs) (Velickovi ˇ c et al., 2018) employ the attention mechanism (Bahdanau ´ et al., 2015) to obtain different and trainable weights for neighbor nodes by learning the attention between their feature vectors and that of the central node. Thus, the aggregator in GATs operates like a learnable weighted mean. Furthermore, Xu et al. (2019b) propose a GCN architecture, denoted Graph Isomorphism Network (GIN), with a sum aggregation that has been shown to have high discriminative power according to the Weisfeiler-Lehman (WL) graph isomorphism test (Weisfeiler & Lehman, 1968). In this work, we propose generalized message aggregation functions, a new family of aggregation functions, that generalizes conventional aggregators including mean, max and sum. With the nature of differentiablity and continuity, generalized message aggregation functions provide a new perspective for designing GCN architectures.
+
+# 3 REPRESENTATION LEARNING ON GRAPHS
+
+Graph Representation. A graph $\mathcal { G }$ is usually defined as a tuple of two sets $\mathcal { G } = ( \nu , \mathcal { E } )$ , where $\mathcal { V } = \{ v _ { 1 } , v _ { 2 } , . . . , v _ { N } \}$ and $\mathcal { E } \subseteq \mathcal { V } \times \mathcal { V }$ are the sets of vertices and edges, respectively. If an edge $e _ { i j } = ( v _ { i } , v _ { j } ) \in \mathcal { E }$ for an undirected graph, $e _ { i j }$ is an edge connecting vertices $v _ { i }$ and $v _ { j }$ ; for a directed graph, $e _ { i j }$ is an edge directed from $v _ { i }$ to $v _ { j }$ . Usually, a vertex $v$ and an edge $e$ in the graph are associated with vertex features $\mathbf { h } _ { v } \in \mathbb { R } ^ { D }$ and edge features $\mathbf { h } _ { e } \in \mathbb { R } ^ { C }$ respectively.1
+
+GCNs for Learning Graph Representation. We define a general graph representation learning operator $\mathcal { F }$ , which takes as input a graph $\mathcal { G }$ and outputs a transformed graph $\mathcal { G } ^ { \prime }$ , i.e. ${ \mathcal { G } } ^ { \prime } = { \mathcal { F } } ( { \mathcal { G } } )$ . The features or even the topology of the graph can be learned or updated after the transformation $\mathcal { F }$ . Typical graph representation learning operators usually learn latent features or representations for graphs such as DeepWalk (Perozzi et al., 2014), Planetoid (Yang et al., 2016), Node2Vec (Grover & Leskovec, 2016), Chebyshev graph CNN (Defferrard et al., 2016), GCN (Kipf & Welling, 2016), Neural Message Passing Network (MPNN) (Gilmer et al., 2017), GraphSage (Hamilton et al., 2017), GAT (Velickovi ˇ c et al., 2018) and GIN (Xu et al., 2019b). In this work, we focus on the GCN family ´ and its message passing framework (Gilmer et al., 2017; Battaglia et al., 2018). To be specific, message passing based on the GCN operator $\mathcal { F }$ operating on vertex $v \in \mathcal V$ at the $l$ -th layer is defined as follows:
+
+$$
+\begin{array} { r l } & { \mathbf { m } _ { v u } ^ { ( l ) } = \rho ^ { ( l ) } ( \mathbf { h } _ { v } ^ { ( l ) } , \mathbf { h } _ { u } ^ { ( l ) } , \mathbf { h } _ { e _ { v u } } ^ { ( l ) } ) , \forall u \in \mathcal { N } ( v ) } \\ & { \mathbf { m } _ { v } ^ { ( l ) } = \zeta ^ { ( l ) } ( \{ \mathbf { m } _ { v u } ^ { ( l ) } \mid u \in \mathcal { N } ( v ) \} ) } \\ & { \mathbf { h } _ { v } ^ { ( l + 1 ) } = \phi ^ { ( l ) } ( \mathbf { h } _ { v } ^ { ( l ) } , \mathbf { m } _ { v } ^ { ( l ) } ) , } \end{array}
+$$
+
+where $\rho ^ { ( l ) } , \zeta ^ { ( l ) }$ , and $\phi ^ { ( l ) }$ are all learnable or differentiable functions for message construction, message aggregation, and vertex update at the $l$ -th layer, respectively. For simplicity, we only consider the case where vertex features are updated at each layer. It is straightforward to extend it to edge features. Message construction function $\boldsymbol { \rho } ^ { ( l ) }$ is applied to vertex features $\mathbf { h } _ { v } ^ { ( l ) }$ of $v$ , its neighbor’s features $\mathbf { h } _ { u } ^ { ( l ) }$ , and the corresponding edge features $\mathbf { h } _ { e _ { v u } }$ to construct an individual message $\bar { \mathbf { m } } _ { v u } ^ { ( l ) }$ for each neighbor $u \in \mathcal { N } ( v )$ . Message aggregation function $\zeta ^ { ( l ) }$ is commonly a permutation invariant set function that takes as input a countable unordered message set $\{ \mathbf { m } _ { v u } ^ { ( l ) } \mid u \in \mathcal { N } ( v ) \}$ , where $\mathbf { m } _ { v u } ^ { ( l ) } \in \mathbb { R } ^ { D }$ , and outputs a reduced or aggregated message $\mathbf { m } _ { v } ^ { ( l ) } \in \mathbb { R } ^ { D }$ . The permutation invariance of $\zeta ^ { ( l ) }$ guarantees the invariance/equivariance to isomorphic graphs (Battaglia et al., 2018). $\zeta ^ { ( l ) }$ can simply be a symmetric function such as mean (Kipf & Welling, 2016), max (Hamilton et al., 2017), or sum (Xu et al., 2019b). Vertex update function $\phi ^ { ( l ) }$ combines the original vertex features $\mathbf { h } _ { v } ^ { ( l ) }$ and the aggregated message $\mathbf { m } _ { v } ^ { ( l ) }$ to obtain the transformed vertex features $\bar { \mathbf { h } } _ { v } ^ { ( l + 1 ) }$ .
+
+# 4 BEYOND MEAN, MAX, AND SUM AGGREGATION FUNCTIONS
+
+Property 1 (Graph Isomorphic Equivariance). If a message aggregation function $\zeta$ is permutation invariant to the message set $\{ \mathbf { m } _ { v u } \mid u \in \mathcal { N } ( v ) \} ,$ , then the message passing based GCN operator $\mathcal { F }$ is equivariant to graph isomorphism, i.e. for any isomorphic graphs $\mathcal { G } _ { 1 }$ and $\mathcal { G } _ { 2 } = \sigma \star \mathcal { G } _ { 1 }$ , ${ \mathcal { F } } ( { \mathcal { G } } _ { 2 } ) =$ $\sigma \star { \mathcal { F } } ( { \mathcal { G } } _ { 1 } )$ , where $\star$ denotes a permutation operator on graphs.
+
+The invariance and equivariance properties on sets or GCNs have been discussed in many recent works. Zaheer et al. (2017) propose DeepSets based on permutation invariance and equivariance to deal with sets as inputs. Maron et al. (2019c) show the universality of invariant GCNs to any continuous invariant function. Keriven & Peyre (2019) further extend it to the equivariant case. ´ Maron et al. (2019b) compose networks by proposing invariant or equivariant linear layers and show that their models are as powerful as any MPNN (Gilmer et al., 2017). In this work, we study permutation invariant functions of GCNs, which enjoy these proven properties.
+
+# 4.1 GENERALIZED MESSAGE AGGREGATION FUNCTIONS
+
+To embrace the properties of invariance and equivariance (Property 1), many works in the graph learning field tend to use simple permutation invariant functions like mean (Kipf & Welling, 2016), max (Hamilton et al., 2017) and sum (Xu et al., 2019b). Inspired by the Weisfeiler-Lehman (WL) graph isomorphism test (Weisfeiler & Lehman, 1968), Xu et al. (2019b) propose a theoretical framework and analyze the representational power of GCNs with mean, max and sum aggregators. Although mean and max aggregators are proven to be less powerful than sum according to the WL test in $\mathrm { { X u } }$ et al., 2019b), they are found to be quite effective in the tasks of node classification (Kipf & Welling, 2016; Hamilton et al., 2017) and 3D point cloud processing (Qi et al., 2017; Wang et al., 2019) To go beyond these simple aggregation functions and study their characteristics, we define generalized aggregation functions in the following.
+
+Definition 2 (Generalized Message Aggregation Functions). We define a generalized message aggregation function $\zeta _ { z } ( \cdot )$ as a function that is parameterized by a continuous variable $_ { z }$ to produce a family of permutation invariant set functions, i.e. $\forall z$ , $\zeta _ { z } ( \cdot )$ is permutation invariant to the order of messages in the set $\{ \mathbf { m } _ { v u } \mid u \in \mathcal { N } ( v ) \}$ .
+
+In order to subsume the popular mean and max aggregations into the generalized space, we further define generalized mean-max aggregation parameterized by a scalar for message aggregation.
+
+Definition 3 (Generalized Mean-Max Aggregation). If there exists a pair of $x$ say $x _ { 1 }$ , $x _ { 2 }$ such that for any message set $\begin{array} { r } { \operatorname* { l i m } _ { x  x _ { 1 } } \zeta _ { x } ( \cdot ) = \mathbf { \bar { M } } \mathrm { e a n } ( \cdot ) ^ { 2 } } \end{array}$ and $\begin{array} { r } { \operatorname* { l i m } _ { x  x _ { 2 } } \zeta _ { x } ( \cdot ) ^ { \circ } = \operatorname { M a x } ( \cdot ) } \end{array}$ , then $\zeta _ { x } ( \cdot )$ is a generalized mean-max aggregation function.
+
+The nice properties of generalized mean-max aggregation functions can be summarized as follows: (1) they provide a large family of permutation invariant aggregation functions; (2) they are continuous and differentiable in $x$ and are potentially learnable; (3) it is possible to interpolate between $x _ { 1 }$ and $x _ { 2 }$ to find a better aggregator than mean and max for a given task. To empirically validate these properties, we propose two families of generalized mean-max aggregation functions based on Definition 3, namely SoftMax aggregation and PowerMean aggregation.
+
+Proposition 4 (SoftMax Aggregation). Given any message set $\{ \mathbf { m } _ { v u } \mid u \in \mathcal { N } ( v ) \}$ , $\mathbf { m } _ { v u } \in \mathbb { R } ^ { D }$ SoftMax $\mathcal { A } g g _ { \beta } ( \cdot )$ is a generalized mean-max aggregation function, where SoftMax $\begin{array} { r l } { \mathcal { A } g g _ { \beta } ( \cdot ) \ = } \end{array}$ $\begin{array} { r } { \sum _ { u \in \mathcal { N } ( v ) } \frac { \exp \left( \beta \mathbf { m } _ { v u } \right) } { \sum _ { i \in \mathcal { N } ( v ) } \exp \left( \beta \mathbf { m } _ { v i } \right) } \cdot \mathbf { m } _ { v u } } \end{array}$ . Here, $\beta$ is a continuous variable called an inverse temperature.
+
+The SoftMax function with a temperature has been studied in many machine learning areas, e.g. Energy-Based Learning (LeCun et al., 2006), Knowledge Distillation (Hinton et al., 2015) and Reinforcement Learning (Gao & Pavel, 2017). Here, for low inverse temperatures $\beta$ , SoftMax $\underline { { \mathbf { A g g } } } _ { \beta } ( \cdot )$ behaves like a mean aggregation. For high inverse temperatures, it approaches a max aggregation. Formally, $\begin{array} { r } { \operatorname* { l i m } _ { \beta \to 0 } \mathrm { S o f t M a x { \_ A g g } } _ { \beta } ( \cdot ) = \mathbf { M e a n } ( \cdot ) } \end{array}$ and $\begin{array} { r } { \operatorname* { l i m } _ { \beta \to \infty } \mathrm { S o f t M a x { \_ } A g g } _ { \beta } ( . ) = \mathbf { M a x } ( . ) } \end{array}$ . It can be regarded as a weighted summation that depends on the inverse temperature $\beta$ and the values of the elements themselves. The full proof of Proposition 4 is in the Appendix.
+
+Proposition 5 (PowerMean Aggregation). Given any message set $\{ \mathbf { m } _ { v u } \mid u \in \mathcal { N } ( v ) \} ,$ , $\begin{array} { r l r } { \mathbf { m } _ { v u } } & { { } \in } & { \mathbb { R } _ { + } ^ { D } } \end{array}$ , PowerMean $A g g _ { p } ( \cdot )$ is $a$ generalized mean-max aggregation function, where PowerMean Aggp(·) = ( 1|N (v)| $\begin{array} { r } { . A g g _ { p } ( . ) \ = \ ( \frac { 1 } { \left| \mathcal { N } ( v ) \right| } \sum _ { u \in \mathcal { N } ( v ) } \mathbf { m } _ { v u } ^ { p } ) ^ { 1 / p } } \end{array}$ . Here, $p$ is a non-zero, continuous variable denoting the $p$ -th power.
+
+Quasi-arithmetic mean (Kolmogorov & Castelnuovo, 1930) was proposed to unify the family of mean functions. Power mean is one member of the Quasi-arithmetic mean family. It is a generalized mean function that includes harmonic mean, geometric mean, arithmetic mean, and quadratic mean. The main difference between Proposition 4 and 5 is that Proposition 5 only holds when message features are all positive, i.e. $\mathbf { m } _ { v u } \in \mathbb { R } _ { + } ^ { D }$ . In particular, we have PowerMean $\mathbf { \bar { A } g g } _ { p = 1 } ( \cdot ) = \mathbf { M e a n } ( \cdot )$ and $\begin{array} { r } { \operatorname* { l i m } _ { p \to \infty } \mathrm { P o w e r M e a n . A g g } _ { p } ( \cdot ) = \mathrm { M a x } ( \cdot ) } \end{array}$ . PowerMean $\underline { { \mathbf { A g g } } } _ { p } ( \cdot )$ becomes the harmonic or the geometric mean aggregation when $p = - 1$ or $p  0$ , respectively. See the Appendix for the proof.
+
+To enhance expressive power according to the WL test (Xu et al., 2019b), we generalize the function space to cover the sum aggregator by introducing another control variable on the degree of vertices.
+
+Proposition 6 (Generalized Mean-Max-Sum Aggregation). Given any generalized mean-max aggregation function $\zeta _ { x } ( \cdot )$ , we can generalize the function to cover sum by combining it with the degree of vertices. For instance, by introducing a variable $y$ , we can compose a generalized meanmax-sum aggregation function as $\left| \mathcal { N } ( v ) \right| ^ { y } \cdot \zeta _ { x } ( \cdot )$ . We can observe that the function becomes a Sum aggregation when $\zeta _ { x } ( \cdot )$ is a Mean aggregation and $y = 1$ . By composing with SoftMax aggregation and PowerMean aggregation, we obtain SoftMaxSum $\mathcal { A } g g _ { ( \beta , y ) } ( \cdot )$ and PowerMeanSum $\mathcal { A } g g _ { ( p , y ) } ( \cdot )$ aggregation functions, respectively.
+
+# 4.2 GENERALIZED AGGREGATION NETWORKS (GEN)
+
+Generalized Message Passing Layer. Based on the Propositions above, we construct a simple message passing based GCN network that satisfies the conditions in Proposition 4 and 5. The key idea is to keep all the message features to be positive, so that generalized mean-max aggregation functions $( \mathrm { S o f t M a x \_ A g g } _ { \beta } ( \cdot )$ and PowerMean $\begin{array} { r } { \mathbf { \nabla } \cdot \mathbf { A } \mathbf { g } \mathbf { g } _ { p } ( \cdot ) . } \end{array}$ ) can be applied. We define the message construction function $\rho ^ { ( l ) }$ as follows:
+
+$$
+\mathbf { m } _ { v u } ^ { ( l ) } = \rho ^ { ( l ) } ( \mathbf { h } _ { v } ^ { ( l ) } , \mathbf { h } _ { u } ^ { ( l ) } , \mathbf { h } _ { e _ { v u } } ^ { ( l ) } ) = \mathrm { R e L U } ( \mathbf { h } _ { u } ^ { ( l ) } + \mathbb { 1 } ( \mathbf { h } _ { e _ { v u } } ^ { ( l ) } ) \cdot \mathbf { h } _ { e _ { v u } } ^ { ( l ) } ) + \epsilon , \forall u \in \mathcal { N } ( v )
+$$
+
+where $\mathrm { R e L U } ( \cdot )$ is a rectified linear unit (Nair & Hinton, 2010) that outputs values to be greater or equal to zero, $\mathbb { 1 } ( \cdot )$ is an indicator function being 1 when edge features exist otherwise 0, and $\epsilon$ is a small positive constant chosen to be $1 0 ^ { - 7 }$ . As the conditions are satisfied, we can choose the message aggregation function $\zeta ^ { ( l ) } ( \cdot )$ to be either $\begin{array} { r } { \mathbf { S o f t M a x \_ A g g } _ { \beta } ( \cdot ) } \end{array}$ , PowerMean $\mathbf { A g g } _ { p } ( \cdot )$ , SoftMaxSum $\mathbf { A g g } _ { ( \beta , y ) } ( \cdot )$ , or PowerMeanSum $\mathsf { \Pi } _ { 1 - } \mathbf { A g g } _ { ( p , y ) } ( \cdot )$ . As for the vertex update function $\phi ^ { ( l ) }$ , we use a simple multi-layer perceptron, where $\phi ^ { ( l ) } = \mathbf { M L P } ( \mathbf { h } _ { v } ^ { ( l ) } + \mathbf { m } _ { v } ^ { ( l ) } )$ .
+
+Skip Connections and Normalization. Skip connections and normalization techniques are important to train deep GCNs. Li et al. (2019b) propose residual GCN blocks with components following the ordering: GraphConv Normalization $ \mathrm { R e L U }  .$ Addition. He et al. (2016b) studied the effect of ordering of ResNet components in CNNs, showing its importance. As recommended in their paper, the output range of the residual function should be $( - \infty , + \infty )$ . Activation functions such as ReLU before addition may impede the representational power of deep models. Therefore, we adopt a pre-activation variant of residual connections for GCNs, which follows the ordering:
+
+Normalization $ \mathrm { R e L U }  \mathrm { G r a p h C o n v }  i$ Addition. Empirically, we find that the pre-activation version performs better. In our architectures, normalization methods such as BatchNorm (Ioffe & Szegedy, 2015) or LayerNorm (Ba et al., 2016) are applied to normalize vertex features.
+
+# 5 EXPERIMENTS
+
+We propose GENeralized Aggregation Networks (GEN) equipped with generalized message aggregators. To evaluate the effectiveness of these aggregators, we perform extensive experiments on the Open Graph Benchmark (OGB) (Hu et al., 2020), which includes a diverse set of challenging and large-scale tasks and datasets. We first conduct a comprehensive ablation study on the task of node property prediction on ogbn-proteins and ogbn-arxiv datasets. Then, we apply our GEN framework on the node property prediction dataset (ogbn-products), three graph property prediction datasets (ogbg-molhiv, ogbg-molpcba and ogbg-ppa), and one link property prediction dataset (ogbl-collab).
+
+# 5.1 EXPERIMENTAL SETUP
+
+Baseline Models. The PlainGCN model stacks GCNs from 3 layers to 112 layers without skip connections. Each GCN layer uses the same message passing operator as in GEN except the aggregation function is replaced by $\mathrm { \ S u m } ( \cdot )$ , Mean $( \cdot )$ , or $\operatorname { M a x } ( \cdot )$ aggregation. LayerNorm or BatchNorm is used in every layer before the ReLU activation function. Similar to Li et al. (2019b), we use ResGCN layers by adding residual connections to PlainGCN following the ordering: GraphGonv Normalization $ \mathrm { \ R e L U  }$ Addition. We construct the pre-activation version of ResGCN by changing the order of residual connections to Normalization → ReLU → GraphGonv Addition. We denote this as ${ \mathrm { R e s G C N } } +$ to differentiate it from ResGCN. The effect of residual connections can be found in Appendix A.
+
+ResGEN. The ResGEN models are designed using the message passing functions described in Section 4.2. The only difference between ResGEN and ${ \mathrm { R e s G C N } } +$ is that generalized message aggregators are used instead of $\operatorname { S u m } ( { \mathord { \cdot } } )$ , Mean $( \cdot )$ , or $\operatorname { M a x } ( \cdot )$ . For simplicity, we study generalized mean-max aggregators $( i . e . \mathrm { \ S o f t M a x \_ A g g } _ { \beta } ( \cdot )$ and PowerMean $\operatorname { A g g } _ { p } ( \cdot ) )$ which are parameterized by only one scalar. To explore the characteristics of the generalized message aggregators, we instantiate them with different hyper-parameters. Here, we freeze the values of $\beta$ to $1 0 ^ { n }$ , where $n \in \{ - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , 4 \}$ and $p$ to $\{ - 1 , 1 0 ^ { - 3 } , 1 , 2 , 3 , 4 , 5 , 1 0 \}$ .
+
+DyResGEN. In contrast to ResGEN, DyResGEN learns variables $\beta$ , $p$ or $y$ dynamically for every layer at every gradient descent step. By learning these variables, we avoid the need to painstakingly search for the best hyper-parameters. In doing so, DyResGEN can learn aggregation functions that adapt to the training process and the dataset. We study the potential of learning these variables for our proposed aggregators: SoftMax $\underline { { \mathbf { A g g } } } _ { \beta } ( \cdot )$ , PowerMean $\mathbf { A g g } _ { p } ( \cdot )$ , SoftMaxSum $\mathbf { A g g } _ { ( \beta , y ) } ( \cdot )$ , and PowerMeanSum $\mathbf { A g g } _ { ( p , y ) } ( \cdot )$ .
+
+Datasets. Traditional graph datasets have been shown limited and unable to provide reliable evaluation and rigorous comparison among methods (Hu et al., 2020; Dwivedi et al., 2020). Reasons include their small-scale nature, non-negligible duplication or leakage rates, unrealistic data splits, etc. Consequently, we conduct our experiments on the recently released datasets of Open Graph Benchmark (OGB) (Hu et al., 2020), which overcome the main drawbacks of commonly used datasets and thus are much more realistic and challenging. OGB datasets cover a variety of real-world applications and span several important domains ranging from social and information networks to biological networks, molecular graphs, and knowledge graphs. They also span a variety of prediction tasks at the level of nodes, graphs, and links/edges. In this work, experiments are performed on three OGB datasets for node property prediction, three OGB datasets for graph property prediction, and one OGB dataset for link property prediction. We introduce these seven datasets briefly in Appendix E.2. More detailed information about OGB datasets can be found in (Hu et al., 2020).
+
+Implementation Details. We first perform ablation studies on the ogbn-proteins and ogbn-arxiv datasets. Then, we evaluate our model on the other datasets and compare the performances with state-of-the-art (SOTA) methods. Since the ogbn-proteins dataset is very dense and comparably large, full-batch training is infeasible when considering very deep GCNs. We simply apply a random partition to generate batches for both mini-batch training and test. We set the number of partitions to
+
+10 for training and 5 for test, and we set the batch size to 1 subgraph. In comparison, the ogbn-arxiv dataset is relatively small, so we conduct experiments via full batch training and test in this case.
+
+# 5.2 RESULTS
+
+Aggregators may Limit the Power of Deep GCNs. Although pre-activation residual connections alleviate the effect of vanishing gradients and enable the training of deep GCNs, the choice of aggregation function is crucial to performance. In Table 1 (a) $R e s G C N { + }$ , we study how conventional aggregators (i.e. Sum, Mean and Max) behave on ogbn-proteins and ogbn-arxiv. We find that not all of them benefit from network depth. The aggregators perform inconsistently among different datasets and cause significant gaps in performance. For instance, the Max aggregator outperforms the other two by a large margin $( \sim 1 \% )$ for all network depths on ogbn-proteins, but reaches unsatisfactory results $( < 7 0 \% )$ and even becomes worse with depth increasing on ogbn-arxiv. The Mean aggregator performs the worst on ogbn-proteins, but the best $( 7 2 . 3 1 \% )$ with 28 layers on ogbn-arxiv.
+
+Table 1: Ablation studies of aggregation functions on the ogbn-proteins and ogbn-arxiv datasets   
+
+<table><tr><td>(a）</td><td>ogbn-proteins</td><td colspan="4"></td><td colspan="4">ogbn-arxiv</td></tr><tr><td>Model</td><td>#Layers</td><td>Sum</td><td>Mean</td><td>Max</td><td>SoftMax</td><td>Sum</td><td>Mean</td><td>Max</td><td>PowerMeanSum</td></tr><tr><td></td><td>3</td><td>82.67</td><td>79.69</td><td>83.47</td><td>83.42</td><td>70.89</td><td>71.17</td><td>69.59</td><td>72.12</td></tr><tr><td></td><td>7</td><td>83.00</td><td>80.84</td><td>84.65</td><td>84.81</td><td>71.17</td><td>71.83</td><td>69.57</td><td>72.31</td></tr><tr><td></td><td>14</td><td>83.33</td><td>82.25</td><td>85.16</td><td>85.29</td><td>71.50</td><td>72.03</td><td>68.97</td><td>72.14</td></tr><tr><td>PPPSSSY</td><td>28</td><td>83.98</td><td>83.28</td><td>85.26</td><td>85.51</td><td>71.32</td><td>72.31</td><td>66.91</td><td>72.40</td></tr><tr><td></td><td>56</td><td>84.48</td><td>83.52</td><td>86.05</td><td>86.12</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td>112</td><td>85.33</td><td>83.40</td><td>85.94</td><td>86.15</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td></td><td>avg.</td><td>83.80</td><td>82.16</td><td>85.09</td><td>85.22</td><td>71.22</td><td>71.83</td><td>68.76</td><td>72.24</td></tr><tr><td>(b)</td><td>ogbn-proteins</td><td></td><td></td><td></td><td></td><td>SoftMax</td><td></td><td></td><td></td></tr><tr><td>Model</td><td>#Layers</td><td>10-3 10-2</td><td></td><td>10-1</td><td>1</td><td>10</td><td>10²</td><td>103</td><td>104</td></tr><tr><td></td><td>3</td><td>79.69</td><td>78.90</td><td>77.80</td><td>81.69</td><td>83.24</td><td>83.16</td><td>83.07</td><td>83.21</td></tr><tr><td>PSeEEN</td><td>7</td><td>80.81</td><td>80.71</td><td>79.83</td><td>83.85</td><td>83.98</td><td>84.66</td><td>84.60</td><td>84.68</td></tr><tr><td></td><td>14</td><td>82.44</td><td>82.14</td><td>81.24</td><td>84.39</td><td>85.13</td><td>84.96</td><td>84.99</td><td>84.85</td></tr><tr><td></td><td>28</td><td>83.13</td><td>82.47</td><td>81.78</td><td>85.08</td><td>85.07</td><td>85.35</td><td>85.80</td><td>85.82</td></tr><tr><td></td><td>56</td><td>83.62</td><td>83.45</td><td>82.86</td><td>85.76</td><td>85.97</td><td>86.20</td><td>85.98</td><td>86.19</td></tr><tr><td></td><td>112</td><td>83.50</td><td>83.61</td><td>83.16</td><td>85.77</td><td>86.38</td><td>86.27</td><td>86.27</td><td>86.30</td></tr><tr><td></td><td>avg.</td><td>82.20</td><td>81.88</td><td>81.11</td><td>84.42</td><td>84.96</td><td>85.10</td><td>85.12</td><td>85.17</td></tr><tr><td>(c)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Model</td><td>ogbn-proteins</td><td></td><td></td><td></td><td></td><td>PowerMean</td><td></td><td></td><td></td></tr><tr><td></td><td>#Layers</td><td>-1</td><td>10-3</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5 82.89</td><td>10</td></tr><tr><td></td><td>3 7</td><td>82.34 83.36</td><td>81.06 81.08</td><td>78.52 81.02</td><td>80.23 83.49</td><td>82.01 83.67</td><td>81.61 84.82</td><td>84.54</td><td>82.89 84.50</td></tr><tr><td>PESESN</td><td>14</td><td>83.73</td><td>80.64</td><td>82.45</td><td>84.15</td><td></td><td>84.64</td><td>85.00</td><td>85.08</td></tr><tr><td></td><td>28</td><td></td><td>80.92</td><td></td><td>84.16</td><td>84.48</td><td>85.87</td><td>85.34</td><td>85.76</td></tr><tr><td></td><td></td><td>84.56</td><td></td><td>82.58</td><td></td><td>85.20</td><td></td><td></td><td></td></tr><tr><td></td><td>56</td><td>84.46</td><td>80.93</td><td>83.49</td><td>85.04</td><td>85.68</td><td>85.90</td><td>85.64</td><td>85.74</td></tr><tr><td></td><td>112</td><td>85.13</td><td>81.10</td><td>83.92</td><td>85.47</td><td>85.70</td><td>86.01</td><td>86.09</td><td>86.31</td></tr><tr><td></td><td>avg.</td><td>83.93</td><td>80.95</td><td>82.00</td><td>83.76</td><td>84.46</td><td>84.81</td><td>84.92</td><td>85.05</td></tr><tr><td>(d）</td><td>ogbn-proteins</td><td colspan="2">SoftMax</td><td colspan="2">SoftMaxSum</td><td colspan="2">PowerMean</td><td colspan="2">PowerMeanSum</td></tr><tr><td>Model</td><td>#Layers</td><td>Fixed</td><td>Learned</td><td>Fixed</td><td>Learned</td><td>Fixed</td><td>Learned</td><td>Fixed</td><td>Learned</td></tr><tr><td></td><td>3</td><td>81.69</td><td>83.42</td><td>83.06</td><td>83.42</td><td>78.52</td><td>82.25</td><td>81.70</td><td>83.71</td></tr><tr><td>BPPSECN</td><td>7</td><td>83.85</td><td>84.81</td><td>84.71</td><td>84.63</td><td>81.02</td><td>84.14</td><td>83.23</td><td>84.62</td></tr><tr><td></td><td>14</td><td>84.39</td><td>85.29</td><td>84.77</td><td>85.03</td><td>82.45</td><td>85.04</td><td>83.96</td><td>84.83</td></tr><tr><td></td><td>28</td><td>85.08</td><td>85.51</td><td>85.64</td><td>85.66</td><td>82.58</td><td>85.04</td><td>84.59</td><td>85.96</td></tr><tr><td></td><td>56</td><td>85.76</td><td>86.12</td><td>85.63</td><td>85.50</td><td>83.49</td><td>85.27</td><td>85.37</td><td>85.81</td></tr><tr><td></td><td>112</td><td>85.77</td><td>86.15</td><td>86.11</td><td>86.13</td><td>83.92</td><td>85.60</td><td>85.71</td><td>86.01</td></tr><tr><td></td><td>avg.</td><td>84.42</td><td>85.22</td><td>84.99</td><td>85.06</td><td>82.00</td><td>84.56</td><td>84.09</td><td>85.16</td></tr></table>
+
+Exploring Generalized Message Aggregators. In Table 1 (b) & (c) ResGEN, we examine SoftMax $\bar { \bf A } \mathrm { g g } _ { \beta } ( \cdot )$ and PowerMean $\operatorname { A g g } _ { p } ( \cdot )$ aggregators on ogbn-proteins by measuring test ROCAUC. Since both are generalized mean-max aggregations, they can theoretically perform at least as good as Mean and Max through interpolation. For SoftMax Agg, when $\beta = 1 0 ^ { - 3 }$ , it performs similarly to Mean aggregation $\mathrm { 8 2 . 2 0 \% }$ vs. $8 2 . 1 6 \%$ ). As $\beta$ increases to $1 0 ^ { 2 }$ , it achieves slightly better performance than Max aggregation. Remarkably, 112-layer ResGEN with SoftMax Agg reaches $\bar { 8 6 . 3 8 \% }$ and $8 6 . 3 0 \%$ ROC-AUC when $\beta = 1 0$ and $\beta = 1 0 ^ { 4 }$ respectively. For PowerMean Agg, we find that it reaches almost the same ROC-AUC as Mean when $p = 1$ (arithmetic mean). We also observe that all other orders of mean except $p = 1 0 ^ { - 3 }$ (akin to geometric mean) achieve better performance than the arithmetic mean. PowerMean Agg with $p = 1 0$ reaches the best ROC-AUC at $\mathrm { \bar { 8 6 . 3 1 \% } }$ with 112 layers. However, due to some numerical issues in PyTorch (Paszke et al., 2019), we are not able to use larger $p$ . These results empirically validate the discussion on existence of better generalized mean-max aggregators beyond mean and max in Section 4.1.
+
+Learning Dynamic Aggregators. Trying out every possible aggregator or searching hyperparameters is computationally expensive. Therefore, we propose DyResGEN to explore the potential of learning dynamic aggregators by learning the parameters $\beta , p .$ , and even $y$ within GEN. Table 1 (d) DyResGEN reports the results of learning $\beta$ , $\beta \& y$ , $p$ and $p \& y$ for SoftMax Agg, SoftMaxSum Agg, PowerMean Agg and PowerMeanSum Agg respectively. In practice, $y$ is bounded from 0 to 1 by a sigmoid function. In all experiments, we initialize the values of $\beta$ , $p$ to 1 and $y$ to 0.5 at the beginning of training. In order to show the improvement of the learning process, we also ablate experiments with fixed initial values. We denote aggregators with fixed initial values as Fixed and learned aggregators as Learned. We see that learning these variables consistently boosts the average performances of all the learned aggregators compared to the fixed initialized counterparts, which shows the effectiveness of learning adaptive aggregators. In particular, when $\beta$ is learned, DyResGEN-SoftMax achieves $8 6 . 1 5 \%$ at 112 layers. We observe that DyResGEN-SoftMax outperforms the best ResGEN-SoftMax $\langle \beta = 1 0 ^ { 4 }$ ) in terms of the average performance $8 5 . 2 2 \% \nu s$ . $8 5 . 1 7 \%$ ). Interesting, we find generalizing the sum aggregation with PowerMean significantly improve the average performance from $8 4 . 5 6 \%$ to $8 5 . 1 6 \%$ . We also put the best learned generalizing message aggregators in Table 1 (a) $R e s G C N { + }$ with gray color for a convenient comparison.
+
+Comparison with SOTA. We apply our GCN models to six other OGB datasets and compare results with the published SOTA method posted on OGB Learderboard at the time of this submission (See Table 2). The methods include Deepwalk (Perozzi et al., 2014), GCN (Kipf & Welling, 2016), GraphSAGE (Hamilton et al., 2017), GIN ( $\mathrm { { X u } }$ et al., 2019b), GIN or GCN with virtual nodes, JKNet (Xu et al., 2019a), GaAN (Zhang et al., 2018), GatedGCN (Bresson & Laurent, 2018), GAT (Velickovi ˇ c et al., 2018), HIMP (Fey et al., 2020), GCNII (Ming Chen et al., 2020), DAGNN (Liu ´ et al., 2020). The provided results on each dataset are obtained by averaging the results from 10 independent runs. It is clear that our proposed GCN models outperform SOTA on all four datasets. In two of these datasets (ogbn-proteins and ogbg-ppa), the improvement is substantial. The implementation details and more experimental results can be found in the Appendix.
+
+Table 2: Comparisons with SOTA.\* denotes that virtual nodes are used.   
+
+<table><tr><td>ogbn-proteins</td><td>GraphSAGE 77.68±0.20</td><td>GCN 72.51 ±0.35</td><td>GaAN 78.03 ±0.73</td><td></td><td></td><td>Ours 86.16 ± 0.16</td></tr><tr><td>ogbn-arxiv</td><td>GraphSAGE 71.49 ± 0.27</td><td>GCN 71.74± 0.29</td><td>GaAN 71.97 ± 0.24</td><td>GCNII 72.74± 0.16</td><td>JKNet DAGNN 72.19 ± 0.21 72.09±0.25</td><td>72.32 ± 0.27</td></tr><tr><td>ogbn-products</td><td>GraphSAGE 78.29 ±0.16</td><td>GCN 75.64± 0.21</td><td>ClusterGCN 78.97 ± 0.33</td><td>GraphSAINT 80.27± 0.26</td><td>GAT 79.45± 0.59</td><td>81.64 ± 0.30</td></tr><tr><td>ogbg-molhiv</td><td>GIN 75.58 ± 1.40</td><td>GCN 76.06 ± 0.97</td><td>GIN* 77.07 ± 1.49</td><td>GCN* 75.99 ± 1.19</td><td>HIMP 78.80 ±0.82</td><td>78.87 ± 1.24</td></tr><tr><td>ogbg-molpcba ogbg-ppa ogbl-collab</td><td>22.66±0.28 68.92 ± 1.00 GraphSAGE</td><td>20.20±0.24 68.39 ± 0.84 GCN</td><td>27.03±0.23 70.37 ± 1.07 DeepWalk</td><td>24.24±0.34 68.57 ± 0.61</td><td>77.12 ± 0.71</td><td>27.81± 0.38* 77.12 ± 0.71</td></tr></table>
+
+# 6 CONCLUSION
+
+In this work, we proposed a differentiable generalized message aggregation function, which defines a family of permutation invariant functions. We identify the choice of aggregation functions is crucial to the performance of deep GCNs. Experiments show that existence of better generalized aggregators beyond mean, max and sum. Empirically, we show the effectiveness of training our proposed deep GEN models, whereby we set a new SOTA on several datasets of the challenging Open Graph Benchmark. We believe the definition of such a generalized aggregation function provides a new view to the design of aggregation functions in GCNs.
+
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+
+# A DISCUSSION ON NETWORK DEPTH
+
+Depth & Residual connections. Experiments in Figure 2 show that residual connections significantly improve the training dynamic of deep GCN models. PlainGCN without skip connections suffers from vanishing gradient and does not gain any improvement from increasing depth. More prominent gains can be observed in ${ \mathrm { R e s G C N } } +$ compared to ResGCN as models go deeper. Notably, ${ \mathrm { R e s G C N } } +$ reaches smallest training loss with 112 layers.This validates the effectiveness of pre-activation residual connections.
+
+![](images/63735486313c5f92cf88443ef013e57bbca2db25f51386f50af1c5217aaa76fc.jpg)  
+Figure 2: Training loss of PlainGCN, ResGCN and ${ \mathrm { R e s G C N } } +$
+
+Depth & Normalization. In our experiments, we find normalization techniques play a crucial role in training deep GCNs. Without normalization, the training of deep network may suffer from vanishing gradient or exploding gradient problem. We apply normalization methods such as BatchNorm (Ioffe & Szegedy, 2015) or LayerNorm (Ba et al., 2016) to normalize vertex features. In addition to this, we also propose a message normalization (MsgNorm) layer to normalize features on the message level, which can significantly boost the performance of networks with under-performing aggregation functions. The main idea of MsgNorm is to normalize the features of the aggregated message $\bar { \mathbf { m } } _ { v } ^ { ( l ) } \in$ $\mathbb { R } ^ { D }$ by combining them with other features during the vertex update phase. Suppose we apply the MsgNorm to a simple vertex update function M $\bar { \mathbf { \Lambda } } \bar { \mathbf { \Lambda } } \bar { \mathbf { P } ( \mathbf { h } _ { v } ^ { ( l ) } + \mathbf { m } _ { v } ^ { ( l ) } ) }$ . The vertex update function becomes as follows:
+
+$$
+\mathbf { h } _ { v } ^ { ( l + 1 ) } = \phi ^ { ( l ) } ( \mathbf { h } _ { v } ^ { ( l ) } , \mathbf { m } _ { v } ^ { ( l ) } ) = \mathbf { M } \mathbf { L } \mathbf { P } ( \mathbf { h } _ { v } ^ { ( l ) } + s \cdot \| \mathbf { h } _ { v } ^ { ( l ) } \| _ { 2 } \cdot \frac { \mathbf { m } _ { v } ^ { ( l ) } } { \| \mathbf { m } _ { v } ^ { ( l ) } \| _ { 2 } } )
+$$
+
+where $\mathrm { M L P ( \cdot ) }$ is a multi-layer perceptron and $s$ is a learnable scaling factor. The aggregated message $\mathbf { m } _ { v } ^ { ( l ) }$ is first normalized by its $\ell _ { 2 }$ norm and then scaled by the $\ell _ { 2 }$ norm of $\mathbf { h } _ { v } ^ { ( l ) }$ by a factor of $s$ . In practice, we set the scaling factor $s$ to be a learnable scalar with an initialized value of 1. Note that when $s = \| \mathbf { m } _ { v } ^ { ( l ) } \| _ { 2 } / \| \mathbf { h } _ { v } ^ { ( l ) } \| _ { 2 }$ , the vertex update function reduces to the original form. In our experiment, we find MsgNorm boosts performance of under-performing aggregation functions such as mean and PowerMean on ogbn-proteins more than $1 \%$ . However, we do not see any significant gain on well-performing aggregation functions such as SoftMax, SoftMaxSum and PowerMeanSum. We leave this for our future investigation.
+
+Depth & Width. In order to gain a larger representational capacity, we can either increase depth or width of networks. In this work, we focus on the depth instead of the width since it is more challenging to train a deeper graph neural network compared to a wider one because of vanishing gradient (Li et al., 2019b) and over-smoothing (Li et al., 2018) problems. Deeper neural networks can learn to extract higher-level features. However, given a certain budget of parameters and computation, a well-designed wider networks can be more accurate and efficient than a deep networks. The tradeoff of depth and width have already studied in CNNs (Zagoruyko & Komodakis, 2016). We believe that it is also important to study the width of GCNs to reduce the computational overhead.
+
+Depth & Receptive Flied & Diameter. There are lots of discussion on whether depth can help for graph neural networks. In our experiments, we find that graph neural networks can gain better performance with proper skip connections, normalization and aggregation functions. A interesting discussion by Rossi et al. (2020) argues that the receptive field of graph neural networks with a few layers can cover the entire graph since most of graph data are ‘small-world’ graphs with small diameter. Depth may be harmful for graph neural networks. In our experiment, we observe a different phenomenon. For instance, ogbn-proteins dataset with a relatively small diameter as 9 can gain improvement with more than 100 layers. However, what is the optimal depth and for what certain kind of graphs depth help more are still mysteries.
+
+# B PROOF FOR PROPOSITION 4
+
+Proof. Suppose we have $N = \left| \mathcal { N } ( v ) \right|$ . We denote the message set as $\textbf { M } = \{ \mathbf { m } _ { 1 } , . . . , \mathbf { m } _ { N } \}$ , mi ∈ RD. We first show for any message set, SoftMax Aggβ(M) = PNj=1 P exp(βmj )Ni=1 exp(βmi) $\rho$ notes a permutation, it is obvious that $\forall \beta \in \mathbb { R }$ anyand $\rho \star { \bf M } = \{ { \bf m } _ { \rho ( 1 ) } , . . . , { \bf m } _ { \rho ( N ) } \}$ $\begin{array} { r } { \sum _ { i = \rho ( 1 ) } ^ { \rho ( N ) } \exp ( \beta \mathbf { m } _ { i } ) ~ = ~ \sum _ { i = 1 } ^ { N } \exp ( \beta \mathbf { m } _ { i } ) } \end{array}$ Pρ(N)j=ρ(1) exp(βmj ) · mj = PNj=1 exp(βmj ) · mj since the Sum function is a permutation invariant function. Thus, we have SoftMax $\mathrm { . A g g } _ { \beta } ( \mathbf { M } ) = \mathrm { S o f t M a x . A g g } _ { \beta } ( \rho \star \mathbf { M } )$ . SoftMax $\mathbf { A } \mathbf { g } \mathbf { g } _ { \beta } ( \cdot )$ satisfies Definition 2.
+
+We now prove SoftMax $\underline { { \mathbf { A g g } } } _ { \beta } ( \cdot )$ satisfies Definition 3, i.e. $\begin{array} { r } { \operatorname* { l i m } _ { \beta \to 0 } \mathrm { S o f t M a x { \_ A g g } } _ { \beta } ( \cdot ) = \mathrm { M e a n } ( \cdot ) } \end{array}$ and $\begin{array} { r } { \operatorname* { l i m } _ { \beta \to \infty } \mathrm { S o f t M a x { \_ A g g } } _ { \beta } ( \cdot ) = \mathbf { M a x } ( \cdot ) } \end{array}$ . For the $k$ -th dimension, we have input message features as $\{ m _ { 1 } ^ { ( k ) } , . . . , m _ { N } ^ { ( k ) } \}$ . $\begin{array} { r } { \operatorname* { l i m } _ { \beta \to 0 } \mathrm { S o f t M a x \_ A g g } _ { \beta } ( \{ m _ { 1 } ^ { ( k ) } , . . . , m _ { N } ^ { ( k ) } \} ) = \sum _ { j = 1 } ^ { N } \frac { \exp ( \beta m _ { j } ^ { ( k ) } ) } { \sum _ { i = 1 } ^ { N } \exp ( \beta m _ { i } ^ { ( k ) } ) } \cdot m _ { j } ^ { ( k ) } = \frac { 1 } { N } \frac { \exp ( \beta m _ { j } ^ { ( k ) } ) } { \sum _ { i = 1 } ^ { N } \exp ( \beta m _ { i } ^ { ( k ) } ) } , } \end{array}$ PNj =1 1N · m(k)j = $\begin{array} { r } { \sum _ { j = 1 } ^ { N } \frac { 1 } { N } \cdot m _ { j } ^ { ( k ) } = \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \cdot m _ { j } ^ { ( k ) } = \mathbf { M e a n } ( \{ m _ { 1 } ^ { ( k ) } , . . . , m _ { N } ^ { ( k ) } \} ) } \end{array}$ i=1 . Suppose we have $c$ i elements that are equal to the maximum value $m ^ { * }$ . When $\beta \to \infty$ , we have:
+
+$$
+\frac { \exp ( \beta m _ { j } ^ { ( k ) } ) } { \sum _ { i = 1 } ^ { N } \exp ( \beta m _ { i } ^ { ( k ) } ) } = \frac { 1 } { \sum _ { i = 1 } ^ { N } \exp ( \beta ( m _ { i } ^ { ( k ) } - m _ { j } ^ { ( k ) } ) ) } = \left\{ 1 / c \quad \mathrm { f o r ~ } m _ { j } ^ { ( k ) } = m ^ { * } \right.
+$$
+
+We obtain $\begin{array} { r l r l r l } { \operatorname* { l i m } _ { \beta \to \infty } \mathrm { S o f t M a x } . \mathrm { A g g } _ { \beta } ( \{ m _ { 1 } ^ { ( k ) } , . . . , m _ { N } ^ { ( k ) } \} ) } & { { } = } & { c \cdot \frac { 1 } { c } \cdot m ^ { * } } & { { } = } & { m ^ { * } } & { { } = } & { - 1 \cdot m ^ { * } , } \end{array}$ $\mathbf { M a x } ( \{ m _ { 1 } ^ { ( k ) } , . . . , m _ { N } ^ { ( k ) } \} )$ N c. It is obvious that the conclusions above generalize to all the dimensions. Therefore, SoftMax $\bar { \bf A g g } _ { \beta } ( \cdot )$
+
+# C PROOF FOR PROPOSITION 5
+
+Proof. Suppose we have $N = \left| \mathcal { N } ( v ) \right|$ . We denote the message set as $\textbf { M } = \{ \mathbf { m } _ { 1 } , . . . , \mathbf { m } _ { N } \}$ , $\mathbf { m } _ { i } \in \mathbb { R } _ { + } ^ { D }$ . We have PowerMean $\begin{array} { r } { { \bf { \underline { { A g g } } } } _ { p } ( { \bf M } ) = ( { \frac { 1 } { N } } \sum _ { i = 1 } ^ { N } { \bf m } _ { i } ^ { p } ) ^ { 1 / p } } \end{array}$ , $p \neq 0$ . Clearly, for any permutation $\rho \star \mathbf { M } = \{ \mathbf { m } _ { \rho ( 1 ) } , . . . , \mathbf { m } _ { \rho ( N ) } \}$ , PowerMean $\mathsf { \Pi } _ { ^ { \mathrm { \tiny ~ l } } } \mathrm { A g g } _ { p } ( \rho \star \mathbf { M } ) \ = \ \mathrm { P o w e r M e a n } \mathrm { \tiny . A g g } _ { p } ( \mathbf { M } ) .$ Hence, PowerMean $\mathbf { A g g } _ { p } ( \cdot )$ satisfies Definition 2. Then we prove PowerMean $\mathbf { A g g } _ { p } ( \cdot )$ satisfies Definition 3 i.e. PowerMean $\mathbf { A g g } _ { p = 1 } ( \cdot ) \ = \ \mathbf { M e a n } ( \cdot )$ and $\begin{array} { r l } { \operatorname* { l i m } _ { p \to \infty } \mathrm { P o w e r M e a n . A g g } _ { p } ( \cdot ) } & { { } = } \end{array}$ $\operatorname { M a x } ( \cdot )$ . For the $k$ -th dimension, we have input message features as $\{ m _ { 1 } ^ { ( k ) } , . . . , m _ { N } ^ { ( k ) } \}$ . PowerMean ${ \mathrm { A g g } } _ { p = 1 , \big ( } \{ m _ { 1 } ^ { ( k ) } , . . . , m _ { N } ^ { ( k ) } \} { \big ) } = { \textstyle { \frac { 1 } { N } } } \sum _ { i = 1 , . . } ^ { N } \cdot m _ { i } ^ { ( k ) } = \operatorname { M e a n } ( \{ m _ { 1 } ^ { ( k ) } , . . . , m _ { N } ^ { ( { \bar { k } } ) } \} { \big ) }$ . Assume elements that are equal to the maximum value $m ^ { * }$ . When $p  \infty$ , we have:
+
+$$
+\begin{array} { l r } { \displaystyle \operatorname* { l i m } _ { p \to \infty } \mathrm { P o w e r M e a n . A g g } _ { p } ( \{ m _ { 1 } ^ { ( k ) } , . . . , m _ { N } ^ { ( k ) } \} ) = ( \frac { 1 } { N } \sum _ { i = 1 } ^ { N } ( m _ { i } ^ { ( k ) } ) ^ { p } ) ^ { 1 / p } = ( \frac { 1 } { N } ( m ^ { * } ) ^ { p } \sum _ { i = 1 } ^ { N } ( \frac { m _ { i } ^ { ( k ) } } { m ^ { * } } ) ^ { p } ) ^ { 1 / p } } \\ { \displaystyle \qquad ( 7 ) } \\ { \displaystyle \qquad = ( \frac { c } { N } ( m ^ { * } ) ^ { p } ) ^ { 1 / p } \frac { m ^ { * } > 0 } { { \bf m } ^ { * } } m ^ { * } } & { \displaystyle ( 8 ) } \end{array}
+$$
+
+We have $\begin{array} { r } { \operatorname* { i m } _ { p \to \infty } \mathrm { P o w e r M e a n . A g g } _ { p } ( \{ m _ { 1 } ^ { ( k ) } , . . . , m _ { N } ^ { ( k ) } \} ) = m ^ { * } = \mathbb { M } \mathrm { a x } ( \{ m _ { 1 } ^ { ( k ) } , . . . , m _ { N } ^ { ( k ) } \} ) } \end{array}$ . The conclusions above hold for all the dimensions. Thus, PowerMean $\mathrm { A g g } _ { p } ( \cdot )$ is a generalized meanmax aggregation function.
+
+# D ANALYSIS OF DYRESGEN
+
+We provide more analysis and some interesting findings of DyResGEN in this section. The experimental results of DyResGEN in this section are obtained on ogbn-proteins dataset. We visualize the learning dynamic of learnable parameters $\beta , \ p$ and $s$ of 7-layer DyResGEN with $\operatorname { S o f t M a x S u m . A g g } _ { ( \beta , y ) } ( \cdot )$ aggregator and PowerMeanSum $\mathbf { \nabla } \_ { \mathbf { A g g } } ^ { } ( \mathbf { \nabla } _ { p , y } ) \big ( \cdot \big )$ aggregator respectively. Learnable parameters $\beta$ and $p$ are initialized as 1 and $y$ are initialized as 0.5. Dropout with a rate of 0.1 is used for each layer to prevent over-fitting. The learning curves of learnable parameters of $\operatorname { S o f t M a x S u m . A g g } _ { ( \beta , y ) } ( \cdot )$ are shown in Figure 3. We observe that both $\beta$ and $y$ change dynamically during the training. The $\beta$ and $y$ parameters of some layers tend to be stable after 1000 training epochs. Exceptionally, the 1-st layer learns a $\beta$ increasingly from 1 to 3.3 which learns a smaller $y \approx 0 . 1$ which make SoftMaxSum $\mathbf { A g g } _ { ( \beta , y ) } ( \cdot )$ behave more like a Max aggregation at the 1-th layer. PowerMean $\underline { { \mathbf { A g g } } } _ { p } ( \cdot )$ aggregator also demonstrates a similar phenomena on learning $y$ in Figure 4. The learned $y$ of the 1-st layer and the last layer trends to be smaller than the initial value.
+
+![](images/00d01eed6ecab44b45fe6a1397b485692a19ff5d07f547c84e26182968e9db1f.jpg)  
+Figure 3: Learning curves of 112-layer DyResGEN with SoftMaxSum $\mathbf { A } \mathbf { g } \mathbf { g } _ { \beta } ( \cdot )$ .
+
+![](images/5fb4021c2d03b98dfc14a8abc638b2dc71ccd17c61c0bd4ff3426d36744cdb85.jpg)  
+Figure 4: Learning curves of 112-layer DyResGEN with PowerMeanSum $\mathbf { A g g } _ { p } ( \cdot )$
+
+# E MORE DETAILS ON THE EXPERIMENTS
+
+In this section, we provide more experimental details on the OGB datasets (ogbn-proteins, ogbnarxiv, ogbn-products, ogbg-molhiv, ogbg-molpcba, ogbg-ppa and ogbl-collab).
+
+# E.1 DETAILS OF DATASETS
+
+Node Property Prediction. Three chosen datasets are dealing with protein-protein association networks (ogbn-proteins), paper citation networks (ogbn-arxiv) and co-purchasing network (ogbnproducts). Ogbn-proteins is an undirected, weighted, and typed (according to species) graph containing 132, 534 nodes and 39, 561, 252 edges. All edges come with 8-dimensional features and each node has an 8-dimensional one-hot feature indicating which species the corresponding protein comes from. Ogbn-arxiv consists of 169, 343 nodes and 1, 166, 243 directed edges. Each node is an arxiv paper represented by a 128-dimensional features and each directed edge indicates the citation direction. As an Amazon products co-purchasing network, ogbn-products is an undirected and unweighted graph which is formed by 2, 449, 029 nodes and 61, 859, 140 edges where nodes are products sold in Amazon that are represented by 100-dimensional features, and edges indicate that the connected nodes are co-purchased. For ogbn-proteins, the prediction task is multi-label and ROC-AUC is used as the evaluation metric. For ogbn-arxiv and ogbn-products, their prediction tasks are both multi-class and evaluated by accuracy.
+
+Graph Property Prediction. Here, we consider three datasets, two of which deals with molecular graphs (ogbg-molhiv and ogbg-molpcba) and the other is biological subgraphs (ogbg-ppa). Ogbgmolhiv has 41, 127 subgraphs and ogbg-molpcba is much bigger which contains 437, 929 subgraphs. For ogbg-ppa, it consists of 158, 100 subgraphs and each subgraph is much denser in comparison to the other two datasets. The tasks of ogbg-molhiv and ogbg-molpcba are both binary classification while the prediction task of ogbg-ppa is multi-class classification. The former two are evaluated by the ROC-AUC and Average Precision (AP) metric separately. Accuracy is used to assess ogbg-ppa.
+
+Link Property Prediction. We select ogbl-collab, an author collaboration network consisting of 235, 868 nodes and 1, 285, 465 edges for link prediction task. Each node in the graph comes with a 128-dimensional feature vector representing an author and edges indicate the collaboration between authors. The task is to predict the future author collaboration relationships given the past collaborations. Each true collaboration is ranked among a set of 100, 000 randomly-sampled negative collaborations, and the ratio of positive edges that are ranked at $K$ -place or above $( H i t s @ k$ , $k$ is 50 here) is counted as the evaluation metric.
+
+# E.2 DETAILS OF RESULTS AND IMPLEMENTATION
+
+For a fair comparison with SOTA methods, we provide results on each dataset by averaging the results from 10 independent runs. We provide the details of the model configuration on each dataset. All models are implemented based on PyTorch Geometric (Fey & Lenssen, 2019) and all experiments are performed on a single NVIDIA V100 32GB.
+
+ogbn-proteins. For both ogbn-proteins and ogbg-ppa, there is no node feature provided. We initialize the features of nodes through aggregating the features of their connected edges by a Sum aggregation, i.e. $\begin{array} { r } { \mathbf { x _ { i } } = \sum _ { j \in \mathcal { N } ( i ) } \mathbf { e } _ { i , j } } \end{array}$ , where $\mathbf { x _ { i } }$ denotes the initialized node features and $\mathbf { e } _ { i , j }$ denotes the input edge features. We train a 112-layer DyResGEN with SoftMax $\underline { { \operatorname { A g g } } } _ { \beta } ( \cdot )$ aggregator. A hidden channel size of 64 is used. A layer normalization and a dropout with a rate of 0.1 are used for each layer. We train the model for 2000 epochs with an Adam optimizer with a learning rate of 0.001.
+
+ogbn-arxiv. We train a 28-layer ResGEN model with SoftMax $\underline { { \mathbf { A g g } } } _ { \beta } ( \cdot )$ aggregator where $\beta$ is fixed as 0.1. We convert this directed graph into undirected and add self-loop. Full batch training and test are applied. A batch normalization is used for each layer. The hidden channel size is 128. We apply a dropout with a rate of 0.5 for each layer. An Adam optimizer with a learning rate of 0.001 is used to train the model for 2000 epochs.
+
+ogbn-products. A 14-layer ResGEN model with SoftMax $\underline { { \mathbf { A g g } } } _ { \beta } ( \cdot )$ aggregator where $\beta$ is fixed as 0.1 is trained for ogbn-products with self-loop added. We apply mini-batch training scenario by randomly partitioning the graph into 10 subgraphs and do full-batch test. For each layer, a batch normalization is used. The hidden channel size is 128. We apply a dropout with a rate of 0.5 for each layer. An Adam optimizer with a learning rate of 0.001 is used to train the model for 1000 epochs.
+
+ogbg-molhiv. We train a 7-layer DyResGEN model with SoftMax $\mathbf { A g g } _ { \beta } ( \cdot )$ aggregator where $\beta$ is learnable. A batch normalization is used for each layer. We set the hidden channel size as 256. A dropout with a rate of 0.2 is used for each layer. An Adam optimizer with a learning rate of 0.0001 are used to train the model for 300 epochs.
+
+ogbg-molpcba. A 14-layer ResGEN model with SoftMax $\underline { { \operatorname { A g g } } } _ { \beta } ( \cdot )$ aggregator where $\beta$ is fixed as 0.1 is trained. In addition, the original model performs message passing over augmented graphs with virtual nodes added. A batch normalization is used for each layer. We set the hidden channel size as 256. A dropout with a rate of 0.5 is used for each layer. An Adam optimizer with a learning rate of 0.01 are used to train the model for 300 epochs.
+
+ogbg-ppa. As mentioned, we initialize the node features via a Sum aggregation. We train a 28- layer ResGEN model with SoftMax $\_ { \_ } \mathrm { A g g } _ { \beta } ( \cdot )$ aggregator where $\beta$ is fixed as 0.01. We apply a layer normalization for each layer. The hidden channel size is set as 128. A dropout with a rate of 0.5 is used for each layer. We use an Adam optimizer with a learning rate of 0.01 to train the model for 200 epochs.
+
+ogbl-collab. The whole model used to train on link prediction task consists of two parts: a 7- layer DyResGEN model with SoftMax $\underline { { \mathbf { A g g } } } _ { \beta } ( \cdot )$ aggregator where $\beta$ is learnable and a 3-layer link predictor model. A batch normalization is used for each layer in DyResGEN model. We set the hidden channel size as 128. An Adam optimizer with a learning rate of 0.001 are used to train the model for 400 epochs.
+
+# F MORE FUTURE WORKS
+
+We believe generalized aggregation functions will open a new view for designing aggregation functions in graph neural networks. Here we discuss some more potential directions as follows:
+
+• Can we learn the parameters of generalized aggregation functions with a mete-learning method such as MAML (Finn et al., 2017)? What is the expressive power border of generalized mean-max-sum aggregation functions with respect to WeisfeilerLehman graph isomorphism test (Xu et al., 2019b)? • Can we design Principal Neighbourhood Aggregation (PNA) (Corso et al., 2020) by combining multiple learnable aggregators from generalized aggregation functions?

parse/train/r1lUl6NFDH/r1lUl6NFDH.md

@@ -0,0 +1,475 @@
+# MIRROR DESCENT VIEW FOR NEURAL NETWORK QUANTIZATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Quantizing large Neural Networks (NN) while maintaining the performance is highly desirable for resource-limited devices due to reduced memory and time complexity. NN quantization is usually formulated as a constrained optimization problem and optimized via a modified version of gradient descent. In this work, by interpreting the continuous parameters (unconstrained) as the dual of the quantized ones, we introduce a Mirror Descent (MD) framework (Bubeck (2015)) for NN quantization. Specifically, we provide conditions on the projections (i.e., mapping from continuous to quantized ones) which would enable us to derive valid mirror maps and in turn the respective MD updates. Furthermore, we discuss a numerically stable implementation of MD by storing an additional set of auxiliary dual variables (unconstrained). This update is strikingly analogous to the popular Straight Through Estimator (STE) based method which is typically viewed as a “trick” to avoid vanishing gradients issue but here we show that it is an implementation method for MD for certain projections. Our experiments on standard classification datasets (CIFAR-10/100, TinyImageNet) with convolutional and residual architectures show that our MD variants obtain fully-quantized networks with accuracies very close to the floating-point networks.
+
+# 1 INTRODUCTION
+
+Despite the success of deep neural networks in various domains, their excessive computational and memory requirements limit their practical usability for real-time applications or in resource-limited devices. Quantization is a prominent technique for network compression, where the objective is to learn a network while restricting the parameters to take values from a small discrete set (usually binary). This leads to a dramatic reduction in memory (a factor of 32 for binary quantization) and inference time – as it enables specialized implementation using bit operations.
+
+Neural Network (NN) quantization is usually formulated as a constrained optimization problem $\mathrm { m i n } _ { \mathbf { x } \in \mathcal { X } } f ( \mathbf { x } )$ , where $f ( \cdot )$ denotes the loss function by abstracting out the dependency on the dataset and $\mathcal { X } \subset \mathbb { R } ^ { \mathrm { r } }$ denotes the set of all possible quantized solutions. Majority of the works in the literature (Hubara et al. (2017); Yin et al. (2018); Ajanthan et al. (2019)) convert this into an unconstrained problem by introducing auxiliary variables $( \tilde { \mathbf { x } } )$ and optimize via (stochastic) gradient descent. Specifically, the objective and the update step take the following form:
+
+$$
+\operatorname* { m i n } _ { { \tilde { \mathbf { x } } } \in \mathbb { R } ^ { \mathbf { r } } } f ( P ( { \tilde { \mathbf { x } } } ) ) , \qquad { \tilde { \mathbf { x } } } ^ { k + 1 } = { \tilde { \mathbf { x } } } ^ { k } - \eta \nabla _ { \tilde { \mathbf { x } } } f ( P ( { \tilde { \mathbf { x } } } ) ) | _ { { \tilde { \mathbf { x } } } = { \tilde { \mathbf { x } } } ^ { k } } \ ,
+$$
+
+where $P : \mathbb { R } ^ { \mathrm { r } }  \mathcal { X }$ is a mapping from the unconstrained space to the quantized space (sometimes called projection) and $\eta > 0$ is the learning rate. In cases where the mapping $P$ is not differentiable, a suitable approximation is employed (Hubara et al. (2017)).
+
+In this work, by noting that the well-known Mirror Descent (MD) algorithm, widely used for online convex optimization (Bubeck (2015)), provides a theoretical framework to perform gradient descent in the unconstrained space (dual space, $\mathbb { R } ^ { \mathrm { r } }$ ) with gradients computed in the quantized space (primal space, $\mathcal { X }$ ), we introduce an MD framework for NN quantization. In essence, MD extends gradient descent to non-Euclidean spaces where Euclidean projection is replaced with a more general projection defined based on the associated distance metric. Briefly, the key ingredient of MD is a concept called mirror map which defines both the mapping between primal and dual spaces and the exact form of the projection. Specifically, in this work, by observing $P$ in Eq. (1) as a mapping from dual space to the primal space, we analytically derive corresponding mirror maps under certain conditions on $P$ This enables us to derive different variants of the MD algorithm useful for NN quantization.
+
+Furthermore, as MD is often found to be numerically unstable (Hsieh et al. (2018)), we discuss a numerically stable implementation of MD by storing an additional set of auxiliary variables similar to the existing methods. As will be shown later, this update is strikingly analogous to the popular Straight Through Estimator (STE) based gradient method (Hubara et al. (2017); Bai et al. (2019)) which is typically viewed as a “trick” to avoid vanishing gradients issue but here we show that it is an implementation method for MD under certain conditions on the mapping $P$ . We believe this connection sheds some light on the practical effectiveness of STE.
+
+We evaluate the merits of our MD variants on CIFAR-10/100 and TinyImageNet classification datasets with convolutional and residual architectures. Our experiments show that the quantized networks obtained by the MD variants yield accuracies very close to the floating-point counterparts while outperforming directly comparable baselines. Finally, we would like to emphasize that even though our formulation does not necessarily extend the theory of MD, we believe showing MD as a suitable framework for NN quantization with superior empirical performance opens up new ways of designing MD-inspired update rules for NNs.
+
+# 2 PRELIMINARIES
+
+We first provide some background on the MD algorithm and NN quantization. Then we discuss the link between them and provide our MD framework for NN quantization.
+
+# 2.1 MIRROR DESCENT
+
+The Mirror Descent (MD) algorithm is first introduced in (Nemirovsky & Yudin (1983)) and it has been extensively studied in the convex optimization literature ever since. In this section we provide a brief overview and we refer the interested reader to Chapter 4 of (Bubeck (2015)). In the context of MD, we consider a problem of the form: (2)
+
+$$
+\operatorname* { m i n } _ { \mathbf { x } \in \mathcal { X } } f ( \mathbf { x } ) \ ,
+$$
+
+where $f : \mathcal { X } \to \mathbb { R }$ is a convex function and $\mathcal { X } \subset \mathbb { R } ^ { \mathrm { r } }$ is a compact convex set. The main concept of MD is to extend gradient descent to a more general non-Euclidean space (Banach space1), thus overcoming the dependency of gradient descent on the Euclidean geometry. The motivation for this generalization is that one might be able to exploit the geometry of the space to optimize much more efficiently. One such example is the simplex constrained optimization where MD converges at a much faster rate than the standard Projected Gradient Descent (PGD).
+
+To this end, since the gradients lie in the dual space, optimization is performed by first mapping the primal point $\mathbf { x } ^ { k } \in \bar { \boldsymbol { B } }$ (quantized space, $\mathcal { X }$ ) to the dual space $B ^ { * }$ (unconstrained space, $\mathbb { R } ^ { \mathrm { r } }$ ), then performing gradient descent in the dual space, and finally mapping back the resulting point to the primal space $\boldsymbol { B }$ . If the new point $\mathbf { x } ^ { k + 1 }$ lie outside of the constraint set $\mathcal { X } \subset B$ , it is projected to the set $\mathcal { X }$ . Both the primal/dual mapping and the projection are determined by the mirror map. Specifically, the gradient of the mirror map defines the mapping from primal to dual and the projection is done via the Bregman divergence of the mirror map. We first provide the definitions for mirror map and Bregman divergence and then turn to the MD updates.
+
+Definition 2.1 (Mirror map). Let ${ \mathcal { C } } \subset \mathbb { R } ^ { \mathrm { r } }$ be a convex open set such that $\boldsymbol { \mathcal { X } } \subset \bar { \boldsymbol { \mathcal { C } } }$ $\bar { \mathcal { C } }$ denotes the closure of set $\mathcal { C }$ ) and $\chi \cap \bar { \mathcal { C } } \neq \emptyset$ . Then, $\Phi : { \mathcal { C } }  \mathbb { R }$ is a mirror map if it satisfies:
+
+1. $\Phi$ is strictly convex and differentiable.   
+2. $\nabla \Phi ( { \mathcal { C } } ) = \mathbb { R } ^ { \mathrm { r } }$ , i.e., $\nabla \Phi$ takes all possible values in $\mathbb { R } ^ { \mathrm { r } }$ .   
+3. $\operatorname* { l i m } _ { \mathbf { x } \to \partial { \mathcal { C } } }$ $\| \nabla \Phi ( \mathbf { x } ) \| = \infty$ ( $\partial \mathcal { C }$ denotes the boundary of $\mathcal { C }$ ), i.e., $\nabla \Phi$ diverges on the boundary of $\mathcal { C }$ .
+
+Definition 2.2 (Bregman divergence). Let $\Phi : { \mathcal { C } }  \mathbb { R }$ be a continuously differentiable, strictly convex function defined on a convex set $\mathcal { C }$ . The Bregman divergence associated with $\Phi$ for points $\mathbf { p } , \mathbf { q } \in \mathcal { C }$ is the difference between the value of $\Phi$ at point $\mathbf { p }$ and the value of the first-order Taylor expansion of $\Phi$ around point q evaluated at point $\mathbf { p }$ , i.e.,
+
+$$
+D _ { \Phi } ( { \bf p } , { \bf q } ) = \Phi ( { \bf p } ) - \Phi ( { \bf q } ) - \left. \nabla \Phi ( { \bf q } ) , { \bf p } - { \bf q } \right. .
+$$
+
+Notice, $D _ { \Phi } ( \mathbf { p } , \mathbf { q } ) \geq 0$ with $D _ { \Phi } ( \mathbf { p } , \mathbf { p } ) = 0$ , and $D _ { \Phi } ( \mathbf { p } , \mathbf { q } )$ is convex on p.
+
+Now we are ready to provide the mirror descent strategy based on the mirror map $\Phi$ . Let $\mathbf { x } ^ { 0 } \in \mathrm { a r g m i n } _ { \mathbf { x } \in \mathcal { X } \cap \mathcal { C } } \bar { \Phi ( \mathbf { x } ) }$ be the initial point. Then, for iteration $k \geq 0$ and step size $\eta > 0$ , the update of the MD algorithm can be written as:
+
+$$
+\begin{array} { r l } & { \nabla \Phi ( \mathbf { y } ^ { k + 1 } ) = \nabla \Phi ( \mathbf { x } ^ { k } ) - \eta \mathbf { g } ^ { k } , \qquad \mathrm { w h e r e ~ } \mathbf { g } ^ { k } \in \partial f ( \mathbf { x } ^ { k } ) \mathrm { ~ a n d ~ } \mathbf { y } ^ { k + 1 } \in \mathcal { C } , } \\ & { \qquad \mathbf { x } ^ { k + 1 } = \underset { \mathbf { x } \in \mathcal { X } \cap \mathcal { C } } { \mathrm { a r g m i n } } D _ { \Phi } ( \mathbf { x } , \mathbf { y } ^ { k + 1 } ) . } \end{array}
+$$
+
+Note that, in Eq. (4), the gradient $\mathbf { g } ^ { k }$ is computed at $\mathbf { x } ^ { k } \in \mathcal { X } \cap \mathcal { C }$ (solution space) but the gradient descent is performed in $\mathbb { R } ^ { \mathrm { r } }$ (unconstrained dual space). Moreover, by simple algebraic manipulation, it is easy to show that the above MD update (4) can be compactly written in a proximal form where the Bregman divergence of the mirror map becomes the proximal term (Beck & Teboulle (2003)):
+
+$$
+\begin{array} { r } { \mathbf { x } ^ { k + 1 } = \underset { \mathbf { x } \in \mathcal { X } \cap \mathcal { C } } { \mathrm { a r g m i n } } \left. \eta \mathbf { g } ^ { k } , \mathbf { x } \right. + D _ { \Phi } ( \mathbf { x } , \mathbf { x } ^ { k } ) . } \end{array}
+$$
+
+Note, if $\begin{array} { r } { \Phi ( \mathbf { x } ) = \frac { 1 } { 2 } \left\| \mathbf { x } \right\| _ { 2 } ^ { 2 } } \end{array}$ , then $\begin{array} { r } { D _ { \Phi } ( \mathbf { x } , \mathbf { x } ^ { k } ) = \frac { 1 } { 2 } \left\| \mathbf { x } - \mathbf { x } ^ { k } \right\| _ { 2 } ^ { 2 } } \end{array}$ , which when plugged back to the above problem and optimized for $\mathbf { x }$ , leads to exactly the same update rule as that of PGD. However, MD allows us to choose various forms of $\Phi$ depending on the problem at hand.
+
+# 2.2 NEURAL NETWORK QUANTIZATION
+
+Neural Network (NN) quantization amounts to training networks with parameters restricted to a small discrete set representing the quantization levels. Here we review two constrained optimization formulations for NN quantization: 1) directly constrain each parameter to be in the discrete set; and 2) optimize the probability of each parameter taking a label from the set of quantization levels.
+
+# 2.2.1 PARAMETER SPACE FORMULATION
+
+Given a dataset $\mathcal { D } = \{ \mathbf { x } _ { i } , \mathbf { y } _ { i } \} _ { i = 1 } ^ { n }$ , NN quantization can be written as:
+
+$$
+\operatorname* { m i n } _ { \mathbf { w } \in \mathcal { Q } ^ { m } } L ( \mathbf { w } ; \mathcal { D } ) : = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ( \mathbf { w } ; ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) ) \ .
+$$
+
+Here, $\ell ( \cdot )$ denotes the input-output mapping composed with a standard loss function (e.g., crossentropy loss), w is the $m$ dimensional parameter vector, and $\mathcal { Q }$ with $| \mathcal { Q } | = d$ is a predefined discrete set representing quantization levels (e.g., $\mathcal { Q } = \{ - 1 , 1 \}$ or $\mathcal { Q } = \{ - 1 , 0 , 1 \}$ ).
+
+The approaches that directly optimize in the parameter space include BinaryConnect (BC) (Courbariaux et al. (2015)) and its variants (Hubara et al. (2017); Rastegari et al. (2016)), where the constraint set is discrete. In contrast, recent approaches (Bai et al. (2019); Yin et al. (2018)) relax this constraint set to be its convex hull:
+
+$$
+\mathrm { c o n v } ( \mathcal { Q } ^ { m } ) = [ q _ { \mathrm { m i n } } , q _ { \mathrm { m a x } } ] ^ { m } ,
+$$
+
+where $q _ { \mathrm { m i n } }$ and $q _ { \mathrm { m a x } }$ represent the minimum and maximum quantization levels, respectively. In this case, a quantized solution is obtained by gradually increasing an annealing hyperparameter.
+
+# 2.2.2 LIFTED PROBABILITY SPACE FORMULATION
+
+Another formulation is based on the Markov Random Field (MRF) perspective to NN quantization recently studied in (Ajanthan et al. (2019)). It treats Eq. (6) as a discrete labelling problem and introduces indicator variables $u _ { j : \lambda } \in \{ 0 , 1 \}$ for each parameter $w _ { j }$ where $j \in \{ 1 , \dots , m \}$ such that $u _ { j : \lambda } = 1$ if and only if $w _ { j } = \lambda \in \mathcal { Q }$ . For convenience, by denoting the vector of quantization levels as $\mathbf { q }$ , a parameter vector $\mathbf { w } \in \mathcal { Q } ^ { m }$ can be written in a matrix vector product as:
+
+$$
+\mathbf { w } = \mathbf { u } \mathbf { q } \ , \quad \mathrm { w h e r e } \quad \mathbf { u } \in \gamma ^ { m } = \left\{ \textbf { u } \Big | \ \sum _ { { u } \atop { u } _ { j : \lambda } \in \{ 0 , 1 \} } \ \forall j , \lambda \ \right\} \ .
+$$
+
+Here, $\mathbf { u }$ is a $m \times d$ matrix (i.e., each row ${ \bf u } _ { j } = \{ u _ { j : \lambda } ~ | ~ \lambda \in \mathcal { Q } \} ;$ ), and $\mathbf { q }$ is a column vector of dimension $d$ . Note that, $\mathbf { u } \in \mathcal { V } ^ { m }$ is an overparametrized (i.e., lifted) representation of $\mathbf { w } \in \mathcal { Q } ^ { m }$ . Now, similar to the relaxation in the parameter space, one can relax the binary constraint in $\mathcal { V } ^ { m }$ to form its convex hull:
+
+$$
+\Delta ^ { m } = \mathrm { c o n v } ( \gamma ^ { m } ) = \left\{ \begin{array} { l l }  \mathbf { u } \left. \begin{array} { l l } { \sum _ { \lambda } u _ { j : \lambda } = 1 , } & { \forall j } \\ { u _ { j : \lambda } \geq 0 , } & { \forall j , \lambda } \end{array} \right. \right\} . \end{array}
+$$
+
+The set $\Delta ^ { m }$ is in fact the Cartesian product of the standard $( d - 1 )$ -probability simplexes embedded in $\mathbb { R } ^ { \mathrm { d } }$ . Therefore, for a feasible point $\mathbf { u } \in \Delta ^ { m }$ , the vector $\mathbf { u } _ { j }$ for each ${ \mathrm { ~ \it ~ j ~ } } ( { \it \ j } )$ -th row of matrix $\mathbf { u }$ ) belongs to the probability simplex $\Delta$ . Hence, we can interpret the value $u _ { j : \lambda }$ as the probability of assigning the discrete label $\lambda$ to the weight $w _ { j }$ . This relaxed optimization can then be written as:
+
+$$
+\operatorname* { m i n } _ { \mathbf { u } \in \Delta ^ { m } } L ( \mathbf { u q } ; \mathcal { D } ) : = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ( \mathbf { u q } ; ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) ) \ .
+$$
+
+In fact, this can be interpreted as finding a probability distribution $\mathbf { u } \in \Delta ^ { m }$ such that the cost $L ( \mathbf { u } )$ is minimized. Note that, the relaxation of $\mathbf { u }$ from $\mathcal { V } ^ { m }$ to $\Delta ^ { m }$ translates into relaxing w from ${ \mathcal { Q } } ^ { m }$ to the convex region $\mathrm { c o n v } ( { \mathcal { Q } } ^ { m } )$ . Even in this case, a discrete solution $\mathbf { u } \in \mathcal { V } ^ { m }$ can be enforced via an annealing hyperparameter or using rounding schemes.
+
+# 3 MIRROR DESCENT FRAMEWORK FOR NETWORK QUANTIZATION
+
+Before introducing the MD formulation, we first write NN quantization as a single objective unifying (6) and (10) as: (11)
+
+$$
+\operatorname* { m i n } _ { \mathbf { x } \in \mathcal { X } } f ( \mathbf { x } ) \ ,
+$$
+
+where $f ( \cdot )$ denotes the loss function by abstracting out the dependency on the dataset $\mathcal { D }$ , and $\mathcal { X }$ denotes the constraint set ( ${ \mathcal { Q } } ^ { m }$ or $\mathcal { V } ^ { m }$ depending on the formulation). Note that, as discussed in Sec. 2.2, many recent NN quantization methods optimize over the convex hull of the constraint set. Following this, we consider the solution space $\mathcal { X }$ in Eq. (11) to be convex and compact. To employ MD, we need to choose a mirror map (refer Definition 2.1) suitable for the problem at hand. In fact, as discussed in Sec. 2.1, mirror map is the core component of an MD algorithm which determines the effectiveness of the resulting MD updates. However, there is no straightforward approach to obtain a mirror map for a given constrained optimization problem, except in certain special cases.
+
+To this end, we observe that the usual approach to optimize the above constrained problem is via a version of projected gradient descent, where the projection is the mapping from the unconstrained auxiliary variables (high-precision) to the quantized space $\mathcal { X }$ . Now, noting the analogy between the purpose of the projection operator and the mirror maps in the MD formulation, we intend to derive the mirror map analogous to a given projection. Precisely, we prove that if the projection is invertible and strictly monotone, a valid mirror map can be derived from the projection itself. This does not necessarily extend the theory of MD as finding a strictly monotone map is as hard as finding the mirror map itself. However, this derivation is interesting as it connects existing PGD type algorithms to their corresponding MD variants. For completeness, we now state it as a theorem for the case $\mathcal { X } \subset \mathbb { R }$ and the multidimensional case can be easily proved with an additional assumption that the vector field $P ^ { - 1 } ( \mathbf { x } )$ is conservative.
+
+Theorem 3.1. Let $\mathcal { X }$ be a compact convex set and $P : \mathbb { R }  { \mathcal { C } }$ be an invertible function where ${ \mathcal { C } } \subset \mathbb { R }$ is a convex open set such that $\mathcal { X } = \bar { \mathcal { C } }$ ( $\bar { \mathcal { C } }$ denotes the closure of $\mathcal { C }$ ). Now, if
+
+1. $P$ is strictly monotonically increasing.
+
+2. $\begin{array} { r } { \operatorname* { l i m } _ { x  \partial { \mathcal C } } \| { \dot { P } } ^ { - 1 } ( x ) \| = \infty } \end{array}$ $\partial \mathcal { C }$ denotes the boundary of $\mathcal { C }$
+
+Then, $\begin{array} { r } { \Phi ( x ) = \int _ { x _ { 0 } } ^ { x } P ^ { - 1 } ( y ) d \ y } \end{array}$ is a valid mirror map.
+
+Proof. This can be proved by noting that $\nabla \Phi ( x ) = P ^ { - 1 } ( x )$ . Please refer to Appendix A.
+
+The MD update based on the mirror map derived from a given projection is illustrated in Fig. 1. Note that, to employ MD to the problem (11), in theory, any mirror map satisfying Definition 2.1 whose domain (the closure of the domain) is a superset of the constraint set $\mathcal { X }$ can be chosen. However, the above theorem provides a method to derive only a subset of all applicable mirror maps, where the closure of the domain of mirror maps is exactly equal to the constraint set $\mathcal { X }$ .
+
+![](images/7a7a5894b6aa7f30af161a65fc5ce32ac2560173aed1fba43134f82d24379115.jpg)  
+Figure 1: MD formulation where mirror map is derived from the projection $P$ . Note, $\mathbf { g } ^ { k }$ is computed in the primal space $( \mathcal { X } )$ but it is directly used to update the auxiliary variables in the dual space.
+
+We now give some example projections useful for NN quantization (tanh for w-space and softmax for u-space) and derive their corresponding mirror maps. Given mirror maps, the MD updates are straightforward based on Eq. (5). Even though we consider differentiable projections, Theorem 3.1 does not require the projection to be differentiable. For the rest of the section, we assume $m = 1$ , i.e., consider projections that are independent for each $j \in \{ 1 , \dots , m \}$ .
+
+Example 3.1 (w-space, binary, tanh). Consider the tanh function, which projects a real value to the interval $[ - 1 , 1 ]$ :
+
+$$
+w = P ( \tilde { w } ) : = \operatorname { t a n h } ( \beta \tilde { w } ) = \frac { \exp ( 2 \beta \tilde { w } ) - 1 } { \exp ( 2 \beta \tilde { w } ) + 1 } ,
+$$
+
+where $\beta > 0$ is the annealing hyperparameter and when $\beta \to \infty$ , tanh approaches the step function. The inverse of the tanh is:
+
+$$
+P ^ { - 1 } ( w ) = { \frac { 1 } { \beta } } \operatorname { t a n h } ^ { - 1 } ( w ) = { \frac { 1 } { 2 \beta } } \log { \frac { 1 + w } { 1 - w } } .
+$$
+
+Note that, $P ^ { - 1 }$ is monotonically increasing for a fixed $\beta$ . Correspondingly, the mirror map from Theorem 3.1 can be written as:
+
+$$
+\Phi ( w ) = \int P ^ { - 1 } ( w ) d w = \frac { 1 } { 2 \beta } \big [ ( 1 + w ) \log ( 1 + w ) + ( 1 - w ) \log ( 1 - w ) \big ] \ .
+$$
+
+Here, the constant from the integration is ignored. It can be easily verified that $\Phi ( w )$ is in fact a valid mirror map. The projection, its inverse and the corresponding mirror map are illustrated in Fig. 2a. Consequently, the resulting MD update (5) takes the following form:
+
+$$
+\begin{array} { r } { w ^ { k + 1 } = \underset { w \in ( - 1 , 1 ) } { \operatorname { a r g m i n } } \langle \eta g ^ { k } , w \rangle + D _ { \Phi } ( w , w ^ { k } ) = \frac { \frac { 1 + w ^ { k } } { 1 - w ^ { k } } \exp ( - 2 \beta \eta g ^ { k } ) - 1 } { \frac { 1 + w ^ { k } } { 1 - w ^ { k } } \exp ( - 2 \beta \eta g ^ { k } ) + 1 } . } \end{array}
+$$
+
+The update formula is derived using the KKT conditions (Boyd & Vandenberghe (2009)). For the detailed derivation please refer to Appendix B. A similar derivation can also be performed for the sigmoid function, where ${ \bar { \mathcal { C } } } = \chi = { \bar { [ 0 , 1 ] } }$ . Note that the sign function has been used for binary quantization in (Courbariaux et al. (2015)) and tanh can be used as a soft version of sign function as pointed out by (Zhang et al. (2015)). Mirror map corresponding to tanh is used for online linear optimization in (Bubeck et al. (2012)) but here we use it for NN quantization.
+
+Example 3.2 (u-space, multi-label, softmax). Now we consider the softmax projection used in Proximal Mean-Field (PMF) (Ajanthan et al. (2019)) to optimize in the lifted probability space. In this case, the projection is defined as $P ( \tilde { \mathbf { u } } ) : = \mathrm { s o f t m a x } ( \beta \tilde { \mathbf { u } } )$ where $P : \mathbb { R } ^ { \mathrm { d } }  \mathcal { C }$ with $\bar { \mathcal { C } } = \dot { \mathcal { X } } = \Delta$ . Here $\Delta$ is the $( d - 1 )$ -dimensional probability simplex and $| { \mathcal { Q } } | = d$ . Note that the softmax projection is not invertible as it is a many-to-one mapping. In particular, it is invariant to translation, i.e.,
+
+$$
+{ \bf u } = \mathrm { s o f t m a x } ( \tilde { \bf u } + c { \bf 1 } ) = \mathrm { s o f t m a x } ( \tilde { \bf u } ) , \qquad \mathrm { w h e r e } \quad u _ { \lambda } = \frac { \exp ( \tilde { u } _ { \lambda } ) } { \sum _ { \mu \in \mathcal { Q } } \exp ( \tilde { u } _ { \mu } ) } ,
+$$
+
+for any scalar $c \in \mathbb { R }$ (1 denotes a vector of all ones). Therefore, the softmax projection does not satisfy Theorem 3.1. However, one could define the inverse of softmax as follows: given $\mathbf { u } \in \Delta$ , find a unique point $\tilde { \mathbf { v } } = \tilde { \mathbf { u } } + c \mathbf { 1 }$ , for a particular scalar $c$ , such that $\mathbf { u } = \mathrm { s o f t m a x } ( \tilde { \mathbf { v } } )$ . Now, by choosing $\begin{array} { r } { c = - \log ( \sum _ { \mu = \mathcal Q } \exp ( \tilde { u } _ { \mu } ) ) } \end{array}$ , softmax can be written as:
+
+$$
+{ \mathbf u } = \mathrm { s o f t m a x } ( \tilde { \mathbf v } ) , \qquad \mathrm { w h e r e } \quad u _ { \lambda } = \mathrm { e x p } ( \tilde { v } _ { \lambda } ) , \quad \forall \lambda \in { \mathcal Q } .
+$$
+
+Now, the inverse of the projection can be written as:
+
+$$
+\tilde { \mathbf { v } } = P ^ { - 1 } ( \mathbf { u } ) = \frac { 1 } { \beta } \operatorname { s o f t m a x } ^ { - 1 } ( \mathbf { u } ) \ , \qquad \mathrm { w h e r e } \quad \tilde { v } _ { \lambda } = \frac { 1 } { \beta } \log ( u _ { \lambda } ) \ , \qquad \forall \lambda \ .
+$$
+
+Indeed, log is a monotonically increasing function and from Theorem 3.1, by summing the integrals, the mirror map can be written as:
+
+$$
+\Phi ( { \mathbf u } ) = \frac { 1 } { \beta } \left[ \sum _ { \lambda } u _ { \lambda } \log ( u _ { \lambda } ) - u _ { \lambda } \right] = - \frac { 1 } { \beta } H ( { \mathbf u } ) - 1 / \beta .
+$$
+
+Here, $\begin{array} { r } { \sum _ { \lambda } u _ { \lambda } = 1 } \end{array}$ as $\mathbf { u } \in \Delta$ , and $H ( \mathbf { u } )$ is the entropy. Interestingly, as the mirror map in this case is the negative entropy (up to a constant), the MD update leads to the well-known Exponentiated
+
+![](images/320d0bb5c966b2c4b8c12529e11395421ebb5dea49c16d7eab355985e548c122.jpg)  
+Figure 2: Plots of tanh and shifted tanh projections, and their inverses corresponding to the tanh projection. Note that, the inverses are monotonically increasing and the mirror map is strictly convex. Moreover, when $\beta \to \infty$ , the projections approaches their respective hard versions.
+
+Gradient Descent (EGD) (or Entropic Descent Algorithm (EDA)) (Beck & Teboulle (2003); Bubeck (2015)). Consequently, the update takes the following form:
+
+$$
+u _ { \lambda } ^ { k + 1 } = \frac { u _ { \lambda } ^ { k } \exp ( - \beta \eta g _ { \lambda } ^ { k } ) } { \sum _ { \mu \in \mathcal { Q } } u _ { \mu } ^ { k } \exp ( - \beta \eta g _ { \mu } ^ { k } ) } \quad \forall \lambda .
+$$
+
+The derivation follows the same approach as in the tanh case above. It is interesting to note that the MD variant of softmax is equivalent to the well-known EGD. Notice, the authors of PMF hinted that PMF is related to EGD but here we have clearly showed that the MD variant of PMF under the above reparametrization (17) is exactly EGD.
+
+Example 3.3 (w-space, multi-label, shifted tanh). Note that, similar to softmax, we wish to extend the tanh projection beyond binary. The idea is to use a function that is an addition of multiple shifted tanh functions. To this end, as an example we consider ternary quantization, with $\mathcal { Q } = \{ - 1 , 0 , 1 \}$ and define our shifted tanh projection $P : \mathbb { R }  { \mathcal { C } }$ as:
+
+$$
+w = P ( \tilde { w } ) = \frac { 1 } { 2 } \big [ \operatorname { t a n h } \left( \beta ( \tilde { w } + 0 . 5 ) \right) + \operatorname { t a n h } \left( \beta ( \tilde { w } - 0 . 5 ) \right) \big ] ,
+$$
+
+where $\beta > 0$ and $w = \bar { \mathcal { C } } = \mathcal { X } = [ - 1 , 1 ]$ . When $\beta \to \infty$ , $P$ approaches a stepwise function with inflection points at $- 0 . 5$ and 0.5 (here, $\pm 0 . 5$ is chosen heuristically), meaning $w$ move towards one of the quantization levels in the set $\mathcal { Q }$ . This behaviour together with its inverse is illustrated in Fig. 2b. Now, one could potentially find the functional form of $\scriptstyle { \bar { P } } ^ { - 1 }$ and analytically derive the mirror map corresponding to this projection. Note that, while Theorem 3.1 provides an analytical method to derive mirror maps, in some cases such as the above, the exact form of mirror map and the MD update might be nontrivial. In such cases, as will be shown subsequently, the MD update can be easily implemented by storing an additional set of auxiliary variables $\tilde { w }$ .
+
+Effect of Annealing. Note that, to ensure a discrete solution, projection $P$ is parametrized by a scalar $\beta$ and it is annealed throughout the optimization. This annealing hyperparameter translates into a time varying mirror map (refer Eqs. (14) and (19)) in our case. Such an adaptive mirror map gradually constrains the solution space $\mathcal { X }$ to its boundary and in the limit enforces a quantized solution. Since this adaptive behaviour can affect the convergence of the algorithm, in our implementation $\beta$ is capped at an arbitrarily chosen maximum value, and empirically, the algorithm converges to fully quantized solutions in all tested cases. We leave the theoretical analysis of annealing for future work.
+
+Numerically Stable form of Mirror Descent. We showed a few examples of valid projections, their corresponding mirror maps, and the final MD updates. Even though, in theory, these updates can be used directly, they are sometimes numerically unstable due to the operations involving multiple logarithms, exponentials and divisions (Hsieh et al. (2018)). To this end, we provide a numerically stable way of performing MD by storing an additional set of auxiliary parameters during training.
+
+A careful look at the Fig. 1 suggests that the MD update with the mirror map derived from Theorem 3.1 can be performed by storing auxiliary variables $\bar { \tilde { \mathbf { x } } } = P ^ { - 1 } ( \mathbf { x } )$ . In fact, once the auxiliary variable $\tilde { \mathbf { x } } ^ { k }$ is updated using gradient $\mathbf { g } ^ { \bar { k } }$ , it is directly mapped back to the constraint set $\mathcal { X }$ via the projection. This is mainly because of the fact that the domain of the mirror maps derived based on the Theorem 3.1 are exactly the same as the constraint set. Formally, with this additional set of variables, one can write the MD update (4) corresponding to the projection $P$ as:
+
+$$
+\begin{array} { r l r } & { \tilde { \mathbf { x } } ^ { k + 1 } = \tilde { \mathbf { x } } ^ { k } - \eta \mathbf { g } ^ { k } , } & { \mathrm { u p d a t e ~ i n ~ t h e ~ d u a l ~ s p a c e } } \\ & { \mathbf { x } ^ { k + 1 } = P ( \tilde { \mathbf { x } } ^ { k + 1 } ) \in \mathcal { X } , } & { \mathrm { p r o j e c t i o n ~ t o ~ t h e ~ p r i m a l ~ s p a c e } } \end{array}
+$$
+
+where $\eta > 0$ , $\mathbf { g } ^ { k } \in \partial f ( \mathbf { x } ^ { k } )$ and $\tilde { \mathbf { x } } ^ { k } = P ^ { - 1 } ( \mathbf { x } ^ { k } )$ . Experimentally we observed these updates to show stable behaviour and performed remarkably well for both the tanh and softmax.
+
+Note, above updates can be seen as optimizing the function $f ( P ( \tilde { { \bf x } } ) )$ using gradient descent where the gradient through the projection (i.e., Jacobian) $J _ { P } = \partial P ( \tilde { \mathbf { x } } ) / \partial \tilde { \mathbf { x } }$ is replaced with the identity matrix. This is exactly the same as the Straight Through Estimator (STE) for NN quantization (following the nomenclature of (Bai et al. (2019); Yin et al. (2018))). Despite being a crude approximation, STE has shown to be highly effective for NN quantization with various network architectures and datasets (Yin et al. (2018); Zhou et al. (2016)). However, a solid understanding of the effectiveness of STE is lacking in the literature except for its convergence analysis in certain special cases (Li et al. (2017); Yin et al. (2019)). In this work, by showing STE based gradient descent as an implementation method of MD under certain conditions on the projection, we provide a justification on the effectiveness of STE. Besides, as shown in Example 3.3, in cases where the MD formulation is nontrivial, the STE based implementation can be used. The pseudocodes of original and numerically stable versions of our MD algorithm for tanh are presented in Appendix B.
+
+Comparison against ProxQuant. The connection between the dual averaging version of MD and STE was recently hinted in ProxQuant (PQ) (Bai et al. (2019)). However, no analysis of whether an analogous mirror map exists to the given projection is provided and their final algorithm is not based on MD. In particular, following our notation, the final update equation of PQ can be written as:
+
+where $\eta > 0$ , and $\mathbf { g } ^ { k } \in \partial f ( \mathbf { x } ^ { k } )$ . Note that, as opposed to MD (refer to Eq. (22)), PQ assumes the point $\mathbf { x } ^ { k }$ and gradient $\mathbf { g } ^ { k }$ are in the same space. Then only the formula $\bar { \mathbf { x } ^ { k } } - \eta \mathbf { g } ^ { k }$ is valid. This would only be true for the Euclidean space. However, as discussed in Sec. 2.1, MD allows gradient descent to be performed on a more general non-Euclidean space by first mapping the primal point $\mathbf { x } ^ { k }$ to a point $\tilde { \mathbf { x } } ^ { k }$ in the dual space via the mirror map. Such an ability enabled theoretical and practical research on MD for the past three decades.
+
+Convergence of MD in the Nonconvex Setting. We would like to point out that MD is originally developed for convex optimization, however, in this paper we directly apply MD to NN quantization where the loss is highly nonconvex and gradient estimates are stochastic, and empirically evaluate its convergence behaviour and performance. Theoretical analysis of MD for nonconvex, stochastic setting is an active research area (Zhou et al. (2017a;b)) and MD has been recently shown to converge in the nonconvex stochastic setting under certain conditions (Zhang & He (2018)). We believe convergence analysis of MD for NNs could constitute to a completely new theoretical paper.
+
+# 4 RELATED WORK
+
+In this work we consider parameter quantization, which is usually formulated as a constrained problem and optimized using a modified projected gradient descent algorithm, where the methods (Courbariaux et al. (2015); Carreira-Perpinán & Idelbayev (2017); Yin et al. (2018); Bai et al. (2019); Ajanthan et al. (2019)) mainly differ in the constraint set, the projection used, and how backpropagation through the projection is performed. Among them, STE based gradient descent is the most popular method as it enables backpropagation through nondifferentiable projections and it has shown to be highly effective in practice (Courbariaux et al. (2015)). In fact, the success of this approach lead to various extensions by including additional layerwise scalars (Rastegari et al. (2016)), relaxing the solution space (Yin et al. (2018)), and even to quantizing activations (Hubara et al. (2017)), and/or gradients (Zhou et al. (2016)). Moreover, there are methods focusing on loss aware quantization (Hou et al. (2017)), quantization for specialized hardware (Esser et al. (2015)), and quantization based on the variational approach (Achterhold et al. (2018); Louizos et al. (2017; 2019)). We have only provided a brief summary of relevant methods and for a comprehensive survey we refer the reader to (Guo (2018)).
+
+# 5 EXPERIMENTS
+
+Due to the popularity of binary neural networks (Courbariaux et al. (2015); Rastegari et al. (2016)), we mainly consider binary quantization and set the quantization levels as $\mathcal { Q } = \{ \bar { - } 1 , 1 \}$ . We would like to point out that we quantize all learnable parameters, meaning our approach results in 32 times less memory compared to the floating-point counterparts.
+
+We evaluate two MD variants corresponding to tanh and softmax projections, on CIFAR-10, CIFAR100 and TinyImageNet2 datasets with VGG-16 and ResNet-18 architectures. We also evaluate the numerically stable versions of our MD variants (denoted with suffix “-S”) performed by storing auxiliary parameters during training as explained in Eq. (22). The results are compared against parameter quantization methods, namely BinaryConnect (BC) (Courbariaux et al. (2015)), ProxQuant (PQ) (Bai et al. (2019)) and Proximal Mean-Field (PMF) (Ajanthan et al. (2019)). In addition, for completeness, we also compare against a standard PGD variant corresponding to the tanh projection (denoted as GD-tanh), i.e., minimizing $f ( \operatorname { t a n h } ( \tilde { \mathbf { x } } ) )$ using gradient descent. The only difference of this to our MD-tanh-S is that, in Eq. (22), the Jacobian of tanh is directly used in the updates. Note that, numerous techniques have emerged with BC as the workhorse algorithm by relaxing constraints such as the layer-wise scalars (Rastegari et al. (2016)), and similar extensions are straightforward even in our case. Briefly, our results indicate that the binary networks obtained by the MD variants yield accuracies very close to the floating-point counterparts while outperforming the baselines.
+
+For all the experiments, standard multi-class cross-entropy loss is used. We crossvalidate the hyperparameters such as learning rate, learning rate scale, rate of increase of annealing hyperparameter $\beta$ , and their respective schedules for all tested methods including the baselines. This extensive crossvalidation improved the accuracies of previous methods by a large margin, e.g., up to $3 \%$ improvement for PMF. We provide the hyperparameter tuning search space and the final hyperparameters in Appendix C. Our algorithm is implemented in PyTorch (Paszke et al. (2017)) and the experiments are performed on NVIDIA Tesla-P100 GPUs. Our code will be released upon publication.
+
+Table 1: Classification accuracies on the test set for different methods for binary quantization. PQ\* denotes performance with biases, fully-connected layers, and shortcut layers in floating-point (original PQ setup) whereas PQ represents full quantization. PMF\* denotes the performance of PMF after crossvalidation similar to our MD-variants and the original results from the paper are denoted as PMF. Note our MD variants obtained accuracies virtually the same as the best performing method and it outperformed the best method by a large margin in much harder TinyImageNet dataset.   
+
+<table><tr><td rowspan="2">Algorithm</td><td rowspan="2">Space</td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td><td rowspan="2">TinyImageNet ResNet-18</td></tr><tr><td>VGG-16</td><td>ResNet-18</td><td>VGG-16</td><td>ResNet-18</td></tr><tr><td></td><td>REF (float)</td><td>W</td><td>93.33</td><td>94.84</td><td>71.50</td><td>76.31</td><td>58.35</td></tr><tr><td>BC</td><td></td><td>W</td><td>89.04</td><td>91.64</td><td>59.13</td><td>72.14</td><td>49.65</td></tr><tr><td></td><td>PQ</td><td>W</td><td>85.41</td><td>90.76</td><td>39.61</td><td>65.13</td><td>44.32</td></tr><tr><td></td><td>PQ*</td><td>W</td><td>90.11</td><td>92.32</td><td>55.10</td><td>68.35</td><td>49.97</td></tr><tr><td></td><td>PMF</td><td>u</td><td>90.51</td><td>92.73</td><td>61.52</td><td>71.85</td><td>51.00</td></tr><tr><td></td><td>PMF*</td><td>u</td><td>91.40</td><td>93.24</td><td>64.71</td><td>71.56</td><td>51.52</td></tr><tr><td></td><td>GD-tanh</td><td>W</td><td>91.47</td><td>93.27</td><td>60.67</td><td>71.46</td><td>51.43</td></tr><tr><td rowspan="5">s.InO</td><td>MD-softmax</td><td>u</td><td>90.47</td><td>91.28</td><td>56.25</td><td>68.49</td><td>46.52</td></tr><tr><td>MD-tanh</td><td>W</td><td>91.64</td><td>92.97</td><td>61.31</td><td>72.13</td><td>54.62</td></tr><tr><td>MD-softmax-S</td><td>u</td><td>91.30</td><td>93.28</td><td>63.97</td><td>72.18</td><td>51.81</td></tr><tr><td>MD-tanh-S</td><td>W</td><td>91.53</td><td>93.18</td><td>61.69</td><td>72.18</td><td>52.32</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+# 5.1 RESULTS
+
+The classification accuracies of binary networks obtained by both variants of our algorithm, namely, MD-tanh and MD-softmax, their numerically stable versions (suffix $^ { 6 6 } { - } S ^ { 7 }$ ) and the baselines BC, PQ, PMF, GD-tanh and the floating point Reference Network (REF) are reported in Table 1. Both the numerically stable MD variants consistently produce better or on par results compared to other binarization methods while narrowing the performance gap between binary networks and floating point counterparts to a large extent, on multiple datasets.
+
+Our stable MD-variant perform slightly better than MD-softmax, whereas for tanh, MD updates either perform on par or sometimes even better than numerically stable version of MD-tanh. We believe, the main reason for this empirical variation in results for our MD-variants is due to numerical instability caused by the floating-point arithmetic of logarithm and exponential functions in Eq. (15) and Eq. (20). Furthermore, even though our two MD-variants, namely MD-softmax and MD-tanh optimize in different spaces, their performance is similar in most cases. This may be explained by the fact that both algorithms belong to the same family where a “soft” projection to the constraint set (in fact the constraints sets are equivalent in this case, refer Sec. 2.2.2) is used and an annealing hyperparameter is used to gradually enforce a quantized solution.
+
+![](images/4e5a71282e90eb317cfad1e9a0d107cbf901eacb7156d47688e3a82763aed61b.jpg)  
+Figure 3: Training curves for binarization for CIFAR-10 (first two) and CIFAR-100 (last two) with ResNet-18. Compared to original MD variants, stable MD variants are less noisy and after the initial exploration phase (up to 60 in CIFAR-10 and 25 epochs CIFAR-100), the validation accuracies rise sharply and show gradual improvement afterwards.
+
+Note, PQ (Bai et al. (2019)) does not quantizee the fully connected layers, biases and shortcut layers. For fair comparison, we crossvalidate PQ with all layers binarized and original PQ settings, and report the results denoted as PQ and $\mathrm { P Q } ^ { * }$ respectively in Table 1. Our MD-variants outperform PQ consistently on multiple datasets in equivalent experimental settings. This clearly shows that entropic or tanh-based regularization with our annealing scheme is superior to a simple “W” shaped regularizer and emphasizes that MD is a suitable framework for quantization.
+
+Furthermore, the superior performance of MD-tanh against GD-tanh and on par or better performance of MD-softmax against PMF for binary quantization empirically validates that MD is useful even in a nonconvex stochastic setting. This hypothesis along with our numerically stable form of MD can be particularly useful to explore other projections which are useful for quantization and/or network compression in general.
+
+The training curves for our MD variants for CIFAR-10 and CIFAR-100 datasets with ResNet-18 are shown in Fig. 3. The original MD variants show unstable behaviour during training which is attributed to the fact that it involves logarithms and exponentials in the update rules. In addition, we believe, the additional annealing hyperparameter also contributes to this instability. Regardless, by storing auxiliary variables, the MD updates are demonstrated to be quite stable. This clear distinction between MD variants emphasizes the significance of practical considerations while implementing MD especially in NN optimization. For more experiments such as training curves comparison to other methods and ternary quantization results please refer to the Appendix C.
+
+# 6 DISCUSSION
+
+In this work, we have introduced an MD framework for NN quantization by deriving mirror maps corresponding to various projections useful for quantization. In addition, we have discussed a numerically stable implementation of MD by storing an additional set of auxiliary variables and showed that this update is strikingly analogous to the popular STE based gradient method. The superior performance of our MD formulation even with simple projections such as tanh and softmax is encouraging and we believe, MD would be a suitable framework for not just NN quantization but for network compression in general. Finally, some theoretical aspects such as the use of time-varying mirror maps and the combination of MD and a stochastic optimizer such as Adam are left unattended in this paper, which we intend to analyze in a future work.
+
+# REFERENCES
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+
+# Appendices
+
+Here, we first provide the proof of the theorem and the technical derivations. Later we give additional experiments and the details of our experimental setting.
+
+# A DERIVING MIRROR MAPS FROM PROJECTIONS
+
+Theorem A.1. Let $\mathcal { X }$ be a compact convex set and $P : \mathbb { R }  { \mathcal { C } }$ be an invertible function where ${ \mathcal { C } } \subset \mathbb { R }$ is a convex open set such that $\mathcal { X } = \bar { \mathcal { C } }$ ( $\bar { \mathcal { C } }$ denotes the closure of $\mathcal { C }$ ). Now, if
+
+1. $P$ is strictly monotonically increasing.   
+2. $\begin{array} { r } { \operatorname* { l i m } _ { x  \partial { \mathcal C } } \| { \dot { P } } ^ { - 1 } ( x ) \| = \infty } \end{array}$ $\partial \mathcal { C }$ denotes the boundary of $\mathcal { C }$ ).
+
+Then, $\begin{array} { r } { \Phi ( x ) = \int _ { x _ { 0 } } ^ { x } P ^ { - 1 } ( y ) d \xi } \end{array}$ is a valid mirror map.
+
+Proof. From the fundamental theorem of calculus, the gradient of $\Phi ( x )$ satisfies, $\nabla \Phi ( x ) = P ^ { - 1 } ( x )$ . Since $P$ is strictly monotonically increasing and invertible, $P ^ { - 1 }$ is strictly monotonically increasing. Therefore, $\Phi ( x )$ is strictly convex and differentiable. Now, from the definition of projection and since it is invertible (i.e., $P ^ { - 1 }$ is one-to-one and onto), $\nabla \Phi ( \mathcal { C } ) = P ^ { - 1 } ( \mathcal { C } ) = \bar { \mathbb { R } }$ . Therefore, together with condition (2), we can conclude that $\begin{array} { r } { \Phi ( x ) = \int _ { x _ { 0 } } ^ { x } P ^ { - 1 } ( y ) d y } \end{array}$ is a valid mirror map (refer Definition 2.2 in the main paper). For the multi-dimensional case, we need an additional condition that the vector field $P ^ { - 1 } ( \mathbf { x } )$ is conservative. Then by the gradient theorem (gra), there exists a mirror map $\begin{array} { r } { \Phi ( \mathbf { x } ) = \int _ { \mathbf { x } _ { 0 } } ^ { \mathbf { x } } P ^ { - 1 } ( \mathbf { y } ) d \mathbf { y } } \end{array}$ for some arbitrary base point $\mathbf { x } _ { \mathrm { 0 } }$ .
+
+# B MD UPDATE DERIVATION FOR THE TANH PROJECTION
+
+We now derive the MD update corresponding to the tanh projection below. From Theorem A.1, the mirror map for the tanh projection can be written as:
+
+$$
+\Phi ( w ) = \int P ^ { - 1 } ( w ) d w = \frac { 1 } { 2 \beta } \big [ ( 1 + w ) \log ( 1 + w ) + ( 1 - w ) \log ( 1 - w ) \big ] \ .
+$$
+
+Correspondingly, the Bregman divergence can be written as:
+
+$$
+\begin{array} { l } { { \displaystyle { \cal D } _ { \Phi } ( w , v ) = \Phi ( w ) - \Phi ( v ) - \Phi ^ { \prime } ( v ) ( w - v ) \ , \quad \mathrm { w h e r e ~ } \Phi ^ { \prime } ( v ) = \frac { 1 } { 2 \beta } \log \frac { 1 + v } { 1 - v } \ , } } \\ { { \displaystyle ~ = \frac { 1 } { 2 \beta } \left[ w \log \frac { ( 1 + w ) ( 1 - v ) } { ( 1 - w ) ( 1 + v ) } + \log ( 1 - w ) ( 1 + w ) - \log ( 1 - v ) ( 1 - v ) \right] \ . } } \end{array}
+$$
+
+Now, consider the proximal form of MD update
+
+$$
+\begin{array} { r } { w ^ { k + 1 } = \underset { { \mathbf x } \in ( - 1 , 1 ) } { \mathrm { a r g m i n } } \left. \eta g ^ { k } , w \right. + D _ { \Phi } ( w , w ^ { k } ) . } \end{array}
+$$
+
+The idea is to find $w$ such that the KKT conditions are satisfied. To this end, let us first write the Lagrangian of Eq. (26) by introducing dual variables $y$ and $z$ corresponding to the constraints $w > - 1$ and $w < 1$ , respectively:
+
+$$
+^ { \mathrm { { \it ~ \alpha } } } ( w , x , y ) = \eta g ^ { k } w + \frac { 1 } { 2 \beta } \left[ w \log \frac { ( 1 + w ) ( 1 - w ^ { k } ) } { ( 1 - w ) ( 1 + w ^ { k } ) } + \log ( 1 - w ) ( 1 + w ) - \log ( 1 - w ^ { k } ) ( 1 - w ^ { k } ) \right] .
+$$
+
+$$
++ y ( - w - 1 ) + z ( w - 1 ) .
+$$
+
+Now, setting the derivatives with respect to $w$ to zero:
+
+$$
+\frac { \partial F } { \partial w } = \eta g ^ { k } + \frac { 1 } { 2 \beta } \log \frac { ( 1 + w ) ( 1 - w ^ { k } ) } { ( 1 - w ) ( 1 + w ^ { k } ) } - y + z = 0 .
+$$
+
+From complementary slackness conditions,
+
+$$
+\begin{array} { r l } { y ( - w - 1 ) = 0 , } & { \quad \mathrm { s i n c e } \quad w > - 1 \Rightarrow y = 0 , } \\ { z ( w - 1 ) = 0 , } & { \quad \mathrm { s i n c e } \quad w < 1 \Rightarrow z = 0 . } \end{array}
+$$
+
+Therefore, Eq. (28) now simplifies to:
+
+$$
+\begin{array} { c } { { \displaystyle \frac { \partial F } { \partial w } = \eta g ^ { k } + \frac { 1 } { 2 \beta } \log \displaystyle \frac { ( 1 + w ) ( 1 - w ^ { k } ) } { ( 1 - w ) ( 1 + w ^ { k } ) } = 0 \ : , } } \\ { { \displaystyle \log \displaystyle \frac { ( 1 + w ) ( 1 - w ^ { k } ) } { ( 1 - w ) ( 1 + w ^ { k } ) } = \exp ( - 2 \beta \eta g ^ { k } ) \ : , } } \\ { { \displaystyle \frac { 1 + w } { 1 - w } = \displaystyle \frac { 1 + w ^ { k } } { 1 - w ^ { k } } \exp ( - 2 \beta \eta g ^ { k } ) \ : , } } \\ { { \displaystyle w = \frac { \frac { 1 + w ^ { k } } { 1 - w ^ { k } } \exp ( - 2 \beta \eta g ^ { k } ) \ : - 1 } { \frac { 1 + w ^ { k } } { 1 + w ^ { k } } \exp ( - 2 \beta \eta g ^ { k } ) \ : + 1 } \ : . } } \end{array}
+$$
+
+The pseudocodes of original (MD-tanh) and numerically stable versions (MD-tanh-S) for tanh are presented in Algorithms 1 and 2 respectively.
+
+# C ADDITIONAL EXPERIMENTS
+
+We first give training curves of all compared methods, and provide ternary quantization results as a proof of concept. Later, we provide experimental details.
+
+Convergence Analysis. The training curves for CIFAR-10 and CIFAR-100 datasets with ResNet-18 are shown in Fig. 4. Notice, after the initial exploration phase (due to low $\beta$ ) the validation accuracies of our MD-tanh-S increase sharply while this steep rise is not observed in regularization methods such as PQ. The training behaviour for both our stable MD-variants (softmax and tanh) is quite similar.
+
+# Algorithm 1 MD-tanh
+
+Require: $K , b , \{ \eta ^ { k } \} , \rho > 1 , \mathcal { D } , L$   
+Ensure: w∗ ∈ Qm   
+1: w0 ∈ IRm, β ← 1 . Initialization   
+2: $\mathbf { w } ^ { 0 }  \operatorname { t a n h } ( \beta \mathbf { w } ^ { 0 } )$ . Projection   
+3: for $k  0 , \ldots , K$ do   
+4: $\mathcal { D } ^ { b } = \{ ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) \} _ { i = 1 } ^ { b } \sim \mathcal { D }$ . Sample a mini-batch   
+5: $\mathbf { g } ^ { k }  \nabla _ { \mathbf { w } } L ( \mathbf { w } ; \mathcal { D } ^ { b } ) | _ { \mathbf { w } = \mathbf { w } ^ { k } }$ . Gradient w.r.t. w at $\mathbf { w } ^ { k }$ (Adam based gradients)   
+6: for j ← 1, . . . , m do   
+1+wkj exp(−2βηkgkj )−1   
+7: wk+1j ← 1+wk . MD update   
+8: end for j1−wkj exp(−2βηkgkj )+1   
+9: $\beta \gets \rho \beta$ . Increase β   
+10: end for   
+11: $\mathbf { w } ^ { * } \gets \mathrm { s i g n } ( \tilde { \mathbf { w } } ^ { K } )$ . Quantization
+
+# Algorithm 2 MD-tanh-S
+
+Require: $K , b , \{ \eta ^ { k } \} , \rho > 1 , \mathcal { D } , L$   
+Ensure: w∗ ∈ Qm   
+1: w˜ 0 ∈ IRm, β ← 1   
+2: for $k  0 , \ldots , K$ do   
+3: $\mathbf { w } ^ { k }  \operatorname { t a n h } ( \beta \tilde { \mathbf { w } } ^ { k } )$   
+4: ${ \mathcal { D } } ^ { b } = \{ ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) \} _ { i = 1 } ^ { b } \sim { \mathcal { D } }$   
+5: gk ← ∇wL(w; Db)w=wk   
+6: w˜ k+1 ← w˜ k − ηkgk   
+7: β ← ρβ   
+8: end for   
+9: $\mathbf { w } ^ { * } \gets \mathrm { s i g n } ( \tilde { \mathbf { w } } ^ { K } )$
+
+. Initialization $\triangleright$ Projection . Sample a mini-batch . Gradient w.r.t. w at $\mathbf { w } ^ { k }$ (Adam based gradients) $\triangleright$ Gradient descent on w˜ . Increase $\beta$ . Quantization
+
+Ternary Quantization. As a proof of concept for our shifted tanh projection (refer Example 3.3), we also show results for ternary quantization with quantization levels $\mathbf { \bar { \mathcal { Q } } } = \{ - 1 , 0 , 1 \}$ in Table 2. Note that the performance improvement of our ternary networks compared to their respective binary networks is marginal as only 0 is included as the $3 ^ { \mathrm { r d } }$ quantization level. In contrast to us, the baseline method PQ (Bai et al. (2019)) optimizes for the quantization levels (differently for each layer) as well in an alternating optimization regime rather than fixing it to $\mathcal { Q } = \{ - 1 , 0 , 1 \}$ . Also, PQ does ternarize the first convolution layer, fully-connected layers and the shortcut layers. We cross-validate hyperparameters for both the original PQ setup and the equivalent setting of our MD-variants where we optimize all the weights and denote them as $\mathrm { P Q } ^ { * }$ and PQ respectively.
+
+Our MD-tanh variant performs on par or sometimes even better in comparison to tanh projection results where gradient is calculated through the projection instead of performing MD. This again empirically validates the hypothesis that MD yields in good approximation for the task of network quantization. The better performance of PQ in their original quantization setup, compared to our approach in CIFAR-10 can be accounted to their non-quantized layers and different quantization levels. We believe, similar explorations are possible with our MD framework as well.
+
+Experimental Details. As mentioned in the main paper the experimental protocol is similar to (Ajanthan et al. (2019)). To this end, the details of the datasets and their corresponding experiment setups are given in Table 3. For CIFAR-10/100 and TinyImageNet, VGG-16 (Simonyan & Zisserman (2015)) and ResNet-18 (He et al. (2016)) architectures adapted for CIFAR dataset are used. In particular, for CIFAR experiments, similar to (Lee et al. (2019)), the size of the fully-connected (FC) layers of VGG-16 is set to 512 and no dropout layers are employed. For TinyImageNet, the stride of the first convolutional layer of ResNet-18 is set to 2 to handle the image size (Huang et al. (2017)). In all the models, batch normalization (Ioffe & Szegedy (2015)) (with no learnable parameters) and
+
+![](images/756d5666aaa419e13f84570773a292285688778778ad1d923abdea49730197d1.jpg)  
+Figure 4: Training curves for binarization for CIFAR-10 (first two columns) and CIFAR-100 (last two columns) with ResNet-18. Compared to BC, our MD-tanh-S and PMF are less noisy and after the initial exploration phase (up to 60 in CIFAR-10 and 25 epochs CIFAR-100), the validation accuracies rise sharply and closely resembles the floating point network afterwards. This steep increase is not observed in regularization methods such as PQ.
+
+Table 2: Classification accuracies on the test set for ternary quantization. $\mathrm { P Q } ^ { * }$ denotes performance with fully-connected layers, first convolution layer and shortcut layers in floating point whereas PQ represent results with all layers quantized. Also, $\mathrm { P Q } ^ { * }$ optimize for the quantization levels as well (different for each layer), in contrast we fix it to $\mathcal { Q } = \{ - 1 , 0 , 1 \}$ . GD-tanh denotes results without using STE and actually calculating the gradient through the projection.   
+
+<table><tr><td></td><td>Algorithm</td><td></td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td><td>TinyImageNet</td></tr><tr><td></td><td></td><td>Space</td><td>VGG-16</td><td>ResNet-18</td><td>VGG-16</td><td>ResNet-18</td><td>ResNet-18</td></tr><tr><td></td><td>REF (float)</td><td>W</td><td>93.33</td><td>94.84</td><td>71.50</td><td>76.31</td><td>58.35</td></tr><tr><td></td><td>PQ</td><td>W</td><td>83.32</td><td>90.50</td><td>32.16</td><td>59.18</td><td>41.46</td></tr><tr><td></td><td>PQ*</td><td>W</td><td>92.20</td><td>93.85</td><td>57.64</td><td>70.98</td><td>45.72</td></tr><tr><td></td><td>GD-tanh</td><td>W</td><td>91.21</td><td>93.20</td><td>53.88</td><td>69.48</td><td>50.65</td></tr><tr><td>0</td><td>MD-softmax-S</td><td>u</td><td>91.69</td><td>93.30</td><td>65.11</td><td>72.01</td><td>52.21</td></tr><tr><td></td><td>MD-tanh-S</td><td>W</td><td>91.70</td><td>93.42</td><td>66.15</td><td>71.29</td><td>52.69</td></tr></table>
+
+ReLU nonlinearity are used. Only for the floating point networks (i.e., REF), we keep the learnable parameters for batch norm. Standard data augmentation (i.e., random crop and horizontal flip) is used.
+
+For both of our MD variants, hyperparameters such as the learning rate, learning rate scale, annealing hyperparameter $\beta$ and its schedule are crossvalidated from the range reported in Table 4 and the chosen parameters are given in the Tables 5 and 6. To generate the plots, we used the publicly available codes of $\mathsf { B C } ^ { 3 }$ , $\mathsf { P Q } ^ { 4 }$ and $\mathrm { P M F } ^ { 5 }$ .
+
+All methods are trained from a random initialization and the model with the best validation accuracy is chosen for each method. Note that, in MD, even though we use an increasing schedule for $\beta$ to enforce a discrete solution, the chosen network may not be fully-quantized (as the best model could be obtained in an early stage of training). Therefore, simple argmax rounding is applied to ensure that the network is fully-quantized.
+
+Table 3: Experiment setup. Here, b is the batch size and $K$ is the total number of iterations for all the methods.   
+
+<table><tr><td>Dataset</td><td>Image</td><td></td><td># class Train / Val. b</td><td>K</td></tr><tr><td>MNIST</td><td>28×28</td><td>10</td><td>50k/10k 100</td><td>20k</td></tr><tr><td>CIFAR-10</td><td>32×32</td><td>10</td><td>45k/5k</td><td>128 100k</td></tr><tr><td>CIFAR-100</td><td>32×32</td><td>100</td><td>45k/5k 128</td><td>100k</td></tr><tr><td>TinyImageNet 64 × 64</td><td></td><td>200</td><td>100k/10k 128</td><td>100k</td></tr></table>
+
+Table 4: The hyperparameter search space for all the experiments. Chosen parameters are given in Tables 5 and 6.   
+
+<table><tr><td>Hyperparameters</td><td>Fine-tuning grid</td></tr><tr><td>learning_rate</td><td>[0.1,0.01, 0.001,0.0001]</td></tr><tr><td>lr_scale</td><td>[0.1, 0.2,0.3,0.5]</td></tr><tr><td>beta_scale</td><td>[1.01,1.02,1.05,1.1,1.2]</td></tr><tr><td>beta_scale_interval</td><td>[100,200,500,1000,2000]</td></tr></table>
+
+Table 5: Hyperparameter settings used for the binary quantization experiments. Here, the learning rate is multiplied by lr_scale after every 30k iterations and annealing hyperparameter $( \beta )$ is multiplied by beta_scale after every beta_scale_interval iterations. We use Adam optimizer with zero weight decay. For PQ, beta_scale denotes regularization rate.   
+
+<table><tr><td rowspan="2"></td><td colspan="8">CIFAR-10 with ResNet-18</td></tr><tr><td>MD-softmax</td><td>MD-tanh</td><td>MD-softmax-S</td><td>MD-tanh-S</td><td>PMF*</td><td>GD-tanh</td><td>BC</td><td>PQ</td></tr><tr><td>learning_rate</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.01</td></tr><tr><td>lr_scale</td><td>0.2</td><td>0.3</td><td>0.3</td><td>0.3</td><td>0.3</td><td>0.3</td><td>0.3</td><td>0.5</td></tr><tr><td>beta_scale</td><td>1.02 200</td><td>1.01</td><td>1.02</td><td>1.02</td><td>1.1</td><td>1.1</td><td>1</td><td>0.0001</td></tr><tr><td>beta_scale_interval</td><td></td><td>100</td><td>200</td><td>200</td><td>1000</td><td>1000</td><td>-</td><td>-</td></tr><tr><td colspan="9">CIFAR-100 with ResNet-18 MD-softmax-S</td></tr><tr><td>learning_rate</td><td>MD-softmax</td><td>MD-tanh</td><td></td><td>MD-tanh-S</td><td>PMF*</td><td>GD-tanh</td><td>BC</td><td>PQ</td></tr><tr><td>lr_scale</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.1</td></tr><tr><td>beta_scale</td><td>0.2</td><td>0.3 1.05</td><td>0.2 1.1</td><td>0.2</td><td>0.3</td><td>0.5 1.01</td><td>0.2</td><td>- 0.001</td></tr><tr><td>beta_scale_interval</td><td>1.05 500</td><td>500</td><td>200</td><td>1.2 500</td><td>1.01 100</td><td>100</td><td>1 1</td><td>-</td></tr><tr><td></td><td colspan="8"></td></tr><tr><td></td><td>MD-softmax</td><td>MD-tanh</td><td>MD-softmax-S</td><td>CIFAR-10 with VGG-16 MD-tanh-S</td><td>PMF*</td><td>GD-tanh</td><td>BC</td><td>PQ</td></tr><tr><td>learning_rate</td><td>0.01</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.0001</td><td>0.01</td></tr><tr><td>lr_scale</td><td>0.2</td><td>0.3</td><td>0.3</td><td>0.2</td><td>0.5</td><td>0.3</td><td>0.3</td><td>0.5</td></tr><tr><td>beta_scale</td><td>1.05</td><td>1.1</td><td>1.2</td><td>1.2</td><td>1.05</td><td>1.1</td><td>-</td><td>0.0001</td></tr><tr><td>beta_scale_interval</td><td>500</td><td>1000</td><td>2000</td><td>2000</td><td>500</td><td>1000</td><td>1</td><td>=</td></tr><tr><td colspan="9">CIFAR-100 with VGG-16</td></tr><tr><td></td><td>MD-softmax</td><td>MD-tanh</td><td>MD-softmax-S</td><td>MD-tanh-S</td><td>PMF*</td><td>GD-tanh</td><td>BC</td><td>PQ</td></tr><tr><td>learning_rate</td><td>0.001</td><td>0.001</td><td>0.0001</td><td>0.001</td><td>0.0001</td><td>0.001</td><td>0.0001</td><td>0.01</td></tr><tr><td>lr_scale beta_scale</td><td>0.3 1.01</td><td>0.3</td><td>0.2</td><td>0.5</td><td>0.5</td><td>0.5</td><td>0.2</td><td>0.5</td></tr><tr><td>beta_scale_interval</td><td>100</td><td>1.05 500</td><td>1.2 500</td><td>1.05</td><td>1.02</td><td>1.1</td><td>1</td><td>0.0001</td></tr><tr><td></td><td></td><td></td><td></td><td>500</td><td>200</td><td>1000</td><td>-</td><td>-</td></tr><tr><td colspan="9">TinyImageNet with ResNet-18</td></tr><tr><td>learning_rate</td><td>MD-softmax</td><td>MD-tanh</td><td>MD-softmax-S</td><td>MD-tanh-S</td><td>PMF*</td><td>GD-tanh</td><td>BC</td><td>PQ</td></tr><tr><td>lr_scale</td><td>0.001 0.2</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.01</td></tr><tr><td></td><td></td><td>0.5</td><td>0.1</td><td>0.1</td><td>0.5</td><td>0.5</td><td>0.5</td><td>-</td></tr><tr><td>beta_scale (ours)</td><td>1.02</td><td>1.2</td><td>1.02</td><td>1.2</td><td>1.01</td><td>1.01</td><td>-</td><td>0.0001</td></tr><tr><td>beta_scale_interval</td><td>200</td><td>2000</td><td>100</td><td>500</td><td>100</td><td>100</td><td>-</td><td>-</td></tr></table>
+
+Table 6: Hyperparameter settings used for the ternary quantization experiments. Here, the learning rate is multiplied by lr_scale after every 30k iterations and annealing hyperparameter $( \beta )$ is multiplied by beta_scale after every beta_scale_interval iterations. We use Adam optimizer except for REF for which SGD with momentum 0.9 is used. For PQ, beta_scale denotes regularization rate.   
+
+<table><tr><td></td><td colspan="6">CIFAR-10 with ResNet-18</td></tr><tr><td></td><td>REF (float)</td><td>MD-softmax-S</td><td>MD-tanh-S</td><td>GD-tanh</td><td>PQ</td><td>PQ*</td></tr><tr><td>learning_rate</td><td>0.1</td><td>0.001</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.01</td></tr><tr><td>lr_scale</td><td>0.3</td><td>0.3</td><td>0.2</td><td>0.5</td><td>0.3</td><td>=</td></tr><tr><td>beta_scale (ours)</td><td>-</td><td>1.05</td><td>1.2</td><td>1.02</td><td>0.0001</td><td>0.0001</td></tr><tr><td>beta_scale_interval</td><td>=</td><td>500</td><td>1000</td><td>500</td><td>=</td><td>=</td></tr><tr><td>weight_decay</td><td>0.0001</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0.0001</td></tr><tr><td colspan="7">CIFAR-100 with ResNet-18</td></tr><tr><td></td><td>REF (float)</td><td>MD-softmax-S</td><td>MD-tanh-S</td><td>GD-tanh</td><td>PQ</td><td>PQ*</td></tr><tr><td>learning_rate</td><td>0.1</td><td>0.001</td><td>0.001</td><td>0.01</td><td>0.01</td><td>0.001</td></tr><tr><td>lr_scale</td><td>0.1</td><td>0.1</td><td>0.5</td><td>0.5</td><td>0.2</td><td></td></tr><tr><td>beta_scale (ours)</td><td>1</td><td>1.1</td><td>1.1</td><td>1.02</td><td>0.0001</td><td>0.0001</td></tr><tr><td>beta_scale_interval</td><td>-</td><td>100</td><td>500</td><td>1000</td><td>-</td><td>- 0.0001</td></tr><tr><td>weight_decay</td><td>0.0001</td><td>0</td><td>0</td><td>0</td><td>0</td><td></td></tr><tr><td colspan="7">CIFAR-10 with VGG-16</td></tr><tr><td>learning_rate</td><td>REF (float)</td><td>MD-softmax-s</td><td>MD-tanh-S</td><td>GD-tanh</td><td>PQ</td><td>PQ*</td></tr><tr><td>lr_scale</td><td>0.1</td><td>0.001</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.1</td></tr><tr><td>beta_scale (ours)</td><td>0.2</td><td>0.3 1.05</td><td>0.3 1.1</td><td>0.3 1.01</td><td>-</td><td>1 0.0001</td></tr><tr><td>beta_scale_interval</td><td>-</td><td>500</td><td>1000</td><td>500</td><td>1e-07</td><td>-</td></tr><tr><td>weight_decay</td><td>- 0.0001</td><td>0</td><td>0</td><td>0</td><td>- 0</td><td>0.0001</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="7">CIFAR-100 with VGG-16 MD-softmax-S</td></tr><tr><td>learning_rate</td><td>REF (float)</td><td></td><td>MD-tanh-S</td><td>GD-tanh</td><td>PQ</td><td>PQ*</td></tr><tr><td>lr_scale</td><td>0.1 0.2</td><td>0.0001 0.3</td><td>0.001 0.5</td><td>0.01 0.2</td><td>0.01</td><td>0.0001</td></tr><tr><td>beta_scale (ours)</td><td></td><td>1.05</td><td>1.1</td><td>1.05</td><td>- 0.0001</td><td>- 0.0001</td></tr><tr><td>beta_scale_interval</td><td>- -</td><td>100</td><td>500</td><td>2000</td><td></td><td>1</td></tr><tr><td>weight_decay</td><td>0.0001</td><td>0</td><td>0</td><td>0</td><td>- 0</td><td>0.0001</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="7">TinyImageNet with ResNet-18 MD-softmax-S</td></tr><tr><td>learning_rate</td><td>REF (float)</td><td></td><td>MD-tanh-S</td><td>GD-tanh</td><td>PQ</td><td>PQ*</td></tr><tr><td>lr_scale</td><td>0.1 0.1</td><td>0.001</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.01</td></tr><tr><td>beta_scale (ours)</td><td></td><td>0.1</td><td>0.1</td><td>0.5</td><td>1</td><td>-</td></tr><tr><td>beta_scale_interval</td><td>1</td><td>1.2</td><td>1.2</td><td>1.05</td><td>0.01</td><td>0.0001</td></tr><tr><td></td><td>-</td><td>500</td><td>2000</td><td>2000</td><td>-</td><td>1</td></tr><tr><td>weight_decay</td><td>0.0001</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0.0001</td></tr></table>

parse/train/r1xjgxBFPB/r1xjgxBFPB.md

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+# CONTINUAL DEEP LEARNING BY FUNCTIONAL REGULARISATION OF MEMORABLE PAST
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Continually learning new skills without forgetting old ones is an important quality for an intelligent system, yet most deep learning methods suffer from catastrophic forgetting of the past. Recent works have addressed this by regularising the network weights, but it is challenging to identify weights crucial to avoid forgetting. A better approach is to directly regularise the network outputs at past inputs, e.g., by using Gaussian processes (GPs), but this is usually computationally challenging. In this paper, we propose a scalable functional-regularisation approach where we regularise only over a few memorable past examples that are crucial to avoid forgetting. Our key idea is to use a GP formulation of deep networks, enabling us to both identify the memorable past and regularise over them. Our method achieves state-of-the-art performance on standard benchmarks and opens a new direction for life-long learning where regularisation methods are naturally combined with memory-based methods.
+
+# 1 INTRODUCTION
+
+The ability to quickly adapt to changing environments is an important quality of intelligent systems. For such quick adaptation, it is important to be able to identify, memorise, and recall useful past experiences when acquiring new ones. Unfortunately, standard deep-learning methods are not good at maintaining previously acquired skills, and can quickly forget them when learning new skills (Kirkpatrick et al., 2017). Such catastrophic forgetting presents a big challenge when deploying deep-learning methods for applications, such as robotics, where new tasks can appear during the training, and data from the previous tasks might be unavailable for retraining.
+
+In recent years, many methods have been proposed to address catastrophic forgetting in deep learning. One of the most popular approaches is to regularise the network weights to keep them close to the weights obtained for the previous tasks/data (Kirkpatrick et al., 2017; Nguyen et al., 2018; Zenke et al., 2017; Ebrahimi et al., 2019; Serra et al., 2018). This is challenging due to the difficulty in identifying the weights that are relevant to past tasks. The exact values of the weights in fact do not matter directly, but rather the network output (Benjamin et al., 2018). Figuring out which weight affects the output is therefore usually difficult. Typically, the Fisher information matrix or covariance matrices over weights are used (Kirkpatrick et al., 2017; Nguyen et al., 2018), but they only partially address the issue.
+
+A better approach is to directly regularise the network outputs, also referred to as functionalregularisation, requiring a memory of past examples (Benjamin et al., 2018; Lopez-Paz & Ranzato, 2017; Rebuffi et al., 2017). However, such methods still lack a mechanism to automatically weight more relevant past memory in the context of the new task, and also do not take uncertainty of the output into account. Methods based on Gaussian processes do this automatically (Titsias et al., 2019), but require optimisation over inducing points and specification of a good kernel, both of which are difficult tasks. In summary, existing methods fall short in building scalable functional-regularisation methods for continual learning.
+
+In this paper, we propose a new functional-regularisation method where we regularise over a few memorable past examples (see Fig. 1). Our approach builds upon a recent method of Khan et al. (2019) that expresses deep networks as Gaussian processes (GPs). We show that the GP formulation not only enables the identification of examples crucial to avoid forgetting, but also computes uncertainty over the network output to appropriately weight the past examples in the light of new ones. Motivated by the GP perspective, we propose a new loss function for function-regularisation with deep networks. A Laplace approximation of this objective enables scalable training. Our work in this paper focusses on avoiding forgetting, but it opens a new direction for life-long learning methods where regularisation methods are naturally combined with memory-based methods.
+
+![](images/cdaa926e6636dcf6f7dee5c08806e67fe60b5b29539c5dcd4a22231c6cce6a76.jpg)  
+Figure 1: This figure illustrates our method. Leftmost figure shows the result of training on task 1. Examples corresponding to memorable-past, shown with big markers, are chosen using a GP formulation of the neural network. These points usually are the ones that support the decision boundary. Middle figure shows the result after task 2 where new network functions are regularised at memorable-past examples to give the same prediction as the previous ones. The resulting green decision boundary classifies both task 1 and 2 well. The rightmost figure shows the result along with memorable-past of each task where the performance over the past tasks is maintained.
+
+Other related works. Broadly, existing work on continual learning can be split into three approaches: inference based, memory/rehearsal based, and model based. Inference based approaches have mostly focused on weight regularisation, with some recent efforts on functional regularisation. Our work falls in the latter category. Memory based approaches either maintain a memory of past data examples (Rebuffi et al., 2017) or train generative models on previous tasks to rehearse pseudo-inputs (Shin et al., 2017). An advantage of our method compared to previous ones is that building memory does not require solving an optimisation problem: the computation simply involves a forward-pass through the network followed by sorting (see Section 3.2).
+
+Similarly to our work, there have also been some efforts in combining the different flavours of approaching continual learning, e.g., VCL plus coresets (Nguyen et al., 2018) and Gradient-Episodic Memory (Lopez-Paz & Ranzato, 2017; Chaudhry et al., 2018). Benjamin et al. (2018) have proposed a similar combination for functional regularisation. In these approaches, two separate methods are usually used for regularisation and memory-building. Recent follow-up work (Aljundi et al., 2019; Chaudhry et al., 2019) has focussed on improving memory-building methods, but this separation has remained the case. In contrast, in our approach, both of these are done within the same GP framework by using the method of Khan et al. (2019).
+
+Finally, model based approaches change the model architecture during training (Rusu et al., 2016), and this can be combined with other approaches (Schwarz et al., 2018). It is possible to use similar features in our GP based framework, which is an interesting future direction to be pursued.
+
+# 2 CONTINUAL LEARNING WITH WEIGHT/FUNCTIONAL REGULARISATION
+
+In deep learning, we minimise loss functions to estimate network weights. For example, in supervised multi-class classification problems, we are given a dataset $\mathcal { D }$ of $N$ input-output pairs with outputs ${ \bf y } _ { i }$ , a vector of $K$ classes, and inputs $\mathbf { x } _ { i }$ , a vector of length $D$ , and our goal is to minimise a loss which takes the following form: $\bar { \ell } ( \mathbf { w } ) + \delta R ( \mathbf { w } )$ , where $\begin{array} { r } { \bar { \ell } ( \mathbf { w } ) : = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \bar { \ell } ( \mathbf { \bar { y } } _ { i } , \mathbf { f } _ { w } ( \mathbf { x } _ { i } ) ) } \end{array}$ with deep neural network $\mathbf { f } _ { w } ( \mathbf { x } ) \in \mathbb { R } ^ { K }$ \`and its weights $\mathbf { w }$ . $\ell ( \mathbf { y } , \hat { \mathbf { y } } )$ \` \` ,denotes a differentiable loss function between an output $\mathbf { y }$ and its prediction ${ \hat { \mathbf { y } } } , R ( \mathbf { w } )$ \` , is a regularisation function (usually an $L _ { 2 }$ -regulariser $R ( \mathbf { w } ) = \mathbf { w } ^ { \top } \dot { \mathbf { w } } )$ and $\delta > 0$ controls the regularisation strength. Standard deep-learning approaches δ >rely on an unbiased stochastic-gradient of the loss $\bar { \ell }$ , which usually requires access to all of the data \`examples for all classes (Bottou, 2010). It is this unbiased, minibatch setting where deep-learning excels and achieves state-of-the-art performance on many benchmark datasets.
+
+In reality, we do not always have access to all the data at once, and it is not possible to obtain unbiased stochastic gradients. New classes may appear during training and old classes may never be seen again. For such settings, vanilla mini-batch stochastic-gradient methods lead to catastrophic forgetting of past information (Kirkpatrick et al., 2017). Our goal in this paper is to design methods that can avoid such catastrophic forgetting. We focus on a particular setting where the classification task is divided into several tasks, e.g., a task may consist of a classification problem over a subset of classes. We assume that the tasks arrive sequentially one after the other. Once the learning is over, we may never see that task again. Such continual-learning settings have been considered in previous works (Kirkpatrick et al., 2017; Nguyen et al., 2018; Zenke et al., 2017), and our goal is to avoid forgetting of old tasks in this setting.
+
+Recent methods have proposed weight-regularisation as a way to combat catastrophic forgetting. The main idea is to keep the new network weights close to the old ones, e.g., when training a task $t$ while given network weights $\mathbf { w } _ { t - 1 }$ trained on task $t - 1$ , we can minimise the following loss: $\bar { \ell } _ { t } ( \mathbf { w } ) + \bar { \delta } ( \mathbf { w } - \mathbf { w } _ { t - 1 } ) ^ { \top } \mathbf { F } _ { t } ( \mathbf { w } - \mathbf { w } _ { t - 1 } )$ , where $\bar { \ell } _ { t } ( \mathbf { w } )$ is the loss defined over all data examples from task $t$ and $\mathbf { F } _ { t }$ δ \`is a preconditioning matrix that favors the weights relevant to the past tasks more than the rest. The Elastic-Weight Consolidation (EWC) method (Kirkpatrick et al., 2017), for example, uses the Fisher information matrix as the pre-conditioner, while Ritter et al. (2018) use the Hessian of the loss, and VCL (Nguyen et al., 2018) essentially employs the precision matrix of the variational approximation to do the same. Such weight-space methods reduce forgetting but do not produce satisfactory results.
+
+The challenge in using weight-regularisation lies in the fact that the exact values of the weights do not really matter due to parametric symmetries (Benjamin et al., 2018; Bishop, 2006). Since only the network outputs matter, an alternative approach is to directly regularise these. Benjamin et al. (2018) propose to use an $L _ { 2 }$ -regulariser over the function values on data examples from past tasks:
+
+$$
+\operatorname* { m i n } _ { w } \bar { \ell } _ { t } ( \mathbf { w } ) + \delta \sum _ { s = 1 } ^ { t - 1 } \sum _ { i \in \mathcal { M } _ { s } } \| f _ { w } ( \mathbf { x } _ { i } ) - f _ { w _ { t - 1 } } ( \mathbf { x } _ { i } ) \| _ { 2 } ^ { 2 } ,
+$$
+
+where $\mathcal { M } _ { s }$ is the set of small examples for task $s$ stored in the working memory (Lopez-Paz & Ranzato, 2017; Rebuffi et al., 2017). One major issue with this approach is that the $L _ { 2 }$ -regulariser weights all the data points equally. In reality, some examples in the memory are more important than others to learn a given task. In addition, the uncertainty of the prediction is also ignored. As we will show, working with distributions over functions allows us to address these two issues.
+
+Gaussian processes (GPs), for example, enable automatic reweighting of old tasks in light of a new one, which happens by using the posterior covariance. Unfortunately, both scalability of GPs as well as the necessity to save many instances of the past makes them impractical. A recent approach by Titsias et al. (2019) attempts to address these issue by employing sparse GP methods with inducing points and using a neural network feature map. However, their approach has some difficulties. Their approach heavily depends on a proper choice of inducing points, which are normally obtained via an ad-hoc procedure. Additionally, they propose using the last layer of the neural network as kernel features, which is limiting as it does not use the whole network’s weights in the kernel.
+
+Ideally, we would want to perform functional regularisation with a deep learning optimiser while borrowing ideas from the GP methods. Our work in this paper takes a step in this direction. Our proposal is to use a GP formulation of neural networks to perform functional regularisation that still allows training with standard deep-learning methods. We also show that the GP view helps in selecting an informative set of memory points.
+
+# 3 FUNCTIONAL-REGULARISATION OF MEMORABLE PAST (FROMP)
+
+We will now describe our proposed method. The first step is to use the GP formulation of Khan et al. (2019) to compute GP-like posteriors over functions. Then, we propose a method to identify a set of memorable past examples for a task. Finally, we describe our functional-regularisation and discuss approximations used to build a scalable method.
+
+# 3.1 FROM DEEP NETWORKS TO GAUSSIAN PROCESS POSTERIORS
+
+Khan et al. (2019) propose an approach to convert deep networks into Gaussian processes. Their main result (see Theorem 1 in their paper) states that, at a local minimiser $\mathbf { W } _ { * }$ of $\begin{array} { r } { \mathbf { \bar { \ell } } ( \mathbf { w } ) + \frac { \delta } { 2 } \mathbf { w } ^ { \top } \mathbf { w } } \end{array}$ , a \`Laplace approximation of the posterior over w is equivalent to the posterior distribution of a linear model. Specifically, they use the following scalable variant of the Laplace approximation:
+
+$$
+p ( \mathbf { w } | \mathcal { D } ) \approx q ( \mathbf { w } ) : = N ( \mathbf { w } | \mu , \Sigma ) , \mathrm { ~ w h e r e ~ } \mu = \mathbf { w } _ { * } \mathrm { ~ a n d ~ } \Sigma ^ { - 1 } = \sum _ { i = 1 } ^ { N } \mathbf { J } ( \mathbf { x } _ { i } ) ^ { \top } \mathbf { A } ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) \mathbf { J } ( \mathbf { x } _ { i } ) + \delta \mathbf { I } _ { P } ,
+$$
+
+where $\mathbf { J } ( \mathbf { x } _ { i } ) : = \nabla _ { w } \mathbf { f } _ { w } ( \mathbf { x } _ { i } ) ^ { \top }$ is a $K \times P$ Jacobian matrix ( $P$ being the number of parameters), and $\pmb { \Lambda } ( \mathbf { x } , \mathbf { y } ) : = \nabla _ { \mathbf { f } \mathbf { f } } ^ { 2 } \ell ( \mathbf { y } , \mathbf { f } )$ is the $K \times K$ Hessian of the loss with $\mathbf { f } = \mathbf { f } _ { w } ( \mathbf { x } )$ , all evaluated at $\mathbf { w } = \mathbf { w } _ { \ast }$ . They ,show that $q ( \mathbf { \dot { w } } )$ ,is equal to the posterior of the following linear model:
+
+$$
+\tilde { \mathbf { y } } = \mathbf { J } ( \mathbf { x } ) \mathbf { w } + \epsilon , \mathrm { w i t h } \epsilon \sim N ( 0 , ( \Lambda ( \mathbf { x } , \mathbf { y } ) ) ^ { - 1 } ) \mathrm { a n d } \mathbf { w } \sim N ( 0 , \delta ^ { - 1 } \mathbf { I } _ { P } ) ,
+$$
+
+where the observations are defined as $\tilde { \mathbf { y } } _ { i } ~ : = ~ \mathbf { J } ( \mathbf { x } _ { i } ) \mathbf { w } _ { * } - \left( \pmb { \Lambda } ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) \right) ^ { - 1 } \mathbf { r } ( \mathbf { x } _ { i } , \mathbf { y } _ { i } )$ with the residual $\mathbf { r } ( \mathbf { x } , \mathbf { y } ) : = \nabla _ { \mathrm { f } } \ell ( \mathbf { y } , \mathbf { f } )$ , ,. They also show that the predictive distribution of this linear model is equiv, \` ,alent to that of a GP regression model defined with a $K \times K$ neural tangent kernel (NTK) (Jacot et al., 2018):
+
+$$
+\begin{array} { r } { \tilde { \mathbf { y } } = \mathbf { f } _ { \mathrm { G P } } ( \mathbf { x } ) + \boldsymbol { \epsilon } , \quad \mathrm { ~ w i t h ~ } \boldsymbol { \epsilon } \sim \mathcal { N } ( 0 , ( \mathbf { \Lambda } ( \mathbf { x } , \mathbf { y } ) ) ^ { - 1 } ) \mathrm { ~ a n d ~ } \mathbf { f } _ { \mathrm { G P } } ( \mathbf { x } ) \sim \mathcal { G P } \left( 0 , \delta ^ { - 1 } \mathbf { J } ( \mathbf { x } ) \mathbf { J } ( \mathbf { x } ^ { \prime } ) ^ { \top } \right) . } \end{array}
+$$
+
+The above result gives a posterior in the defined observation space of $\tilde { \mathbf { y } }$ . In the case where we can find a map from $\tilde { \mathbf { y } }$ to $\mathbf { y }$ , the above inference problem allows for reparameterisation into the original data space. Therefore, we will obtain a GP that serves as an approximation to the neural network posterior. Below, we demonstrate this for binary classification with sigmoid link function and Bernoulli likelihood (see Appendix A for details and extension to multi-class classification).
+
+We parameterise the Bernoulli likelihood by $\mathfrak { p } ( \mathbf { x } ) : = \sigma ( \mathrm { f } _ { w _ { * } } ( \mathbf { x } ) )$ where $\sigma$ is the sigmoid function (note that we now have scalars $\{ \mathrm { f } , \mathrm { y } , \ldots \}$ σ σas we are dealing with a single output). The residual and , , ...loss Hessian for this model are given by $\operatorname { r } ( \mathbf { x } , \mathbf { y } ) = \operatorname { p } ( \mathbf { x } ) - \mathbf { y }$ and $\Lambda ( \mathbf { x } , \mathbf { y } ) = \mathsf { p } ( \mathbf { x } ) ( 1 - \mathsf { p } ( \mathbf { x } ) )$ , respectively, where $\mathsf { y } \in \{ 0 , 1 \}$ , ,. Because residuals are linear in y and the Hessian is independent of y, we can ,substitute the definitions of $\tilde { \bf y }$ and $\mathbf { r } ( \mathbf { x } , \mathbf { y } )$ into Eq. 3, and rearrange for y,
+
+$$
+\mathbf { y } = \underbrace { \mathbf { p } ( \mathbf { x } ) + \Lambda ( \mathbf { x , y } ) \mathbf { J } ( \mathbf { x } ) ( \mathbf { w } - \mathbf { w } _ { * } ) } _ { : = \mathbf { f } _ { \mathrm { i n } } ( \mathbf { X } ) } + \tau , \ \mathrm { ~ w i t h ~ } \tau \sim \mathcal { N } ( 0 , \Lambda ( \mathbf { x , y } ) ) \ \mathrm { ~ a n d ~ } \mathbf { w } \sim \mathcal { N } ( 0 , \delta ^ { - 1 } \mathbf { I } _ { P } ) .
+$$
+
+The Laplace approximation in Eq. 2 is the posterior of this model after observing data $\mathcal { D }$ . We can equivalently write the posterior predictive as a GP (Rasmussen, 2003) with mean and covariance
+
+$$
+\begin{array} { r l } & { \qquad \mathrm { m } ( \mathbf { x } ) : = \mathbb { E } _ { q ( w ) } \left[ \mathrm { f } _ { \mathrm { l i n } } ( \mathbf { x } ) \right] = \mathrm { p } ( \mathbf { x } , \mathbf { y } ) + \Lambda ( \mathbf { x } , \mathbf { y } ) \mathbf { J } ( \mathbf { x } ) ( \mu - \mathbf { w } _ { \ast } ) = \mathrm { p } ( \mathbf { x } ) , } \\ & { \qquad \mathrm { k } ( \mathbf { x } , \mathbf { x } ^ { \prime } ) : = \mathbb { E } _ { q ( w ) } \left[ ( \mathrm { f } _ { \mathrm { l i n } } ( \mathbf { x } ) - \mathrm { m } ( \mathbf { x } ) ) \left( \mathrm { f } _ { \mathrm { l i n } } ( \mathbf { x } ^ { \prime } ) - \mathrm { m } ( \mathbf { x } ^ { \prime } ) \right) \right] } \\ & { \qquad = \Lambda ( \mathbf { x } , \mathbf { y } ) \mathbf { J } ( \mathbf { x } ) \mathbb { E } _ { q ( w ) } \left[ ( \mathbf { w } - \mathbf { w } _ { \ast } ) ( \mathbf { w } - \mathbf { w } _ { \ast } ) ^ { \top } \right] \mathbf { J } ( \mathbf { x } ^ { \prime } ) ^ { \top } \Lambda ( \mathbf { x } ^ { \prime } , \mathbf { y } ^ { \prime } ) } \\ & { \qquad = \Lambda ( \mathbf { x } , \mathbf { y } ) \mathbf { J } ( \mathbf { x } ) \Sigma \mathbf { J } ( \mathbf { x } ^ { \prime } ) ^ { \top } \Lambda ( \mathbf { x } ^ { \prime } , \mathbf { y } ^ { \prime } ) . } \end{array}
+$$
+
+A GP defined with the above mean and covariance function can be viewed as an approximation to the posterior process over network outputs. Throughout the paper, we will denote the process obtained at a weight $\mathbf { W } _ { * }$ by a distribution $q _ { w _ { * } } ( \mathbf { f } )$ where f is a vector of $\operatorname { f } ( \mathbf { x } )$ evaluated at many different inputs $\mathbf { X }$ . We will use this GP posterior predictive for functional regularisation.
+
+# 3.2 MEMORABLE PAST
+
+In the previous section, we derived a GP posterior approximation over the network outputs. Now, we propose a method to obtain a small set of examples that are crucial to avoid forgetting. We first note that the posterior GP mean in Eq. 4 corresponds to a kernel Ridge regression that requires the computation of $( \delta ^ { - 1 } \mathbf { K } + \mathbf { { A } } ^ { - 1 } ) ^ { - 1 } \tilde { \mathbf { y } }$ where $\pmb { \Lambda }$ is a block-diagonal matrix containing all $\mathbf { \Delta } \Lambda _ { i } : = \mathbf { \Delta } \Lambda ( \mathbf { x } _ { i } , \mathbf { y } _ { i } )$ and $\mathbf { K }$ δ ,is the NTK defined in Eq. 4. The predictions therefore strongly depend on the eigenvalues of the preconditioning matrix. Selection of important data examples therefore boils down to selecting
+
+# Algorithm 1: Functional Regularisation of Memorable Past (FROMP)
+
+![](images/74d94103ae588172889f704c1d1cba78090bbde4ef4de4682ccb6b5590f6b649.jpg)  
+Figure 2: A pseudo-code for our FROMP algorithm. The additional computations on top of Adam are in lines 4 and 7, where we add the contribution from the functional-regularisation term. After every task, we update the memorable past in Eq. 9-11 which involves a matrix inversion of size $M$ , and a forward pass through the network to compute $\mathbf { \Delta } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda }$ .
+
+important columns of this matrix, such that the predictions remain unchanged. The theory of leverage score sampling (Alaoui & Mahoney, 2015; Bach, 2013) suggests picking the points proportional to the leverage score defined to be the diagonal of the following matrix: ${ \bf K } ( { \bf K } + \delta { \bf A } ^ { - 1 } ) ^ { - 1 }$ . Typically, δthis matrix is difficult to compute and methods are employed to approximately obtain the leverage score (Alaoui & Mahoney, 2015).
+
+For deep-learning applications too, exact computation of leverage score is very difficult. We instead propose a simple solution. Since the eigenvalues of the previous matrix heavily depend on $\delta { \bf A } _ { i } ^ { - 1 }$ , we can pick the data examples by simply sorting $\mathbf { \Lambda } _ { \mathbf { \Lambda } } \mathbf { \Lambda } _ { \mathbf { \Lambda } }$ . For the inverse of $K + \delta \mathbf { { A } } ^ { - 1 }$ δto have high eigenvalues, we should favour examples with smaller values of $\boldsymbol { \Lambda } _ { i } ^ { - 1 }$ δ. Therefore, we simply sort the $\mathbf { \Delta } \Lambda _ { i }$ and pick the top $M$ examples as the most relevant ones. This simple solution is very effective for our particular problem because $\mathbf { \Delta } \Lambda _ { i }$ are noise variances for the data examples, obtained by using an already trained network. These are second derivatives of the loss for data examples, and so also reflect the sensitivity of the decision boundaries if a particular data point is perturbed. Therefore, they tend to reflect the relevance of data examples. An example is shown in Figure 1 where we clearly see that our solution picks the examples lying close to decision boundary. Computation of $\mathbf { \Delta } \Lambda _ { i }$ requires us to run the forward pass to get the $\ell ( \mathbf { y } _ { i } , \hat { \mathbf { y } } _ { i } )$ and then compute its second derivative with respect to $\hat { \mathbf { y } } _ { i }$ \` ,, both of which are cheap operations. Throughout the paper, examples chosen by this method are referred to as the memorable past examples, and denoted $\boldsymbol { \mathcal { M } } _ { t }$ for task $t$ .
+
+# 3.3 FUNCTIONAL-REGULARISATION
+
+So far, we described the construction of the GP posterior predictive over the function space, as well as the construction of a set of memorable-past examples. We are now ready to describe our objective function where we employ these two to perform functional regularisation.
+
+Suppose that we are given network weights $\mathbf { w } _ { t - 1 }$ that are obtained by training over data examples from task $t - 1$ . Our goal then is to train a network with weights w such that its performance on memorable past $\boldsymbol { \mathcal { M } } _ { 1 : t - 1 }$ is unchanged. We denote the vector of function outputs over these examples by $\mathbf { a } _ { 1 : t - 1 }$ . A straightforward idea is to directly optimise the weights w such that the predictions using $q _ { w } ( \mathbf { f } _ { t } )$ are good on current tasks while the predictive distribution $q _ { w } ( \mathbf { a } _ { 1 : t - 1 } )$ is close to $q _ { w _ { t - 1 } } ( \mathbf { a } _ { 1 : t - 1 } )$ . Since the number of tasks can be very large, we choose to regularise each task separately, i.e., we will match $q _ { w } ( { \bf a } _ { s } )$ with $q _ { w _ { t - 1 } } ( \mathbf { a } _ { s } )$ separately for all tasks $s \ < \ t$ , in line with Titsias et al. (2019). <These choices give us the following objective function with trade-off parameter $\tau$ :
+
+$$
+\operatorname* { m i n } _ { w } \quad \tau \mathbb { E } _ { q _ { w } ( f _ { t } ) } \Big [ \sum _ { i \in \mathcal { D } _ { t } } \ell ( y _ { i } , f ( { \mathbf x } _ { i } ) ) \Big ] + \sum _ { s = 1 } ^ { t - 1 } \mathbb { D } _ { K L } [ q _ { w } ( { \mathbf a } _ { s } ) \| { q } _ { w _ { t - 1 } } ( { \mathbf a } _ { s } ) ] .
+$$
+
+Table 1: Train accuracy of FROMP and batch-trained Adam (upper bound on performance) on variations of a toy 2D binary classification dataset, with mean and standard deviations over 10 runs (3 runs for Adam). FROMP performs well across variations. See Appendix D.2 for visualisations.   
+
+<table><tr><td>Dataset variation</td><td>FROMP</td><td>Batch Adam</td></tr><tr><td>10x less data (400 per task)</td><td>99.9% ± 0.0</td><td>99.7% ±0.2</td></tr><tr><td>10x more data (40000 per task)</td><td>96.9% ± 3.0</td><td>99.7% ± 0.0</td></tr><tr><td>Introduced 6th task</td><td>97.8% ± 3.3</td><td>99.6% ± 0.1</td></tr><tr><td>Increased std dev of each class distribution</td><td>96.0% ± 2.4</td><td>96.9% ± 0.4</td></tr><tr><td>2 tasks have overlapping data</td><td>90.1% ± 0.8</td><td>91.1% ± 0.3</td></tr></table>
+
+A major issue with this objective is that computing gradients wrt. the posterior predictive $q _ { w }$ will be computationally heavy as it involves gradients of the Jacobian and $\mathbf { \Lambda } _ { \mathbf { \Lambda } } \mathbf { \Lambda } _ { \mathbf { \Lambda } }$ , which involve higher order derivatives. During training on a task, we have no distribution on parameters (cf. Laplace approximation) and can treat it as fixed. Therefore, taking the gradient wrt. the parameter $\mathbf { w }$ of the above objective will only involve differentiating the mean function $\mathbf { m } ( \mathbf { x } _ { i } )$ formed by the neural network, and we have equivalence to the following objective:
+
+$$
+\operatorname* { m i n } _ { \boldsymbol { w } } \quad \tau \sum _ { i \in \mathcal { D } _ { t } } \ell ( y _ { i } , f _ { w } ( \mathbf { x } _ { i } ) ) + \frac { 1 } { 2 } \sum _ { s = 1 } ^ { t - 1 } ( \mathbf { m } _ { s , w } - \mathbf { m } _ { s , w _ { t - 1 } } ) ^ { \top } \mathbf { K } _ { w _ { t - 1 } , s } ^ { - 1 } ( \mathbf { m } _ { s , w } - \mathbf { m } _ { s , w _ { t - 1 } } ) ,
+$$
+
+where $\mathbf { m } _ { s , w _ { t - 1 } }$ and ${ \bf K } _ { w _ { t - 1 } , s }$ are the vector and matrices containing the corresponding mean and co, ,variance function of the network weights $\mathbf { w } _ { t - 1 }$ evaluated at the memorable past for task $s$ . Eqs 6 and 7 are used to calculate $\mathbf { m } _ { s , w _ { t - 1 } }$ and ${ \bf K } _ { w _ { t - 1 } , s }$ respectively. Essentially, this gives us something similar ,to Eq. 1 but instead of the $L _ { 2 }$ ,distance, a kernel is used to improve weighting of examples. The final FROMP algorithm is summarised in Algorithm 1. Please see App. B for details on scaling to the multi-class setting.
+
+The additional time complexity of this algorithm above vanilla Adam is $O ( M P K t )$ per optimisation step, and $O ( [ M P K + \bar { M ^ { 3 } } ] t + \bar { N } P K )$ once per task. $O ( N P K )$ is for updating the Laplace approximation $\pmb { \Sigma }$ at the end of training on a task, as the Jacobian for each input training point needs to be calculated (as in Eq. 2). This is similar to EWC (Kirkpatrick et al., 2017). Note that, as we use a diagonal approximation to $\pmb { \Sigma }$ , inverting it is fast. $O ( M P K t )$ is for calculating each of the $t$ matrices ${ \bf K } _ { w _ { t - 1 } , s }$ , and $O ( M ^ { 3 } t )$ is for inverting them. This is done once before training on each task (Eq. 7). ,Finally, during optimisation of the objective function, we need $O ( M P K )$ for differentiating the mean function $\mathbf { m } ( \mathbf { x } _ { i } )$ . All of these additional costs are small for small $M$ , except for $O ( N P K )$ , which can be reduced by randomly sampling a subset of the task’s data. Note that it is possible to be stochastic in sampling a subset of previous tasks’ memory during every update, removing the time complexity dependency on $t$ . Doing this gives the same overall complexity per step as in Adam (as $M$ is usually of the same order as the mini-batch size).
+
+# 4 EXPERIMENTS
+
+We run the proposed method FROMP on toy datasets, permuted MNIST and split MNIST (LeCun et al., 1998; Goodfellow et al., 2013), and a split version of CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2009). We optimise the objective in Eq. 9 using Adam (Kingma & Ba, 2015) with parameter $\beta _ { 1 } = 0 . 9 9$ and further use gradient clipping to speed up training. To identify the individual benefits β .of the kernel in the loss (Eq. 9) as well as the selection of data points, we compare the following four methods: FROMP uses kernel and the proposed example selection, while FRORP instead uses randomly chosen examples. Further, we replace the kernel by an identity matrix leading to functional $L _ { 2 }$ regularisation and call the corresponding methods with and without our example selection technique FROMP- $L _ { 2 }$ and FRORP- $L _ { 2 }$ .
+
+# 4.1 TOY DATASET
+
+We first test FROMP on many variations of a toy 2D binary classification dataset like that in Figure 1. We want to test its performance when exposed to different datasets of varying difficulty. We use a
+
+![](images/4da7f8f7504945c3efe091d92ad338addb1376d1c86b5bfd6c018330f3ad863c.jpg)  
+Figure 3: Permuted MNIST: added kernels across classes (with subtracted diagonal for visualisation purposes), and performance as a function of memory size (average accuracy after 10 tasks). Memorable past examples lead to a more uniform kernel structure that prevents weighting previously overfit examples too highly, e.g. task 1 in the random selection exhibits strong correlation and low variance. As we reduce the number of examples in memory, FROMP gracefully reduces accuracy.
+
+2-hidden layer MLP (with 20 hidden units in each layer) for all experiments. Appendix D.3 details hyperparameter selection. In Appendix D.1 we show the brittleness and inconsistent behaviour of weight-space regularisation methods. In contrast, FROMP performs extremely well across many variations, showing consistently good results (see Table 1 and Appendix D.2 for visualisations).
+
+Table 2: The average validation accuracy on Permuted-MNIST (10 tasks) and Split-MNIST. $\mathrm { \Delta ^ { * } \mathrm { 2 0 0 p / t } ^ { \mathrm { * } } }$ denotes that 200 examples are selected for each task. We report mean and standard deviations over 5 runs, and use results from Nguyen et al. (2018) for baselines. FROMP is state-of-the-art. Additionally, as we reduce the number of points (Figure 3), FROMP gracefully reduces accuracy, due to the clever choice of memorable past and the use of kernels in the functional regularisation.   
+
+<table><tr><td>Method</td><td>Permuted MNIST</td><td> Split MNIST</td></tr><tr><td>DLP (Smola et al., 2003)</td><td>82%</td><td>61.2%</td></tr><tr><td>EWC (Kirkpatrick et al., 2017)</td><td>84%</td><td>63.1%</td></tr><tr><td>SI (Zenke et al., 2017)</td><td>86%</td><td>98.9%</td></tr><tr><td>Improved VCL (Swaroop et al., 2019)</td><td>93%± 1</td><td>98.4% ± 0.4</td></tr><tr><td>+ random Coreset</td><td>94.6% ± 0.3 (200 p/t)</td><td>98.2% ± 0.4 (40 p/t)</td></tr><tr><td>FRCL-RND (Titsias et al., 2019)</td><td>94.2% ± 0.1 (200 p/t)</td><td>96.7% ± 1.0 (40 p/t)</td></tr><tr><td>FRCL-TR (Titsias et al., 2019)</td><td>94.3% ± 0.1 (200 p/t)</td><td>97.4% ± 0.6 (40 p/t)</td></tr><tr><td>FRORP-L2</td><td>87.9% ± 0.7 (200 p/t)</td><td>98.5% ± 0.2 (40 p/t)</td></tr><tr><td>FROMP-L2</td><td>94.6% ± 0.1 (200 p/t)</td><td>98.7% ± 0.1 (40 p/t)</td></tr><tr><td>FRORP</td><td>94.6% ± 0.1 (200 p/t)</td><td>99.0% ± 0.1 (40 p/t)</td></tr><tr><td>FROMP</td><td>94.9% ± 0.1 (200 p/t)</td><td>99.0% ± 0.1 (40 p/t)</td></tr></table>
+
+# 4.2 PERMUTED AND SPLIT MNIST
+
+Permuted MNIST consists of a series of tasks where each task is a fixed permutation of pixels to the entire labelled MNIST dataset. Like in previous work (Nguyen et al., 2018; Kirkpatrick et al., 2017; Zenke et al., 2017; Titsias et al., 2019), we implement a fully connected single-head network with two hidden layers, and report performance after 10 tasks. Each hidden layer consists of 100 hidden units and ReLU activation functions. We set the learning rate to 0 001, batch size to 128, and learn each task for 10 epochs. We use $\tau = 1$ .for both FROMP and FROMP- $L _ { 2 }$ (see Eq. 9).
+
+The Split MNIST experiment was introduced by Zenke et al. (2017) and consists of five binary classification tasks built from MNIST: 0/1, 2/3, 4/5, 6/7, and $^ { 8 / 9 }$ . We use a fully connected multi-head neural network with two hidden layers, each with 256 hidden units and ReLU activation functions. Following the settings of previous work, we select 40 memorable points per task. The learning rate is set to 0 0001, batch size to 128, and we learn each task for 15 epochs. We find optimal parameters $\tau = 1$ . and $\tau = 1 0$ for FROMP and FROMP- $L _ { 2 }$ respectively.
+
+![](images/b4369110b5fb362aec49d546d839c2a65e92d215201756fe5754562ccfc00bc6.jpg)  
+Figure 4: Split MNIST: randomly selected samples and memorable past in comparison. 10 samples per class and method. In line with the toy example in Fig. 1, memorable past examples appear to be closer to the decision boundary of a classifier: many samples of the memorable past are possibly hard to distinguish from other classes.
+
+We report the final average accuracy for both benchmarks across all the tasks in Table 2. The proposed method achieves better performance than the weight-space methods EWC and VCL, as well as compared to the function-space method FRCL that is based on a GP formulation. Further, the benchmarks show superior performance of both the approach to select important examples (Sec. 3.2) and the functional regularisation using the kernel (Sec. 3.3). Memorable examples improve performance of the naive and efficient FRORP- $. L _ { 2 }$ method by more than $6 \%$ on permuted MNIST and by $0 . 2 \%$ on the split MNIST. Standard deviation is also reduced in both cases. FROMP does not profit much from memorable examples compared to FRORP, probably because the performance is already close to the maximum achievable. Furthermore, Fig. 3c shows that our selection method greatly reduces the number of memorable points required: the $L _ { 2 }$ algorithm with random points requires over 100 points to match the performance when using 20 carefully selected points, and 200 points to match performance with 40, respectively. When combined with kernel-based functional regularisation, we obtain the best-performing method, particularly when memory size is small. In line with the toy example in Fig. 1, Fig. 4 illustrates the memorable past examples compared to randomly selected examples. The memorable past examples appear to be more special cases, and may therefore lie closer to the decision boundary.
+
+Figs. 3a and 3b show the summed kernels across all the classes in permuted MNIST for a random and memorable set of points. For visualisation purposes, the diagonal is suppressed. Note that random points lead to less uniform weighting in the kernel, making it even more important in functional regularisation, leading to better performance (FRORP vs FROMP). All memorable points are important and the kernel is more uniform. In Fig. 3a, the kernel’s weighting of task 1 is very different from other tasks, leading to different magnitudes in the functional regularisation among tasks and therefore to eventual forgetting. The kernel further tells us that the tasks are correlated, as expected.
+
+# 4.3 SPLIT CIFAR
+
+Split CIFAR is far more complex than the MNIST experiments and consists of 6 tasks. The first task is the full CIFAR-10 dataset, followed by 5 tasks, each consisting of 10 consecutive classes from CIFAR-100. We follow the SI paper (Zenke et al., 2017) for our model architecture, using a multi-head CNN with 4 convolutional layers, followed by 2 dense layers with dropout. We use learning rate 0 0001 and batch size 256. All tasks are learned for 80 epochs and we use a memory .of 10–200 examples. We use $\tau = 0 . 1$ and $\tau = 0 . 0 5$ for FROMP and FROMP-L2 respectively. In τ . τ .addition to continual learning baselines, we report the performance of networks trained from scratch on each task. These cannot profit from forward/backward transfer. We also report the performance of a network jointly trained on all tasks. This is an upper bound on performance.
+
+![](images/015e41d3d81cc334976fa2e2763903197c635e3c2f47fa535d4f92bb83957db2.jpg)  
+Figure 5: Split CIFAR: performance for each task, and performance variation as function of memory size, after training on the final task. We run all methods 5 times and report the mean and standard error. For baselines, we train from scratch on each task and jointly on all tasks achieving $7 3 . 6 \% \pm 0 . 4$ and $7 8 . 1 \% \pm 0 . 3$ . ., respectively. The left figure reports results for 200 memory examples per task. . .The final average validation accuracy of FROMP is $7 6 . 2 \% \pm 0 . 4$ , FROMP- $L _ { 2 }$ is $7 4 . 6 \% \pm 0 . 4$ , SI is $7 3 . 5 \% \pm 0 . 5$ .(result from Zenke et al. (2017)), EWC is $7 1 . 6 \% \pm 0 . 9$ , VCL $^ +$ . . random coreset is $6 7 . 4 \% \pm 1 . 4$ . . .. FROMP outperforms all other methods, and is close to the performance of the jointly . .trained model, particularly in tasks 4-6. Additionally, as we reduce the memory size, FROMP still performs well, even with only 20 examples per task.
+
+The experimental results in Fig. 5a show that FROMP outperforms other continual learning methods by a notable margin. The weight-space methods employed as baselines either cannot learn later tasks to a high accuracy (EWC, SI), or forget previous tasks (VCL). Interestingly, averaged over all tasks, FROMP also out-performs ‘from scratch’ training and achieves performance close to a jointly training on all tasks by a margin of less than $2 \%$ . Note that on tasks 4-6, FROMP achieves the same accuracy as the jointly trained model, showing no forgetting. We also calculate the backward transfer metric BWT from Lopez-Paz & Ranzato (2017). We find that FROMP has a score of $- 2 . 6 \pm 0 . 9$ , which is joint best with EWC’s score of $- 2 . 3 \pm 1 . 4$ (VCL $^ +$ coresets has $- 9 . 2 \pm 1 . 8 )$ . .). Although . . . .EWC performs well in backward transfer, it does so at the cost of forward transfer. We calculate a forward transfer metric as the average of the improvement in accuracy on a new task over an independently trained model on that task (see App. C for more precise definitions of these metrics). FROMP achieves a forward transfer of $6 . 1 \pm 0 . 7$ , whereas EWC has $0 . 1 7 \pm 0 . 9$ and $\mathrm { V C L + }$ coresets has $1 . 8 \pm 3 . 1$ . . . .. Although we only focussed on preventing catastrophic forgetting, we find evidence . .of forward transfer, a key requirement in continual learning. Overall, given final average accuracy, backward transfer and forward transfer, FROMP clearly outperforms the other baselines.
+
+In contrast to the rather simple MNIST benchmarks, both the benefit of selecting memorable points as well as using the kernel are clearly visible in Fig. 5b. If we only memorise a few examples, the performance gap due to using the kernel is around $4 \%$ . The selection of memorable points according to our metric leads to an increase in performance of around $7 \%$ . Applying both kernel and memorable point selection increases the performance by up to $11 \%$ . Additionally, standard deviation is reduced when using memorable points or the kernel. It is clear that both parts of the proposed algorithm are vital in achieving state-of-the-art performance on this benchmark.
+
+# 5 DISCUSSION
+
+We propose FROMP, a scalable function-regularisation approach for continual learning. FROMP uses a GP formulation of neural networks to select memorable past examples, regularising them using a kernel, and achieving state-of-the-art performance across benchmarks. This work enables a new way of combining regularisation methods and memory-based methods in continual learning. Future research could investigate other ways of selecting a memorable past (e.g. fixed memory size), more efficient ways of calculating kernel matrices, and the case where data does not arrive in tasks.
+
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+
+# A DETAILS ON DEEP NETWORKS TO GAUSSIAN PROCESSES
+
+In this section, we detail the map that is presented in Sec. 3.1 to go from the posterior of a deep neural network to a Gaussian process. The map for linear regression and logistic regression can be found in the appendix of Khan et al. (2019). We recap the map for logistic regression and, in addition, walk through the case of softmax regression here. Recall the linear model from Eq. 3:
+
+$$
+\begin{array} { r l } & { \mathrm { M o d e l : } \quad \tilde { \mathbf { y } } = \mathbf { J } ( \mathbf { x } ) \mathbf { w } + \boldsymbol { \epsilon } , \quad \mathrm { w i t h } \ \boldsymbol { \epsilon } \sim \mathcal { N } ( 0 , ( \mathbf { A } ( \mathbf { x } , \mathbf { y } ) ) ^ { - 1 } ) \mathrm { a n d } \ \mathbf { w } \sim \mathcal { N } ( 0 , \delta ^ { - 1 } \mathbf { I } _ { P } ) } \\ & { \quad \mathrm { D a t a : } \quad \tilde { \mathbf { y } } _ { i } : = \mathbf { J } ( \mathbf { x } _ { i } ) \mathbf { w } _ { * } - \left( \mathbf { A } ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) \right) ^ { - 1 } \mathbf { r } ( \mathbf { x } _ { i } , \mathbf { y } _ { i } ) . } \end{array}
+$$
+
+Inference in this model yields the deep neural network posterior $q ( \mathbf { w } )$ in Eq. 2.
+
+Logistic regression: as mentioned in Sec. 3.1, the parameter of the Bernoulli likelihood is given by applying the sigmoid link function to $\operatorname { f } ( \mathbf { x } )$ and we defined $\mathfrak { p } ( \mathbf { x } ) : = \sigma ( \mathrm { f } _ { w _ { \ast } } ( \mathbf { x } ) )$ . Further, the derivative and Hessian of loss were particularised as $\operatorname { r } ( \mathbf { x } , \mathbf { y } ) = \operatorname { p } ( \mathbf { x } ) - \mathbf { y }$ and $\Lambda ( \mathbf { x } , \mathbf { y } ) = \mathsf { p } ( \mathbf { x } ) ( 1 - \mathsf { p } ( \mathbf { x } ) )$ . We write $\varepsilon \sim { \cal N } ( 0 , 1 )$ , ,for standard noise and plug the quantities model (Eq. 10):
+
+$$
+{ \bf { J } } ( { \bf { x } } ) { { \bf { w } } _ { \ast } } - \frac { { \bf { p } } ( { \bf { x } } ) - { \bf { y } } } { \Lambda ( { \bf { x } } , { \bf { y } } ) } = { \bf { J } } ( { \bf { x } } ) { \bf { w } } + ( \Lambda ( { \bf { x } } , { \bf { y } } ) ) ^ { - \frac { 1 } { 2 } } \varepsilon _ { i } ,
+$$
+
+which can be re-arranged to a new model that has an equivalent posterior, but where inference is done by observing original data points $\mathcal { D }$ :
+
+$$
+\mathbf { y } = \mathbf { p } ( \mathbf { x } ) + \Lambda ( \mathbf { x } , \mathbf { y } ) \mathbf { J } ( \mathbf { x } ) ( \mathbf { w } - \mathbf { w } _ { * } ) + \tau , ~ \mathrm { w i t h } ~ \tau \sim N ( 0 , \Lambda ( \mathbf { x } , \mathbf { y } ) ) ~ \mathrm { a n d } ~ \mathbf { w } \sim N ( 0 , \delta ^ { - 1 } \mathbf { I } _ { P } ) .
+$$
+
+Softmax regression is a simple extension to the logistic regression case: we have a categorical likelihood with parameter $\mathbf { p } ( \mathbf { x } ) : = \mathrm { s o f t m a x } ( \mathbf { f } _ { w _ { * } } ( \mathbf { x } ) )$ and labels ${ \bf y } _ { i }$ that are standard one-hot encoded basis vectors, i.e. have a single 1 at some position and otherwise zeros. Then we have $\mathbf { r } ( \mathbf { x } , \mathbf { y } ) =$ $\mathbf { p } ( \mathbf { x } ) - \mathbf { y }$ and $\mathbf { \boldsymbol { \Lambda } } ( \mathbf { \boldsymbol { x } } , \mathbf { \boldsymbol { y } } ) = \mathrm { d i a g } ( \mathbf { \boldsymbol { p } } ( \mathbf { \boldsymbol { x } } ) ) - \mathbf { \boldsymbol { p } } ( \mathbf { \boldsymbol { x } } ) \mathbf { \boldsymbol { p } } ( \mathbf { \boldsymbol { x } } ) ^ { \top }$ . Unfortunately, $\mathbf { \Delta } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda }$ is of rank $K - 1$ ,and therefore , ,we cannot uniquely invert it. The theory of Khan et al. (2019) therefore does not allow this because it requires $\mathbf { \delta } \mathbf { A } ( \mathbf { x } , \mathbf { y } ) \succ 0$ . However, we can still write a model in line with Eq. 13 that we expect to ,have a posterior close to that of the deep neural network:
+
+$$
+\mathbf { y } = \mathbf { p } ( \mathbf { x } ) + \Lambda ( \mathbf { x } , \mathbf { y } ) \mathbf { J } ( \mathbf { x } ) ( \mathbf { w } - \mathbf { w } _ { * } ) + \tau ~ \mathrm { w i t h } ~ \tau \sim N ( 0 , \Lambda ( \mathbf { x } , \mathbf { y } ) ) ~ \mathrm { a n d } ~ \mathbf { w } \sim N ( 0 , \delta ^ { - 1 } \mathbf { I } _ { P } ) .
+$$
+
+Conversion to GP can be achieved by taking the expectation and covariance of the linear model. To obtain the GP prior, we take the expectation of the linear model (10) over the prior $p ( \mathbf { w } )$ . To obtain the posterior, we take the expectation over $q ( \mathbf { w } )$ . Since we are interested in the posterior process, the latter is of interest and calculated for the logistic case in Sec. 3.1. The softmax regression case works equivalently but yields a $K \times K$ kernel, which can be represented as a symmetric tensor, or a $N K { \times } N K$ matrix over data set $\mathcal { D }$ with $N$ examples. To avoid growing complexity with more classes, we model the GPs for each class as being independent, as discussed in Appendix B.
+
+# B REDUCING COMPLEXITY IN THE MULTICLASS SETTING
+
+For the softmax regression with $K$ classes, we build an individual GP for each class, in order to avoid growing complexity with more classes. We utilise $\mathbf { y } ^ { ( k ) }$ to denote the $k$ -th item of $\mathbf { y }$ in Eq. (14). Then $\mathbf { \bar { y } } ^ { ( k ) }$ will be:
+
+$$
+\mathbf { y } ^ { ( k ) } = \mathbf { p } ^ { ( k ) } + \mathbf { A } ( \mathbf { x } , \mathbf { y } ) ^ { ( k ) } \mathbf { J } ( \mathbf { x } ) ( \mathbf { w } - \mathbf { w } _ { \ast } ) + \tau ^ { ( k ) } .
+$$
+
+Here $\mathbf { \Delta } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda \Lambda } \mathbf { \Lambda } \mathbf { \Lambda } \mathbf { \Lambda \Lambda } \mathbf { \Lambda } \mathbf { \Lambda \Lambda } \mathbf { \Lambda \Lambda } \mathbf \Lambda \Lambda \mathbf { } \Lambda \Lambda \mathbf { \Lambda } \Lambda \Lambda \mathbf \Lambda \Lambda \Lambda \Lambda \mathbf { } \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \mathbf \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \ c \ c \ c \ c \ c \ c \ c \ c \ c \ c \ c \ c \ c \ c \ c \ c$ denotes the $k$ -th row of the Hessian matrix. We transform the $K$ models to function ,space and treat them as independent. This allows us to split the KL-divergence in Eq. 8 into $K$ individual divergences, and we obtain the simplified final objective,
+
+$$
+\operatorname* { m i n } _ { \boldsymbol { w } } \quad \tau \sum _ { i \in \mathcal { D } _ { t } } \ell ( y _ { i } , f _ { w } ( \mathbf { x } _ { i } ) ) + \frac { 1 } { 2 } \sum _ { s = 1 } ^ { t - 1 } \sum _ { k = 1 } ^ { K } ( \mathbf { m } _ { s , w } ^ { ( k ) } - \mathbf { m } _ { s , w _ { t - 1 } } ^ { ( k ) } ) ^ { \top } [ \mathbf { K } _ { w _ { t - 1 } , s } ^ { ( k ) } ] ^ { - 1 } ( \mathbf { m } _ { s , w } ^ { ( k ) } - \mathbf { m } _ { s , w _ { t - 1 } } ^ { ( k ) } ) .
+$$
+
+# C FURTHER DETAIL ON CONTINUAL LEARNING METRICS REPORTED
+
+We report a backward transfer metric and a forward transfer metric on split CIFAR. The backward transfer metric is exactly as defined in Lopez-Paz & Ranzato (2017). The forward transfer metric is a measure of how well the method uses previously seen knowledge to improve classification accuracy on newly seen tasks. Let there be a total of $T$ tasks. Let $R _ { i , j }$ be the classification accuracy of the model on task $t _ { j }$ after observing the last sample from task $t _ { i }$ ,. Let $R _ { i } ^ { \mathrm { i n d } }$ be the classification accuracy of an independent model trained only on task $i$ . Then,
+
+$$
+\begin{array} { r l r } & { } & { \mathrm { B a c k w a r d ~ T r a n s f e r , B W T } = \displaystyle \frac { 1 } { T - 1 } \sum _ { i = 1 } ^ { T - 1 } R _ { T , i } - R _ { i , i } , } \\ & { } & { \mathrm { F o r w a r d ~ T r a n s f e r } = \displaystyle \frac { 1 } { T - 1 } \sum _ { i = 2 } ^ { T } R _ { i , i } - R _ { i } ^ { \mathrm { i n d } } . } \end{array}
+$$
+
+# D FURTHER TOY DATA EXPERIMENTS
+
+This section provides further information and visualisations of toy 2D datasets, as well as hyperparameter settings for VCL.
+
+D.1 WEIGHT-SPACE REGULARISATION’S INCONSISTENT BEHAVIOUR
+
+Table 3: Train accuracy of FROMP, VCL (no coresets), VCL $^ +$ coresets and batch-trained Adam (an upper bound on performance) on a toy 2D binary classification dataset, with mean and standard deviations over 5 runs for VCL and batch Adam, and 10 runs for FROMP. ‘VCL’ is without coresets. VCL-RP and FRORP have the same (random) coreset selections. VCL-MP is provided with ‘ideal’ coreset points as chosen by an independent run of FROMP. VCL (no coreset) does very poorly, forgetting previous tasks. VCL $^ +$ coresets is brittle with high standard deviations, while FROMP is stable.   
+
+<table><tr><td>FROMP</td><td>FRORP</td><td>VCL-RP</td><td>VCL-MP</td><td>VCL</td><td>Batch Adam</td></tr><tr><td>99.6% ± 0.2</td><td>98.5% ± 0.6</td><td>92% ±10</td><td>85%±14</td><td>68%±8</td><td>99.70% ± 0.03</td></tr></table>
+
+Table 3 summarises the performance (measured by train accuracy) of FROMP and ${ \mathrm { V C L } } +$ coresets on a toy dataset similar to that in Figure 1. FROMP is very consistent, while VCL (with coresets) is extremely brittle: it can perform well sometimes (1 run out of 5), but usually does not (4 runs out of 5). This is regardless of coreset points chosen for VCL. Without coresets, VCL forgets many past tasks, with very low performance. Previous work (Farquhar & Gal, 2019) has argued that this is inevitable for weight-regularisation-only methods such as VCL and EWC.
+
+We now 3 runs with different random seeds of VCL-MP from Table 3: the coreset is chosen from an independent run of FROMP, with datapoints all on the task boundary. This selection of coreset is intuitively better than a random coreset selection. Please note that the results we show here are not specific to coreset selection. Any coreset selection (whether random or otherwise) all show the same inconsistency when VCL is run on them. This behaviour is not specific to coreset choice.
+
+![](images/ff1e52f38925211222f014ef43b74938c3a6d9cedd3d3ab0f916b1f0e7635f0b.jpg)  
+Figure 6: Three runs of VCL-MP on toy 2D data. These are the middle performing 3 runs out of 5 runs with different random seeds. VCL’s inconsistent behaviour is clear.
+
+This section visualises the different dataset variations presented in Table 1. We pick the middle performing FROMP run (out of 5) and batch Adam run to show.
+
+![](images/90863bb11f7283da4561f10637aee968ede083ede8f99683ff840adda4292b87.jpg)  
+Figure 7: FROMP (middle performing of 5 runs) and batch Adam on a dataset $1 0 \mathrm { x }$ smaller (400 points per task).
+
+![](images/94b9965140412a3f0acc6899aef7e0f913194e7848fffd22c44f00b12dfdce7b.jpg)  
+Figure 8: FROMP (middle performing of 5 runs), left, and batch Adam, right, on a dataset $1 0 \mathrm { x }$ larger (40,000 points per task).
+
+![](images/487747e7e51c8e481e933f5c248e4ea7d931a3ee6cbdf9ddec742cf8d09b6041.jpg)  
+Figure 9: FROMP (middle performing of 5 runs), left, and batch Adam, right, on a dataset with a new, easy, 6th task.
+
+![](images/2a27ed7655503fe4d2177a038b503044f0df72a3578dbf7745628e4e43b5947c.jpg)  
+Figure 10: FROMP (middle performing of 5 runs), left, and batch Adam, right, on a dataset with increased standard deviations of each class’ points, making classification tougher.
+
+![](images/0c5f6487a6cec61cdd6becc9344b865e3804264f9e743edf6567afca03d9af72.jpg)  
+Figure 11: FROMP (middle performing of 5 runs), left, and batch Adam, right, on a dataset with 2 tasks having overlapping data, which is not separable.
+
+# D.3 VCL AND FROMP HYPERPARAMETER SETTINGS FOR TOY DATASETS
+
+FROMP. We optimised the number of epochs, Adam learning rate, and batch size. We optimised by running various settings for 5 runs and picking the settings with largest mean train accuracy on the toy dataset in Figure 1. We found the best settings were: number of epoch $\scriptstyle = 5 0$ , batch $\mathrm { s i z e } { = } 2 0$ , learning rate ${ \mathrm { : = 0 . 0 1 } }$ . The hyperparameters were then fixed across all toy data experimental runs, including across dataset variations (number of epochs was appropriately scaled by 10 if dataset size was scaled by 10).
+
+VCL $^ +$ coresets. We optimised the number of epochs, the number of coreset epochs (because VCL $^ +$ coresets trains on non-coreset data first, then on coreset data just before test-time: see Nguyen et al. (2018)), learning rate (we use Adam to optimise the means and standard deviations of each parameter), batch size, and prior variance. We optimised by running various settings for 5 runs and picking the settings with largest mean train accuracy. We found the best settings were: number of epochs ${ \ o } { = } 2 0 0$ , number of coreset epoch ${ \ o } { = } 2 0 0$ , a standard normal prior (varianc ${ \mathrm { ; = } } 1$ ), batch size $scriptstyle = 4 0$ , learning rate $= 0 . 0 1$ . VCL is slow to run (an order of magnitude longer) compared to all other methods (FROMP and batch Adam).

parse/train/r1xjgxBFPB/r1xjgxBFPB_content_list.json

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+    {
+        "type": "text",
+        "text": "CONTINUAL DEEP LEARNING BY FUNCTIONAL REGULARISATION OF MEMORABLE PAST ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Continually learning new skills without forgetting old ones is an important quality for an intelligent system, yet most deep learning methods suffer from catastrophic forgetting of the past. Recent works have addressed this by regularising the network weights, but it is challenging to identify weights crucial to avoid forgetting. A better approach is to directly regularise the network outputs at past inputs, e.g., by using Gaussian processes (GPs), but this is usually computationally challenging. In this paper, we propose a scalable functional-regularisation approach where we regularise only over a few memorable past examples that are crucial to avoid forgetting. Our key idea is to use a GP formulation of deep networks, enabling us to both identify the memorable past and regularise over them. Our method achieves state-of-the-art performance on standard benchmarks and opens a new direction for life-long learning where regularisation methods are naturally combined with memory-based methods. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            468,
+            336,
+            484
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The ability to quickly adapt to changing environments is an important quality of intelligent systems. For such quick adaptation, it is important to be able to identify, memorise, and recall useful past experiences when acquiring new ones. Unfortunately, standard deep-learning methods are not good at maintaining previously acquired skills, and can quickly forget them when learning new skills (Kirkpatrick et al., 2017). Such catastrophic forgetting presents a big challenge when deploying deep-learning methods for applications, such as robotics, where new tasks can appear during the training, and data from the previous tasks might be unavailable for retraining. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In recent years, many methods have been proposed to address catastrophic forgetting in deep learning. One of the most popular approaches is to regularise the network weights to keep them close to the weights obtained for the previous tasks/data (Kirkpatrick et al., 2017; Nguyen et al., 2018; Zenke et al., 2017; Ebrahimi et al., 2019; Serra et al., 2018). This is challenging due to the difficulty in identifying the weights that are relevant to past tasks. The exact values of the weights in fact do not matter directly, but rather the network output (Benjamin et al., 2018). Figuring out which weight affects the output is therefore usually difficult. Typically, the Fisher information matrix or covariance matrices over weights are used (Kirkpatrick et al., 2017; Nguyen et al., 2018), but they only partially address the issue. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A better approach is to directly regularise the network outputs, also referred to as functionalregularisation, requiring a memory of past examples (Benjamin et al., 2018; Lopez-Paz & Ranzato, 2017; Rebuffi et al., 2017). However, such methods still lack a mechanism to automatically weight more relevant past memory in the context of the new task, and also do not take uncertainty of the output into account. Methods based on Gaussian processes do this automatically (Titsias et al., 2019), but require optimisation over inducing points and specification of a good kernel, both of which are difficult tasks. In summary, existing methods fall short in building scalable functional-regularisation methods for continual learning. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a new functional-regularisation method where we regularise over a few memorable past examples (see Fig. 1). Our approach builds upon a recent method of Khan et al. (2019) that expresses deep networks as Gaussian processes (GPs). We show that the GP formulation not only enables the identification of examples crucial to avoid forgetting, but also computes uncertainty over the network output to appropriately weight the past examples in the light of new ones. Motivated by the GP perspective, we propose a new loss function for function-regularisation with deep networks. A Laplace approximation of this objective enables scalable training. Our work in this paper focusses on avoiding forgetting, but it opens a new direction for life-long learning methods where regularisation methods are naturally combined with memory-based methods. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/cdaa926e6636dcf6f7dee5c08806e67fe60b5b29539c5dcd4a22231c6cce6a76.jpg",
+        "image_caption": [
+            "Figure 1: This figure illustrates our method. Leftmost figure shows the result of training on task 1. Examples corresponding to memorable-past, shown with big markers, are chosen using a GP formulation of the neural network. These points usually are the ones that support the decision boundary. Middle figure shows the result after task 2 where new network functions are regularised at memorable-past examples to give the same prediction as the previous ones. The resulting green decision boundary classifies both task 1 and 2 well. The rightmost figure shows the result along with memorable-past of each task where the performance over the past tasks is maintained. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Other related works. Broadly, existing work on continual learning can be split into three approaches: inference based, memory/rehearsal based, and model based. Inference based approaches have mostly focused on weight regularisation, with some recent efforts on functional regularisation. Our work falls in the latter category. Memory based approaches either maintain a memory of past data examples (Rebuffi et al., 2017) or train generative models on previous tasks to rehearse pseudo-inputs (Shin et al., 2017). An advantage of our method compared to previous ones is that building memory does not require solving an optimisation problem: the computation simply involves a forward-pass through the network followed by sorting (see Section 3.2). ",
+        "bbox": [
+            174,
+            433,
+            823,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Similarly to our work, there have also been some efforts in combining the different flavours of approaching continual learning, e.g., VCL plus coresets (Nguyen et al., 2018) and Gradient-Episodic Memory (Lopez-Paz & Ranzato, 2017; Chaudhry et al., 2018). Benjamin et al. (2018) have proposed a similar combination for functional regularisation. In these approaches, two separate methods are usually used for regularisation and memory-building. Recent follow-up work (Aljundi et al., 2019; Chaudhry et al., 2019) has focussed on improving memory-building methods, but this separation has remained the case. In contrast, in our approach, both of these are done within the same GP framework by using the method of Khan et al. (2019). ",
+        "bbox": [
+            174,
+            550,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Finally, model based approaches change the model architecture during training (Rusu et al., 2016), and this can be combined with other approaches (Schwarz et al., 2018). It is possible to use similar features in our GP based framework, which is an interesting future direction to be pursued. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 CONTINUAL LEARNING WITH WEIGHT/FUNCTIONAL REGULARISATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            742,
+            800,
+            758
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In deep learning, we minimise loss functions to estimate network weights. For example, in supervised multi-class classification problems, we are given a dataset $\\mathcal { D }$ of $N$ input-output pairs with outputs ${ \\bf y } _ { i }$ , a vector of $K$ classes, and inputs $\\mathbf { x } _ { i }$ , a vector of length $D$ , and our goal is to minimise a loss which takes the following form: $\\bar { \\ell } ( \\mathbf { w } ) + \\delta R ( \\mathbf { w } )$ , where $\\begin{array} { r } { \\bar { \\ell } ( \\mathbf { w } ) : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\bar { \\ell } ( \\mathbf { \\bar { y } } _ { i } , \\mathbf { f } _ { w } ( \\mathbf { x } _ { i } ) ) } \\end{array}$ with deep neural network $\\mathbf { f } _ { w } ( \\mathbf { x } ) \\in \\mathbb { R } ^ { K }$ \\`and its weights $\\mathbf { w }$ . $\\ell ( \\mathbf { y } , \\hat { \\mathbf { y } } )$ \\` \\` ,denotes a differentiable loss function between an output $\\mathbf { y }$ and its prediction ${ \\hat { \\mathbf { y } } } , R ( \\mathbf { w } )$ \\` , is a regularisation function (usually an $L _ { 2 }$ -regulariser $R ( \\mathbf { w } ) = \\mathbf { w } ^ { \\top } \\dot { \\mathbf { w } } )$ and $\\delta > 0$ controls the regularisation strength. Standard deep-learning approaches δ >rely on an unbiased stochastic-gradient of the loss $\\bar { \\ell }$ , which usually requires access to all of the data \\`examples for all classes (Bottou, 2010). It is this unbiased, minibatch setting where deep-learning excels and achieves state-of-the-art performance on many benchmark datasets. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In reality, we do not always have access to all the data at once, and it is not possible to obtain unbiased stochastic gradients. New classes may appear during training and old classes may never be seen again. For such settings, vanilla mini-batch stochastic-gradient methods lead to catastrophic forgetting of past information (Kirkpatrick et al., 2017). Our goal in this paper is to design methods that can avoid such catastrophic forgetting. We focus on a particular setting where the classification task is divided into several tasks, e.g., a task may consist of a classification problem over a subset of classes. We assume that the tasks arrive sequentially one after the other. Once the learning is over, we may never see that task again. Such continual-learning settings have been considered in previous works (Kirkpatrick et al., 2017; Nguyen et al., 2018; Zenke et al., 2017), and our goal is to avoid forgetting of old tasks in this setting. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recent methods have proposed weight-regularisation as a way to combat catastrophic forgetting. The main idea is to keep the new network weights close to the old ones, e.g., when training a task $t$ while given network weights $\\mathbf { w } _ { t - 1 }$ trained on task $t - 1$ , we can minimise the following loss: $\\bar { \\ell } _ { t } ( \\mathbf { w } ) + \\bar { \\delta } ( \\mathbf { w } - \\mathbf { w } _ { t - 1 } ) ^ { \\top } \\mathbf { F } _ { t } ( \\mathbf { w } - \\mathbf { w } _ { t - 1 } )$ , where $\\bar { \\ell } _ { t } ( \\mathbf { w } )$ is the loss defined over all data examples from task $t$ and $\\mathbf { F } _ { t }$ δ \\`is a preconditioning matrix that favors the weights relevant to the past tasks more than the rest. The Elastic-Weight Consolidation (EWC) method (Kirkpatrick et al., 2017), for example, uses the Fisher information matrix as the pre-conditioner, while Ritter et al. (2018) use the Hessian of the loss, and VCL (Nguyen et al., 2018) essentially employs the precision matrix of the variational approximation to do the same. Such weight-space methods reduce forgetting but do not produce satisfactory results. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The challenge in using weight-regularisation lies in the fact that the exact values of the weights do not really matter due to parametric symmetries (Benjamin et al., 2018; Bishop, 2006). Since only the network outputs matter, an alternative approach is to directly regularise these. Benjamin et al. (2018) propose to use an $L _ { 2 }$ -regulariser over the function values on data examples from past tasks: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/c2c164ce870d85b405f71af7c7a672dc3e818acf59d59ffdb3fb2527853a0d37.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { w } \\bar { \\ell } _ { t } ( \\mathbf { w } ) + \\delta \\sum _ { s = 1 } ^ { t - 1 } \\sum _ { i \\in \\mathcal { M } _ { s } } \\| f _ { w } ( \\mathbf { x } _ { i } ) - f _ { w _ { t - 1 } } ( \\mathbf { x } _ { i } ) \\| _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            513
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { M } _ { s }$ is the set of small examples for task $s$ stored in the working memory (Lopez-Paz & Ranzato, 2017; Rebuffi et al., 2017). One major issue with this approach is that the $L _ { 2 }$ -regulariser weights all the data points equally. In reality, some examples in the memory are more important than others to learn a given task. In addition, the uncertainty of the prediction is also ignored. As we will show, working with distributions over functions allows us to address these two issues. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Gaussian processes (GPs), for example, enable automatic reweighting of old tasks in light of a new one, which happens by using the posterior covariance. Unfortunately, both scalability of GPs as well as the necessity to save many instances of the past makes them impractical. A recent approach by Titsias et al. (2019) attempts to address these issue by employing sparse GP methods with inducing points and using a neural network feature map. However, their approach has some difficulties. Their approach heavily depends on a proper choice of inducing points, which are normally obtained via an ad-hoc procedure. Additionally, they propose using the last layer of the neural network as kernel features, which is limiting as it does not use the whole network’s weights in the kernel. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Ideally, we would want to perform functional regularisation with a deep learning optimiser while borrowing ideas from the GP methods. Our work in this paper takes a step in this direction. Our proposal is to use a GP formulation of neural networks to perform functional regularisation that still allows training with standard deep-learning methods. We also show that the GP view helps in selecting an informative set of memory points. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 FUNCTIONAL-REGULARISATION OF MEMORABLE PAST (FROMP) ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We will now describe our proposed method. The first step is to use the GP formulation of Khan et al. (2019) to compute GP-like posteriors over functions. Then, we propose a method to identify a set of memorable past examples for a task. Finally, we describe our functional-regularisation and discuss approximations used to build a scalable method. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1 FROM DEEP NETWORKS TO GAUSSIAN PROCESS POSTERIORS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Khan et al. (2019) propose an approach to convert deep networks into Gaussian processes. Their main result (see Theorem 1 in their paper) states that, at a local minimiser $\\mathbf { W } _ { * }$ of $\\begin{array} { r } { \\mathbf { \\bar { \\ell } } ( \\mathbf { w } ) + \\frac { \\delta } { 2 } \\mathbf { w } ^ { \\top } \\mathbf { w } } \\end{array}$ , a \\`Laplace approximation of the posterior over w is equivalent to the posterior distribution of a linear model. Specifically, they use the following scalable variant of the Laplace approximation: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6c441d78101c29111d2736dd3c18cb5f57c6fbf55ad39abb144a16274b327c98.jpg",
+        "text": "$$\np ( \\mathbf { w } | \\mathcal { D } ) \\approx q ( \\mathbf { w } ) : = N ( \\mathbf { w } | \\mu , \\Sigma ) , \\mathrm { ~ w h e r e ~ } \\mu = \\mathbf { w } _ { * } \\mathrm { ~ a n d ~ } \\Sigma ^ { - 1 } = \\sum _ { i = 1 } ^ { N } \\mathbf { J } ( \\mathbf { x } _ { i } ) ^ { \\top } \\mathbf { A } ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) \\mathbf { J } ( \\mathbf { x } _ { i } ) + \\delta \\mathbf { I } _ { P } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            191,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { J } ( \\mathbf { x } _ { i } ) : = \\nabla _ { w } \\mathbf { f } _ { w } ( \\mathbf { x } _ { i } ) ^ { \\top }$ is a $K \\times P$ Jacobian matrix ( $P$ being the number of parameters), and $\\pmb { \\Lambda } ( \\mathbf { x } , \\mathbf { y } ) : = \\nabla _ { \\mathbf { f } \\mathbf { f } } ^ { 2 } \\ell ( \\mathbf { y } , \\mathbf { f } )$ is the $K \\times K$ Hessian of the loss with $\\mathbf { f } = \\mathbf { f } _ { w } ( \\mathbf { x } )$ , all evaluated at $\\mathbf { w } = \\mathbf { w } _ { \\ast }$ . They ,show that $q ( \\mathbf { \\dot { w } } )$ ,is equal to the posterior of the following linear model: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/820c3bc8a882b07395a6209fef8491437362bcc093f95777343f5af85d3d3b07.jpg",
+        "text": "$$\n\\tilde { \\mathbf { y } } = \\mathbf { J } ( \\mathbf { x } ) \\mathbf { w } + \\epsilon , \\mathrm { w i t h } \\epsilon \\sim N ( 0 , ( \\Lambda ( \\mathbf { x } , \\mathbf { y } ) ) ^ { - 1 } ) \\mathrm { a n d } \\mathbf { w } \\sim N ( 0 , \\delta ^ { - 1 } \\mathbf { I } _ { P } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the observations are defined as $\\tilde { \\mathbf { y } } _ { i } ~ : = ~ \\mathbf { J } ( \\mathbf { x } _ { i } ) \\mathbf { w } _ { * } - \\left( \\pmb { \\Lambda } ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) \\right) ^ { - 1 } \\mathbf { r } ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } )$ with the residual $\\mathbf { r } ( \\mathbf { x } , \\mathbf { y } ) : = \\nabla _ { \\mathrm { f } } \\ell ( \\mathbf { y } , \\mathbf { f } )$ , ,. They also show that the predictive distribution of this linear model is equiv, \\` ,alent to that of a GP regression model defined with a $K \\times K$ neural tangent kernel (NTK) (Jacot et al., 2018): ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0daf9faa9a6b2500386cf9afa66e65c14b79b6c952181f78d39f09b25ddb3e63.jpg",
+        "text": "$$\n\\begin{array} { r } { \\tilde { \\mathbf { y } } = \\mathbf { f } _ { \\mathrm { G P } } ( \\mathbf { x } ) + \\boldsymbol { \\epsilon } , \\quad \\mathrm { ~ w i t h ~ } \\boldsymbol { \\epsilon } \\sim \\mathcal { N } ( 0 , ( \\mathbf { \\Lambda } ( \\mathbf { x } , \\mathbf { y } ) ) ^ { - 1 } ) \\mathrm { ~ a n d ~ } \\mathbf { f } _ { \\mathrm { G P } } ( \\mathbf { x } ) \\sim \\mathcal { G P } \\left( 0 , \\delta ^ { - 1 } \\mathbf { J } ( \\mathbf { x } ) \\mathbf { J } ( \\mathbf { x } ^ { \\prime } ) ^ { \\top } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The above result gives a posterior in the defined observation space of $\\tilde { \\mathbf { y } }$ . In the case where we can find a map from $\\tilde { \\mathbf { y } }$ to $\\mathbf { y }$ , the above inference problem allows for reparameterisation into the original data space. Therefore, we will obtain a GP that serves as an approximation to the neural network posterior. Below, we demonstrate this for binary classification with sigmoid link function and Bernoulli likelihood (see Appendix A for details and extension to multi-class classification). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We parameterise the Bernoulli likelihood by $\\mathfrak { p } ( \\mathbf { x } ) : = \\sigma ( \\mathrm { f } _ { w _ { * } } ( \\mathbf { x } ) )$ where $\\sigma$ is the sigmoid function (note that we now have scalars $\\{ \\mathrm { f } , \\mathrm { y } , \\ldots \\}$ σ σas we are dealing with a single output). The residual and , , ...loss Hessian for this model are given by $\\operatorname { r } ( \\mathbf { x } , \\mathbf { y } ) = \\operatorname { p } ( \\mathbf { x } ) - \\mathbf { y }$ and $\\Lambda ( \\mathbf { x } , \\mathbf { y } ) = \\mathsf { p } ( \\mathbf { x } ) ( 1 - \\mathsf { p } ( \\mathbf { x } ) )$ , respectively, where $\\mathsf { y } \\in \\{ 0 , 1 \\}$ , ,. Because residuals are linear in y and the Hessian is independent of y, we can ,substitute the definitions of $\\tilde { \\bf y }$ and $\\mathbf { r } ( \\mathbf { x } , \\mathbf { y } )$ into Eq. 3, and rearrange for y, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/055c948c4be9da4747002290d26466219e20e1370fbc960ea97d00fcf9bd1455.jpg",
+        "text": "$$\n\\mathbf { y } = \\underbrace { \\mathbf { p } ( \\mathbf { x } ) + \\Lambda ( \\mathbf { x , y } ) \\mathbf { J } ( \\mathbf { x } ) ( \\mathbf { w } - \\mathbf { w } _ { * } ) } _ { : = \\mathbf { f } _ { \\mathrm { i n } } ( \\mathbf { X } ) } + \\tau , \\ \\mathrm { ~ w i t h ~ } \\tau \\sim \\mathcal { N } ( 0 , \\Lambda ( \\mathbf { x , y } ) ) \\ \\mathrm { ~ a n d ~ } \\mathbf { w } \\sim \\mathcal { N } ( 0 , \\delta ^ { - 1 } \\mathbf { I } _ { P } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The Laplace approximation in Eq. 2 is the posterior of this model after observing data $\\mathcal { D }$ . We can equivalently write the posterior predictive as a GP (Rasmussen, 2003) with mean and covariance ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3ea5bc104c2c8bbe22fcbdc576abd8cb92b317361732628c32d5bac92c14e0c3.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\qquad \\mathrm { m } ( \\mathbf { x } ) : = \\mathbb { E } _ { q ( w ) } \\left[ \\mathrm { f } _ { \\mathrm { l i n } } ( \\mathbf { x } ) \\right] = \\mathrm { p } ( \\mathbf { x } , \\mathbf { y } ) + \\Lambda ( \\mathbf { x } , \\mathbf { y } ) \\mathbf { J } ( \\mathbf { x } ) ( \\mu - \\mathbf { w } _ { \\ast } ) = \\mathrm { p } ( \\mathbf { x } ) , } \\\\ & { \\qquad \\mathrm { k } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ) : = \\mathbb { E } _ { q ( w ) } \\left[ ( \\mathrm { f } _ { \\mathrm { l i n } } ( \\mathbf { x } ) - \\mathrm { m } ( \\mathbf { x } ) ) \\left( \\mathrm { f } _ { \\mathrm { l i n } } ( \\mathbf { x } ^ { \\prime } ) - \\mathrm { m } ( \\mathbf { x } ^ { \\prime } ) \\right) \\right] } \\\\ & { \\qquad = \\Lambda ( \\mathbf { x } , \\mathbf { y } ) \\mathbf { J } ( \\mathbf { x } ) \\mathbb { E } _ { q ( w ) } \\left[ ( \\mathbf { w } - \\mathbf { w } _ { \\ast } ) ( \\mathbf { w } - \\mathbf { w } _ { \\ast } ) ^ { \\top } \\right] \\mathbf { J } ( \\mathbf { x } ^ { \\prime } ) ^ { \\top } \\Lambda ( \\mathbf { x } ^ { \\prime } , \\mathbf { y } ^ { \\prime } ) } \\\\ & { \\qquad = \\Lambda ( \\mathbf { x } , \\mathbf { y } ) \\mathbf { J } ( \\mathbf { x } ) \\Sigma \\mathbf { J } ( \\mathbf { x } ^ { \\prime } ) ^ { \\top } \\Lambda ( \\mathbf { x } ^ { \\prime } , \\mathbf { y } ^ { \\prime } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A GP defined with the above mean and covariance function can be viewed as an approximation to the posterior process over network outputs. Throughout the paper, we will denote the process obtained at a weight $\\mathbf { W } _ { * }$ by a distribution $q _ { w _ { * } } ( \\mathbf { f } )$ where f is a vector of $\\operatorname { f } ( \\mathbf { x } )$ evaluated at many different inputs $\\mathbf { X }$ . We will use this GP posterior predictive for functional regularisation. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2 MEMORABLE PAST ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the previous section, we derived a GP posterior approximation over the network outputs. Now, we propose a method to obtain a small set of examples that are crucial to avoid forgetting. We first note that the posterior GP mean in Eq. 4 corresponds to a kernel Ridge regression that requires the computation of $( \\delta ^ { - 1 } \\mathbf { K } + \\mathbf { { A } } ^ { - 1 } ) ^ { - 1 } \\tilde { \\mathbf { y } }$ where $\\pmb { \\Lambda }$ is a block-diagonal matrix containing all $\\mathbf { \\Delta } \\Lambda _ { i } : = \\mathbf { \\Delta } \\Lambda ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } )$ and $\\mathbf { K }$ δ ,is the NTK defined in Eq. 4. The predictions therefore strongly depend on the eigenvalues of the preconditioning matrix. Selection of important data examples therefore boils down to selecting ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: Functional Regularisation of Memorable Past (FROMP) ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/74d94103ae588172889f704c1d1cba78090bbde4ef4de4682ccb6b5590f6b649.jpg",
+        "image_caption": [
+            "Figure 2: A pseudo-code for our FROMP algorithm. The additional computations on top of Adam are in lines 4 and 7, where we add the contribution from the functional-regularisation term. After every task, we update the memorable past in Eq. 9-11 which involves a matrix inversion of size $M$ , and a forward pass through the network to compute $\\mathbf { \\Delta } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "important columns of this matrix, such that the predictions remain unchanged. The theory of leverage score sampling (Alaoui & Mahoney, 2015; Bach, 2013) suggests picking the points proportional to the leverage score defined to be the diagonal of the following matrix: ${ \\bf K } ( { \\bf K } + \\delta { \\bf A } ^ { - 1 } ) ^ { - 1 }$ . Typically, δthis matrix is difficult to compute and methods are employed to approximately obtain the leverage score (Alaoui & Mahoney, 2015). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For deep-learning applications too, exact computation of leverage score is very difficult. We instead propose a simple solution. Since the eigenvalues of the previous matrix heavily depend on $\\delta { \\bf A } _ { i } ^ { - 1 }$ , we can pick the data examples by simply sorting $\\mathbf { \\Lambda } _ { \\mathbf { \\Lambda } } \\mathbf { \\Lambda } _ { \\mathbf { \\Lambda } }$ . For the inverse of $K + \\delta \\mathbf { { A } } ^ { - 1 }$ δto have high eigenvalues, we should favour examples with smaller values of $\\boldsymbol { \\Lambda } _ { i } ^ { - 1 }$ δ. Therefore, we simply sort the $\\mathbf { \\Delta } \\Lambda _ { i }$ and pick the top $M$ examples as the most relevant ones. This simple solution is very effective for our particular problem because $\\mathbf { \\Delta } \\Lambda _ { i }$ are noise variances for the data examples, obtained by using an already trained network. These are second derivatives of the loss for data examples, and so also reflect the sensitivity of the decision boundaries if a particular data point is perturbed. Therefore, they tend to reflect the relevance of data examples. An example is shown in Figure 1 where we clearly see that our solution picks the examples lying close to decision boundary. Computation of $\\mathbf { \\Delta } \\Lambda _ { i }$ requires us to run the forward pass to get the $\\ell ( \\mathbf { y } _ { i } , \\hat { \\mathbf { y } } _ { i } )$ and then compute its second derivative with respect to $\\hat { \\mathbf { y } } _ { i }$ \\` ,, both of which are cheap operations. Throughout the paper, examples chosen by this method are referred to as the memorable past examples, and denoted $\\boldsymbol { \\mathcal { M } } _ { t }$ for task $t$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.3 FUNCTIONAL-REGULARISATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "So far, we described the construction of the GP posterior predictive over the function space, as well as the construction of a set of memorable-past examples. We are now ready to describe our objective function where we employ these two to perform functional regularisation. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Suppose that we are given network weights $\\mathbf { w } _ { t - 1 }$ that are obtained by training over data examples from task $t - 1$ . Our goal then is to train a network with weights w such that its performance on memorable past $\\boldsymbol { \\mathcal { M } } _ { 1 : t - 1 }$ is unchanged. We denote the vector of function outputs over these examples by $\\mathbf { a } _ { 1 : t - 1 }$ . A straightforward idea is to directly optimise the weights w such that the predictions using $q _ { w } ( \\mathbf { f } _ { t } )$ are good on current tasks while the predictive distribution $q _ { w } ( \\mathbf { a } _ { 1 : t - 1 } )$ is close to $q _ { w _ { t - 1 } } ( \\mathbf { a } _ { 1 : t - 1 } )$ . Since the number of tasks can be very large, we choose to regularise each task separately, i.e., we will match $q _ { w } ( { \\bf a } _ { s } )$ with $q _ { w _ { t - 1 } } ( \\mathbf { a } _ { s } )$ separately for all tasks $s \\ < \\ t$ , in line with Titsias et al. (2019). <These choices give us the following objective function with trade-off parameter $\\tau$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dc5cd7d0e3ae355357e5fd4b6429503044115b7e4c66d2527ae2e4291e9ce0d6.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { w } \\quad \\tau \\mathbb { E } _ { q _ { w } ( f _ { t } ) } \\Big [ \\sum _ { i \\in \\mathcal { D } _ { t } } \\ell ( y _ { i } , f ( { \\mathbf x } _ { i } ) ) \\Big ] + \\sum _ { s = 1 } ^ { t - 1 } \\mathbb { D } _ { K L } [ q _ { w } ( { \\mathbf a } _ { s } ) \\| { q } _ { w _ { t - 1 } } ( { \\mathbf a } _ { s } ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/d7d3e051f2896428b9c335bd496a8a9b9ad422657d45d1b0e65def5fdd701164.jpg",
+        "table_caption": [
+            "Table 1: Train accuracy of FROMP and batch-trained Adam (upper bound on performance) on variations of a toy 2D binary classification dataset, with mean and standard deviations over 10 runs (3 runs for Adam). FROMP performs well across variations. See Appendix D.2 for visualisations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset variation</td><td>FROMP</td><td>Batch Adam</td></tr><tr><td>10x less data (400 per task)</td><td>99.9% ± 0.0</td><td>99.7% ±0.2</td></tr><tr><td>10x more data (40000 per task)</td><td>96.9% ± 3.0</td><td>99.7% ± 0.0</td></tr><tr><td>Introduced 6th task</td><td>97.8% ± 3.3</td><td>99.6% ± 0.1</td></tr><tr><td>Increased std dev of each class distribution</td><td>96.0% ± 2.4</td><td>96.9% ± 0.4</td></tr><tr><td>2 tasks have overlapping data</td><td>90.1% ± 0.8</td><td>91.1% ± 0.3</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A major issue with this objective is that computing gradients wrt. the posterior predictive $q _ { w }$ will be computationally heavy as it involves gradients of the Jacobian and $\\mathbf { \\Lambda } _ { \\mathbf { \\Lambda } } \\mathbf { \\Lambda } _ { \\mathbf { \\Lambda } }$ , which involve higher order derivatives. During training on a task, we have no distribution on parameters (cf. Laplace approximation) and can treat it as fixed. Therefore, taking the gradient wrt. the parameter $\\mathbf { w }$ of the above objective will only involve differentiating the mean function $\\mathbf { m } ( \\mathbf { x } _ { i } )$ formed by the neural network, and we have equivalence to the following objective: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/f2e2cb27bf037213d85e2474ee82c29d97027a048fee2883556f53c788c17b56.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\boldsymbol { w } } \\quad \\tau \\sum _ { i \\in \\mathcal { D } _ { t } } \\ell ( y _ { i } , f _ { w } ( \\mathbf { x } _ { i } ) ) + \\frac { 1 } { 2 } \\sum _ { s = 1 } ^ { t - 1 } ( \\mathbf { m } _ { s , w } - \\mathbf { m } _ { s , w _ { t - 1 } } ) ^ { \\top } \\mathbf { K } _ { w _ { t - 1 } , s } ^ { - 1 } ( \\mathbf { m } _ { s , w } - \\mathbf { m } _ { s , w _ { t - 1 } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { m } _ { s , w _ { t - 1 } }$ and ${ \\bf K } _ { w _ { t - 1 } , s }$ are the vector and matrices containing the corresponding mean and co, ,variance function of the network weights $\\mathbf { w } _ { t - 1 }$ evaluated at the memorable past for task $s$ . Eqs 6 and 7 are used to calculate $\\mathbf { m } _ { s , w _ { t - 1 } }$ and ${ \\bf K } _ { w _ { t - 1 } , s }$ respectively. Essentially, this gives us something similar ,to Eq. 1 but instead of the $L _ { 2 }$ ,distance, a kernel is used to improve weighting of examples. The final FROMP algorithm is summarised in Algorithm 1. Please see App. B for details on scaling to the multi-class setting. ",
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+    },
+    {
+        "type": "text",
+        "text": "The additional time complexity of this algorithm above vanilla Adam is $O ( M P K t )$ per optimisation step, and $O ( [ M P K + \\bar { M ^ { 3 } } ] t + \\bar { N } P K )$ once per task. $O ( N P K )$ is for updating the Laplace approximation $\\pmb { \\Sigma }$ at the end of training on a task, as the Jacobian for each input training point needs to be calculated (as in Eq. 2). This is similar to EWC (Kirkpatrick et al., 2017). Note that, as we use a diagonal approximation to $\\pmb { \\Sigma }$ , inverting it is fast. $O ( M P K t )$ is for calculating each of the $t$ matrices ${ \\bf K } _ { w _ { t - 1 } , s }$ , and $O ( M ^ { 3 } t )$ is for inverting them. This is done once before training on each task (Eq. 7). ,Finally, during optimisation of the objective function, we need $O ( M P K )$ for differentiating the mean function $\\mathbf { m } ( \\mathbf { x } _ { i } )$ . All of these additional costs are small for small $M$ , except for $O ( N P K )$ , which can be reduced by randomly sampling a subset of the task’s data. Note that it is possible to be stochastic in sampling a subset of previous tasks’ memory during every update, removing the time complexity dependency on $t$ . Doing this gives the same overall complexity per step as in Adam (as $M$ is usually of the same order as the mini-batch size). ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "We run the proposed method FROMP on toy datasets, permuted MNIST and split MNIST (LeCun et al., 1998; Goodfellow et al., 2013), and a split version of CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2009). We optimise the objective in Eq. 9 using Adam (Kingma & Ba, 2015) with parameter $\\beta _ { 1 } = 0 . 9 9$ and further use gradient clipping to speed up training. To identify the individual benefits β .of the kernel in the loss (Eq. 9) as well as the selection of data points, we compare the following four methods: FROMP uses kernel and the proposed example selection, while FRORP instead uses randomly chosen examples. Further, we replace the kernel by an identity matrix leading to functional $L _ { 2 }$ regularisation and call the corresponding methods with and without our example selection technique FROMP- $L _ { 2 }$ and FRORP- $L _ { 2 }$ . ",
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+    {
+        "type": "text",
+        "text": "4.1 TOY DATASET ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first test FROMP on many variations of a toy 2D binary classification dataset like that in Figure 1. We want to test its performance when exposed to different datasets of varying difficulty. We use a ",
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+        "img_path": "images/4da7f8f7504945c3efe091d92ad338addb1376d1c86b5bfd6c018330f3ad863c.jpg",
+        "image_caption": [
+            "Figure 3: Permuted MNIST: added kernels across classes (with subtracted diagonal for visualisation purposes), and performance as a function of memory size (average accuracy after 10 tasks). Memorable past examples lead to a more uniform kernel structure that prevents weighting previously overfit examples too highly, e.g. task 1 in the random selection exhibits strong correlation and low variance. As we reduce the number of examples in memory, FROMP gracefully reduces accuracy. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "2-hidden layer MLP (with 20 hidden units in each layer) for all experiments. Appendix D.3 details hyperparameter selection. In Appendix D.1 we show the brittleness and inconsistent behaviour of weight-space regularisation methods. In contrast, FROMP performs extremely well across many variations, showing consistently good results (see Table 1 and Appendix D.2 for visualisations). ",
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+    {
+        "type": "table",
+        "img_path": "images/38ef4b2b7c711c2dca3e29ba4f8e886b42cce23911e7662b178af0938dcf0d90.jpg",
+        "table_caption": [
+            "Table 2: The average validation accuracy on Permuted-MNIST (10 tasks) and Split-MNIST. $\\mathrm { \\Delta ^ { * } \\mathrm { 2 0 0 p / t } ^ { \\mathrm { * } } }$ denotes that 200 examples are selected for each task. We report mean and standard deviations over 5 runs, and use results from Nguyen et al. (2018) for baselines. FROMP is state-of-the-art. Additionally, as we reduce the number of points (Figure 3), FROMP gracefully reduces accuracy, due to the clever choice of memorable past and the use of kernels in the functional regularisation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Permuted MNIST</td><td> Split MNIST</td></tr><tr><td>DLP (Smola et al., 2003)</td><td>82%</td><td>61.2%</td></tr><tr><td>EWC (Kirkpatrick et al., 2017)</td><td>84%</td><td>63.1%</td></tr><tr><td>SI (Zenke et al., 2017)</td><td>86%</td><td>98.9%</td></tr><tr><td>Improved VCL (Swaroop et al., 2019)</td><td>93%± 1</td><td>98.4% ± 0.4</td></tr><tr><td>+ random Coreset</td><td>94.6% ± 0.3 (200 p/t)</td><td>98.2% ± 0.4 (40 p/t)</td></tr><tr><td>FRCL-RND (Titsias et al., 2019)</td><td>94.2% ± 0.1 (200 p/t)</td><td>96.7% ± 1.0 (40 p/t)</td></tr><tr><td>FRCL-TR (Titsias et al., 2019)</td><td>94.3% ± 0.1 (200 p/t)</td><td>97.4% ± 0.6 (40 p/t)</td></tr><tr><td>FRORP-L2</td><td>87.9% ± 0.7 (200 p/t)</td><td>98.5% ± 0.2 (40 p/t)</td></tr><tr><td>FROMP-L2</td><td>94.6% ± 0.1 (200 p/t)</td><td>98.7% ± 0.1 (40 p/t)</td></tr><tr><td>FRORP</td><td>94.6% ± 0.1 (200 p/t)</td><td>99.0% ± 0.1 (40 p/t)</td></tr><tr><td>FROMP</td><td>94.9% ± 0.1 (200 p/t)</td><td>99.0% ± 0.1 (40 p/t)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.2 PERMUTED AND SPLIT MNIST ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Permuted MNIST consists of a series of tasks where each task is a fixed permutation of pixels to the entire labelled MNIST dataset. Like in previous work (Nguyen et al., 2018; Kirkpatrick et al., 2017; Zenke et al., 2017; Titsias et al., 2019), we implement a fully connected single-head network with two hidden layers, and report performance after 10 tasks. Each hidden layer consists of 100 hidden units and ReLU activation functions. We set the learning rate to 0 001, batch size to 128, and learn each task for 10 epochs. We use $\\tau = 1$ .for both FROMP and FROMP- $L _ { 2 }$ (see Eq. 9). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The Split MNIST experiment was introduced by Zenke et al. (2017) and consists of five binary classification tasks built from MNIST: 0/1, 2/3, 4/5, 6/7, and $^ { 8 / 9 }$ . We use a fully connected multi-head neural network with two hidden layers, each with 256 hidden units and ReLU activation functions. Following the settings of previous work, we select 40 memorable points per task. The learning rate is set to 0 0001, batch size to 128, and we learn each task for 15 epochs. We find optimal parameters $\\tau = 1$ . and $\\tau = 1 0$ for FROMP and FROMP- $L _ { 2 }$ respectively. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/b4369110b5fb362aec49d546d839c2a65e92d215201756fe5754562ccfc00bc6.jpg",
+        "image_caption": [
+            "Figure 4: Split MNIST: randomly selected samples and memorable past in comparison. 10 samples per class and method. In line with the toy example in Fig. 1, memorable past examples appear to be closer to the decision boundary of a classifier: many samples of the memorable past are possibly hard to distinguish from other classes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We report the final average accuracy for both benchmarks across all the tasks in Table 2. The proposed method achieves better performance than the weight-space methods EWC and VCL, as well as compared to the function-space method FRCL that is based on a GP formulation. Further, the benchmarks show superior performance of both the approach to select important examples (Sec. 3.2) and the functional regularisation using the kernel (Sec. 3.3). Memorable examples improve performance of the naive and efficient FRORP- $. L _ { 2 }$ method by more than $6 \\%$ on permuted MNIST and by $0 . 2 \\%$ on the split MNIST. Standard deviation is also reduced in both cases. FROMP does not profit much from memorable examples compared to FRORP, probably because the performance is already close to the maximum achievable. Furthermore, Fig. 3c shows that our selection method greatly reduces the number of memorable points required: the $L _ { 2 }$ algorithm with random points requires over 100 points to match the performance when using 20 carefully selected points, and 200 points to match performance with 40, respectively. When combined with kernel-based functional regularisation, we obtain the best-performing method, particularly when memory size is small. In line with the toy example in Fig. 1, Fig. 4 illustrates the memorable past examples compared to randomly selected examples. The memorable past examples appear to be more special cases, and may therefore lie closer to the decision boundary. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Figs. 3a and 3b show the summed kernels across all the classes in permuted MNIST for a random and memorable set of points. For visualisation purposes, the diagonal is suppressed. Note that random points lead to less uniform weighting in the kernel, making it even more important in functional regularisation, leading to better performance (FRORP vs FROMP). All memorable points are important and the kernel is more uniform. In Fig. 3a, the kernel’s weighting of task 1 is very different from other tasks, leading to different magnitudes in the functional regularisation among tasks and therefore to eventual forgetting. The kernel further tells us that the tasks are correlated, as expected. ",
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+    {
+        "type": "text",
+        "text": "4.3 SPLIT CIFAR ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Split CIFAR is far more complex than the MNIST experiments and consists of 6 tasks. The first task is the full CIFAR-10 dataset, followed by 5 tasks, each consisting of 10 consecutive classes from CIFAR-100. We follow the SI paper (Zenke et al., 2017) for our model architecture, using a multi-head CNN with 4 convolutional layers, followed by 2 dense layers with dropout. We use learning rate 0 0001 and batch size 256. All tasks are learned for 80 epochs and we use a memory .of 10–200 examples. We use $\\tau = 0 . 1$ and $\\tau = 0 . 0 5$ for FROMP and FROMP-L2 respectively. In τ . τ .addition to continual learning baselines, we report the performance of networks trained from scratch on each task. These cannot profit from forward/backward transfer. We also report the performance of a network jointly trained on all tasks. This is an upper bound on performance. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/015e41d3d81cc334976fa2e2763903197c635e3c2f47fa535d4f92bb83957db2.jpg",
+        "image_caption": [
+            "Figure 5: Split CIFAR: performance for each task, and performance variation as function of memory size, after training on the final task. We run all methods 5 times and report the mean and standard error. For baselines, we train from scratch on each task and jointly on all tasks achieving $7 3 . 6 \\% \\pm 0 . 4$ and $7 8 . 1 \\% \\pm 0 . 3$ . ., respectively. The left figure reports results for 200 memory examples per task. . .The final average validation accuracy of FROMP is $7 6 . 2 \\% \\pm 0 . 4$ , FROMP- $L _ { 2 }$ is $7 4 . 6 \\% \\pm 0 . 4$ , SI is $7 3 . 5 \\% \\pm 0 . 5$ .(result from Zenke et al. (2017)), EWC is $7 1 . 6 \\% \\pm 0 . 9$ , VCL $^ +$ . . random coreset is $6 7 . 4 \\% \\pm 1 . 4$ . . .. FROMP outperforms all other methods, and is close to the performance of the jointly . .trained model, particularly in tasks 4-6. Additionally, as we reduce the memory size, FROMP still performs well, even with only 20 examples per task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The experimental results in Fig. 5a show that FROMP outperforms other continual learning methods by a notable margin. The weight-space methods employed as baselines either cannot learn later tasks to a high accuracy (EWC, SI), or forget previous tasks (VCL). Interestingly, averaged over all tasks, FROMP also out-performs ‘from scratch’ training and achieves performance close to a jointly training on all tasks by a margin of less than $2 \\%$ . Note that on tasks 4-6, FROMP achieves the same accuracy as the jointly trained model, showing no forgetting. We also calculate the backward transfer metric BWT from Lopez-Paz & Ranzato (2017). We find that FROMP has a score of $- 2 . 6 \\pm 0 . 9$ , which is joint best with EWC’s score of $- 2 . 3 \\pm 1 . 4$ (VCL $^ +$ coresets has $- 9 . 2 \\pm 1 . 8 )$ . .). Although . . . .EWC performs well in backward transfer, it does so at the cost of forward transfer. We calculate a forward transfer metric as the average of the improvement in accuracy on a new task over an independently trained model on that task (see App. C for more precise definitions of these metrics). FROMP achieves a forward transfer of $6 . 1 \\pm 0 . 7$ , whereas EWC has $0 . 1 7 \\pm 0 . 9$ and $\\mathrm { V C L + }$ coresets has $1 . 8 \\pm 3 . 1$ . . . .. Although we only focussed on preventing catastrophic forgetting, we find evidence . .of forward transfer, a key requirement in continual learning. Overall, given final average accuracy, backward transfer and forward transfer, FROMP clearly outperforms the other baselines. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In contrast to the rather simple MNIST benchmarks, both the benefit of selecting memorable points as well as using the kernel are clearly visible in Fig. 5b. If we only memorise a few examples, the performance gap due to using the kernel is around $4 \\%$ . The selection of memorable points according to our metric leads to an increase in performance of around $7 \\%$ . Applying both kernel and memorable point selection increases the performance by up to $11 \\%$ . Additionally, standard deviation is reduced when using memorable points or the kernel. It is clear that both parts of the proposed algorithm are vital in achieving state-of-the-art performance on this benchmark. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We propose FROMP, a scalable function-regularisation approach for continual learning. FROMP uses a GP formulation of neural networks to select memorable past examples, regularising them using a kernel, and achieving state-of-the-art performance across benchmarks. This work enables a new way of combining regularisation methods and memory-based methods in continual learning. Future research could investigate other ways of selecting a memorable past (e.g. fixed memory size), more efficient ways of calculating kernel matrices, and the case where data does not arrive in tasks. ",
+        "bbox": [
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+    {
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+        "text": "Siddharth Swaroop, Cuong V Nguyen, Thang D Bui, and Richard E Turner. Improving and understanding variational continual learning. arXiv preprint arXiv:1905.02099, 2019. ",
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+    {
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+        "text": "Michalis K Titsias, Jonathan Schwarz, Alexander G de G Matthews, Razvan Pascanu, and Yee Whye Teh. Functional regularisation for continual learning using gaussian processes. arXiv preprint arXiv:1901.11356, 2019. ",
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+        "text": "Friedemann Zenke, Ben Poole, and Surya Ganguli. Continual learning through synaptic intelligence. In ICML, pp. 3987–3995. JMLR. org, 2017. ",
+        "bbox": [
+            173,
+            512,
+            823,
+            541
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A DETAILS ON DEEP NETWORKS TO GAUSSIAN PROCESSES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            101,
+            687,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this section, we detail the map that is presented in Sec. 3.1 to go from the posterior of a deep neural network to a Gaussian process. The map for linear regression and logistic regression can be found in the appendix of Khan et al. (2019). We recap the map for logistic regression and, in addition, walk through the case of softmax regression here. Recall the linear model from Eq. 3: ",
+        "bbox": [
+            173,
+            135,
+            825,
+            193
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7454731eafba3194264aa251465b604db38d779bd6c51718a9dd59275183aa3a.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { M o d e l : } \\quad \\tilde { \\mathbf { y } } = \\mathbf { J } ( \\mathbf { x } ) \\mathbf { w } + \\boldsymbol { \\epsilon } , \\quad \\mathrm { w i t h } \\ \\boldsymbol { \\epsilon } \\sim \\mathcal { N } ( 0 , ( \\mathbf { A } ( \\mathbf { x } , \\mathbf { y } ) ) ^ { - 1 } ) \\mathrm { a n d } \\ \\mathbf { w } \\sim \\mathcal { N } ( 0 , \\delta ^ { - 1 } \\mathbf { I } _ { P } ) } \\\\ & { \\quad \\mathrm { D a t a : } \\quad \\tilde { \\mathbf { y } } _ { i } : = \\mathbf { J } ( \\mathbf { x } _ { i } ) \\mathbf { w } _ { * } - \\left( \\mathbf { A } ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) \\right) ^ { - 1 } \\mathbf { r } ( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            200,
+            746,
+            246
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Inference in this model yields the deep neural network posterior $q ( \\mathbf { w } )$ in Eq. 2. ",
+        "bbox": [
+            173,
+            252,
+            686,
+            268
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Logistic regression: as mentioned in Sec. 3.1, the parameter of the Bernoulli likelihood is given by applying the sigmoid link function to $\\operatorname { f } ( \\mathbf { x } )$ and we defined $\\mathfrak { p } ( \\mathbf { x } ) : = \\sigma ( \\mathrm { f } _ { w _ { \\ast } } ( \\mathbf { x } ) )$ . Further, the derivative and Hessian of loss were particularised as $\\operatorname { r } ( \\mathbf { x } , \\mathbf { y } ) = \\operatorname { p } ( \\mathbf { x } ) - \\mathbf { y }$ and $\\Lambda ( \\mathbf { x } , \\mathbf { y } ) = \\mathsf { p } ( \\mathbf { x } ) ( 1 - \\mathsf { p } ( \\mathbf { x } ) )$ . We write $\\varepsilon \\sim { \\cal N } ( 0 , 1 )$ , ,for standard noise and plug the quantities model (Eq. 10): ",
+        "bbox": [
+            173,
+            273,
+            825,
+            332
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f4f400cc5915d5e30286a48d12690640026bd6e50e3fdf2c5efc6fa2b4dcef8c.jpg",
+        "text": "$$\n{ \\bf { J } } ( { \\bf { x } } ) { { \\bf { w } } _ { \\ast } } - \\frac { { \\bf { p } } ( { \\bf { x } } ) - { \\bf { y } } } { \\Lambda ( { \\bf { x } } , { \\bf { y } } ) } = { \\bf { J } } ( { \\bf { x } } ) { \\bf { w } } + ( \\Lambda ( { \\bf { x } } , { \\bf { y } } ) ) ^ { - \\frac { 1 } { 2 } } \\varepsilon _ { i } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            339,
+            643,
+            373
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "which can be re-arranged to a new model that has an equivalent posterior, but where inference is done by observing original data points $\\mathcal { D }$ : ",
+        "bbox": [
+            173,
+            382,
+            823,
+            411
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/358ee339d8c413d423fb9f66dad71cdff44ed4c15629d51dcc49da605a9896a7.jpg",
+        "text": "$$\n\\mathbf { y } = \\mathbf { p } ( \\mathbf { x } ) + \\Lambda ( \\mathbf { x } , \\mathbf { y } ) \\mathbf { J } ( \\mathbf { x } ) ( \\mathbf { w } - \\mathbf { w } _ { * } ) + \\tau , ~ \\mathrm { w i t h } ~ \\tau \\sim N ( 0 , \\Lambda ( \\mathbf { x } , \\mathbf { y } ) ) ~ \\mathrm { a n d } ~ \\mathbf { w } \\sim N ( 0 , \\delta ^ { - 1 } \\mathbf { I } _ { P } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            422,
+            759,
+            440
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Softmax regression is a simple extension to the logistic regression case: we have a categorical likelihood with parameter $\\mathbf { p } ( \\mathbf { x } ) : = \\mathrm { s o f t m a x } ( \\mathbf { f } _ { w _ { * } } ( \\mathbf { x } ) )$ and labels ${ \\bf y } _ { i }$ that are standard one-hot encoded basis vectors, i.e. have a single 1 at some position and otherwise zeros. Then we have $\\mathbf { r } ( \\mathbf { x } , \\mathbf { y } ) =$ $\\mathbf { p } ( \\mathbf { x } ) - \\mathbf { y }$ and $\\mathbf { \\boldsymbol { \\Lambda } } ( \\mathbf { \\boldsymbol { x } } , \\mathbf { \\boldsymbol { y } } ) = \\mathrm { d i a g } ( \\mathbf { \\boldsymbol { p } } ( \\mathbf { \\boldsymbol { x } } ) ) - \\mathbf { \\boldsymbol { p } } ( \\mathbf { \\boldsymbol { x } } ) \\mathbf { \\boldsymbol { p } } ( \\mathbf { \\boldsymbol { x } } ) ^ { \\top }$ . Unfortunately, $\\mathbf { \\Delta } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda }$ is of rank $K - 1$ ,and therefore , ,we cannot uniquely invert it. The theory of Khan et al. (2019) therefore does not allow this because it requires $\\mathbf { \\delta } \\mathbf { A } ( \\mathbf { x } , \\mathbf { y } ) \\succ 0$ . However, we can still write a model in line with Eq. 13 that we expect to ,have a posterior close to that of the deep neural network: ",
+        "bbox": [
+            173,
+            457,
+            825,
+            555
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cf889a9fbc343ca09cbc2243b9e28a5186f2e059a597073a34b766917fe68d93.jpg",
+        "text": "$$\n\\mathbf { y } = \\mathbf { p } ( \\mathbf { x } ) + \\Lambda ( \\mathbf { x } , \\mathbf { y } ) \\mathbf { J } ( \\mathbf { x } ) ( \\mathbf { w } - \\mathbf { w } _ { * } ) + \\tau ~ \\mathrm { w i t h } ~ \\tau \\sim N ( 0 , \\Lambda ( \\mathbf { x } , \\mathbf { y } ) ) ~ \\mathrm { a n d } ~ \\mathbf { w } \\sim N ( 0 , \\delta ^ { - 1 } \\mathbf { I } _ { P } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            225,
+            566,
+            769,
+            584
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Conversion to GP can be achieved by taking the expectation and covariance of the linear model. To obtain the GP prior, we take the expectation of the linear model (10) over the prior $p ( \\mathbf { w } )$ . To obtain the posterior, we take the expectation over $q ( \\mathbf { w } )$ . Since we are interested in the posterior process, the latter is of interest and calculated for the logistic case in Sec. 3.1. The softmax regression case works equivalently but yields a $K \\times K$ kernel, which can be represented as a symmetric tensor, or a $N K { \\times } N K$ matrix over data set $\\mathcal { D }$ with $N$ examples. To avoid growing complexity with more classes, we model the GPs for each class as being independent, as discussed in Appendix B. ",
+        "bbox": [
+            173,
+            601,
+            825,
+            700
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B REDUCING COMPLEXITY IN THE MULTICLASS SETTING",
+        "text_level": 1,
+        "bbox": [
+            174,
+            723,
+            674,
+            741
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For the softmax regression with $K$ classes, we build an individual GP for each class, in order to avoid growing complexity with more classes. We utilise $\\mathbf { y } ^ { ( k ) }$ to denote the $k$ -th item of $\\mathbf { y }$ in Eq. (14). Then $\\mathbf { \\bar { y } } ^ { ( k ) }$ will be: ",
+        "bbox": [
+            174,
+            756,
+            825,
+            799
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ea4131b7fba6b5a8057ee8c12f711ac6ced80c889202a027427b59553e773fea.jpg",
+        "text": "$$\n\\mathbf { y } ^ { ( k ) } = \\mathbf { p } ^ { ( k ) } + \\mathbf { A } ( \\mathbf { x } , \\mathbf { y } ) ^ { ( k ) } \\mathbf { J } ( \\mathbf { x } ) ( \\mathbf { w } - \\mathbf { w } _ { \\ast } ) + \\tau ^ { ( k ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            801,
+            640,
+            821
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here $\\mathbf { \\Delta } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda \\Lambda } \\mathbf { \\Lambda } \\mathbf { \\Lambda \\Lambda } \\mathbf { \\Lambda \\Lambda } \\mathbf \\Lambda \\Lambda \\mathbf { } \\Lambda \\Lambda \\mathbf { \\Lambda } \\Lambda \\Lambda \\mathbf \\Lambda \\Lambda \\Lambda \\Lambda \\mathbf { } \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\mathbf \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\Lambda \\ c \\ c \\ c \\ c \\ c \\ c \\ c \\ c \\ c \\ c \\ c \\ c \\ c \\ c \\ c \\ c$ denotes the $k$ -th row of the Hessian matrix. We transform the $K$ models to function ,space and treat them as independent. This allows us to split the KL-divergence in Eq. 8 into $K$ individual divergences, and we obtain the simplified final objective, ",
+        "bbox": [
+            174,
+            828,
+            823,
+            872
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/21c82bea5f3e61e1474eb6c549ae4f827a7af136242b9fed5aa0a1ecbc1851f7.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\boldsymbol { w } } \\quad \\tau \\sum _ { i \\in \\mathcal { D } _ { t } } \\ell ( y _ { i } , f _ { w } ( \\mathbf { x } _ { i } ) ) + \\frac { 1 } { 2 } \\sum _ { s = 1 } ^ { t - 1 } \\sum _ { k = 1 } ^ { K } ( \\mathbf { m } _ { s , w } ^ { ( k ) } - \\mathbf { m } _ { s , w _ { t - 1 } } ^ { ( k ) } ) ^ { \\top } [ \\mathbf { K } _ { w _ { t - 1 } , s } ^ { ( k ) } ] ^ { - 1 } ( \\mathbf { m } _ { s , w } ^ { ( k ) } - \\mathbf { m } _ { s , w _ { t - 1 } } ^ { ( k ) } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            204,
+            881,
+            764,
+            928
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C FURTHER DETAIL ON CONTINUAL LEARNING METRICS REPORTED ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            751,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We report a backward transfer metric and a forward transfer metric on split CIFAR. The backward transfer metric is exactly as defined in Lopez-Paz & Ranzato (2017). The forward transfer metric is a measure of how well the method uses previously seen knowledge to improve classification accuracy on newly seen tasks. Let there be a total of $T$ tasks. Let $R _ { i , j }$ be the classification accuracy of the model on task $t _ { j }$ after observing the last sample from task $t _ { i }$ ,. Let $R _ { i } ^ { \\mathrm { i n d } }$ be the classification accuracy of an independent model trained only on task $i$ . Then, ",
+        "bbox": [
+            173,
+            146,
+            825,
+            234
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6218b9e3c393c46b82a471f5c22ed8848b18c6d177b54880f9a0a6edbb1da01b.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { } & { \\mathrm { B a c k w a r d ~ T r a n s f e r , B W T } = \\displaystyle \\frac { 1 } { T - 1 } \\sum _ { i = 1 } ^ { T - 1 } R _ { T , i } - R _ { i , i } , } \\\\ & { } & { \\mathrm { F o r w a r d ~ T r a n s f e r } = \\displaystyle \\frac { 1 } { T - 1 } \\sum _ { i = 2 } ^ { T } R _ { i , i } - R _ { i } ^ { \\mathrm { i n d } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            279,
+            663,
+            367
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D FURTHER TOY DATA EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            405,
+            501,
+            421
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This section provides further information and visualisations of toy 2D datasets, as well as hyperparameter settings for VCL. ",
+        "bbox": [
+            173,
+            449,
+            823,
+            479
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D.1 WEIGHT-SPACE REGULARISATION’S INCONSISTENT BEHAVIOUR",
+        "bbox": [
+            174,
+            518,
+            660,
+            534
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/92588ed1e1e9c7429f6e049e1ade57733b1c6a43fe9836f0f2ed72b33eefec7a.jpg",
+        "table_caption": [
+            "Table 3: Train accuracy of FROMP, VCL (no coresets), VCL $^ +$ coresets and batch-trained Adam (an upper bound on performance) on a toy 2D binary classification dataset, with mean and standard deviations over 5 runs for VCL and batch Adam, and 10 runs for FROMP. ‘VCL’ is without coresets. VCL-RP and FRORP have the same (random) coreset selections. VCL-MP is provided with ‘ideal’ coreset points as chosen by an independent run of FROMP. VCL (no coreset) does very poorly, forgetting previous tasks. VCL $^ +$ coresets is brittle with high standard deviations, while FROMP is stable. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>FROMP</td><td>FRORP</td><td>VCL-RP</td><td>VCL-MP</td><td>VCL</td><td>Batch Adam</td></tr><tr><td>99.6% ± 0.2</td><td>98.5% ± 0.6</td><td>92% ±10</td><td>85%±14</td><td>68%±8</td><td>99.70% ± 0.03</td></tr></table>",
+        "bbox": [
+            215,
+            689,
+            774,
+            727
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 3 summarises the performance (measured by train accuracy) of FROMP and ${ \\mathrm { V C L } } +$ coresets on a toy dataset similar to that in Figure 1. FROMP is very consistent, while VCL (with coresets) is extremely brittle: it can perform well sometimes (1 run out of 5), but usually does not (4 runs out of 5). This is regardless of coreset points chosen for VCL. Without coresets, VCL forgets many past tasks, with very low performance. Previous work (Farquhar & Gal, 2019) has argued that this is inevitable for weight-regularisation-only methods such as VCL and EWC. ",
+        "bbox": [
+            173,
+            762,
+            825,
+            847
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We now 3 runs with different random seeds of VCL-MP from Table 3: the coreset is chosen from an independent run of FROMP, with datapoints all on the task boundary. This selection of coreset is intuitively better than a random coreset selection. Please note that the results we show here are not specific to coreset selection. Any coreset selection (whether random or otherwise) all show the same inconsistency when VCL is run on them. This behaviour is not specific to coreset choice. ",
+        "bbox": [
+            174,
+            853,
+            823,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/ff1e52f38925211222f014ef43b74938c3a6d9cedd3d3ab0f916b1f0e7635f0b.jpg",
+        "image_caption": [
+            "Figure 6: Three runs of VCL-MP on toy 2D data. These are the middle performing 3 runs out of 5 runs with different random seeds. VCL’s inconsistent behaviour is clear. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            282,
+            99,
+            714,
+            361
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "This section visualises the different dataset variations presented in Table 1. We pick the middle performing FROMP run (out of 5) and batch Adam run to show. ",
+        "bbox": [
+            171,
+            580,
+            825,
+            608
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/90863bb11f7283da4561f10637aee968ede083ede8f99683ff840adda4292b87.jpg",
+        "image_caption": [
+            "Figure 7: FROMP (middle performing of 5 runs) and batch Adam on a dataset $1 0 \\mathrm { x }$ smaller (400 points per task). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            238,
+            736,
+            767,
+            859
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/94b9965140412a3f0acc6899aef7e0f913194e7848fffd22c44f00b12dfdce7b.jpg",
+        "image_caption": [
+            "Figure 8: FROMP (middle performing of 5 runs), left, and batch Adam, right, on a dataset $1 0 \\mathrm { x }$ larger (40,000 points per task). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            117,
+            766,
+            253
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/487747e7e51c8e481e933f5c248e4ea7d931a3ee6cbdf9ddec742cf8d09b6041.jpg",
+        "image_caption": [
+            "Figure 9: FROMP (middle performing of 5 runs), left, and batch Adam, right, on a dataset with a new, easy, 6th task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            238,
+            433,
+            766,
+            554
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/2a27ed7655503fe4d2177a038b503044f0df72a3578dbf7745628e4e43b5947c.jpg",
+        "image_caption": [
+            "Figure 10: FROMP (middle performing of 5 runs), left, and batch Adam, right, on a dataset with increased standard deviations of each class’ points, making classification tougher. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            731,
+            766,
+            862
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/0c5f6487a6cec61cdd6becc9344b865e3804264f9e743edf6567afca03d9af72.jpg",
+        "image_caption": [
+            "Figure 11: FROMP (middle performing of 5 runs), left, and batch Adam, right, on a dataset with 2 tasks having overlapping data, which is not separable. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            238,
+            126,
+            764,
+            250
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "D.3 VCL AND FROMP HYPERPARAMETER SETTINGS FOR TOY DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            337,
+            696,
+            351
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "FROMP. We optimised the number of epochs, Adam learning rate, and batch size. We optimised by running various settings for 5 runs and picking the settings with largest mean train accuracy on the toy dataset in Figure 1. We found the best settings were: number of epoch $\\scriptstyle = 5 0$ , batch $\\mathrm { s i z e } { = } 2 0$ , learning rate ${ \\mathrm { : = 0 . 0 1 } }$ . The hyperparameters were then fixed across all toy data experimental runs, including across dataset variations (number of epochs was appropriately scaled by 10 if dataset size was scaled by 10). ",
+        "bbox": [
+            174,
+            362,
+            825,
+            445
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "VCL $^ +$ coresets. We optimised the number of epochs, the number of coreset epochs (because VCL $^ +$ coresets trains on non-coreset data first, then on coreset data just before test-time: see Nguyen et al. (2018)), learning rate (we use Adam to optimise the means and standard deviations of each parameter), batch size, and prior variance. We optimised by running various settings for 5 runs and picking the settings with largest mean train accuracy. We found the best settings were: number of epochs ${ \\ o } { = } 2 0 0$ , number of coreset epoch ${ \\ o } { = } 2 0 0$ , a standard normal prior (varianc ${ \\mathrm { ; = } } 1$ ), batch size $scriptstyle = 4 0$ , learning rate $= 0 . 0 1$ . VCL is slow to run (an order of magnitude longer) compared to all other methods (FROMP and batch Adam). ",
+        "bbox": [
+            173,
+            453,
+            825,
+            564
+        ],
+        "page_idx": 15
+    }
+]

parse/train/rJzLciCqKm/rJzLciCqKm.md

@@ -0,0 +1,572 @@
+# LEARNING FROM POSITIVE AND UNLABELED DATA WITH A SELECTION BIAS
+
+Masahiro Kato1,2, Takeshi Teshima1,2, and Junya Honda1,2
+
+1The University of Tokyo, Tokyo, Japan 2RIKEN, Tokyo, Japan {mkato, teshima}@ms.k.u-tokyo.ac.jp, honda@edu.k.u-tokyo.ac.jp
+
+# ABSTRACT
+
+We consider the problem of learning a binary classifier only from positive data and unlabeled data (PU learning). Recent methods of PU learning commonly assume that the labeled positive data are identically distributed as the unlabeled positive data. However, this assumption is unrealistic in many instances of PU learning because it fails to capture the existence of a selection bias in the labeling process. When the data has selection bias, it is difficult to learn the Bayes optimal classifier by conventional methods of PU learning. In this paper, we propose a method to partially identify the classifier. The proposed algorithm learns a scoring function that preserves the order induced by the class posterior under mild assumptions, which can be used as a classifier by setting an appropriate threshold. Through experiments, we show that the method outperforms previous methods for PU learning on various real-world datasets.
+
+# 1 INTRODUCTION
+
+We consider a situation that there are only positive and unlabeled data, and train a binary classifier only from them (PU learning). This problem arises in various practical situations, such as information retrieval and outlier detection (Elkan & Noto, 2008; Ward et al., 2009; Scott & Blanchard, 2009; Blanchard et al., 2010; Li et al., 2009; Nguyen et al., 2011). One of the milestones of PU learning is Elkan & Noto (2008), who proposed a practically useful algorithm with theoretical analysis, and there is subsequent research called unbiased PU learning (du Plessis & Sugiyama, 2014; du Plessis et al., 2015) where an unbiased estimator of the classification risk is minimized.
+
+We focus on the case-control scenario (a.k.a. the problem setting based on two samples of data; Ward et al., 2009; Niu et al., 2016). In this scenario, positive data are obtained separately from unlabeled data, and unlabeled data are sampled from the whole population. As Elkan & Noto (2008) explained, we cannot identify a classifier without an assumption on how positive data are labeled. Therefore “selected completely at random” (SCAR) is traditionally assumed, i.e., the positive labeled data are identically distributed as the positive unlabeled data (Elkan & Noto, 2008; du Plessis et al., 2015). The assumption of SCAR is, however, unrealistic in many instances of PU learning, e.g., a patient’s electronic health record (Bekker & Davis, 2018a) and a recommendation system (Marlin & Zemel, 2009; Schnabel et al., 2016). In these cases, there is a selection bias (Heckman, 1979; Manski, 2008; Angrist & Pischke, 2008); the distribution of the positive data may differ between the labeled data and the unlabeled data.
+
+Several recent related works have proposed alternative assumptions (Bekker & Davis, 2018a;b). However, in order to weaken SCAR, they impose other additional assumptions. In this work, we consider a more natural assumption such that SCAR becomes its special case. We assume that $p ( o = + 1 | x , y = + 1 )$ and $p ( y = + 1 | x )$ induce the same ordering on the input space $\mathcal { X }$ , where $y \in \{ - 1 , + 1 \}$ is the data label and $o = + 1$ (resp. $o = 0$ ) denotes the event that the data is observed (resp. not observed). We call this property the invariance of order. In the real-world application, there are many situations with a selection bias which follows the invariance of order. Among them, we list the following two examples.
+
+# Example 1: (Anomaly Detection)
+
+The goal of anomaly detection is to find anomaly data in an unlabeled dataset based on another dataset that consists only of anomaly data. When the anomaly data is collected, the more likely a datum is an anomaly, the more likely it is noticed and gets labeled.
+
+Example 2: (Face Recognition)
+
+The goal of this task is to identify a user from a set of face images based on some pictures of the user. In this case, the positive data are the face images identified with the user, and the unlabeled data consist of all the unidentified face images. The user may more likely provide pictures in which the face can be seen clearly, while in the unlabeled data there may be many unclear images.
+
+Our problem setting is similar to the problem called learning from instance-dependent noisy labels (Du & Cai, 2015; Bootkrajang, 2016). In this problem setting, positive and negative data are available, but the labels are subject to the noise that flips the label with instance-dependent probability. Besides, there are existing works putting assumption similar to the invariance order (Du & Cai, 2015; Bootkrajang, 2016). We explain the difference between their works and ours around Assumption 1 in Section 2.1. We name our problem setting PU learning with a Selection Bias (PUSB).
+
+In this paper, we propose a novel framework to deal with this problem setting. The experimental results show that our proposed method is appropriate for real-world applications compared to existing approaches for PU learning.
+
+# 2 PROBLEM SETTING OF PU LEARNING WITH A SELECTION BIAS
+
+We consider a binary classification problem to classify $\pmb { x } \in \mathcal { X } \subset \mathbb { R } ^ { d }$ into one of the two classes $\{ - 1 , + 1 \}$ . We assume that there exists a joint distribution $p ( { \pmb x } , y , o )$ , where $y \in \{ - 1 , + 1 \}$ is the class label of $_ { \textbf { \em x } }$ , and $o \in \{ 0 , + 1 \}$ is the observation status of $y$ (observed if $o = + 1$ and unobserved if $o = 0$ ). In other words, $_ { \textbf { \em x } }$ is labeled if $o = + 1$ , and it is unlabeled otherwise.
+
+In PU learning, there are two distinguished sampling schemes called one sample and two samples of data (Niu et al., 2016). They are also called the censoring scenario and case-control scenario, respectively (Elkan & Noto, 2008). In the censoring scenario, a set of unlabeled data is sampled from the marginal density $p ( { \pmb x } )$ . Then, if a data point $_ { \textbf { \em x } }$ is positive, it gets labeled with probability $p ( o = + 1 | x , \bar { y } = + 1 )$ ; if $_ { \textbf { \em x } }$ is negative, it is never labeled. In the case-control scenario, a set of positive data is drawn from the positive conditional density $p ( { \pmb x } | y = + 1 )$ and a set of unlabeled data is drawn from $p ( { \pmb x } )$ . As Niu et al. (2016) stated, the case-control scenario is slightly more general than the censoring scenario setting. It is because the censoring scenario assumes the access to samples generated by $p ( { \pmb x } )$ , $p ( { \pmb x } | o = + 1 )$ , and $p ( { \pmb x } | o = 0 )$ whereas the case-control scenario only assumes the access to samples generated by $p ( { \pmb x } )$ and $p ( { \pmb x } | o = + 1 )$ ). Therefore we focus on the case-control scenario.
+
+Suppose that we have a positive dataset $\{ { \pmb x } _ { i } \} _ { i = 1 } ^ { n }$ and an unlabeled dataset $\{ { \pmb x } _ { i } ^ { \prime } \} _ { i = 1 } ^ { n ^ { \prime } }$
+
+$$
+\{ \pmb { x } _ { i } \} _ { i = 1 } ^ { n } \stackrel { \mathrm { i . i . d . } } { \sim } p ( \pmb { x } | y = + 1 , o = + 1 ) , \{ \pmb { x } _ { i } ^ { \prime } \} _ { i = 1 } ^ { n ^ { \prime } } \stackrel { \mathrm { i . i . d . } } { \sim } p ( \pmb { x } ) .
+$$
+
+We assume that negative data are never labeled.
+
+Note that we do not assume SCAR. Therefore, $p ( { \pmb x } | y = + 1 )$ may differ from $p ( { \pmb x } | y = + 1 , o = + 1$ ).   
+In the case they differ, we say that there is a selection bias.
+
+The quantity $\pi = p ( y = + 1 ) $ is called the class-prior. In our problem setting, we assume that $\pi$ is known. For example, in anomaly detection, the percentage of anomaly in the whole batch of products can be reported based on past experiences. Although there are various methods for estimating the class-prior in the traditional framework of the case-control scenario (du Plessis et al., 2016; Ramaswamy et al., 2016; Jain et al., 2016; Kato et al., 2018), we cannot estimate the class-prior in our problem setting under a theoretical guarantee. In Section 5, we show how misspecified class priors affect the performance of a classifier. As we explain later, even if we do not know the class-prior, we only have to change the last step of our algorithm. In summary, our goal is to obtain a classifier $h : \mathcal { X }  \{ - 1 , 1 \}$ only from $\{ { \pmb x } _ { i } \} _ { i = 1 } ^ { n }$ , $\{ { \pmb x } _ { i } ^ { \prime } \} _ { i = 1 } ^ { n ^ { \prime } }$ , and $\pi$ under a weaker assumption than SCAR.
+
+# 2.1 IDENTIFICATION STRATEGY
+
+As stated by Elkan & Noto (2008), even if the class prior is given, we cannot estimate $p ( y = + 1 | x )$ only from positive data and unlabeled data without any assumption in PU learning. In the case-control scenario, a standard assumption is SCAR, i.e. $p ( { \pmb x } | y = + 1 , o = + 1 ) = p ( { \pmb x } | y = + 1 , o = 0 )$ , so that $p ( y = + 1 | x )$ can be estimated from the data in principle. However, in many instances of PU learning, the SCAR assumption is unreasonable as discussed before. Therefore, we relax SCAR and accommodate a selection bias. We can see how SCAR makes the class posterior identifiable in the following equation:
+
+$$
+p ( y = + 1 | x ) = \frac { p ( \pmb { x } , y = + 1 ) } { p ( \pmb { x } ) } = \frac { p ( \pmb { x } | y = + 1 ) \pi } { p ( \pmb { x } ) } \overset { = } \underset { \mathrm { s c a R } } { \underbrace { = } } \frac { p ( \pmb { x } | y = + 1 , o = + 1 ) \pi } { p ( \pmb { x } ) } .
+$$
+
+Here, $p ( { \pmb x } | y = + 1 , o = + 1 )$ can be estimated from the sample. This estimate can be used to obtain an estimate of $p ( { \pmb x } | y = + 1 )$ and hence that of $p ( y = + 1 | x )$ if we assume SCAR. However, without assuming SCAR, $p ( { \pmb x } | y = + 1 )$ ) may differ from $p ( { \pmb x } | y = + 1 , o = + 1 )$ ), and $p ( y = + 1 | x )$ is not identifiable.
+
+Therefore, instead of estimating $p ( y = + 1 | x )$ , we consider extracting some useful information of $p ( y = + 1 | x )$ to learn a classifier. This kind of approach is known as “partial identification” (Manski, 2008) in statistics and economics. First, we introduce an assumption that is weaker than SCAR.
+
+Assumption 1 (Invariance of Order). For any $\pmb { x } _ { i } , \pmb { x } _ { j } \in \mathcal { X }$ , we have
+
+$$
+p ( y = + 1 | \boldsymbol { x } _ { i } ) \leq p ( y = + 1 | \boldsymbol { x } _ { j } ) \Leftrightarrow p ( o = + 1 | \boldsymbol { x } _ { i } ) \leq p ( o = + 1 | \boldsymbol { x } _ { j } ) .
+$$
+
+Although Assumption 1 does not allow one to construct an unbiased estimator of the risk functional, we try to partially identify $p ( y = + 1 | x )$ under this assumption. Our problem setting can be regarded as a generalization of the traditional case-control scenario because SCAR is a special case of the invariance of order.
+
+# 2.2 RELATED WORKS
+
+A similar assumption can be found in the literature of learning from instance-dependent noisy labels (Du & Cai, 2015; Bootkrajang, 2016), which considers a probabilistic label flipping that is proportional to $p ( y = + 1 | x )$ . However, in order to apply methods of learning from noisy labels to PU learning, we need to assume the censoring scenario and these methods cannot be applied to our problem setting based on the case-control scenario. The censoring scenario is a special case of learning from noisy labels where only negative data is contaminated, i.e., some positive labels flip to negative labels. Thus, in the censoring scenario, unlabeled data can be regarded as negative-labeled data contaminated by positive data. On the other hand, in the case-control scenario, unlabeled data is generated from the marginal distribution $p ( { \pmb x } )$ , i.e., we cannot observe samples generated from $p ( { \pmb x } | o = 0 )$ . Therefore, our problem setting, namely the case-control scenario with invariance of order, is different from the existing works of learning from instance-dependent noisy labels. In addition, our method is also applicable to the censoring scenario when the invariance of order holds because the unlabeled data of the case-control scenario can be made from positive and unlabeled data of the censoring scenario.
+
+In Example 1, $p ( y = + 1 | x )$ is the probability that a given input $_ { \textbf { \em x } }$ is anomaly, while $p ( o = + 1 | x )$ is the probability that a given input $_ { \textbf { \em x } }$ gets labeled in the dataset. In Example 2, a positively labeled data is an image $_ { \textbf { \em x } }$ that is known to belong to a user. Here, $p ( o = + 1 | x )$ is the probability that the user provides the picture $_ { \textbf { \em x } }$ as a training datum.
+
+# 3 STRATEGY FOR PARTIAL IDENTIFICATION AND CLASSIFICATION
+
+As discussed in Section 2.1, we cannot estimate $p ( y = + 1 | x )$ when there is a selection bias even if the class prior is given. Our idea of partial identification is based on the following theorem with the density ratio
+
+$$
+r ( { \pmb x } ) = \frac { p ( { \pmb x } | y = + 1 , o = + 1 ) } { p ( { \pmb x } ) } .
+$$
+
+# Algorithm 1 Conceptual Algorithm in Population
+
+Input: $p ( { \pmb x } | y = + 1 )$ , $p ( { \pmb x } )$ and the class-pror $\pi$ .   
+Using $p ( { \pmb x } | y = + 1 )$ and $p ( { \pmb x } )$ , calculate $r ( { \pmb x } )$ by minimization of either (4) or (7).   
+Using $r ( { \pmb x } )$ , calculate $\theta _ { \pi }$ in (2).   
+Using the density ratio $r ( { \pmb x } )$ and the threshold $\theta _ { \pi }$ , obtain a classifier $h ( \pmb { x } ) = \mathtt { s i g n } ( r ( \pmb { x } ) - \theta _ { \pi } )$ .
+
+Theorem 1 (Order Preserving Property of the score Function). Suppose that Assumption $I$ holds. Then, for any $\pmb { x } _ { i } , \pmb { x } _ { j } \in \mathcal { X }$ ,
+
+$$
+p ( y = + 1 | { \boldsymbol x } _ { i } ) \leq p ( y = + 1 | { \boldsymbol x } _ { j } ) \Leftrightarrow r ( { \boldsymbol x } _ { i } ) \leq r ( { \boldsymbol x } _ { j } ) .
+$$
+
+A proof is provided in Appendix A.
+
+Even though we cannot estimate $p ( y = + 1 | \cdot )$ , Theorem 1 implies that we can still extract the total order in $\mathcal { X }$ induced by $p ( y = + 1 | \cdot )$ if we can estimate $r ( \cdot )$ . Therefore, we propose to estimate $r$ and use it as a score function that captures the total order induced by $p ( y = + 1 | \cdot )$ . After obtaining an estimator of $r$ (denoted by $\hat { r }$ ), we set a threshold $\theta \in \mathbb { R }$ and use $h ( \pmb { x } ) = \mathtt { s i g n } ( r ( \pmb { x } ) - \theta )$ as a classifier. There are various ways of determining the threshold. For instance, we put labels from data with the highest density ratio under a constraint on the number of data to which we can put labels (Hido et al., 2011). Here, we introduce one useful principle for choosing $\theta$ . We consider using a threshold $\theta _ { \pi }$ defined by the following equation,
+
+$$
+\pi = \int \mathbf { 1 } [ r ( \pmb { x } ) \geq \theta _ { \pi } ] p ( \pmb { x } ) d \pmb { x } .
+$$
+
+The intuition behind the definition of $\theta _ { \pi }$ is that the proportion of the positive data in the test data points should not deviate so much from the class-prior. This intuition becomes clearer in Section 4.3.
+
+In Section 4, we discuss detailed methods for estimating $r$ and setting $\theta$ based on data. Our approach is summarized in the form of a pseudo-code in Algorithm 1. In the rest of this section, we theoretically justify $\theta _ { \pi }$ defined in (2).
+
+Property of $\theta _ { \pi }$ : Let us consider the case where a classifier is given as $h ( \pmb { x } ) = \mathtt { s i g n } ( r ( \pmb { x } ) - \theta )$ . Then four population quantities, true positives (TP), true negatives (TN), false positives (FP), and false negatives (FN) (Lipton et al., 2014), that depend on $r ( \cdot )$ and $\theta$ is written as follows:
+
+$$
+\begin{array} { l r } { { T P = \displaystyle \int _ { \{ { \bf x } : r ( { \bf x } ) \geq \theta } \} } p ( y = + 1 | x ) p ( { \bf x } ) d x , } & { { F P = \displaystyle \int _ { \{ { \bf x } : r ( { \bf x } ) \geq \theta } \} } p ( y = - 1 | x ) p ( x ) d x , } \\ { { T N = \displaystyle \int _ { \{ { \bf x } : r ( { \bf x } ) < \theta } \} } p ( y = - 1 | x ) p ( { \bf x } ) d x , } & { { F N = \displaystyle \int _ { \{ { \bf x } : r ( { \bf x } ) < \theta \} } p ( y = + 1 | x ) p ( x ) d x . } } \end{array}
+$$
+
+Then, the precision and the recall of a classifier are expressed as precision $\begin{array} { r l r } { \mathrm { ~ } } & { { } } & { = \mathrm { ~ \left( \frac { { \cal T } P } { { \cal T } P + { \cal F } P } \right) } } \end{array}$ and $\begin{array} { r } { \mathrm { r e c a l l } = \left( \frac { T P } { T P + F N } \right) } \end{array}$ , respectively. For $\theta _ { \pi }$ , we have the following result.
+
+Theorem 2. If we use $\theta = \theta _ { \pi }$ , then precision $=$ recall holds.
+
+A proof is shown in Appendix B. The threshold $\theta _ { \pi }$ is known as precision–recall breakeven point (BEP) (Sammut & Webb, 2010), which makes the precision and the recall the same. BEP is originally used to evaluate a generic classification model with a score function and a threshold. Besides, we can interpret BEP as a point which balances a prediction result; as explained by Powers (2015), a classifier using BEP as a threshold puts the same cost to the false positives and false negatives. Knowing BEP is also useful for deciding on a threshold which put unbalanced weight on the precision and the recall because we can tell if we are weighing precision more or recall more.
+
+# 4 ALGORITHM
+
+Here, we propose two directions for estimating r(x) = p(x|y=+1,o=+1)p(x) under the assumption of invariance of order, namely minimizing a pseudo classification risk and direct density ratio estimation. We discuss how to set $\theta$ based on the data. A pseudo-code of our algorithm is shown in Algorithm 2.
+
+# Algorithm 2 PUSB
+
+Input: A class-pror sitive dataset . $\{ { \pmb x } _ { i } \} _ { i = 1 } ^ { n }$ , an unlabeled dataset $\{ { \pmb x } _ { i } ^ { \prime } \} _ { i = 1 } ^ { n ^ { \prime } }$ , a test dataset $\{ \pmb { x } _ { i } ^ { \mathrm { t e } } \} _ { i = 1 } ^ { n ^ { \mathrm { t e } } }$ and the $\pi$   
+Using $\{ { \pmb x } _ { i } \} _ { i = 1 } ^ { n }$ and $\{ { \pmb x } _ { i } ^ { \prime } \} _ { i = 1 } ^ { n ^ { \prime } }$ , estimate $r ( { \pmb x } )$ by any of (5), (6) or (8) and obtain $\hat { r } ( { \pmb x } )$ .   
+Using $\hat { r } ( { \pmb x } )$ , estimate $\theta _ { \pi }$ by (9) and obtain $\hat { \theta } _ { \pi }$ .   
+Using an estimator $\hat { r } ( { \pmb x } )$ and $\hat { \theta }$ , obtain a classifier $h ( \pmb { x } ) = \mathtt { s i g n } ( \hat { r } ( \pmb { x } ) - \hat { \theta } _ { \pi } ) .$
+
+# 4.1 ESTIMATION OF $r$ BY MINIMIZING PSEUDO CLASSIFICATION RISK
+
+First, we introduce the minimization of the pseudo classification risk. The idea is to minimize the classification risk used in traditional PU learning (du Plessis et al., 2014; 2015) as if there is no selection bias. Under a selection bias, we cannot construct unbiased risk function, but the minimizer can be substituted for the density ratio in (1).
+
+Conventional PU risk formulation: Let $\ell : \mathbb { R } \times \{ \pm 1 \} \to \mathbb { R } ^ { + }$ be a loss function, where $\mathbb { R } ^ { + }$ is the set of non-negative real values, and $\mathcal { F }$ be the set of measurable functions from $\mathcal { X }$ to $[ \epsilon , 1 - \epsilon ]$ , where $\epsilon \in ( 0 , 1 / 2 )$ is a small positive value. This constant $\epsilon$ is introduced to make the following optimization problem well-defined.
+
+du Plessis et al. (2015) showed that the classification risk of $f \in { \mathcal { F } }$ in the traditional PU problem setting with SCAR can be expressed as
+
+$$
+R _ { \mathrm { P U } } ( f ) = \pi \mathbb { E } _ { \mathrm { p } } [ \ell ( f ( X ) , + 1 ) ] - \pi \mathbb { E } _ { \mathrm { p } } [ \ell ( f ( X ) , - 1 ) ] + \mathbb { E } _ { \mathrm { u } } [ \ell ( f ( X ) , - 1 ) ] ,
+$$
+
+where $\mathbb { E } _ { \mathrm { p } }$ and $\mathbb { E } _ { \mathrm { u } }$ are the expectations over $p ( { \pmb x } | y = + 1 )$ and $p ( { \pmb x } )$ , respectively. When there is no selection bias, we can replace the expectations with the corresponding sample averages to obtain an unbiased estimator of the classification risk.
+
+The pseudo classification risk: In our problem setting, we only have samples from $p ( { \pmb x } | { \pmb y } =$ $+ 1 , o = + 1 \rangle$ and not from $p ( { \pmb x } | y = + 1 )$ . Therefore, we cannot use our sample to obtain an empirical version of $R _ { \mathrm { P U } }$ . However, we still consider the pseudo classification risk of $f \in { \mathcal { F } }$ :
+
+$$
+\begin{array} { r } { R _ { \mathrm { P U } } ^ { \mathrm { b i a s } } ( f , \ell ) = \pi \mathbb { E } _ { \mathrm { p } } ^ { \mathrm { b i a s } } [ \ell ( f ( X ) , + 1 ) ] - \pi \mathbb { E } _ { \mathrm { p } } ^ { \mathrm { b i a s } } [ \ell ( f ( X ) , - 1 ) ] + \mathbb { E } _ { \mathrm { u } } [ \ell ( f ( X ) , - 1 ) ] , } \end{array}
+$$
+
+where $\mathbb { E } _ { \mathrm { p . } } ^ { \mathtt { b i a s } }$ is the expectation over $p ( \pmb { x } | y = + 1 , o = + 1 )$ . We call this functional the pseudo classification risk because it is not the true classification risk. An unbiased estimator for the pseudo classification risk can be obtained by replacing the expectations with the corresponding sample averages even if there is a selection bias. For the loss function, we use the logarithmic loss: $\ell ( f ( \pmb { x } ) , + 1 ) ) = - \log ( f ( \pmb { x } ) )$ and $\ell ( f ( \pmb { x } ) , - 1 ) = - \log ( 1 - f ( \pmb { x } ) )$ . In this case, the pseudo classification risk of $f \in { \mathcal { F } }$ becomes
+
+$$
+R _ { \mathrm { P U } } ^ { \mathrm { b i a s } } ( f ) = - \pi \mathbb { E } _ { \mathrm { p } } ^ { \mathrm { b i a s } } [ \log ( f ( X ) ) ] + \pi \mathbb { E } _ { p } ^ { \mathrm { b i a s } } [ \log ( 1 - f ( X ) ) ] - \mathbb { E } _ { \mathrm { u } } [ \log ( 1 - f ( X ) ) ] .
+$$
+
+Justification for minimizing the pseudo classification risk: For the pseudo classification risk with the logarithmic loss function, the following theorem justifies its use. Let us denote a minimizer of (4) by $f ^ { * }$ , that is,
+
+$$
+f ^ { * } \in \arg \operatorname* { m i n } _ { f \in \mathcal { F } } R _ { \mathrm { P U } } ^ { \mathrm { b i a s } } ( f ) .
+$$
+
+For the minimizer of (4), we show the following theorem.
+
+Theorem 3. It holds almost everywhere that
+
+$$
+f ^ { * } ( \pmb { x } ) = \left\{ \begin{array} { l l } { \epsilon } & { ( \pmb { x } \notin D _ { 1 } ) , } \\ { \frac { \pi p ( \pmb { x } | y = + 1 , o = + 1 ) } { p ( \pmb { x } ) } } & { ( \pmb { x } \in D _ { 1 } \cap D _ { 2 } ) , } \\ { 1 - \epsilon } & { ( \pmb { x } \notin D _ { 2 } ) , } \end{array} \right.
+$$
+
+where $D _ { 1 } \ = \ \{ { \pmb x } | \pi p ( { \pmb x } | y = 1 , o = + 1 ) \ \geq \ \epsilon p ( { \pmb x } ) \}$ and $D _ { 2 } \ = \ \{ { \pmb x } | \pi p ( { \pmb x } | y = 1 , o \ = \ + 1 ) \ \leq$ $( 1 - \epsilon ) p ( { \pmb x } ) \}$ .
+
+A proof is provided in Appendix C. Theorem 3 implies that the minimization of the empirical version of the pseudo classification risk allows us to estimate $r$ .
+
+Empirical Estimation: When we train a classifier with training samples, we can naively replace the expectations with the corresponding sample averages. For a hypothesis set $\mathcal { H }$ , which is a set of measurable functions, let us define the following risk minimization problem,
+
+$$
+\hat { f } _ { 1 } = \underset { f \in \mathcal { H } } { \arg \operatorname* { m i n } } \left[ - \pi \hat { \mathbb { E } } _ { p } ^ { \mathrm { b i a s } } [ \log ( f ( X ) ) ] + \pi \hat { \mathbb { E } } _ { p } ^ { \mathrm { b i a s } } [ \log ( 1 - f ( X ) ) ] - \hat { \mathbb { E } } _ { u } [ \log ( 1 - f ( X ) ) ] + \mathcal { R } ( f ) \right] ,
+$$
+
+where $\hat { \mathbb { E } } _ { p } ^ { \mathrm { { b i a s } } }$ denotes the averaging operator over positive data with a selection bias, $\hat { \mathbb { E } } _ { u }$ denotes the averaging over the unlabeled data, and $\mathcal { R }$ is a regularization term. du Plessis et al. (2015) showed that, under SCAR, the empirical version of the risk becomes unbiased toward the classification risk.
+
+However, Kiryo et al. (2017) pointed out that unbiased PU learning does not work with deep neural networks. Minimizing an empirical risk of (3) with deep neural networks easily causes over-fitting because the risk is not bounded from below by 0. In order to implement PU learning with deep neural networks, Kiryo et al. (2017) proposed the following non-negative risk,
+
+$$
+\begin{array} { r } { \hat { \bar { \mathfrak { c } } } _ { 2 } = \underset { f \in \mathcal { H } } { \mathrm { a r g } } \operatorname* { m i n } \left[ - \pi \hat { \mathbb { E } } _ { p } ^ { \mathrm { b i a s } } [ \mathrm { l o g } ( f ( X ) ) ] + \left( \pi \hat { \mathbb { E } } _ { p } ^ { \mathrm { b i a s } } [ \mathrm { l o g } ( 1 - f ( X ) ) ] - \hat { \mathbb { E } } _ { u } [ \mathrm { l o g } ( 1 - f ( X ) ) ] \right) _ { + } + \mathcal { R } ( f ) \right] , } \end{array}
+$$
+
+where $( \cdot ) _ { + } : = \operatorname* { m a x } \{ 0 , \cdot \}$ . After obtaining $\hat { f }$ , we construct an estimator of the density ratio $r$ by $\begin{array} { r } { \hat { r } = \frac { 1 } { \pi } \hat { f } _ { 1 } } \end{array}$ or $\begin{array} { r } { \hat { r } = \frac { 1 } { \pi } \hat { f } _ { 2 } } \end{array}$ .
+
+# 4.2 ESTIMATION OF $r$ BY DIRECT DENSITY RATIO ESTIMATION
+
+For another approach, we consider estimating the density ratio $\begin{array} { r } { r ( \pmb { x } ) = \frac { p ( \pmb { x } | y = + 1 , o = + 1 ) } { p ( \pmb { x } ) } } \end{array}$ directly. We can estimate the probability density functions of the numerator and the denominator. However, as known as Vapnik’s principle, we should avoid solving more difficult intermediate problems than the target problem. Sugiyama et al. (2012) summarized methods estimating the density ratio directly. Among existing methods, we employ Least-squares importance fitting (LSIF), which uses the squared loss for density-ratio function fitting. The reason for this choice is that there is an algorithm called unconstrained Least-Squares Importance Fitting (uLSIF) with a computational advantage. We can obtain the closed-form solution just by solving the linear equations. Thus, uLSIF is numerically stable when it is regularized properly. Moreover, the leave-one-out cross-validation score for uLSIF can also be computed analytically, which significantly improves the computational efficiency in model selection.
+
+Here, we introduce the formulation of LSIF. Let $s$ be the class of non-negative measurable functions $s : \mathcal { X } \to \mathbb { R } ^ { + }$ . We consider minimizing the following squared error between $s$ and $r$ :
+
+$$
+R _ { \mathrm { D R } } ( s ) = \mathbb { E } _ { \mathrm { u } } [ ( s ( X ) - r ( X ) ) ^ { 2 } ] = \mathbb { E } _ { \mathrm { u } } [ ( r ( X ) ) ^ { 2 } ] - 2 \mathbb { E } _ { \mathrm { p } } ^ { \mathrm { b i a s } } [ s ( X ) ] + \mathbb { E } _ { \mathrm { u } } [ ( s ( X ) ) ^ { 2 } ] .
+$$
+
+The first term of the last equation does not affect the result of minimization and we can ignore the term, i.e., the density ratio is estimated through the following minimization problem:
+
+$$
+s ^ { * } = \underset { s \in \cal S } { \arg \operatorname* { m i n } } R _ { \mathrm { D R } } ( s ) = \underset { s \in \cal S } { \arg \operatorname* { m i n } } \left[ \frac { 1 } { 2 } \mathbb { E } _ { \mathrm { u } } [ ( s ( X ) ) ^ { 2 } ] - \mathbb { E } _ { \mathrm { p } } ^ { \tt b i a s } [ s ( X ) ] \right] .
+$$
+
+Empirical Estimation: As mentioned above, to minimize the empirical version of (7), we use uLSIF (Kanamori et al., 2009). Given a hypothesis class $\mathcal { H }$ , we obtain $\hat { r }$ by
+
+$$
+\hat { r } = \underset { s \in \mathcal { H } } { \arg \operatorname* { m i n } } \left[ \frac { 1 } { 2 } \hat { \mathbb { E } } _ { \mathrm { u } } [ ( s ( X ) ) ^ { 2 } ] - \hat { \mathbb { E } } _ { \mathrm { p } } ^ { \mathrm { b i a s } } [ s ( X ) ] + \mathcal { R } ( s ) \right] ,
+$$
+
+where $\mathcal { R }$ is a regularization term.
+
+# 4.3 ESTIMATION OF $\theta _ { \pi }$
+
+We consider replacing the threshold defined by (2) with samples. By using the test inputs or held-out training data, $\{ \pmb { x } _ { i } ^ { \mathrm { t e } } \} _ { i = 1 } ^ { n ^ { \mathrm { t e } } } \sim p ( \pmb { x } )$ be calcu, we find $\hat { \theta } _ { \pi }$ ed only fromthat satisfies
+
+Table 1: Dataset statistics (Pos. frac.: Positive fraction, Dim: Dimension).   
+
+<table><tr><td>Dataset</td><td>#of samples</td><td>Pos.frac.</td><td>Dim.</td></tr><tr><td>mushrooms</td><td>8,124</td><td>0.517</td><td>112</td></tr><tr><td>shuttle</td><td>58.000</td><td>0.786</td><td>9</td></tr><tr><td>pageblocks</td><td>5,473</td><td>0.898</td><td>10</td></tr><tr><td>usps</td><td>9,298</td><td>0.524</td><td>256</td></tr><tr><td>connect-4</td><td>67,557</td><td>0.658</td><td>126</td></tr><tr><td>spambase</td><td>4.601</td><td>0.394</td><td>57</td></tr><tr><td>MNIST</td><td>70.000</td><td>0.511</td><td>784</td></tr><tr><td>CIFAR-10</td><td>60,000</td><td>0.400</td><td>3,072</td></tr></table>
+
+the following equation,
+
+$$
+\lceil \pi n ^ { \mathrm { t e } } \rceil = \sum _ { i = 1 } ^ { n ^ { \mathrm { t e } } } { \bf 1 } [ \hat { r } ( { \bf x } _ { i } ^ { \mathrm { t e } } ) > \hat { \theta } _ { \pi } ] .
+$$
+
+Here, we used the knowledge of $\pi$ , the class-prior. This choice of $\hat { \theta } _ { \pi }$ amounts to classifying top- $\pi$ test data as positive after ranking the inputs by $\hat { r }$ .
+
+# 5 EXPERIMENTS
+
+In this section, we report experimental results which were conducted using synthetic data and realworld datasets1). We used seven classification datasets, mushrooms, shuttle, pageblocks, usps, connect-4, spambase, and MNIST, from UCI repository2), $\mathsf { C I F A R } - \mathsf { 1 0 } ^ { 3 } )$ and a document dataset obtained from SwissProt (Boeckmann et al., $2 0 0 3 ) ^ { 4 ) }$ . MNIST and $\mathtt { C I F A R - 1 0 }$ originally have 10 and 10 classes, respectively, and we constructed the positive and negative datasets from them as follows: MNIST was preprocessed in such a way that 0, 2, 4, 6, 8 constitute the positive class, while 1, 3, 5, 7, 9 constitute the negative class; for CIFAR-10, the positive dataset is formed by ‘airplane’, ‘automobile’, ‘ship’ and ‘truck’, and the negative dataset is formed by ‘bird’, ‘cat’, ‘deer’, ‘dog’, ‘frog’ and ‘horse’. Except for the document dataset, we show the details of datasets in Table 1 and made positive data with a selection bias based on estimators of $p ( y = + 1 | x )$ as we show in each experiments. For six datasets of the UCI repository and the CIFAR-10, we made positive datasets with a selection bias artificially, but, for the document dataset, we have an unlabeled dataset and a positive dataset, which is gathered for classifying the labels in the unlabeled dataset.
+
+We call unbiased PU learning proposed by du Plessis et al. (2015) “PU”, unbiased PU learning with a threshold estimated by (9) “PUSB”, uLSIF with a threshold estimated by (9) “DRSB”, nonnegative PU learning “nnPU” and nonnegative PU learning with a threshold estimated by (9) “nnPUSB”.
+
+For the hypothesis class $\mathcal { H }$ in the density ratio estimation (8), we use the linear-in-parameter model:
+
+$$
+\begin{array} { r } { \mathcal { H } : = \left. \ : s ( \pmb { x } ) = \beta ^ { \top } \varphi ( \pmb { x } ) \right. \beta \in \mathbb { R } ^ { m + 1 } \} , } \end{array}
+$$
+
+where $\pmb { \varphi } ( \pmb { x } ) = [ 1 , \varphi _ { 1 } ( \pmb { x } ) , . . . , \varphi _ { m } ( \pmb { x } ) ] ^ { \top }$ is a vector of basis functions. For basis functions, we used the Gaussian kernel located at sample points $\varphi _ { \ell } ( \pmb { x } ) = \exp \left( - \| \pmb { x } - \pmb { c } _ { \ell } \| ^ { 2 } / ( 2 \sigma ^ { 2 } ) \right)$ , where $\{ \pmb { c } _ { 1 } , . . . , \pmb { c } _ { m } \} =$ $\left\{ { \pmb x } _ { 1 } , . . . , { \pmb x } _ { n } , { \pmb x } _ { 1 } ^ { \prime } , . . . , { \pmb x } _ { n ^ { \prime } } ^ { \prime } \right\}$ and $m = n + n ^ { \prime }$ . For the hypothesis class $\mathcal { H }$ in the risk minimization of PU learning (5), we use the following model with the sigmoid function:
+
+$$
+\mathcal { H } : = \left. \left. f ( \pmb { x } ) = \frac { 1 } { 1 + \exp ( - \beta ^ { \top } \varphi ( \pmb { x } ) ) } \right| \beta \in \mathbb { R } ^ { m + 1 } \right. .
+$$
+
+In this case, the loss is the same as the logistic loss and unbiased PU learning becomes convex. For the hypothesis class $\mathcal { H }$ in the risk minimization of nonnegative PU learning (6), we use deep neural networks. The specifications of deep neural networks are given in the following sections for each dataset. We mainly used the same structure proposed in Kiryo et al. (2017) in order to compare the performances. For the regularization term $\mathcal { R }$ , we used the $\ell _ { 2 }$ norm of the parameters scaled by a positive scalar $\lambda$ . For the linear models, hyperparameters were selected via cross-validation.
+
+![](images/1a7f8f021d0a9575101d41519d64f2420d93de04a02a65669e2a49c7e1d4d2fb.jpg)  
+Figure 1: Two Gaussians: The horizontal axis is the value of $_ { \textbf { \em x } }$ and the vertical axis is the probability density. The vertical lines represent the decision boundaries of the classifiers. The distribution of positive data, negative data, unlabeled data and $p ( \pmb { x } | o = + 1 , y = + 1 )$ are plotted.
+
+# 5.1 TEST WITH SYNTHETIC DATA
+
+This experiment shows the classifier given by PUSB. We used samples from a mixture distribution of the following two class-conditional distributions:
+
+$$
+p ( { \pmb x } | y = + 1 ) = \mathcal { N } ( 1 , 2 ^ { 2 } ) \mathrm { a n d } p ( { \pmb x } | y = - 1 ) = \mathcal { N } ( - 1 , 2 ^ { 2 } ) ,
+$$
+
+where ${ \mathcal { N } } ( \mu , \sigma ^ { 2 } )$ denotes the univariate normal distribution with mean $\mu$ and variance $\sigma ^ { 2 }$ . A positive dataset with a selection bias was sampled from
+
+$$
+p ( \pmb { x } | o = + 1 , y = + 1 ) \propto ( p ( y = + 1 | \pmb { x } ) ) ^ { 1 0 } .
+$$
+
+We generated 1, 000 positive samples and $1 0 , 0 0 0$ unlabeled samples. We made two datasets with different class-priors $\pi = 0 . 3$ and $\pi = 0 . 7$ . Figure 1 shows classifiers constructed by PU and PUSB along with the Bayes optimal classifier $\mathtt { s i g n } ( \bar { p } ( y = + 1 | x ) - 1 / 2 )$ . The classifier of PUSB is closer to the Bayes optimal classifier than that of PU.
+
+# 5.2 TEST WITH BENCHMARK DATA
+
+Here, we investigate the experimental performance in detail.
+
+Linear-in-parameter model: We used the mushrooms, shuttle, pageblocks, usps, connect-4 and spambase datasets. First, we estimated $p ( y = + 1 | x )$ using the logistic regression with the same linear model. Then, we obtained the labeled positive data by labeling some instances of the positive data following
+
+$$
+p ( o = + 1 | x , y = + 1 ) \propto ( p ( y = + 1 | x ) ) ^ { 2 0 } .
+$$
+
+Then, we trained a classifier by minimizing the empirical risk of PU learning (4) and the density ratio estimation (7).
+
+For each binary labeled dataset, we made 12 different pairs of positive and unlabeled data with 4 different class-priors, $\{ 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 \}$ , and 3 different numbers of unlabeled data, $\{ 8 0 0 , 1 6 0 0 , 3 2 0 0 \}$ . The number of positive data was fixed at 400. We used 1000 test data sampled from the same distribution as the unlabeled data. We ran the experiments 100 times and calculated the mean and standard deviation for the test dataset with PU, PUSB, and DRSB. The results are shown in Table 2. The classifiers obtained by our algorithm always show preferable performance to existing methods.
+
+Neural network model: We used the MNIST and CIFAR-10 datasets. For MNIST, a 3-layer multilayer perceptron (MLP) with ReLU activation (Nair & Hinton, 2010) was used. For CIFAR-10, an all convolutional net (Springenberg et al., 2015) was used. Details of the network structure are shown in Appendix D.
+
+Table 2: The error rate of classification in test data $( \% )$ are shown for the different class-priors and the different number of samples. For all experiments, the linear-in-parameter model was used. Best and equivalent methods (under $5 \%$ t-test) are bold.   
+
+<table><tr><td colspan="2"></td><td colspan="3">PU</td><td colspan="3">PUSB</td><td colspan="3">DRSB</td></tr><tr><td>Dataset</td><td>π</td><td>800</td><td>1600</td><td>3200</td><td>800</td><td>1600</td><td>3200</td><td>800</td><td>1600</td><td>3,200</td></tr><tr><td>mushrooms</td><td>0.2</td><td>5.1(.076)</td><td>5.0(.055)</td><td>5.5 (.083)</td><td>4.1 (.007)</td><td>3.8 (.007)</td><td>4.0 (.007)</td><td>10.7 (.010)</td><td>9.9 (.011)</td><td>9.9 (.012)</td></tr><tr><td></td><td>0.4</td><td>7.0 (.012)</td><td>6.6(.010)</td><td>6.8(.025)</td><td>6.9 (.012)</td><td>6.7 (.010)</td><td>6.8 (.010)</td><td>15.8 (.014)</td><td>15.0 (.016)</td><td>15.2 (.012)</td></tr><tr><td></td><td>0.6</td><td>10.8 (.020)</td><td>11.3 (.015)</td><td>11.3 (.016)</td><td>8.2(.013)</td><td>8.5 (.010)</td><td>8.3 (.011)</td><td>18.5 (.014)</td><td>18.8(.012)</td><td>18.5 (.013)</td></tr><tr><td></td><td>0.8</td><td>21.1 (.022)</td><td>21.6(.021)</td><td>21.7 (.016)</td><td>8.1(.015)</td><td>8.0 (.012)</td><td>7.6(.014)</td><td>14.2 (.013)</td><td>14.3 (.014)</td><td>14.4(.011)</td></tr><tr><td>shuttle</td><td>0.2</td><td>23.8 (.006)</td><td>23.8 (.007)</td><td>23.8(.007)</td><td>5.2 (.008)</td><td>5.0 (.007)</td><td>5.0 (.007)</td><td>5.1(.008)</td><td>5.1(.008)</td><td>5.1 (.007)</td></tr><tr><td></td><td>0.4</td><td>15.5 (.023)</td><td>15.5 (.019)</td><td>15.0 (.016)</td><td>7.8 (.011)</td><td>7.7 (.011)</td><td>7.6(.012)</td><td>6.7 (.025)</td><td>7.6 (.029)</td><td>7.2 (.027)</td></tr><tr><td></td><td>0.6</td><td>26.1 (.018)</td><td>26.4 (.017)</td><td>26.4(.016)</td><td>11.1 (.015)</td><td>10.8 (.012)</td><td>11.0 (.015)</td><td>7.8 (.028)</td><td>7.4 (.028)</td><td>7.1(.028)</td></tr><tr><td></td><td>0.8</td><td>14.6(.006)</td><td>14.6 (.005)</td><td>14.8 (.007)</td><td>12.3 (.014)</td><td>12.4 (.014)</td><td>12.3 (.012)</td><td>7.9 (.021)</td><td>7.6(.023)</td><td>7.4 (.022)</td></tr><tr><td>pageblocks</td><td>0.2</td><td>39.5 (.009)</td><td>39.6 (.010)</td><td>39.6(.008)</td><td>41.6 (.021)</td><td>42.5 (.022)</td><td>43.7 (.019)</td><td>23.1 (.016)</td><td>23.1 (.015)</td><td>22.2 (.012)</td></tr><tr><td></td><td>0.4</td><td>56.5 (.011)</td><td>56.7 (.011)</td><td>56.6(.008)</td><td>33.7 (.020)</td><td>33.7 (.019)</td><td>33.5 (.024)</td><td>23.6 (.012)</td><td>23.7(.014)</td><td>23.6 (.014)</td></tr><tr><td></td><td>0.6</td><td>22.4 (.023)</td><td>23.4(.033)</td><td>28.1(.028)</td><td>20.9 (.016)</td><td>21.1 (.016)</td><td>21.5 (.020)</td><td>18.7 (.011)</td><td>18.6(.012)</td><td>18.1 (.017)</td></tr><tr><td></td><td>0.8</td><td>19.8 (.003)</td><td>20.0 (.001)</td><td>20.0 (.001)</td><td>13.9 (.022)</td><td>13.6 (.023)</td><td>16.7 (.043)</td><td>15.4 (.011)</td><td>14.8 (.014)</td><td>14.3 (.019)</td></tr><tr><td>usps</td><td>0.2</td><td>9.0 (.011)</td><td>8.5 (.009)</td><td>8.2 (.010)</td><td>8.0 (.009)</td><td>7.7 (.007)</td><td>7.4 (.009)</td><td>18.2 (.016)</td><td>19.6 (.013)</td><td>19.7 (.014)</td></tr><tr><td></td><td>0.4</td><td>10.5 (.012)</td><td>10.3 (.013)</td><td>10.0 (.010)</td><td>10.5 (.013)</td><td>10.2 (.012)</td><td>10.0 (.010)</td><td>30.5 (.029)</td><td>30.2(.022)</td><td>29.9 (.023)</td></tr><tr><td></td><td>0.6</td><td>12.5 (.015)</td><td>12.3 (.015)</td><td>12.1 (.013)</td><td>11.2 (.016)</td><td>10.9 (.015)</td><td>10.6 (.013)</td><td>32.9 (.026)</td><td>33.2(.031)</td><td>33.1(.030)</td></tr><tr><td></td><td>0.8</td><td>19.8 (.020)</td><td>19.8 (.016)</td><td>19.5 (.017)</td><td>10.3 (.016)</td><td>9.8 (.014)</td><td>9.6 (.014)</td><td>25.9 (.028)</td><td>25.3(.027)</td><td>25.8 (.030)</td></tr><tr><td>connect-4</td><td>0.2</td><td>22.3 (0.017)</td><td>21.9 (.014)</td><td>21.6 (.013)</td><td>21.8 (.013)</td><td>21.5 (.012)</td><td>21.2 (.011)</td><td>26.7(.012)</td><td>26.6(.012)</td><td>26.4(.011)</td></tr><tr><td></td><td>0.4</td><td>31.2 (.015)</td><td>31.0 (.015)</td><td>30.7(.016)</td><td>31.2 (.015)</td><td>31.0 (.015)</td><td>30.7 (.016)</td><td>39.4 (.016)</td><td>39.6(.018)</td><td>39.2 (.016)</td></tr><tr><td></td><td>0.6</td><td>32.0 (.017)</td><td>32.1 (.013)</td><td>31.9 (.015</td><td>31.7 (.016)</td><td>31.7 (.013)</td><td>31.4 (.016)</td><td>40.9 (.015)</td><td>41.1 (.017)</td><td>41.0 (.017)</td></tr><tr><td>spambase</td><td>0.8</td><td>33.6(.018)</td><td>33.5 (.016)</td><td>33.4(.017)</td><td>24.2 (.013)</td><td>23.9 (.013)</td><td>23.8(.013)</td><td>29.8 (.011)</td><td>29.6(.013)</td><td>29.5 (.012)</td></tr><tr><td></td><td>0.2</td><td>20.1 (0.002)</td><td>20.1(.002)</td><td>20.1(.003)</td><td>13.6 (.013)</td><td>13.8 (.014)</td><td>14.0 (.014)</td><td>18.1 (.026)</td><td>18.3 (.025)</td><td>17.9 (.023)</td></tr><tr><td></td><td>0.4</td><td>36.2 (.024)</td><td>35.9 (.025)</td><td>30.7 (.024)</td><td>19.0 (.020)</td><td>18.7 (.021)</td><td>18.9 (.018)</td><td>27.4(.042)</td><td>27.8 (.043)</td><td>27.7 (.039)</td></tr><tr><td></td><td>0.6</td><td>40.0 (.001)</td><td>39.9 (.001)</td><td>31.9 (.001</td><td>20.8 (.019)</td><td>20.7(.018)</td><td>20.0 (.017)</td><td>31.2 (.037)</td><td>30.3 (.035)</td><td>31.2 (.035)</td></tr><tr><td></td><td>0.8</td><td>20.0 (.000)</td><td>20.0 (.000)</td><td>20.0 (.000)</td><td>18.0 (.015)</td><td>17.7 (.013)</td><td>17.3 (.013)</td><td>24.5 (.058)</td><td>23.9 (.017)</td><td>24.2(.017)</td></tr></table>
+
+![](images/4abdbc81ce3765c2307a5af29663956f7e1478f0894b3011136fc1a66d75bdf9.jpg)  
+Figure 2: Experimental results of training deep neural networks. Left: MNIST; Center: $\mathrm { C I F A R } { - } 1 0$ ; Right: RealData: All measures are calculated for test data sampled from the marginal distribution $p ( { \pmb x } )$ . The horizontal axis is the epoch of training the network, the vertical axes of the top figures are the error rates and the vertical axes of the bottom figures are the precision and the recall.
+
+First, we estimated $p ( y = + 1 | x )$ using the logistic regression with the same network structure using the positive and negative datasets in the unlabeled dataset. Next, from the positive dataset, we resampled positive dataset with an observation, which follows
+
+$$
+p ( o = + 1 | x , y = + 1 ) \propto ( p ( y = + 1 | x ) ) ^ { 1 0 } .
+$$
+
+Then, we trained a classifier by minimizing (6) with the model defined above. We used 10, 000 test data sampled from the same distribution as the unlabeled data. We ran the experiments 100 times and calculated the mean of the error rate, the standard deviation of the error rate, the mean of recall and the mean of precision for each epoch in training with nnPU and nnPUSB. The results are shown on the left side and center of Figure 2. In the upper row, we show the mean and standard deviation of the error rate. In the lower row, we show the mean of recall and the mean of precision. As shown in Figure 2, the mean of the error rate of our algorithm is lower and the variance is also lower than the existing method. As we discussed in Section 4.3, $\mathrm { F P = F N }$ is also empirically observed.
+
+![](images/41a01b15646e66b58ee3428f9690f4d8ab6fff5b7bfe94a371ef3efa5092b484.jpg)  
+Figure 3: Results of the second experiment of Section 5.4 using the RealData dataset with the estimated class prior 0.1095 (the true class prior is 0.0709). The horizontal axes are the epochs of training the network, the vertical axis of the right figure is the error rate and the vertical axis of the left figure is the value of the precision and the recall.
+
+# 5.3 TEST WITH REAL-WORLD DATA
+
+In the previous sections, we artificially made positive data with a selection bias. Here we demonstrate the effectiveness of our algorithm in real-world data with a selection bias. We used a document dataset based on the SwissProt database released by Elkan & Noto (2008). We call this dataset RealData. The dataset originally contained 2,453 labeled positive examples (P) and 4,906 unlabeled examples (U). The unlabeled examples were labeled later by Das et al. (2007). As a result, the dataset is likely to have a natural observation bias in the $\mathrm { \bf P }$ data while it allows an access to the ground-truth labels for all data. Out of the U data, 348 examples are positive and the rest are negative. The class prior is $\pi = 3 4 8 / 4 , 9 0 6 = 0 . 0 7 0 9$ . We used Bag-of-Words to represent the documents as 78, 894-dimensional vectors.
+
+Details of the network structure are shown in Appendix D. We trained a classifier using positive and unlabeled data. Then, after finding a threshold estimated by (9), we evaluated the same evaluation measures as the previous experiments by classifying U, i.e., we regarded the unlabeled data as test data. As shown on the right of Figure 2, the result of our algorithm outperforms the existing method.
+
+# 5.4 TEST FOR UNKNOWN CLASS PRIOR
+
+In order to evaluate how our algorithm works for the case that the class prior is unknown, we empirically tested our algorithm with an estimator of the class prior. We used the RealData dataset in Section 5.3, whose class prior is 0.0709. For the class prior estimator, we used the KM2 method by Ramaswamy et al. (2016), which is considered to be the state-of-the-art method in the case-control scenario under SCAR. The estimated class prior was 0.1095 and the result is shown in Figure 3. We can see from the figure that our method works well with an estimated class prior in so far as RealData dataset is concerned. We also show results for the classifiers trained under misspecified class priors in Appendix E, which also shows that our method works stably for misspecified class priors.
+
+# 6 CONCLUSION
+
+In this paper, we proposed a novel framework for PU learning with a selection bias in positive data. We put the assumption of the invariance of order and showed the density ratio of labeled positive data and unlabeled data has the same order as the class-conditional distribution for inputs. Based on this result, we proposed a method based on partial identification in which we first estimate the density ratio and then use it as a classifier by setting a threshold. We conducted experiments to confirm the effectiveness of our approach. As we showed in the experiments, our method outperforms previous PU methods on real-world data.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by JSPS KAKENHI 16H00881 and the AIP challenge program, Japan.
+
+# REFERENCES
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+
+# A PROOF OF THEOREM 1
+
+Proof. We assumed that no negative data can be labeled. As a result, we have $p ( o = + 1 | x , y =$ $- 1 ) = 0$ for arbitrary $x \in \mathcal { X }$ . Therefore,
+
+$$
+\begin{array} { c } { { p ( o = + 1 | x ) = p ( y = + 1 | x ) p ( o = + 1 | x , y = + 1 ) + p ( y = - 1 | x ) p ( o = + 1 | x , y = - 1 ) } } \\ { { { } } } \\ { { = p ( y = + 1 | x ) p ( o = + 1 | x , y = + 1 ) . } } \end{array}
+$$
+
+By Bayes’ theorem, we can expand the density ratio in (1) as follows:
+
+$$
+{ \begin{array} { r l } & { \cdot ( \mathbf { x } ) = { \frac { p ( x | o = + 1 , y = + 1 ) } { p ( \mathbf { x } ) } } } \\ & { \quad = { \frac { p ( y = + 1 | x , o = + 1 ) p ( \mathbf { x } | o = + 1 ) } { p ( y = + 1 | o = + 1 ) } } { \frac { 1 } { p ( \mathbf { x } ) } } } \\ & { \quad = p ( y = + 1 | x , o = + 1 ) { \frac { 1 } { p ( y = + 1 | o = + 1 ) } } { \frac { p ( \mathbf { x } | o = + 1 ) } { p ( \mathbf { x } ) } } } \\ & { \quad = { \frac { p ( y = + 1 | x ) p ( o = + 1 | x , y = + 1 ) } { p ( o = + 1 | x ) } } { \frac { 1 } { p ( y = + 1 | o = + 1 ) } } { \frac { p ( o = + 1 ) p ( x | o = + 1 ) } { p ( o = + 1 ) } } . } \end{array} }
+$$
+
+Because $\begin{array} { r } { \frac { p ( y = + 1 | \pmb { x } ) p ( o = + 1 | \pmb { x } , y = + 1 ) } { p ( o = + 1 | \pmb { x } ) } = 1 } \end{array}$ from (11), this is equivalent to
+
+$$
+\begin{array} { r } { r ( { \pmb x } ) = C p ( o = + 1 | { \pmb x } ) , } \end{array}
+$$
+
+where $\begin{array} { r } { C = \frac { 1 } { p ( y = + 1 , o = + 1 ) } } \end{array}$ . Hence, if Assumption 1 holds, for any $\pmb { x } _ { i } , \pmb { x } _ { j } \in \mathcal { X }$
+
+$$
+p ( y = + 1 | { \boldsymbol { \mathbf { x } } } _ { i } ) \leq p ( y = + 1 | { \boldsymbol { \mathbf { x } } } _ { j } ) \Leftrightarrow r ( { \boldsymbol { \mathbf { x } } } _ { i } ) \leq r ( { \boldsymbol { \mathbf { x } } } _ { j } ) .
+$$
+
+# B PROOF OF THEOREM 2
+
+Before proving Theorem 2, we consider a threshold defined as follows:
+
+$$
+\pi = \int \mathbf { 1 } [ p ( y = + 1 | \pmb { x } ) \geq \gamma ] p ( \pmb { x } ) d \pmb { x } .
+$$
+
+Then, we state the following lemma on the relationship between $\gamma$ and $\theta _ { \pi }$
+
+Lemma 1. The equation $\mathbf { \boldsymbol { x } } \in \mathcal { X }$ $\begin{array} { r } { \ell , ( r ( { \pmb x } ) - \theta _ { \pi } ) ( p ( { \pmb y } = + 1 | { \pmb x } ) - \gamma ) \ge 0 } \end{array}$ holds almost everywhere with respect to $p ( { \pmb x } )$ .
+
+Proof. By (2) and (12), the following equation also hold,
+
+$$
+\int \mathbf { 1 } [ r ( \pmb { x } ) > \theta _ { \pi } ] p ( \pmb { x } ) d \pmb { x } = \int \mathbf { 1 } [ p ( y = + 1 | \pmb { x } ) > \gamma ] p ( \pmb { x } ) d \pmb { x } .
+$$
+
+Hence, we can derive
+
+$$
+\int ( \mathbf { 1 } [ r ( \pmb { x } ) > \theta _ { \pi } ] - \mathbf { 1 } [ p ( y = + 1 | \pmb { x } ) > \gamma ] ) p ( \pmb { x } ) d \pmb { x } = 0 .
+$$
+
+This is equivalent to
+
+$$
+\int ( \mathbf { 1 } [ ( r ( x ) - \theta _ { \pi } ) ( p ( y = + 1 | x ) - \gamma ) < 0 ] p ( x ) d x = 0 .
+$$
+
+From this equation, $( r ( { \pmb x } ) - \theta _ { \pi } ) ( p ( { \pmb y } = + 1 | { \pmb x } ) - \gamma ) \geq 0$ holds almost surely.
+
+Using Lemma 1 we prove Theorem 2.
+
+Proof. $\begin{array} { r } { \pi = \int \mathbf { 1 } [ p ( y = + 1 | \pmb { x } ) \geq \gamma ] p ( \pmb { x } ) d \pmb { x } } \end{array}$ is equivalent to
+
+$$
+\int p ( y = + 1 | x ) p ( x ) d x = \int \mathbf { 1 } [ p ( y = + 1 | x ) \geq \gamma ] p ( x ) d x ,
+$$
+
+where the left hand side is equal to
+
+$$
+\int _ { \{ x | p ( y = + 1 | x ) < \gamma \} } p ( y = + 1 | x ) p ( x ) d x + \int _ { \{ x | p ( y = + 1 | x ) \geq \gamma \} } p ( y = + 1 | x ) p ( x ) d x ,
+$$
+
+and the right hand side is equal to
+
+$$
+\int _ { \{ x | p ( y = + 1 | x ) \geq \gamma \} } p ( y = + 1 | x ) p ( x ) d x + \int _ { \{ x | p ( y = + 1 | x ) \geq \gamma \} } p ( y = - 1 | x ) p ( x ) d x .
+$$
+
+Hence, (12) is equivalent to the following equation,
+
+$$
+\int _ { \{ x | p ( y = + 1 | x ) < \gamma \} } p ( y = + 1 | x ) p ( x ) d x = \int _ { \{ x | p ( y = + 1 | x ) \geq \gamma \} } p ( y = - 1 | x ) p ( x ) d x .
+$$
+
+From the definitions of TP, FP, TN, and FN, the left hand side of the above equation is equal to $F P$ and the right hand side of the above equation is equal to $F N$ . Hence, $F P = F N$ . We showed $F P = F N$ for a threshold $\gamma$ , but we can also insist that the same result holds when we use $r ( { \pmb x } )$ as a score function and $\theta _ { \pi }$ as a threshold. According to Lemma 1, the sign of $p ( y = + 1 + x ) - \gamma$ and $r ( { \pmb x } ) - \theta _ { \pi }$ are the same almost everywhere with respect to $p ( { \pmb x } )$ . Therefore,
+
+$$
+\begin{array} { l } { \displaystyle \int _ { \{ x | p ( y = + 1 | x ) < \gamma \} } p ( y = + 1 | x ) p ( x ) d x = \int _ { \{ x | r ( x ) < \theta _ { \pi } \} } p ( y = + 1 | x ) p ( x ) d x } \\ { \displaystyle \int _ { \{ x | p ( y = + 1 | x ) \geq \gamma \} } p ( y = - 1 | x ) p ( x ) d x = \int _ { \{ x | r ( x ) \geq \theta _ { \pi } \} } p ( y = - 1 | x ) p ( x ) d x . } \end{array}
+$$
+
+In these equations, the right hand sides mean $F P$ and $F N$ of a score function $r ( { \pmb x } )$ with a threshold $\theta _ { \pi }$ , respectively. Because precision $\begin{array} { r } { \mathbf { \Phi } = \left( \frac { T P } { T P + F P } \right) } \end{array}$ and $\begin{array} { r } { \mathrm { r e c a l l } = \left( \frac { T P } { T P + F N } \right) } \end{array}$ , $F P = F N$ means that precision $=$ recall. □
+
+# C PROOF OF THEOREM 3
+
+Proof. We first consider minimizing $R _ { \mathrm { P U } } ^ { \tt b i a s }$ in the space of all functions from $\mathcal { X }$ to $[ \epsilon , 1 - \epsilon ]$ instead of minimizing it in . Later, we will see that the minimizer matches $f ^ { * }$ as stated in Theorem 3 which belongs to $\mathcal { F }$ . The minimization of
+
+$$
+R _ { \mathtt { P U } } ^ { \mathtt { b i a s } } ( f ) = \int \big ( - \pi p ( x | y = + 1 , o = + 1 ) ( \log f ( x ) - \log ( 1 - f ( x ) ) ) - \log ( 1 - f ( x ) ) p ( x ) \big ) d x
+$$
+
+over all functions $f$ taking values in $[ \epsilon , 1 - \epsilon ]$ is reduced to the following point-wise minimization problem
+
+$$
+\operatorname * { a r g m i n } _ { z \in [ \epsilon , 1 - \epsilon ] } C ( z , x ) : = - \pi p ( x | y = + 1 , o = + 1 ) ( \log z - \log ( 1 - z ) ) - \log ( 1 - z ) p ( x ) .
+$$
+
+Denoting the solution by $z ^ { * }$ , the Karush-Kuhn-Tucker (KKT) condition of this minimization problem is
+
+$$
+\begin{array} { l l } & { \pi p ( \pmb { x } | y = + 1 , o = + 1 ) \left( \cfrac { 1 } { z ^ { * } } + \cfrac { 1 } { 1 - z ^ { * } } \right) - \cfrac { p ( \pmb { x } ) } { 1 - z ^ { * } } - \lambda + \mu = 0 , } \\ & { \lambda ( z ^ { * } - 1 - \epsilon ) = 0 , \mu z ^ { * } = 0 , } \\ & { \lambda , \mu \geq 0 , } \\ & { z ^ { * } \in [ \epsilon , 1 - \epsilon ] , } \end{array}
+$$
+
+where $\lambda$ and $\mu$ are the Lagrange multipliers. The first equation is equivalent to
+
+$$
+( \mu - \lambda ) ( z ^ { * } ) ^ { 2 } + ( p ( { \pmb x } ) + \lambda - \mu ) z ^ { * } - \pi p ( { \pmb x } | y = + 1 , o = + 1 ) = 0 .
+$$
+
+We investigate the solution by dividing the KKT condition into the following four cases.
+
+1. If we assume $\lambda = \mu = 0$ , then the KKT condition is reduced to
+
+$$
+\begin{array} { l } { { z ^ { * } } = \frac { \pi p ( \pmb { x } | y = + 1 , o = + 1 ) } { p ( \pmb { x } ) } , } \\ { { z ^ { * } } \in [ \epsilon , 1 - \epsilon ] . } \end{array}
+$$
+
+In this case, $z ^ { * } \in [ \epsilon , 1 - \epsilon ]$ is equivalent to $\pmb { x } \in D _ { 1 } \cap D _ { 2 }$ .
+
+2. If we assume $\lambda > 0$ and $\mu = 0$ , the KKT condition is reduced to
+
+$$
+\begin{array} { l } { z ^ { \ast } = 1 - \epsilon , } \\ { \lambda = \frac { ( 1 - \epsilon ) p ( { \pmb x } ) - \pi p ( { \pmb x } | y = + 1 , o = + 1 ) } { ( 1 - \epsilon ) ^ { 2 } + ( 1 - \epsilon ) } . } \end{array}
+$$
+
+In this case, $\lambda > 0$ is equivalent to $\pmb { x } \notin D _ { 2 }$ .
+
+3. If we assume $\lambda = 0$ and $\mu > 0$ , the KKT condition is reduced to
+
+$$
+\begin{array} { l } { { z ^ { * } = \epsilon , } } \\ { { \mu = \frac { p ( { \pmb x } ) \epsilon - \pi p ( { \pmb x } | y = + 1 , o = + 1 ) } { \epsilon ^ { 2 } - \epsilon } . } } \end{array}
+$$
+
+In this case, $\lambda > 0$ is equivalent to $\pmb { x } \notin D _ { 1 }$
+
+4. If $\lambda , \mu > 0$ , there is no feasible solution.
+
+In summary, the solution for the optimization problem arg $\begin{array} { r } { \operatorname* { m i n } _ { z \in [ \epsilon , 1 - \epsilon ] } C ( z , \pmb { x } ) } \end{array}$ is
+
+$$
+z ^ { * } = \left\{ \begin{array} { l l } { \epsilon } & { ( x \notin D _ { 1 } ) , } \\ { \frac { \pi p ( x \mid y = + 1 , o = + 1 ) } { p ( x ) } } & { ( x \in D _ { 1 } \cap D _ { 2 } ) , } \\ { 1 - \epsilon } & { ( x \notin D _ { 2 } ) . } \end{array} \right.
+$$
+
+Finally, we define $f ^ { * } ( x ) : = \arg \operatorname* { m i n } _ { z \in [ \epsilon , 1 - \epsilon ] } C ( z , x )$ . It can be confirmed that $f ^ { * } \in { \mathcal { F } }$ because $p ( { \pmb x } | y = 1 , o = + 1 )$ and $p ( { \pmb x } )$ are measurable and $f ^ { * }$ takes values in $[ \epsilon , 1 - \epsilon ]$ . Therefore, the solution of the original optimization problem
+
+$$
+\arg \operatorname* { m i n } _ { f \in \mathcal { F } } R _ { \mathtt { P U } } ^ { \mathtt { b i a s } } ( f )
+$$
+
+is equal to $f ^ { * }$ almost everywhere.
+
+# D NETWORK STRUCTURE USED IN SECTIONS 5.2 AND SECTIONS 5.3
+
+In Section 5.2, we used the MNIST and CIFAR-10 datasets. The model for the MNIST dataset was a 3-layer multilayer perceptron (MLP) with ReLU (Nair & Hinton, 2010) (more specifically, 784-100-1). The model for the CIFAR-10 dataset was an all convolutional net (Springenberg et al., 2015): $( 3 2 \times 3 2 \times 3 )$ $| - [ C ( 3 \times 3 , 9 6 ) ] \times 2 - C ( 3 \times 3 , 9 6 , 2 ) - [ C ( 3 \times 3 , 1 9 2 ) ] \times 2 - C ( 3 \times 3 , 1 9 2 , 2 ) - C ( 3 \times 3 , 1 9 2 , 2 ) + \ldots$ 3, 192) $\begin{array} { r } { - C ( 1 \times 1 , 1 9 2 ) - C ( 1 \times 1 , 1 0 ) - 1 0 0 0 - 1 0 0 0 - 1 , } \end{array}$ where the input is a $3 2 \times 3 2$ RGB image, $C ( 3 \times$ 3, 96) means 96 channels of $3 \times 3$ convolutions followed by ReLU, $[ \cdot ] \times 2$ means there are two such layers, $C ( 3 \times 3 , 9 6 , 2 )$ means a similar layer but with stride 2, etc.; it is one of the best architectures for CIFAR-10. Batch normalization (Ioffe & Szegedy, 2015) was applied before hidden layers.
+
+In Section 5.3, the model for this dataset was a 5-layer multilayer perceptron (MLP) with ReLU (more specifically, 78894-300-300-300-300-1).
+
+# E EXPERIMENTAL RESULTS OF THE FIRST EXPERIMENT OF SECTION 5.4
+
+In the first experiment, we trained a classifier under misspecified class priors, $0 . 0 2 0 9 ( = 0 . 0 7 0 9 -$ 0.0500), $0 . 1 \hat { 2 0 } 9 ( = 0 . 0 7 0 9 + 0 . 0 5 0 0 )$ , $0 . 1 7 0 9 ( = 0 . 0 7 0 9 + 0 . 1 0 0 0 )$ , and $0 . 2 7 0 9 ( = 0 . 0 7 0 9 + 0 . 2 0 0 0 )$ . The results are shown in Figures 4–6. In Figure 4, the test error in each misspecified class priors are shown. In Figure 5, the precision and recall in each misspecified class prior is shown. In both the existing method and our method, misspecified class priors had a bad influence. However, our method was less influenced by the misspecification and showed better performance than the existing method. Besides, the difference between the precision and recall of our algorithm is narrower than that of the existing method as expected. In order to discuss how misspecified class prior affects the precision and recall, we show the difference the recall from the precision and show the values in Figure 6. This result shows how the difference broadens as the class prior is misspecified worse.
+
+![](images/dcb222732638b8bd515c2e0ecf62447e3fb7a116b57ae300905bfc0c69a893e7.jpg)  
+Figure 4: Experimental results of training deep neural networks using the RealData dataset with misspecified class priors (the true class prior is 0.0709). Upper Left: $\pi = 0 . 0 2 0 9$ ; Upper Right: $\pi = 0 . 1 2 0 9$ ; Lower Left: $\pi = 0 . 1 7 0 9$ ; Lower Right: $\pi = 0 . 2 7 0 9$ : All measures are calculated for test data sampled from the marginal distribution $p ( { \pmb x } )$ . The horizontal line is epoch of training the network, the vertical line of the upper is error rate.
+
+![](images/9312ac621c68bd9c132fc3c28448a793ae71c0a365ac058ea48bbb20a01077f9.jpg)  
+Figure 5: Experimental results of training deep neural networks using the RealData dataset with misspecified class priors (the true class prior is 0.0709). Upper Left: $\pi = 0 . 0 2 0 9$ ; Upper Right: $\pi = 0 . 1 2 0 9$ ; Lower Left: $\pi = 0 . 1 7 0 9$ ; Lower Right: $\pi = 0 . 2 7 0 9$ : All measures are calculated for test data sampled from the marginal distribution $p ( { \pmb x } )$ . The horizontal line is epoch of training the network, the vertical line of the upper is value of the precision and the recall.
+
+![](images/ec30e7532cc9ec532c3d0bf3ec388044735229dec00b28a30e3dbc888e970ec6.jpg)  
+Figure 6: Experimental results of training deep neural networks using the RealData dataset, whose true class prior is 0.0709, with misspecified class priors, 0.0209, 0.1209, 0.1709, and 0.2709.: All measures are calculated for test data sampled from the marginal distribution $p ( { \pmb x } )$ . The horizontal line is epoch of training the network, the vertical line of the right graph is value of the precision − recall.

parse/train/ssohLcmn4-r/ssohLcmn4-r.md

@@ -0,0 +1,389 @@
+# Choose a Transformer: Fourier or Galerkin
+
+Shuhao Cao Department of Mathematics and Statistics Washington University in St. Louis s.cao@wustl.edu
+
+# Abstract
+
+In this paper, we apply the self-attention from the state-of-the-art Transformer in Attention Is All You Need [88] for the first time to a data-driven operator learning problem related to partial differential equations. An effort is put together to explain the heuristics of, and to improve the efficacy of the attention mechanism. By employing the operator approximation theory in Hilbert spaces, it is demonstrated for the first time that the softmax normalization in the scaled dot-product attention is sufficient but not necessary. Without softmax, the approximation capacity of a linearized Transformer variant can be proved to be comparable to a Petrov-Galerkin projection layer-wise, and the estimate is independent with respect to the sequence length. A new layer normalization scheme mimicking the Petrov-Galerkin projection is proposed to allow a scaling to propagate through attention layers, which helps the model achieve remarkable accuracy in operator learning tasks with unnormalized data. Finally, we present three operator learning experiments, including the viscid Burgers’ equation, an interface Darcy flow, and an inverse interface coefficient identification problem. The newly proposed simple attention-based operator learner, Galerkin Transformer, shows significant improvements in both training cost and evaluation accuracy over its softmax-normalized counterparts.
+
+# 1 Introduction
+
+Partial differential equations (PDEs) arise from almost every multiphysics and biological systems, from the interaction of atoms to the merge of galaxies, from the formation of cells to the change of climate. Scientists and engineers have been working on approximating the governing PDEs of these physical systems for centuries. The emergence of the computer-aided simulation facilitates a cost-friendly way to study these challenging problems. Traditional methods, such as finite element/difference [20, 22], spectral methods [12], etc., leverage a discrete structure to reduce an infinite dimensional operator map to a finite dimensional approximation problem. Meanwhile, in the field practice of many scientific disciplines, substantial data for PDE-governed phenomena available on discrete grids enable modern black-box models like Physics-Informed Neural Network (PINN) [71, 62, 49] to exploit measurements on collocation points to approximate PDE solutions.
+
+Nonetheless, for traditional methods or data-driven function learners such as PINN, given a PDE, the focus is to approximate a single instance, for example, solving for an approximated solution for one coefficient with a fixed boundary condition. A slight change to this coefficient invokes a potentially expensive re-training of any data-driven function learners. In contrast, an operator learner aims to learn a map between infinite-dimensional function spaces, which is much more difficult yet rewarding. A well-trained operator learner can evaluate many instances without re-training or collocation points, thus saving valuable resources, and poses itself as a more efficient approach in the long run. Data-driven resolution-invariant operator learning is a booming new research direction [60, 5, 56, 64, 90, 57, 61, 91, 37, 74]. The pioneering model, DeepONet [60], attributes architecturally to a universal approximation theorem for operators [18]. Fourier Neural Operator (FNO) [57] notably shows an awing state-of-the-art performance outclassing classic models such as the one in [100] by orders of magnitudes in certain benchmarks.
+
+Under a supervised setting, an operator learner is trained with the operator’s input functions and their responses to the inputs as targets. Since both functions are sampled at discrete grid points, this is a special case of a seq2seq problem [81]. The current state-of-the-art seq2seq model is the Transformer first introduced in [88]. As the heart and soul of the Transformer, the scaled dot-product attention mechanism is capable of unearthing the hidden structure of an operator by capturing long-range interactions. Inspired by many insightful pioneering work in Transformers [50, 19, 75, 84, 96, 97, 95, 59, 76, 66], we have modified the attention mechanism minimally yet in a mathematically profound manner to better serve the purpose of operator learning.
+
+Among our new Hilbert space-inspired adaptations of the scaled dot-product attention, the first and foremost change is: no softmax, or the approximation thereof. In the vanilla attention [88], the softmax succeeding the matrix multiplication convexifies the weights for combining different positions’ latent representations, which is regarded as an indispensable ingredient in the positive kernel interpretation of the attention mechanism [84]. However, softmax acts globally in the sequence length dimension for each row of the attention matrix, and further adds to the quadratic complexity of the attention in the classic Transformer. Theory-wise, instead of viewing “row $\approx$ word” in the Natural Language Processing (NLP) tradition, the columns of the query/keys/values are seen as sampling of functions in Hilbert spaces on discretized grids. Thus, taking the softmax away allows us to verify a discrete Ladyzhenskaya–Babuška–Brezzi (LBB) condition, which further amounts to the proof that the newly proposed Galerkin-type attention can explicitly represent a Petrov-Galerkin projection, and this approximation capacity is independent of the sequence length (Theorem 4.3).
+
+Numerically, the softmax-free models save valuable computational resources, outperforming the ones with the softmax in terms of training FLOP and memory consumption (Section 5). Yet in an ablation study, the training becomes unstable for softmax-free models (Table 8). To remedy this, a new Galerkin projection-type layer normalization scheme is proposed to act as a cheap diagonal alternative to the normalizations explicitly derived in the proof of the Petrov-Galerkin interpretation (equation (40)). Since a learnable scaling can now be propagated through the encoder layers, the attentionbased operator learner with this new layer normalization scheme exhibits better comprehension of certain physical properties associated with the PDEs such as the energy decay. Combining with other approximation theory-inspired tricks including a diagonally dominant rescaled initialization for the projection matrices and a layer-wise enrichment of the positional encodings, the evaluation accuracies in various operator learning tasks are boosted by a significant amount.
+
+Main contributions. The main contributions of this work are summarized as follows.
+
+• Attention without softmax. We propose a new simple self-attention operator and its linear variant without the softmax normalization. Two new interpretations are offered, together with the approximation capacity of the linear variant proved comparable to a Petrov-Galerkin projection.
+
+• Operator learner for PDEs. We combine the newly proposed attention operators with the current best state-of-the-art operator learner Fourier Neural Operator (FNO) [57] to significantly improve its evaluation accuracy in PDE solution operator learning benchmark problems. Moreover, the new model is capable of recovering coefficients based on noisy measurements that traditional methods or FNO cannot accomplish.
+
+• Experimental results. We present three benchmark problems to show that operator learners using the newly proposed attentions are superior in computational/memory efficiency, as well as in accuracy versus those with the conventional softmax normalization. The PyTorch codes to reproduce our results are available as an open-source software.
+
+# 2 Related Works
+
+Operator learners related to PDEs. In [4, 5], certain kernel forms of the solution operator of parametric PDEs are approximated using graph neural networks. The other concurrent notable approach is DeepONet [60, 61]. [56] further improves the kernel approach by exploiting the multilevel grid structure. [57] proposes a discretization-invariant operator learner to achieve a state-of-the-art performance in certain benchmark problems. [90, 91] proposed a DeepONet roughly equivalent to an additive attention, similar to the one in the Neural Turing Machine (NMT) in [7]. Model/dimension reduction combined with neural nets is another popular approach to learn the solution operator for parametric PDEs [10, 64, 55, 24]. Deep convolutional neural networks (DCNN) are widely applied to learn the solution maps with a fixed discretization size [1, 9, 40, 36, 35, 100, 86]. Recently, DCNN has been successfully applied in various inverse problems [35, 47] such as Electrical Impedance Tomography (EIT). To our best knowledge, there is no work on data-driven approaches to an inverse interface coefficient identification for a class of coefficients with random interface geometries.
+
+Attention mechanism and variants. Aside from the ground-breaking scaled dot-product attention in [88], earlier [7] proposed an additive content-based attention, however, with a vanishing gradient problem due to multiple nonlinearity composition. [25] shows the first effort in removing the softmax normalization in [7] after the projection, however, it still uses a Sigmoid nonlinearity before the additive interpolation propagation stage, and performs worse than its softmax counterpart. The current prevailing approach to linearize the attention leverages the assumption of the existence of a feature map to approximate the softmax kernel [50, 19, 70]. Another type of linearization exploits the low-rank nature of the matrix product using various methods such as sampling or projection [73, 11, 79, 92], or fast multipole decomposition [65]. The conjecture in [75] inspires us to remove the softmax overall. [76] first proposed the inverse sequence length scaling normalization for a linear complexity attention without the softmax, however, the scaling normalization has not been extensively studied in examples and performs worse.
+
+Various studies on Transformers. The kernel interpretation in [84] inspires us to reformulate the attention using the Galerkin projection. [95, Theorem 2] gives a theoretical foundation of removing the softmax normalization to formulate the Fourier-type attention. The Nyström approximation [97] essentially acknowledges the similarity between the attention matrix and an integral kernel. [96, 66, 59] inspires us to try different layer normalization and the rescaled diagonally dominant initialization schemes. The practices of enriching the latent representations with the positional encoding recurrently in our work trace back to [2, 26], and more recently, contribute to the success of AlphaFold 2 [48], as it is rewarding to exploit the universal approximation if the target has a dependence ansatz in the coordinate frame and/or transformation group but hard to be explicitly quantified. Other studies on adapting the attention mechanisms to conserve important physical properties are in [82, 31, 44].
+
+# 3 Operator learning related to PDEs
+
+Closely following the setup in [56, 57], we consider a data-driven model to approximate a denselydefined operator $T : \mathcal { H } _ { 1 } \to \mathcal { H } _ { 2 }$ between two Hilbert spaces with an underlying bounded spacial domain $\Omega \subset \mathbb { R } ^ { m }$ . The operator $T$ to be learned is usually related to certain physical problems, of which the formulation is to seek the solution to a PDE of the following two types.
+
+Parametric PDE: given coefficient $a \in { \mathcal { A } }$ , and source $f \in \mathcal { V }$ , find $u \in \mathcal X$ such that $L _ { a } ( u ) = f$ .
+
+(i) To approximate the nonlinear mapping from the varying parameter $a$ to the solution with a fixed right-hand side, $T : \mathcal { A }  \mathcal { X } , \ a \mapsto u$ .   
+(ii) The inverse coefficient identification problem to recover the coefficient from a noisy measurement $\tilde { u }$ of the steady-state solution $u$ , in this case, $T : \mathcal { X }  \mathcal { A } , \ \tilde { u } \mapsto a$ .
+
+Nonlinear initial value problem: given $u _ { 0 } \in \mathcal { H } _ { 0 }$ , find $u \in C ( [ 0 , T ] ; \mathcal { H } )$ such that $\partial _ { t } u + N ( u ) = 0$ .
+
+(iii) Direct inference from the initial condition to the solution. $T : \mathcal { H } _ { 0 }  \mathcal { H } , u _ { 0 } ( \cdot ) \mapsto u ( t _ { 1 } , \cdot )$ with $t _ { 1 } \gg \Delta t$ with $t _ { 1 }$ much greater than the step-size in traditional explicit integrator schemes.
+
+Using (i) as an example, based on the given $N$ observations $\{ a ^ { ( j ) } , u ^ { ( j ) } \} _ { j = 1 } ^ { N }$ and their approximations $\{ a _ { h } ^ { ( j ) } , u _ { h } ^ { ( j ) } \}$ u(j)h } defined at a discrete grid of size h  1, the goal of our operator learning problem is to build an approximation $T _ { \theta }$ to $T$ , such that $T _ { \theta } ( a _ { h } )$ is a good approximation to $u = L _ { a } ^ { - 1 } f = : T ( a ) \approx$ $u _ { h }$ independent of the mesh size $h$ , where $a _ { h }$ and $u _ { h }$ are in finite dimensional spaces $\mathbb { A } _ { h } , \mathbb { X } _ { h }$ on this grid. We further assume that $a ^ { ( j ) } \sim \nu$ for a measure $\nu$ compactly supported on $\mathcal { A }$ , and the sampled data form a reasonably sized subset of $\mathcal { A }$ representative of field applications. The loss ${ \mathcal { I } } ( \theta )$ is
+
+$$
+\mathcal { I } ( \theta ) : = \mathbb { E } _ { a \sim \nu } \left[ \| \left( T _ { \theta } ( a ) - u \| _ { \mathcal { H } } ^ { 2 } + \mathfrak { G } ( a , u ; \theta ) \right] \right.
+$$
+
+and in practice is approximated using the sampled observations on a discrete grid
+
+$$
+\mathcal { I } ( \boldsymbol { \theta } ) \approx \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \Big \{ \big \| \big ( T _ { \theta } \big ( a _ { h } ^ { ( j ) } \big ) - u _ { h } ^ { ( j ) } \big \| _ { \mathcal { H } } ^ { 2 } + \mathfrak { G } \big ( a _ { h } ^ { ( j ) } , u _ { h } ^ { ( j ) } ; \boldsymbol { \theta } \big ) \Big \} .
+$$
+
+In example (i), $\lVert \cdot \rVert _ { \mathcal { H } }$ is the standard $L ^ { 2 }$ -norm, and $\mathfrak { G } ( a , u ; \theta )$ serves as a regularizer with strength $\gamma$ and is problem-dependent. In Darcy flow where $L _ { a } : = - \nabla \cdot ( a \nabla ( \cdot ) )$ , it is $\gamma \Vert a \nabla ( T _ { \theta } ( a ) - u ) \Vert _ { L ^ { 2 } ( \Omega ) } ^ { 2 }$ , since $u \in H ^ { 1 + \alpha } ( \Omega )$ ( $\vert \alpha > 0$ depends on the regularity of $a$ ) and $a \nabla u \in H ( \mathrm { d i v } ; \Omega )$ a priori. For the evaluation metric, we drop the $\mathfrak { G } ( a , u ; \theta )$ term, and monitor the minimization of (2) using $\| \cdot \| _ { \mathcal { H } }$ .
+
+# 4 Attention-based operator learner
+
+Feature extractor. We assume the functions in both inputs and targets are sampled on a uniform grid. In an operator learning problem on $\Omega \subset \mathbb { R } ^ { 1 }$ , a simple feedforward neural network (FFN) is used as the feature extractor that is shared by every position (grid point).
+
+Interpolation-based CNN. If $\Omega \subset \mathbb { R } ^ { 2 }$ , inspired by the multilevel graph kernel network in [56], we use two 3-level interpolation-based CNNs (CiNN) as the feature extractor, but also as the downsampling and upsampling layer, respectively, in which we refer to restrictions/prolongations between the coarse/fine grids both as interpolations. For the full details of the network structure please refer to Appendix B.
+
+Recurrent enrichment of positional encoding. The Cartesian coordinates of the grid, on which the attention operator’s input latent representation reside, are concatenated as additional feature dimension(s) to the input, as well as to each latent representation in every attention head.
+
+Problem-dependent decoder. The decoder is a problem-dependent admissible network that maps the learned representations from the encoder back to the target dimension. For smooth and regular solutions in $\mathring { H } ^ { 1 + \alpha } ( \Omega )$ , we opt for a 2-layer spectral convolution that is the core component in [57]. A simple pointwise feedforward neural network (FFN) is used for nonsmooth targets in $L ^ { \infty } ( \Omega )$ .
+
+# 4.1 Simple self-attention encoder
+
+![](images/bfa066e663aac42c93a9459c93ff58f6eea9ff8c0559bdb2ed24a09688549726.jpg)  
+Figure 1: Comparison of the vanilla attention [88] with the Galerkin-type simple self-attention in a single head; (a) in the standard softmax attention, the softmax is applied row-wise after the matrix product matmul; (b) a mesh-weighted normalization allows an integration-based interpretation.
+
+The encoder contains a stack of identical simple attention-based encoder layers. For simplicity, we consider a single attention head that maps $\mathbf { y } ^ { \mathbf { ^ { \prime } } } \in \mathbb { R } ^ { n \times d }$ to another element in $\mathbb { R } ^ { n \times d }$ , and define the
+
+trainable projection matrices, and the latent representations $Q / K / V$ as follows.
+
+$$
+\begin{array} { r } { Q : = \mathbf { y } W ^ { Q } , \quad K : = \mathbf { y } W ^ { K } , \quad V : = \mathbf { y } W ^ { V } . } \end{array}
+$$
+
+We propose the following simple attention that (i) uses a mesh (inverse sequence length)-weighted normalization without softmax, (ii) allows a scaling to propagate through the encoder layers.
+
+$$
+\mathrm { A t t n } _ { \mathrm { s p } } : \mathbb { R } ^ { n \times d } \to \mathbb { R } ^ { n \times d } , \quad \widetilde { \mathbf { y } } \gets \mathbf { y } + \mathrm { A t t n } _ { \dagger } ( \mathbf { y } ) , \quad \mathbf { y } \mapsto \widetilde { \mathbf { y } } + g ( \widetilde { \mathbf { y } } ) ,
+$$
+
+ewhere the head-wise normalizations are applied pre-dot-product: for $\dag \in \{ \mathfrak { f } , \mathfrak { g } \}$
+
+$$
+\begin{array} { r l } { \mathrm { ( F o u r i e r - t y p e ~ a t t e n t i o n ) } \quad } & { \mathbf { z } = \mathrm { A t t n } _ { \mathfrak { f } } ( \mathbf { y } ) : = ( \widetilde { Q } \widetilde { K } ^ { \top } ) V / n , } \\ { \mathrm { ( G a l e r k i n – t y p e ~ a t t e n t i o n ) } \quad } & { \mathbf { z } = \mathrm { A t t n } _ { \mathfrak { g } } ( \mathbf { y } ) : = Q ( \widetilde { K } ^ { \top } \widetilde { V } ) / n , } \end{array}
+$$
+
+and $\widetilde { \diamond }$ denotes a trainable non-batch-based normalization. As in the classic Transformer [88], and einspired by the Galerkin projection interpretation, we choose $\widetilde { \diamond }$ as the layer normalization $\operatorname { L n } ( \diamond )$ , and $g ( \cdot )$ e as the standard 2-layer FFN identically applied on every position (grid point). In simple attentions, the weight for each row of $V$ , or column of $Q$ in the linear variant, is not all positive anymore. This can be viewed as a cheap alternative to the cosine similarity-based attention.
+
+Remark 4.1. If we apply the regular layer normalization rule that eliminates any scaling:
+
+$\mathbf { y } \mapsto \mathrm { L n } \bigl ( \mathbf { y } + \mathrm { A t t n } _ { \dagger } ( \mathbf { y } ) + g \bigl ( \mathrm { L n } ( \mathbf { y } + \mathrm { A t t n } _ { \dagger } ( \mathbf { y } ) ) \bigr ) \bigr )$ , where $\mathrm { A t t n } _ { \dagger } ( \mathbf { y } ) : = { Q } ( K ^ { \top } V ) / n ,$ then this reduces to the efficient attention first proposed in [76].
+
+# 4.1.1 Structure-preserving feature map as a function of positional encodings
+
+Consider an operator learning problem with an underlying domain $\Omega \subset \mathbb { R } ^ { 1 }$ . $\{ x _ { i } \} _ { i = 1 } ^ { n }$ denotes the set of grid points in the discretized $\Omega$ such that the weight $1 / n \ : = \ : h$ is the mesh size. Let $\zeta _ { q } ( \cdot ) , \phi _ { k } ( \cdot ) , \psi _ { v } ( \cdot ) : \Omega \to \mathbb { R } ^ { 1 \times d }$ denote the feature maps of $Q , K , V$ , i.e., the $i$ -th row of $Q , K , V$ written as $q _ { i } = \zeta _ { q } ( x _ { i } )$ , $\pmb { k } _ { i } = \phi _ { k } ( \boldsymbol { x } _ { i } )$ , ${ \pmb v } _ { i } = \psi _ { v } ( x _ { i } )$ . They are, in the NLP convention, viewed as the feature (embedding) vector at the $i$ -th position, respectively. The inter-position topological structure such as continuity/differentiability in the same feature dimension is learned thus not explicit. The following ansatz for $Q / K / V$ in the same attention head is fundamental to our new interpretations.
+
+Assumption 4.2. The columns of $Q / K / V$ , respectively, contain the vector representations of the learned basis functions spanning certain subspaces of the latent representation Hilbert spaces.
+
+Using $V \in \mathbb { R } ^ { n \times d }$ with a full column rank as an example, its columns contain potentially a set of bases $\{ v _ { j } ( \cdot ) \} _ { j = 1 } ^ { d }$ evaluated at the grid points (degrees of freedom, or DoFs). Similarly, the learned bases whose DoFs form the columns of $Q , K$ are denoted as $\{ q _ { j } ( \cdot ) \} _ { j = 1 } ^ { d }$ , $\{ k _ { j } ( \cdot ) \} _ { j = 1 } ^ { d }$ , as well as $\{ z _ { j } ( \cdot ) \} _ { j = 1 } ^ { d }$ for the outputs in (5) and (6). To be specific, the $j$ -th column of $V$ , denoted by $v ^ { j }$ , then stands for a vector representation of the $j$ -th basis function evaluated at each grid point, i.e., its $l$ -th position stands for $\mathbf { \dot { \rho } } ( \pmb { v } ^ { j } ) _ { l } = v _ { j } ( x _ { l } )$ . Consequently, the row ${ \pmb v } _ { i } = ( v _ { 1 } ( x _ { i } ) , \dots , v _ { d } ( x _ { i } ) )$ can be alternatively viewed as the evaluation of a vector latent basis function at $x _ { i }$ .
+
+# 4.1.2 Fourier-type attention of a quadratic complexity
+
+![](images/7494af8b8700a81cff4ea0e5c3e597fb4d50eb492e6f8664d1a90769ba749eb6.jpg)  
+Figure 2: A dissection of Fourier-type attention’s output. Both matmuls have complexity $O ( n ^ { 2 } d )$ .
+
+In the Fourier-type attention (5), $Q , K$ are assumed to be normalized for simplicity, the $j$ -th column $( 1 \leq j \leq d )$ in the $i$ -th row $1 \leq i \leq n )$ of $\mathbf { z }$ is computed by (see Figure 2):
+
+$$
+\begin{array} { r l } & { ( { \boldsymbol { z } } _ { i } ) _ { j } = h ( { \boldsymbol { Q } } { \boldsymbol { K } } ^ { \top } ) _ { i \bullet } { \boldsymbol { v } } ^ { j } = h \big ( \mathbf { { q } } _ { i } \cdot { \boldsymbol { k } } _ { 1 } , \dots , { \boldsymbol { q } } _ { i } \cdot { \boldsymbol { k } } _ { l } , \dots , { \boldsymbol { q } } _ { i } \cdot { \boldsymbol { k } } _ { n } \big ) ^ { \top } \cdot { \boldsymbol { v } } ^ { j } } \\ & { \qquad = h \displaystyle \sum _ { l = 1 } ^ { n } ( { \boldsymbol { q } } _ { i } \cdot { \boldsymbol { k } } _ { l } ) ( { \boldsymbol { v } } ^ { j } ) _ { l } \approx \int _ { { \Omega } } \big ( \zeta _ { q } ( { \boldsymbol { x } } _ { i } ) \cdot \phi _ { k } ( \boldsymbol { \xi } ) \big ) v _ { j } ( \boldsymbol { \xi } ) \mathrm { d } \boldsymbol { \xi } , } \end{array}
+$$
+
+where the $h$ -weight facilitates the numerical quadrature interpretation of the inner product. Concatenating columns $1 \leq j \leq d$ yields the $i$ -row $z _ { i }$ of the output $\mathbf { z }$ : $\begin{array} { r } { z _ { i } \approx \int _ { \Omega } \big ( \zeta _ { q } ( x _ { i } ) \cdot \phi _ { k } ( \xi ) \big ) \psi _ { v } ( \xi ) \mathrm { d } \xi } \end{array}$ . Therefore, without the softmax nonlinearity, the local dot-product attention output at $i$ -th row computes approximately an integral transform with a non-symmetric learnable kernel function $\kappa ( x , \bar { \xi } ) : = \bar { \zeta _ { q } } ( x ) \phi _ { k } ( \xi )$ evaluated at $x _ { i }$ , whose approximation property has been studied in [95, Theorem 2], yet without the logits technicality due to the removal of the softmax normalization.
+
+After the skip-connection, if we further exploit the learnable nature of the method and assume $W ^ { V } = \mathrm { d i a g } \{ \delta _ { 1 } , \cdot \cdot \cdot , \delta _ { d } \}$ such that $\delta _ { j } \neq 0$ for $1 \leq j \leq d$ , under Assumption 4.2:
+
+$$
+\delta _ { j } ^ { - 1 } v _ { j } ( x ) \approx z _ { j } ( x ) - \int _ { \Omega } \kappa ( x , \xi ) v _ { j } ( \xi ) \mathrm { d } \xi , \quad \mathrm { f o r ~ } j = 1 , \cdots , d , \mathrm { a n d } x \in \{ x _ { i } \} _ { i = 1 } ^ { n } .
+$$
+
+This is the forward propagation of the Fredholm equation of the second-kind for each $v _ { j } ( \cdot )$ . When using an explicit orthogonal expansion such as Fourier to solve for $\{ v _ { j } ( \cdot ) \} _ { j = 1 } ^ { d }$ , or to seek for a better set of $\{ v _ { j } ( \cdot ) \}$ in our case, it is long known being equivalent to the Nyström’s method with numerical integrations [8] (similar to the $h = 1 / n$ weighted sum). Therefore, the successes of the random Fourier features in [19, 70] and the Nyströmformer’s approximation [97] are not surprising.
+
+Finally, we name this type of simple attention “Fourier” is due to the striking resemblance between the scaled dot-product attention and a Fourier-type kernel [30] integral transform, since eventually the target resides in a Hilbert space with an underlying spacial domain $\Omega$ , while the latent representation space parallels a “frequency” domain on $\Omega ^ { * }$ . This also bridges the structural similarity of the scaled dot-product attention with the Fourier Neural Operator [57] where the Fast Fourier Transform (FFT) can be viewed as a non-learnable change of basis.
+
+# 4.1.3 Galerkin-type attention of a linear complexity
+
+![](images/9916700d4f8c8a4a5264df5a640c0a25de25096e8fbcd6fd19999e8dc1d8ec7c.jpg)  
+Figure 3: A dissection of Galerkin-type attention’s output. Both matmuls have complexity $O ( n d ^ { 2 } )$
+
+For the Galerkin-type simple attention in (6), $K , V$ are assumed to be normalized for simplicity, we first consider the $i$ -th entry in the $j$ -th column $z ^ { j }$ of $\mathbf { z }$ (see Figure 3):
+
+$$
+( \boldsymbol { z } ^ { j } ) _ { i } = h \mathbf { \boldsymbol { q } } _ { i } ^ { \intercal } \cdot ( K ^ { \intercal } V ) _ { \bullet j } ,
+$$
+
+which is the inner product of the $i$ -th row of $Q$ and the $j$ -th column of $K ^ { \top } V$ . Thus,
+
+$$
+z ^ { j } = h \left( \begin{array} { c c c c } { { | } } & { { | } } & { { | } } & { { | } } \\ { { q _ { 1 } } } & { { q _ { 2 } } } & { { \cdots } } & { { q _ { n } } } \\ { { | } } & { { | } } & { { | } } & { { | } } \end{array} \right) ^ { \top } ( K ^ { \top } V ) _ { \bullet j } = h \left( ( K ^ { \top } V ) _ { \bullet j } ^ { \top } \left( \begin{array} { c c c } { { } } & { { q ^ { 1 } } } & { { } } \\ { { } } & { { \vdots } } & { { \overline { { { \bf \sigma } } } } } \\ { { } } & { { } } & { { q ^ { d } } } \end{array} \right) \right) ^ { \top }
+$$
+
+This reads as: $( K ^ { \top } V ) _ { \bullet j }$ contains the coefficients for the linear combination of the vector representations $\{ \pmb q ^ { l } \} _ { l = 1 } ^ { d }$ of the bases stored in $Q$ ’s column space to form the output $\mathbf { z }$ . Meanwhile, the $j$ -th column $( K ^ { \top } V ) _ { \bullet j }$ of $K ^ { \top } V$ consists the inner product of $j$ -th column of $V$ with every column of $K$ .
+
+$$
+z ^ { j } = h \sum _ { l = 1 } ^ { d } q ^ { l } ( K ^ { \top } V ) _ { l j } , \quad \mathrm { w h e r e ~ } ( K ^ { \top } V ) _ { \bullet j } = \left( k ^ { 1 } \cdot v ^ { j } , k ^ { 2 } \cdot v ^ { j } , \cdot \cdot \cdot , k ^ { d } \cdot v ^ { j } \right) ^ { \top } .
+$$
+
+As a result, using Assumption 4.2, and for simplicity the latent Hilbert spaces $\mathcal { Q } , \kappa , \nu$ are assumed to be defined on the same spacial domain $\Omega$ , i.e., $k _ { l } ( \cdot )$ , $v _ { j } ( \cdot )$ evaluated at every $x _ { i }$ are simply their vector representations $k ^ { l }$ $1 \leq l \leq d )$ and $v ^ { j }$ , we have the functions represented by the columns of the output $\mathbf { z }$ can be then compactly written as: rewriting $\langle v _ { j } , k _ { l } \rangle : = ( \dot { K } ^ { \top } V ) _ { l j }$
+
+$$
+z _ { j } ( x ) : = \sum _ { l = 1 } ^ { d } \langle v _ { j } , k _ { l } \rangle q _ { l } ( x ) , { \mathrm { ~ f o r ~ } } j = 1 , \cdots , d , { \mathrm { ~ a n d ~ } } x \in \{ x _ { i } \} _ { i = 1 } ^ { n } ,
+$$
+
+where the bilinear form $\langle \cdot , \cdot \rangle : \mathcal { V } \times \mathcal { K } \to \mathbb { R }$ . (13) can be also written in a componentwise form:
+
+$$
+z _ { j } ( x _ { i } ) : = ( z ^ { j } ) _ { i } = h \sum _ { l = 1 } ^ { d } ( \pmb { k } ^ { l } \cdot \pmb { v } ^ { j } ) ( \pmb { q } ^ { l } ) _ { i } \approx \sum _ { l = 1 } ^ { d } \left( \int _ { \Omega } v _ { j } ( \xi ) k _ { l } ( \xi ) \mathrm { d } \xi \right) q _ { l } ( x _ { i } ) .
+$$
+
+Therefore, when different subspac $\{ \circ _ { j } ( \cdot ) \} _ { j = 1 } ^ { d } , \diamond \in \{ q , k , v \}$ consist approximations to threespaces as the column spaces of ets of bases for potentiallyand the test space as that $Q$ of $K$ , respectively, the forward propagation of the Galerkin-type attention is a recast of a learnable Petrov–Galerkin-type projection (cf. Appendix D.1) for every basis represented by the columns of $V$ . While the form of (14) suggests the orthonormality of the basis represented by $Q , K , V$ , as well as being of full column ranks, the learnable nature of the method suggests otherwise (see Appendix D). At last, we have the following strikingly simple yet powerful approximation result.
+
+Theorem 4.3 (Céa-type lemma, simplified version). Consider a Hilbert space $\mathcal { H }$ defined on a bounded domain $\Omega \subset \mathbb { R } ^ { m }$ discretized by $n$ grid points, and $f \in \mathcal H$ . $\mathbf { y } \in \mathbb { R } ^ { n \times d }$ is the current latent representation for $n > d > m$ and full column rank. $\mathbb { Q } _ { h } \subset \mathcal { Q } \subset \mathcal { H }$ and $\mathbb { V } _ { h } \subset \mathcal { V } \subset \mathcal { H }$ are the latent approximation subspaces spanned by basis functions with the columns of $Q$ and $V$ in (3) as degrees of freedom, respectively, and $0 < \dim \mathbb { Q } _ { h } = r \leq \dim \mathbb { V } _ { h } = d .$ . Let $\mathfrak { b } ( \cdot , \cdot ) : \mathcal { V } \times \mathcal { Q } \to \mathbb { R }$ be a continuous bilinear form, and if for any fixed $q \in \mathbb { Q } _ { h }$ the functional norm of $\mathfrak { b } ( \cdot , q )$ is bounded below by $c > 0$ , then there exists a learnable map $g _ { \boldsymbol { \theta } } ( \cdot )$ that is the composition of the Galerkin-type attention operator with an updated set of projection matrices $\{ W ^ { Q } , \mathbf { \dot { W } } ^ { K } , W ^ { V } \}$ , and a pointwise universal approximator, such that for $f _ { h } \in \mathbb { Q } _ { h }$ being the best approximation of $f i n \parallel \cdot \parallel _ { \mathcal { H } }$ it holds:
+
+$$
+\| f - g _ { \theta } ( \mathbf { y } ) \| _ { \mathcal { H } } \leq c ^ { - 1 } \operatorname* { m i n } _ { q \in \mathbb { Q } _ { h } } \operatorname* { m a x } _ { v \in \mathbb { V } _ { h } } \frac { | \mathfrak { b } ( v , f _ { h } - q ) | } { \| v \| _ { \mathcal { H } } } + \| f - f _ { h } \| _ { \mathcal { H } } .
+$$
+
+Remarks on and interpretations of the best approximation result. Theorem 4.3 states that the Galerkin-type attention has the architectural capacity to represent a quasi-optimal approximation in $\| \cdot \| _ { \mathcal { H } }$ in the current subspace $\mathbb { Q } _ { h }$ . For the mathematically rigorous complete set of notations and the full details of the proof we refer the readers to Appendix D.3. Even though Theorem 4.3 is presented for a single instance of $f \in \mathcal H$ for simplicity, the proof shows that the attention operator is fully capable of simultaneously approximating a collection of functions (Appendix D.3.4).
+
+Estimate (15) comes with great scalability with respect to the sequence length in that it all boils down to whether $c$ is independent of $n$ in the lower bound of $\| \mathsf { b } _ { \cdot } ( \cdot , q ) \| _ { \mathbb { V } _ { h } ^ { \prime } }$ . The existence of an $n$ -independent lower bound is commonly known as the discrete version of the Ladyzhenskaya–Babuška–Brezzi (LBB) condition [21, Chapter 6.12], also referred as the Banach-Necas-Babuška (BNB) condition in ˇ Galerkin methods on Banach spaces [29, Theorem 2.6].
+
+As the cornerstone of the approximation to many PDEs, the discrete LBB condition establishes the surjectivity of a map from $\mathbb { V } _ { h }$ to $\mathbb { Q } _ { h }$ . In a simplified context (15) above of approximating functions using this linear attention variant $Q$ : values, query, $V$ : keys), it roughly translates to: for an incoming “query” (function $f$ in a Hilbert space), to deliver its best approximator in “value” (trial function space), the “key” (test function space) has to be sufficiently rich such that there exists a key to unlock every possible value.
+
+Dynamic basis update. Another perspective is to interpret the Galerkin-type dot-product attention (14) as a change of basis: essentially, the new set of basis is the column space of $Q$ , and how to linearly combine the bases in $Q$ is based on the inner product (response) of the corresponding feature dimension’s basis in $V$ against every basis in $K$ . From this perspective $Q$ : values, $K$ : keys, $V$ : query), we have the following result of a layer-wise dynamical change of basis: through testing against the “keys”, a latent representation is sought such that “query” (input trial space) and “values” (output trial space) can achieve the minimum possible difference under a functional norm; for details and the proof please refer to Appendix D.3.4.
+
+Theorem 4.4 (layer-wise dynamic basis update, simple version). Under the same assumption as Theorem 4.3, it is further assumed that $\mathfrak { b } ( \cdot , q )$ is bounded below on $\mathbb { K } _ { h } \subset \mathcal { K } = \mathcal { V } \subset \mathcal { H }$ and $\mathfrak { a } ( \cdot , \cdot ) : \mathcal { V } \times \mathcal { K } \to \bar { \mathbb { R } }$ is continuous. Then, there exists a set of projection matrices to update the value space $\{ \widetilde { q } _ { l } ( \cdot ) \} _ { l = 1 } ^ { d } \subset \mathbb { Q } _ { h } = \mathrm { s p a n } \{ q _ { l } ( \cdot ) \} _ { l = 1 } ^ { d } ,$ , for $z _ { j } \in \mathbb { Q } _ { h }$ $\ ' j = 1 , \cdots , d )$ obtained through the basis update rule (14), it holds
+
+$$
+\left\| \mathfrak { a } ( v _ { j } , \cdot ) - \mathfrak { b } ( \cdot , z _ { j } ) \right\| _ { \mathbb { K } _ { h } ^ { \prime } } \leq \operatorname* { m i n } _ { q \in \mathbb { Q } _ { h } } \operatorname* { m a x } _ { k \in \mathbb { K } _ { h } } \frac { | \mathfrak { a } ( v _ { j } , k ) - \mathfrak { b } ( k , q ) | } { \| k \| _ { K } } .
+$$
+
+The role of feed-forward networks and positional encodings in the dynamic basis update. Due to the presence of the concatenated coordinates $\mathbf { x } : = \| _ { i = 1 } ^ { n } \bar { x _ { i } } \in \mathbb { R } ^ { n \times m }$ to the latent representation y, the pointwise subnetwork $g _ { s } ( \cdot ) : \mathbb { R } ^ { n \times m }  \mathbb { R } ^ { n \times d }$ of the nonlinear universal approximator (FFN) in each attention block is one among many magics of the attention mechanism. In every attention layer, the basis functions in $\mathbb { Q } _ { h } / \mathbb { K } _ { h } / \bar { \mathbb { V } } _ { h }$ are being constantly enriched by $\operatorname { s p a n } \{ w _ { j } \in \mathbb { X } _ { h } : w _ { j } ( x _ { i } ) =$ $( \bar { g _ { s } } ( \mathbf { x } ) ) _ { i j } , 1 \le j \le d \} \subset \mathcal { H }$ , thus being dynamically updated to try to capture how an operator of interest responses to the subset of inputs. Despite the fact that the FFNs, when being viewed as a class of functions, bear no linear structure within, the basis functions produced this way act as a building block to characterize a linear space for a learnable projection. This heuristic shows to be effective when the target is assumed to be a function of the (relative) positional encodings (coordinates, transformation groups, etc.), in that this is incorporated in many other attention-based learners with applications in physical sciences [82, 31, 44, 48].
+
+# 5 Experiments
+
+In this section we perform a numerical study the proposed Fourier Transformer (FT) with the Fourier-type encoder, and the Galerkin Transformer (GT) with the Galerkin-type encoder, in various PDE-related operator learning tasks. The models we compare our newly proposed models with are the operator learners with the simple attention replaced by the standard softmax normalized scaled dot-product attention (ST) [88], and a linear variant (LT) [76] in which two independent softmax normalizations are applied on $Q , K$ separately.2 The data are obtained courtesy of the PDE benchmark under the MIT license.3 For full details of the training/evaluation and model structures please refer to Appendix C.
+
+Instead of the standard Xavier uniform initialization [34], inspired by the interpretations of Theorem 4.3 in Appendix D.3.4, we modify the initialization for the projection matrices slightly as follows
+
+$$
+W _ { \mathrm { i n i t } } ^ { \diamond }  \eta U + \delta I , ~ \mathrm { f o r } \diamond \in \{ Q , K , V \} ,
+$$
+
+where $U = ( x _ { i j } )$ is a random matrix using the Xavier initialization with gain 1 such that $x _ { i j } \sim$ $\mathcal { U } ( [ - \sqrt { 3 / d } , \sqrt { 3 / d } ] )$ , and $\delta$ is a small positive number. In certain operator learning tasks, we found that this tiny modification boosts the evaluation performance of models by up to $5 0 \%$ (see Appendix C.2) and improves the training stability acting as a cheap remedy to the lack of a softmax normalization. We note that similar tricks have been discovered concurrently in [23].
+
+Unsurprisingly, when compared the memory usage and the speed of the networks (Table 1), the Fourier-type attention features a $40 \% - 5 0 \%$ reduction in memory versus the attention with a softmax normalization. The Galerkin attention-based models have a similar memory profile with the standard linear attention, it offers up to a $120 \%$ speed boost over the linear attention in certain tests.
+
+Table 1: The memory usage/FLOP/complexity comparison of the models. Batch size: 4; the CUDA mem (GB): the sum of the self_cuda_memory_usage; GFLOP: Giga FLOP for 1 backpropagation (BP); both are from the PyTorch autograd profiler for 1 BP averaging from 1000 BPs; the mem (GB) is recorded from nvidia-smi of the memory allocated for the active Python process during profiling; the speed (iteration per second) is measured during training; the exponential operation is assumed to have an explicit complexity of $c _ { e } > 1$ [14].   
+
+<table><tr><td rowspan="2"></td><td colspan="4">Example 1: n = 8192</td><td colspan="4">Encoders only: n = 8192,d = 128, =10</td><td rowspan="2">Computational complexity of the dot-product per layer</td></tr><tr><td>Mem</td><td>CUDA Mem</td><td>Speed</td><td>GFLOP</td><td>Mem</td><td>CUDA Mem</td><td>Speed</td><td>GFLOP</td></tr><tr><td>ST</td><td>18.39</td><td>31.06</td><td>5.02</td><td>1393</td><td>18.53</td><td>31.34</td><td>4.12</td><td>1876</td><td>O(n²ced)</td></tr><tr><td>FT</td><td>10.05</td><td>22.92</td><td>6.10</td><td>1138</td><td>10.80</td><td>22.32</td><td>5.46</td><td>1610</td><td>O(n²d）</td></tr><tr><td>LT</td><td>2.55</td><td>2.31</td><td>12.70</td><td>606</td><td>2.73</td><td>2.66</td><td>10.98</td><td>773</td><td>O(n(d² +ced))</td></tr><tr><td>GT</td><td>2.36</td><td>1.93</td><td>27.15</td><td>275</td><td>2.53</td><td>2.33</td><td>19.20</td><td>412</td><td>O(nd²）</td></tr></table>
+
+The baseline models for each example are the best operator learner to-date, the state-of-the-art Fourier Neural Operator (FNO) in [57] but without the original built-in batch normalization. All attention-based models match the parameter quota of the baseline, and are trained using the loss in (2) with the same 1cycle scheduler [78] for 100 epochs. For fairness, we have also included the results for the standard softmax normalized models (ST and LT) using the new layer normalization scheme in (5) and (6). We have retrained the baseline with the same 1cycle scheduler using the code provided in [57], and listed the original baseline results using a step scheduler of 500 epochs of training from [57] Example 5.1 and Example 5.2, respectively.
+
+# 5.1 Example 1: viscous Burgers’ equation
+
+In this example, we consider a benchmark problem of the viscous Burgers’ equation with a periodic boundary condition on $\Omega : = ( 0 , 1 )$ in [57]. The nonlinear operator to be learned is the discrete approximations to the solution operator $T : C _ { p } ^ { 0 } ( \Omega ) \cap L ^ { 2 } ( \Omega ) \stackrel { \cdot } { \to } C _ { p } ^ { 0 } ( \Omega ) \cap H ^ { 1 } ( \Omega )$ , $u _ { 0 } ( \cdot ) \mapsto u ( \cdot , 1 )$ . The initial condition $u _ { 0 } ( \cdot )$ ’s are sampled following a Gaussian Random Field (GRF).
+
+The result can be found in Table 2a. All attention-based operator learners achieve a resolutioninvariant performance similar with FNO1d in [57]. The new Galerkin projection-type layer normalization scheme significantly outperforms the regular layer normalization rule in this example, in which both inputs and targets are unnormalized. For full details please refer to Appendix C.2.
+
+# 5.2 Example 2: Darcy flow
+
+In this example, we consider another well-known benchmark $- \nabla \cdot ( a \nabla u ) = f$ for $u \in H _ { 0 } ^ { 1 } ( \Omega )$ from [10, 57, 56, 64], and the operator to be learned is the approximations to $T : L ^ { \infty } ( \Omega ) \to H _ { 0 } ^ { 1 } ( \Omega ) , a \mapsto u$ in which $a$ is the coefficient with a random interface geometry, and $u$ is the weak solution. Here $L ^ { \infty } ( \Omega )$ is a Banach space and cannot be compactly embedded in $L ^ { 2 } ( \Omega )$ (a Hilbert space), we choose to avoid this technicality as the finite dimensional approximation space can be embedded in $L ^ { 2 } ( \Omega )$ given that $\Omega$ is compact.
+
+The result can be found in Table 2b. As the input/output are normalized, in contrast to Example 5.1, the Galerkin projection-type layer normalization scheme does not significantly outperform the regular layer normalization rule in this example. The attention-based operator learners achieve on average $30 \%$ to $50 \%$ better evaluation results than the baseline FNO2d (only on the fine grid) using the same trainer. For full details please refer to Appendix C.3.
+
+Table 2: (a) Evaluation relative error $( \times 1 0 ^ { - 3 } )$ of Burgers’ equation 5.1. (b) Evaluation relative error $( \times 1 0 ^ { - 2 } )$ of Darcy interface problem 5.2.   
+
+<table><tr><td colspan="4">(a)</td></tr><tr><td></td><td>n=512</td><td>n=2048</td><td>n=8192</td></tr><tr><td>FNO1d [57]</td><td>15.8</td><td>14.6</td><td>13.9</td></tr><tr><td>FNO1d 1cycle</td><td>4.373</td><td>4.126</td><td>4.151</td></tr><tr><td>FT regular Ln</td><td>1.400</td><td>1.477</td><td>1.172</td></tr><tr><td>GT regular Ln</td><td>2.181</td><td>1.512</td><td>2.747</td></tr><tr><td>ST regular Ln</td><td>1.927</td><td>2.307</td><td>1.981</td></tr><tr><td>LT regular Ln</td><td>1.813</td><td>1.770</td><td>1.617</td></tr><tr><td>FT Ln on Q,K</td><td>1.135</td><td>1.123</td><td>1.071</td></tr><tr><td>GT Ln on K,V</td><td>1.203</td><td>1.150</td><td>1.025</td></tr><tr><td>ST Ln on Q,K</td><td>1.271</td><td>1.266</td><td>1.330</td></tr><tr><td>LT Ln on K,V</td><td>1.139</td><td>1.149</td><td>1.221</td></tr></table>
+
+<table><tr><td colspan="3">(b)</td></tr><tr><td></td><td>nf,nc =141,43</td><td>nf,nc= 211,61</td></tr><tr><td>FNO2d [57]</td><td>1.09</td><td>1.09</td></tr><tr><td>FNO2d 1cycle</td><td>1.419</td><td>1.424</td></tr><tr><td>FT regular Ln</td><td>0.838</td><td>0.847</td></tr><tr><td>GT regular Ln</td><td>0.894</td><td>0.856</td></tr><tr><td>ST regular Ln</td><td>1.075</td><td>1.131</td></tr><tr><td>LT regular Ln</td><td>1.024</td><td>1.130</td></tr><tr><td>FT Ln on Q,K</td><td>0.873</td><td>0.921</td></tr><tr><td>GT Ln on K,V</td><td>0.839</td><td>0.844</td></tr><tr><td>STLn on Q,K</td><td>0.946</td><td>0.959</td></tr><tr><td>LT Ln on K,V</td><td>0.875</td><td>0.970</td></tr></table>
+
+# 5.3 Example 3: inverse coefficient identification for Darcy flow
+
+In this example, we consider an inverse coefficient identification problem based on the same data used in Example 5.2. The input (solution) and the target (coefficient) are reversed from Example 5.2, and the noises are added to the input. The inverse problems in practice are a class of important tasks in many scientific disciplines such as geological sciences and medical imaging but much more difficult due to poor stability [51]. We aim to learn an approximation to an ill-posed operator $T : H _ { 0 } ^ { 1 } ( \Omega ) \to L ^ { \infty } ( \bar { \Omega } ) , u + \epsilon N _ { \nu } ( u ) \mapsto a$ , where $N _ { \nu } ( u )$ stands for noises related to the sampling distribution and the data. $\epsilon = 0 . 0 1$ means $1 \%$ of noise added in both training and evaluation, etc.
+
+The result can be found in Table 3. It is not surprising that FNO2d, an excellent smoother which filters higher modes in the frequency domain, struggles in this example to recover targets consisting of high-frequency traits (irregular interfaces) from low-frequency prevailing data (smooth solution due to ellipticity). We note that, the current state-of-the-art methods [16] for inverse interface coefficient identification need to carry numerous iterations to recover a single instance of a simple coefficient with a regular interface, provided that a satisfactory denoising has done beforehand. The attentionbased operator learner has capacity to unearth structurally how this inverse operator’s responses on a subset, with various benefits articulated in [56, 57, 5, 64, 10].
+
+Table 3: Evaluation relative error $( \times 1 0 ^ { - 2 } ,$ ) of the inverse problem 5.3.   
+
+<table><tr><td rowspan="3"></td><td colspan="3">nf,nc= 141,36</td><td colspan="3">nf,nc = 211,71</td></tr><tr><td>e=0</td><td>∈=0.01</td><td>∈=0.1</td><td>∈=0</td><td>∈=0.01</td><td>∈=0.1</td></tr><tr><td>FNO2d (only n f)</td><td>13.71</td><td>13.78</td><td>15.12</td><td>13.93</td><td>13.96</td><td>15.04</td></tr><tr><td>FNO2d (only nc)</td><td>14.17</td><td>14.31</td><td>17.30</td><td>13.60</td><td>13.69</td><td>16.04</td></tr><tr><td>FT regular Ln</td><td>1.799</td><td>2.467</td><td>6.814</td><td>1.563</td><td>2.704</td><td>8.110</td></tr><tr><td>GT regular Ln</td><td>2.026</td><td>2.536</td><td>6.659</td><td>1.732</td><td>2.775</td><td>8.024</td></tr><tr><td>ST regular Ln</td><td>2.434</td><td>3.106</td><td>7.431</td><td>2.069</td><td>3.365</td><td>8.918</td></tr><tr><td>LT regular Ln</td><td>2.254</td><td>3.194</td><td>9.056</td><td>2.063</td><td>3.544</td><td>9.874</td></tr><tr><td>FT Ln on Q,K</td><td>1.921</td><td>2.717</td><td>6.725</td><td>1.523</td><td>2.691</td><td>8.286</td></tr><tr><td>GT Ln on K,V</td><td>1.944</td><td>2.552</td><td>6.689</td><td>1.651</td><td>2.729</td><td>7.903</td></tr><tr><td>ST Ln on Q,K</td><td>2.160</td><td>2.807</td><td>6.995</td><td>1.889</td><td>3.123</td><td>8.788</td></tr><tr><td>LT Ln on K,V</td><td>2.360</td><td>3.196</td><td>8.656</td><td>2.136</td><td>3.539</td><td>9.622</td></tr></table>
+
+# 6 Conclusion
+
+We propose a general operator learner based on a simple attention mechanism. The network is versatile and is able to approximate both the PDE solution operator and the inverse coefficient identification operator. The evaluation accuracy on the benchmark problems surpasses the current best state-ofthe-art operator learner Fourier Neural Operator (FNO) in [57]. However, we acknowledge the limitation of this work: (i) similar to other operator learners, the subspace, on which we aim to learn the operator’s responses, may be infinite dimensional, but the operator must exhibit certain low-dimensional attributes (e.g., smoothing property of the higher frequencies in GRF); (ii) it is not efficient for the attention operator to be applied at the full resolution for a 2D problem, and this limits the approximation to a nonsmooth subset such as functions in $L ^ { \infty }$ ; (iii) due to the order of the matrix product, the proposed linear variant of the scaled dot-product attention is non-causal thus can only apply to encoder-only applications.
+
+# 7 Broader Impact
+
+Our work introduces the state-of-the-art self-attention mechanism the first time to PDE-related operator learning problems. The new interpretations of attentions invite numerical analysts to work on a more complete and delicate approximation theory of the attention mechanism. We have proved the Galerkin-type attention’s approximation capacity in an ideal Hilbertian setting. Numerically, the new attention-based operator learner has capacity to approximate the difficult inverse coefficient identification problem with an extremely noisy measurements, which was not attainable using traditional iterative methods for nonlinear mappings. Thus, our method may pose a huge positive impact in geoscience, medical imaging, etc. Moreover, traditionally the embeddings in Transformerbased NLP models map the words to a high dimensional space, but the topological structure in the same feature dimension between different positions are learned thereby not efficient. Our proof provides a theoretical guide for the search of feature maps that preserve, or even create, structures such as differentiability or physical invariance. Thus, it may contribute to the removal of the softmax nonlinearity to speed up significantly the arduous training or pre-training of larger encoder-only models such as BERT [27], etc. However, we do acknowledge that our research may negatively impact on the effort of building a cleaner future for our planet, as inverse problems are widely studied in reservoir detection, and we have demonstrated that the attention-based operator learner could potentially help to discover new fossil fuel reservoirs due to its capacity to infer the coefficients from noisy measurements.
+
+# Acknowledgments and Disclosure of Funding
+
+The hardware to perform this work is kindly donated by Andromeda Saving Fund. The first author was supported in part by the National Science Foundation under grants DMS-1913080 and DMS-2136075. No additional revenues are related to this work. We would like to thank the anonymous reviewers and the area chair for the suggestions on improving this article. We would like to thank Dr. Long Chen (Univ of California Irvine) for the inspiration of and encouragement on the initial conceiving of this paper, as well as numerous constructive advices on revising this paper, not mentioning his persistent dedication of making publicly available tutorials [17] on writing beautiful vectorized code. 4 We would like to thank Dr. Ari Stern (Washington Univ in St. Louis) for the help on the relocation during the COVID-19 pandemic. We would like to thank Dr. Likai Chen (Washington Univ in St. Louis) for the invitation to the Stats and Data Sci seminar at WashU that resulted the reboot of this study. 5 We would like to thank Dr. Ruchi Guo (Univ of California Irvine) and Dr. Yuanzhe Xi (Emory Univ) for the invaluable feedbacks on the choice of the numerical experiments. We would like to thank the Kaggle community, including but not limited to Jean-François Puget (CPMP $@$ Kaggle) for sharing a simple Graph Transformer in TensorFlow,6 Murakami Akira (mrkmakr $@$ Kaggle) for sharing a Graph Transformer with a CNN feature extractor in Tensorflow, 7 and Cher Keng Heng (hengck23 $@$ Kaggle) for sharing a Graph Transformer in PyTorch.8 We would like to thank daslab $@$ Stanford, OpenVaccine, and Eterna for hosting the COVID-19 mRNA Vaccine competition and Deng Lab (Univ of Georgia) for collaborating in this competition. We would like to thank CHAMPS (Chemistry and Mathematics in Phase Space) for hosting the $J$ -coupling quantum chemistry competition and Corey Levinson (Eligo Energy, LLC) for collaborating in this competition. We would like to thank Zongyi Li (Caltech) for sharing some early dev code in the updated PyTorch fft interface and the comments on the viscosity of the Burgers’ equation. We would like to thank Ziteng Pang (Univ of Michigan) and Tianyang Lin (Fudan Univ) to update us with various references on Transformers. We would like to thank Joel Schlosser (Facebook) to incorporate our change to the PyTorch transformer module to simplify our testing pipeline. We would be grateful to the PyTorch community for selflessly code sharing, including Phil Wang(lucidrains $@$ github) and Harvard NLP group [52]. We would like to thank the chebfun [28] for integrating powerful tools into a simple interface to solve PDEs. We would like to thank Dr. Yannic Kilcher (ykilcher $@$ twitter) and Dr. Hung-yi Lee (National Taiwan Univ) for frequently covering the newest research on Transformers in video formats. We would also like to thank the Python community [87, 68] for sharing and developing the tools that enabled this work, including PyTorch [69], NumPy [39], SciPy [89], Plotly [45] Seaborn [93], Matplotlib [43], and the Python team for Visual Studio Code. We would like to thank draw.io [46] for providing an easy and powerful interface for producing vector format diagrams. For details please refer to the documents of every function that is not built from the ground up in our open-source software library.9
+
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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] In Section 6 and in each experiment’s detailed description in Appendix C.3 and C.4.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] In Section 7.   
+(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] The key assumption of the new interpretations of the attention mechanism is in Assumption 4.2. Due to the page limit, the assumptions and settings are fully elaborated in a mathematical rigorous fashion in Appendix D for the proof of Theorem 4.3.   
+(b) Did you include complete proofs of all theoretical results? [Yes] In 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] The data are publicly available. For instruction to reproduce our result please refer to README.md in https://github.com/scaomath/galerkin-transformer/ tree/main/examples.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We include all details in the supplemental material Appendix B and C.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We found that our method is numerically seed-invariant, and we have reported the error bands in Appendix C.   
+(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] Please refer to Appendix C.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] Multiple creators of or contributing work to the dataset used in our work are cited [57, 5, 56, 64]. The code base we have implemented our model upon is in the footnote of Section 5, and cited in the Acknowledgement [52]. In the publicly available code repository, in the document of every function, we have credited every author we can find for even a small code snippet, if that function is not built from ground up by us.   
+(b) Did you mention the license of the assets? [Yes] In Section 5.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] The full code base to replicate our results is publicly available as an open-source software.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] The data is obtained from https://github.com/zongyi-li/ fourier_neural_operator with the MIT license as well as the consent from the author per a personal communication.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]