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+# Collaborating with Humans without Human Data
+
+DJ Strouse⇤, Kevin R. McKee, Matt Botvinick, Edward Hughes, Richard Everett⇤ DeepMind {strouse, kevinrmckee, botvinick, edwardhughes, reverett}@deepmind.com
+
+# Abstract
+
+Collaborating with humans requires rapidly adapting to their individual strengths, weaknesses, and preferences. Unfortunately, most standard multi-agent reinforcement learning techniques, such as self-play (SP) or population play (PP), produce agents that overfit to their training partners and do not generalize well to humans. Alternatively, researchers can collect human data, train a human model using behavioral cloning, and then use that model to train “human-aware” agents (“behavioral cloning play”, or BCP). While such an approach can improve the generalization of agents to new human co-players, it involves the onerous and expensive step of collecting large amounts of human data first. Here, we study the problem of how to train agents that collaborate well with human partners without using human data. We argue that the crux of the problem is to produce a diverse set of training partners. Drawing inspiration from successful multi-agent approaches in competitive domains, we find that a surprisingly simple approach is highly effective. We train our agent partner as the best response to a population of self-play agents and their past checkpoints taken throughout training, a method we call Fictitious Co-Play (FCP). Our experiments focus on a two-player collaborative cooking simulator that has recently been proposed as a challenge problem for coordination with humans. We find that FCP agents score significantly higher than SP, PP, and BCP when paired with novel agent and human partners. Furthermore, humans also report a strong subjective preference to partnering with FCP agents over all baselines.
+
+# 1 Introduction
+
+Generating agents which collaborate with novel partners is a longstanding challenge for Artificial Intelligence (AI) [4, 16, 37, 52]. Achieving ad-hoc, zero-shot coordination [31, $\bar { 6 6 } ]$ is especially important in situations where an AI must generalize to novel human partners [6, 61]. Many successful approaches have employed human models, either constructed explicitly [14, 35, 53] or learnt implicitly [12, 60]. By contrast, recent work in competitive domains has shown that it is possible to reach humanlevel using model-free reinforcement learning (RL) without human data, via self-play [8, 9, 63, 64]. This begs the question: Can model-free RL without human data generate agents that can collaborate with novel humans?
+
+We seek an answer to this question in the space of common-payoff games, where all agents work towards a shared goal and receive the same reward. Self-play (SP), in which an agent learns from repeated games played against copies of itself, does not produce agents that generalize well to novel co-players [10, 11, 21, 44]. Intuitively, this is because agents trained in self-play only ever need to coordinate with themselves, and so make for brittle and stubborn collaborators with new partners who act differently. Population play (PP) trains a population of agents, all of whom interact with each other $\pmb { \| 3 9 \| }$ . While PP can generate agents capable of cooperation with humans in competitive team games $\textcircled { 1 3 4 } \textcircled { 1 }$ , it still fails to produce robust partners for novel humans in pure common-payoff settings [12]. PP in common-payoff settings naturally encourages agents to play the same way, reducing strategic diversity and producing agents not so different from self-play $\dot { \mathbb { B } } \dot { \mathbb { 4 } } \mathbb { I }$ .
+
+![](images/70b2b8c1e9478aed36abef766805e05c03561ba29533d4a7de7ad061fa2d261f.jpg)  
+Figure 1: In this work, we evaluate a variety of agent training methods (Section 2) in zero-shot coordination with agents (Section 4). We then run a human-agent collaborative study designed to elicit human preferences over agents (Section 5)
+
+Our approach starts with the intuition that the key to producing robust agent collaborators is exposure to diverse training partners. We find that a surprisingly simple strategy is effective in generating sufficient diversity. We train $N$ self-play agents varying only their random seed for neural network initialization. Periodically during training, we save agent “checkpoints” representing their strategy at that point in time. Then, we train an agent partner as the best-response to both the fully-trained agents and their past checkpoints. The different checkpoints simulate different skill levels, and the different random seeds simulate breaking symmetries in different ways. We refer to this agent training procedure as Fictitious $\mathbf { C o }$ -Play (FCP) for its relationship to fictitious self-play [7, 27, 28, 69].
+
+We evaluate FCP in a fully-observable two-player common-payoff collaborative cooking simulator. Based on the game Overcooked $\boldsymbol { \left[ \left[ 2 5 \right] \right] }$ , it has recently been proposed as a coordination challenge for AI [12, 50, 70]. State-of-the-art performance in producing agents capable of generalization to novel humans was achieved in $[ \mathbb { 1 2 } ]$ via behavioral cloning (BC) of human data. More precisely, BC was used to produce models that can stand in as human proxies during training in simulation, a method we call behavioral cloning play (BCP). We demonstrate that FCP outperforms BCP in generalizing to both novel agent and human partners, and that humans express a significant preference for partnering with FCP over BCP. Our method avoids the cost and potential privacy concerns of collecting human data for training, while achieving better outcomes for humans at test time.
+
+We summarize the novel contributions of this paper as follows:
+
+1. We propose Fictitious Co-Play (FCP) to train agents capable of zero-shot coordination with humans (Section 2.1).   
+2. We demonstrate that FCP agents generalize better than SP, PP, and BCP in zero-shot coordination with a variety of held-out agents (Section 4.2).   
+3. We propose a rigorous human-agent interaction study with behavioral analysis and participant feedback (Section $\boxed { 5 . 1 }$ .   
+4. We demonstrate that FCP significantly outperforms the BCP state-of-the-art, both in task score and in human partner preference (Section $5 . 2 )$ .
+
+# 2 Methods
+
+# 2.1 Fictitious Co-Play (FCP)
+
+Diverse training conditions have been shown to make agents more robust, from environmental variations (i.e. domain randomization $\textcircled { 1 5 4 } , \textcircled { 5 6 } , \textcircled { 6 7 } \textcircled { 1 }$ to heterogeneity in training partners $\left[ \left[ 6 9 \right] \right]$ . We seek to train agents that are robust partners for humans in common-payoff games, and so extend this line of work to that setting.
+
+One important challenge in collaborating with novel partners is dealing with symmetries $\textcircled { \left| 3 1 \right| }$ . For example, two agents A and B facing each other may move past each other by A going left and B going right, or vice versa. Both are valid solutions, but a good agent partner will adaptively switch between these conventions if a human clearly prefers one over the other. A second important challenge is dealing with variations in skill level. Good agent partners should be able to assist both highly-skilled partners, as well as partners who are still learning.
+
+![](images/2372b76b2e16cbfb2db5439982c244e9b3e60b30aff99699aa5043d431598f6f.jpg)  
+Figure 2: The four agent training methods we evaluate in this work. Self-play (SP) where an agent learns with itself, population-play (PP) where a population of agents are co-trained together, and behavioral cloning play (BCP) where data from human games is used to create a behaviorally cloned agent with which an RL agent is then trained. In our method, Fictitious Co-Play (FCP), $N$ self-play agents are trained independently and checkpointed throughout training. An agent is then trained to best respond to the entire population of SP agents and their checkpoints.
+
+Fictitious co-play (FCP) is a simple two-stage approach for training agents that overcomes both of these challenges (Figure $\bigstar$ right). In the first stage, we train a diverse pool of partners. To allow the pool to represent different symmetry breaking conventions, we train $N$ partner agents in self-play. Since these partners are trained independently, they can arrive at different arbitrary conventions for breaking symmetries. To allow the pool to represent different skill levels, we use multiple checkpoints of each self-play partner throughout training. The final checkpoint represents a fully-trained “skillful” partner, while earlier checkpoints represent less skilled partners. Notably, by using multiple checkpoints per partner, this additional diversity in skill incurs no extra training cost.
+
+In the second stage, we train an FCP agent as the best response to the pool of diverse partners created in the first stage. Importantly, the partner parameters are frozen and thus FCP must learn to adapt to partners, rather than expect partners to adapt to it. In this way, FCP agents are prepared to follow the lead of human partners, and learn a general policy across a range of strategies and skills. We call our method “fictitious” co-play for its relationship to fictitious self-play in which competitive agents are trained with past checkpoints (in that case, to avoid strategy cycling) [7, 27, 28, 39, 69].
+
+# 2.2 Baselines and ablations
+
+We compare FCP agents to the three baseline training methods listed below, each varying only in their set of training partners, with the RL algorithm and architecture consistent across all agents:
+
+1. Self-play (SP), where agents learn solely through interaction with themselves.   
+2. Population-play (PP), where a population of agents are co-trained through random pairings.   
+3. Behavioral cloning play (BCP), where an agent is trained with a BC model of a human [12].
+
+We also evaluate three variations on FCP to better understand the conditions for its success:
+
+1. To test the importance of including past checkpoints in training, we evaluate an ablation of FCP in which agents are trained only with the converged checkpoints of their partners $\mathrm { F C P } _ { - T }$ for “FCP minus time”). 2. To test whether FCP would benefit from additional diversity in its partner population, we evaluate an augmentation of FCP in which the population of SP partners varies not just in random seed, but also in architecture $\operatorname { F C P } _ { + A }$ for “FCP plus architectural variation”). 3. To test whether architectural variation can serve as a full replacement for playing with past checkpoints, we evaluate the combination of both modifications $( \mathrm { F C P } _ { - T , + A } )$ .
+
+# 2.3 Environment
+
+Following prior work on zero-shot coordination in human-agent interaction, we study the Overcooked environment (see Figure 3) [12, 13, 38, 50, 70]. We draw particular inspiration from the environment in Carroll et al. [12]. For full details, see Appendix A.
+
+In this environment, players are placed into a gridworld kitchen as chefs and tasked with delivering as many cooked dishes of tomato soup as possible within an episode. This involves a series of sequential high-level actions to which both players can contribute: collecting tomatoes, depositing them into cooking pots, letting the tomatoes cook into soup, collecting a dish, getting the soup, and delivering it. Upon a successful delivery, both players are rewarded equally.
+
+To effectively complete the task, players must learn to navigate the kitchen and interact with objects in the correct order, all while maintaining awareness of their partner’s behavior to coordinate with them. This environment therefore presents the challenges of both movement and strategic coordination.
+
+Each player observes an egocentric RGB view of the world, and at every step can perform one of six actions: stand still, move {up, down, left, right}, interact. The behavior ofLÈ¡áyҏįŘįºµò‹µÒ¡y® interact varies based on the cell which the player is facing (e.g. place tomato on counter).
+
+![](images/48e8af75423e0fa5e5c23e2088c49716c4c6044ee3579f26cdd9911dbb4b4925.jpg)  
+Figure 3: The Overcooked environment: a two-player common-payoff game in which players must coordinate to cook and deliver soup.
+
+![](images/2b3253fcaac0677dbb15cc1ed5aa0663e561ff52605ba15bd61605f96c716872.jpg)  
+Figure 4: Layouts: the kitchens which agents and humans play in, each emphasizing different coordination strategies. Highlighted in bold are the terms used to refer to each in the rest of this paper.
+
+# 2.4 Implementation details
+
+Here we highlight several key implementation details for our training methods. For full details, including the architectures, hyperparameters, and compute used, please see Appendix B.
+
+For our reinforcement learning agents, we use the V-MPO $\pmb { \Vert 6 5 \Vert }$ algorithm along with a ResNet [26] plus LSTM $\mathbb { \left| \bigstar \bigstar \right\| }$ architecture which we found led to optimal behavior across all layouts. Agents are trained using a distributed set of environments running in parallel $\textcircled { 1 1 7 }$ , each sampling two agents from the training population to play together every episode.
+
+Both PP and FCP are trained with a population size of $N = 3 2$ agents which are sampled uniformly. For FCP, we use 3 checkpoints for each agent, therefore incurring no additional training burden: (1) at initialization (i.e. a low-skilled agent), (2) at the end of training (i.e. a fully-trained expert agent), and (3) at the middle of training, defined as when the agent reaches $50 \%$ of its final reward (i.e. an average-skilled agent). When varying architecture for the training partners of the $\operatorname { F C P } _ { + A }$ and $\mathrm { F C P } _ { - T , + A }$ variants, we vary whether the partners use memory (i.e. LSTM vs not) and the width of their policy and value networks (i.e. 16 vs 256). In total, we train 8 agents for each of the 4 combinations, leaving the total population size of $N = 3 2$ unchanged, ensuring a fair comparison.
+
+To train agents via behavioral cloning $\left[ \left[ 5 8 \right] \right]$ , we use the open-source Acme $\pmb { \mathbb { B } } 0 \|$ to learn a policy from human gameplay data. Specifically, we collected 5 human-human trajectories of length 1200 time steps for each of the 5 layouts, resulting in 60k total environment steps. We divide this data in half and train two BC agents: (1) a partner for training a BCP agent, and (2) a “human proxy” partner for agent-agent evaluation. Following Carroll et al. $\mathbb { \overline { { \lVert \lambda \rVert } } }$ , we use a set of feature-based observations for the agents (as opposed to RGB) and generate comparable results: performance is higher on 3 layouts (asymmetric, cramped, and ring) but poorer on the other 2 (circuit and forced).
+
+# 3 Related work
+
+Ad-hoc team play There is a large and diverse body of literature on ad-hoc team-play $ { \mathbb { B } } ,  { \mathbb { G } } 6  { \mathbb { I } }$ , also known as zero-shot coordination $[ \overbrace { 3 \mathrm { 1 } } ]$ . Prior work based in game-theoretic settings has suggested the benefits of planning $\mathbb { [ [  \mathrm { { 7 1 } ] }  }$ , online learning $\mathbb { \left[ \left. 5 1 \right] \right. }$ , and novel solution concepts $\bar { \mathbb { D } }$ , to name a few examples. More recently, multi-agent deep reinforcement learning has provided the tools to scale to more complex gridworld or continuous control settings, leading to work on hierarchical social planning $\widehat { \left\| 3 6 \right\| }$ , adapting to existing social conventions $\checkmark$ , trajectory diversity $\lVert \boldsymbol { \mathsf { A } } \boldsymbol { \mathsf { S } } \rVert$ , and theory of mind [14]. Ad-hoc team-play among novel agent partners is also an object of active study in the emergent communication literature [10, 11, 43]. This prior work has tended to focus on generalization to held-out agent partners as a proxy for human co-players.
+
+Collaborative play with novel humans has been evaluated more actively in the context of training agent assistants; see for instance [57, 68]. To our knowledge, our FCP agents represent the stateof-the-art in coordinating with novel human partners on an equal footing of capabilities in a rich gridworld environment, as measured by the challenge tasks in Carroll et al. [12].
+
+Diversity in multi-agent reinforcement learning In multi-agent reinforcement learning, agents that train with behaviorally diverse populations of game partners tend to demonstrate stronger performance than their self-play counterparts. For example, across a range of multi-agent games, generalization to held-out populations can be improved by training larger and more diverse populations [13, 42, 50]. In mixed-motive settings, cooperation among agents can be encouraged through social diversity, such as in player preferences and rewards [3, 47, 49]. Similarly, competitiveness can be optimized through selective matchmaking between increasingly diverse agents $\pm \boxed { 1 2 4 } \boxed { 3 9 } \boxed { 6 9 }$ .
+
+Despite the increased focus on improving multi-agent performance, evaluation has typically been constrained to agent-agent settings. High-performing agents have infrequently been evaluated with humans, particularly in non-competitive domains $\dot { \left. \overline { { \dot { \left. \dot { \theta } \right\| } } } \right. }$ . We add to this growing literature, showing that training with diversity is a powerful approach for effective human-agent collaboration.
+
+Human-agent interaction In recent years, increased attention has been directed toward designing machine learning agents capable of collaborating with humans [41, 57, 68, 72] (see also $\textcircled { 1 1 6 } \textcircled { }$ for a broader review on Cooperative AI). Tylkin et al. $\lVert \overline { { 6 8 } } \rVert$ is particularly notable in also demonstrating that partially trained agents can be useful learning targets for human helpers, although in a different domain (cooperative Atari). Our method, FCP, can be seen as extending theirs by training with multiple “skill levels” and random seeds, rather than just one, which we demonstrate to be crucial to our agents’ performance (Tables 1 and 2 and Figure 7b).
+
+A key preceding entry in this research area is Carroll et al. $[ \mathbb { 1 2 } ]$ , who similarly investigated humanagent coordination in Overcooked. We use their method (BCP) as a baseline throughout our experiments (Section $\boxed { 2 . 2 }$ . Relative to BCP, our approach removes the need for the expensive step of human data collection for agent training. Furthermore, through our novel human-agent experimental design, we go beyond objective performance metrics to compare the subjective preferences that agents generate. For a detailed comparison of methods and results, see Appendix E.
+
+# 4 Zero-shot coordination with agents
+
+In this section, we evaluate our FCP agent, its ablations, and the baselines with held-out agents.
+
+# 4.1 Evaluation method: collaborative evaluation with agent partners
+
+Our primary concern in this work is generalization to novel human partners (as investigated in Section $\textcircled{5}$ . However, just as collecting human-human data for behavioral cloning is expensive, so too is evaluating agents with humans. Consequently, we instead use generalization to held-out agent partners as a cheap proxy of performance with humans. This is then used to guide our model selection process, allowing us to be more targeted with the agents we select for our human-agent evaluations.
+
+We evaluate with three held-out populations:
+
+1. A BC model trained on human data, $H _ { \mathrm { p r o x y } }$ , intended as a proxy of generalization to humans, as done by Carroll et al. $[ \overbrace { | 1 2 | }$ . 2. A set of self-play agents varying in seed, architecture, and training time (specifically, heldout seeds of the $N = 3 2$ partners trained for the $\mathrm { F C P } _ { + A }$ agent; see Section 2.4). These are intended to test generalization to a diverse yet still skillful population. 3. Randomly initialized agents intended to test generalization to low-skill partners.
+
+For all results, we report the average number of deliveries made by both players within an episode, aggregated across the 5 different layouts from Figure $\boxed { 4 }$ (with the per-layout results reported in Appendix $\underline { { \overline { { ( \mathrm { C . 2 } ) } } } }$ . We estimate mean and standard deviation across 5 random seeds. For each seed, we evaluate the agent with all members of the held-out population for 10 episodes per agent-partner pair.
+
+# 4.2 Results
+
+# Finding 1: FCP significantly outperforms all baselines
+
+To begin, we compare our FCP agent and the baselines when partnered with the three held-out populations introduced above. As can be seen in Figure $\boxed { 5 }$ FCP significantly outperforms all baselines when partnered with all three held-out populations. Notably, it performs better than BCP with $H _ { \mathrm { p r o x y } }$ even though BCP trains with such a model and FCP does not. Similar to Carroll et al. $\mathbb { \lVert 1 2 \rVert }$ , we find that BCP significantly outscores SP.
+
+When paired with a randomly initialized partner which behaves suboptimally, we see an even greater difference between FCP and the baselines. Given that FCP is trained with non-held-out versions of such agents, it may not be surprising that it does so well with partners that behave poorly. However, what is surprising is how brittle the other training methods are. This suggests that they may not perform well with humans who are not highly skilled players, which we will see in Section 5.
+
+![](images/8083dcac415b6ef9624e1aa4f428a0a33c3e9d47ce8903df3798f3e80ef0427c.jpg)  
+Figure 5: Agent-agent collaborative evaluation: Performance of each agent when partnered with each of the held-out populations (Section $4 . 1 )$ in episodes of length $T = 5 4 0$ . Importantly, FCP scores higher than all baselines with a variety of test partners. Error bars represent standard deviation over five random training seeds. Plots aggregate data across kitchen layouts; results calculated by individual layout can be found in Appendix C.2.
+
+Finding 2: Training with past checkpoints is the most beneficial variation for performance Next, we investigate how the different training partner variations influence FCP’s performance. In particular, we separately ablate the past checkpoints $( T )$ and architecture $( A )$ variations, evaluating them with the same partners as in Figure 5. The results of this evaluation are presented in Table 1. Comparing the FCP and $\mathrm { F C P } _ { - T }$ columns, we see that removing past checkpoints from training significantly reduces performance. Comparing the FCP and $\operatorname { F C P } _ { + A }$ columns, we see that adding architectural variation to the training population offers no improvement over training with past
+
+<table><tr><td>Partner</td><td>FCP</td><td>FCP-T</td><td>FCP+A</td><td>FCP-T,+A</td></tr><tr><td>Hproxy</td><td>10.6± 0.5</td><td>4.7± 0.4</td><td>9.9±0.6</td><td>7.0±0.8</td></tr><tr><td>Diverse SP</td><td>11.2 ± 0.1</td><td>6.9 ± 0.1</td><td>11.1 ± 0.4</td><td>8.6 ± 0.4</td></tr><tr><td>Random</td><td>8.6± 0.2</td><td>1.0 ± 0.1</td><td>8.4±0.4</td><td>3.2 ± 0.5</td></tr></table>
+
+Table 1: Ablation results: Performance of each variation of FCP – training with past partner checkpoints $T$ for time) and adding partner variation in architecture $( A )$ . Scores are mean deliveries with standard deviation over 5 random seeds. Notably, we find that the inclusion of past checkpoints is essential for strong performance $( \mathrm { F C P } > \mathrm { F C P } _ { - T }$ ), and additionally including architectural variation does not improve performance $( \mathrm { F C P } \approx \mathrm { F C P } _ { + A . }$ ). However, architectural variation is better than no variation, improving performance when past checkpoints are not available $( \mathrm { F C P } _ { - T , + A } > \mathrm { F C P } _ { - T , }$ ).
+
+checkpoints. However, comparing the $\mathrm { F C P } _ { - T }$ and $\mathrm { F C P } _ { - T , + A }$ columns, we see that without training with past checkpoints, architectural variation in the population does improve performance.
+
+# 5 Zero-shot coordination with humans
+
+Ultimately, our goal is to develop agents capable of coordinating with novel human partners. In this section, we run an online study to evaluate our FCP agent and the baseline agents in collaborative play with human partners.
+
+![](images/5856cbdd90eeef2a7a6093668472063862a759746c339aa0324e78efb5d67b22.jpg)  
+Figure 6: Human-agent collaborative study: For our human-agent collaboration study, we recruited participants online to play games with FCP and baseline agents. Participants played a randomized sequence of episodes with different agent partners and kitchen layouts. After every two episodes, participants reported the direction and strength of their preference between their last two partners.
+
+# 5.1 Evaluation method: collaborative evaluation with human participants
+
+To test how effectively FCP’s performance generalizes to human partners, we recruited participants from Prolific $\mathbb { 1 1 8 } , \lvert 5 5 \rvert$ for an online collaboration study $N = 1 1 4$ ; $3 7 . 7 \%$ female, $5 9 . 6 \%$ male, $1 . 8 \%$ nonbinary; median age between 25–34 years). We used a within-participant design for the study: each participant played with a full cohort of agents (i.e. generated through every training method). This design allowed us to evaluate both objective performance as well as subjective preferences.
+
+Participants first read game instructions and played a short tutorial episode guiding them through the dish preparation sequence (see Appendix $\underline { { \overline { { \mathbb { D . 1 . 1 } } } } }$ for instruction text and study screenshots). Participants then played 20 episodes with a randomized sequence of agent partners and kitchen layouts. Episodes lasted $T = 3 0 0$ steps (1 minute) each. After every two episodes, participants reported their preference over the agent partners from those episodes on a five-point Likert-type scale. After playing all 20 episodes, participants completed a debrief questionnaire collecting standard demographic information and open-ended feedback on the study. Our statistical analysis below primarily relies upon the repeated-measures analysis of variance (ANOVA) method. See Appendix D for additional details of our study design and analysis, including independent ethical review.
+
+# 5.2 Results
+
+Finding 1: FCP coordinates best with humans, achieving the highest score across maps To begin, we compare the objective team performance supported by our FCP and baseline agents. The strong FCP performance observed in agent-agent play generalizes to human-agent collaboration:
+
+the FCP-human teams significantly outperform all other agent-human teams, achieving the highest average scores across maps, every $p < 0 . 0 0 1$ (Figure $\lvert \overline { { 7 \mathrm { a } } } \rvert$ , while performing as well as or better than the other teams on each individual map (see Appendix $\mathbf { D } . 3 )$ . Echoing the results from our agent-agent ablation experiments (Table $^ { 1 ) }$ , the inclusion of past checkpoints in training proves critical to FCP’s strong performance, $p < 0 . 0 { \overline { { 0 1 } } }$ (Figure $\textcircled { 7 6 }$ . Similar to Carroll et al. $\mathbb { \lVert \rVert }$ , we find that BCP outscores SP when collaborating with human players, $p < 0 . 0 0 1$ .
+
+# Finding 2: Participants prefer FCP over all baselines
+
+FCP’s strong collaborative performance carries over to our participants’ subjective partner preferences. Participants expressed a significant preference for FCP partners over all other agents, including BCP, with every $p < 0 . 0 5$ (Figure $\bar { 7 \mathrm { c } } )$ . Notably, while human-BCP and human-PP teams did not significantly differ in their completed deliveries, participants reported significantly preferring BCP over PP, $p = 0 . 0 0 3$ , highlighting the informativeness of our subjective analysis.
+
+![](images/14e048153d10325b3617f94b393393ed3ab6cf3e38198f8791acf0fc83b051df.jpg)  
+Figure 7: Human-agent collaborative evaluation: Evaluation and preference metrics from humanagent play in episodes of length $T = 3 0 0$ . Error bars represents $9 5 \%$ confidence intervals, calculated over episodes. Plots aggregate data across kitchen layouts; results calculated by individual layout can be found in Appendix D.3.
+
+# 5.3 Exploratory behavioral analysis
+
+To better understand how the human-agent scores and preferences may have arisen, here we analyze the resulting action trajectories of each human and agent player in our experiment.
+
+![](images/65959c0ab23a18c0f8de72bc451d199e7d9d9cc2698e7ca2ac4f320e26dac829.jpg)
+
+Figure 8: Behavioral analysis: (a) FCP is able to move most frequently $3 5 \%$ of the time), corresponding to the best movement coordination with human partners. (b) FCP exhibits the most equal preferences over cooking pots (0.11 difference), aligning with human preferences. Values are calculated as the absolute difference in preferences between the two pots; 1 indicates that the player only uses one of the two available pots, while 0 indicates that the player uses both pots equally.
+
+# Finding 1: FCP exhibits the best movement coordination with humans
+
+First, we investigate how much each player moves in an episode (Figure $\textcircled { 8 \mathrm { a } }$ , where moving in a higher fraction of timesteps may suggest fewer collisions and thus better coordination with a partner. Notably, we observe two results: (1) humans rarely move, a behavior which is out-of-distribution for typical training methods (e.g. SP, PP) but is seen in the training distribution for BCP and FCP.
+
+(2) FCP moves the most on all layouts other than Forced, suggesting it is better at coordinating its movement strategy with its partner. This result was also reported by human participants, for example: “I noticed that some of my partners seemed to know they needed to move around me, while others seemed to get ‘stuck’ until I moved out of their way” (see Appendix D for more examples).
+
+# Finding 2: FCP’s preferences over cooking pots aligns best with that of humans
+
+Next, we investigate whether there was a preference for a specific cooking pot in the layouts which included two cooking pots (Figure $\textcircled { 8 6 }$ . To do this, we calculate the difference in the number of times each pot was used by each player, where a high value indicates a strong preference for one pot and a low value indicates more equal preference for the two pots.
+
+As can be seen in the FCP column, our agent typically has the most aligned preferences with that of humans (0.11 for FCP to 0.14 for humans). Behaviorally speaking, this means that our agent prefers one cooking pot over the other $5 5 . 5 \%$ of the time (i.e. a 0.11 point difference). In contrast, all other agents have a strong preference for a single pot. This is a non-adaptive strategy which generalizes poorly to typical human behavior of using both pots, leading to worse performance.
+
+# 6 Discussion
+
+Summary In this work, we investigated the challenging problem of zero-shot collaboration with humans without using human data in the training pipeline. To accomplish this, we introduced Fictitious Co-Play (FCP) – a surprisingly simple yet effective method based on creating a diverse set of training partners. We found that FCP agents scored significantly higher than all baselines when partnered with both novel agent and human partners. Furthermore, through a rigorous human-agent experimental design, we also found that humans reported a strong subjective preference to partnering with FCP agents over all baselines.
+
+Limitations and future work Our method currently relies on the manual process of initially training and selecting a diverse set of partners. This is not only time consuming, but also prone to researcher biases that may negatively influence the behavior of the created agents. Additionally, while we found FCP with a partner population size of $N = 3 2$ sufficient here, for more complex games, FCP may require an unrealistically large partner population size to represent sufficiently diverse strategies. To address these concerns, methods for automatically generating partner diversity for common-payoff games may be important. Possibilities include adaptive population matchmaking as been used in competitive zero-sum games $\mathbb { \lVert 6 9 \rVert }$ , as well as auxiliary objectives that explicitly encourage behavioral diversity [19, 45, 46].
+
+Our method requires a known and fixed reward function. We also focus on one domain in order to compare with prior work which has argued that human-in-the-loop training is necessary. Consequently, the resulting agents are only designed to adaptively collaborate on a single task, and not to infer human preferences in general $\textcircled { 1 1 } \textcircled { 3 3 } \textcircled { 5 9 }$ . Moreover, if a task’s reward function is poorly aligned with how humans approach the task, our method may well produce subpar partners, as would any method without access to human data. Thus, additional domains and tasks should be studied to better understand how our method generalizes. Targeted experiments to test specific forms of generalization may be especially helpful in this regard $\overline { { \| 3 8 \| } }$ , as could approaches that procedurally generate environment layouts requiring diverse solutions $\lVert 2 2 \rVert$
+
+Finally, it may be possible to produce even stronger agent assistants by combining the strengths of FCP (i.e. diversity) and BCP (i.e. human-like play). Indeed, Knott et al. $\pmb { \Vert 3 8 \Vert }$ recently demonstrated that modifying BCP to train with multiple BC partners produces more robust collaboration with held-out agents, a finding that would be interesting to test with human partners.
+
+Societal impact A challenge for this line of work is ensuring agent behavior is aligned with human values (i.e. the AI value alignment problem [23, 59]). Our method has no guarantees that the resulting policy aligns with the preferences, intentions, or welfare of its potential partners. It likewise does not exclude the possibility that the target being optimized for is harmful (e.g. if the agent’s partner expresses preferences or intentions to harm others). This could therefore produce negative societal effects either if training leads to poor alignment or if agents are optimized for harmful metrics.
+
+One potential strategy for mitigating these risks is the use of human preference data [15]. Such data could be used to fine-tune and filter trained agents before deployment, encouraging better alignment with human values. A key question in this line of research is how human preference data should be aggregated—or selected, in the case of expert preferences—when our aim is to create socially aligned agents (i.e. agents that are sufficiently aligned for everyone). Relatedly, targeted research on human beliefs and perceptions of AI $\pm 8 \jmath$ , and how they steer human-agent interaction, would help inform agent design for positive societal impact. For instance, developers could incorporate specific priors into agents to reinforce tendencies for fair outcomes $\pm \mathbb { Z } 0 . \pm \mathbb { B } 2 \mathbb { I }$ .
+
+Conclusion We proposed a method which is both effective at collaborating with humans and simple to implement. We also presented a rigorous and general methodology for evaluating with humans and eliciting their preferences. Together, these establish a strong foundation for future research on the important challenge of human-agent collaboration for benefiting society.
+
+# Acknowledgements
+
+The authors would like to thank Mary Cassin for creating the game sprite art; Rohin Shah, Thore Graepel, and Iason Gabriel for feedback on the draft; Lucy Campbell-Gillingham, Tina Zhu, and Saffron Huang for support in evaluating agents with humans; and Max Kleiman-Weiner, Natasha Jaques, Marc Lanctot, Mike Bowling, and Dan Roberts for useful discussions.
+
+# Funding disclosure
+
+This work was funded solely by DeepMind. The authors declare no competing interests.
+
+# References
+
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+    {
+        "type": "text",
+        "text": "Collaborating with Humans without Human Data ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "DJ Strouse⇤, Kevin R. McKee, Matt Botvinick, Edward Hughes, Richard Everett⇤ DeepMind {strouse, kevinrmckee, botvinick, edwardhughes, reverett}@deepmind.com ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+    },
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+        "type": "text",
+        "text": "Collaborating with humans requires rapidly adapting to their individual strengths, weaknesses, and preferences. Unfortunately, most standard multi-agent reinforcement learning techniques, such as self-play (SP) or population play (PP), produce agents that overfit to their training partners and do not generalize well to humans. Alternatively, researchers can collect human data, train a human model using behavioral cloning, and then use that model to train “human-aware” agents (“behavioral cloning play”, or BCP). While such an approach can improve the generalization of agents to new human co-players, it involves the onerous and expensive step of collecting large amounts of human data first. Here, we study the problem of how to train agents that collaborate well with human partners without using human data. We argue that the crux of the problem is to produce a diverse set of training partners. Drawing inspiration from successful multi-agent approaches in competitive domains, we find that a surprisingly simple approach is highly effective. We train our agent partner as the best response to a population of self-play agents and their past checkpoints taken throughout training, a method we call Fictitious Co-Play (FCP). Our experiments focus on a two-player collaborative cooking simulator that has recently been proposed as a challenge problem for coordination with humans. We find that FCP agents score significantly higher than SP, PP, and BCP when paired with novel agent and human partners. Furthermore, humans also report a strong subjective preference to partnering with FCP agents over all baselines. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Generating agents which collaborate with novel partners is a longstanding challenge for Artificial Intelligence (AI) [4, 16, 37, 52]. Achieving ad-hoc, zero-shot coordination [31, $\\bar { 6 6 } ]$ is especially important in situations where an AI must generalize to novel human partners [6, 61]. Many successful approaches have employed human models, either constructed explicitly [14, 35, 53] or learnt implicitly [12, 60]. By contrast, recent work in competitive domains has shown that it is possible to reach humanlevel using model-free reinforcement learning (RL) without human data, via self-play [8, 9, 63, 64]. This begs the question: Can model-free RL without human data generate agents that can collaborate with novel humans? ",
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+    {
+        "type": "text",
+        "text": "We seek an answer to this question in the space of common-payoff games, where all agents work towards a shared goal and receive the same reward. Self-play (SP), in which an agent learns from repeated games played against copies of itself, does not produce agents that generalize well to novel co-players [10, 11, 21, 44]. Intuitively, this is because agents trained in self-play only ever need to coordinate with themselves, and so make for brittle and stubborn collaborators with new partners who act differently. Population play (PP) trains a population of agents, all of whom interact with each other $\\pmb { \\| 3 9 \\| }$ . While PP can generate agents capable of cooperation with humans in competitive team games $\\textcircled { 1 3 4 } \\textcircled { 1 }$ , it still fails to produce robust partners for novel humans in pure common-payoff settings [12]. PP in common-payoff settings naturally encourages agents to play the same way, reducing strategic diversity and producing agents not so different from self-play $\\dot { \\mathbb { B } } \\dot { \\mathbb { 4 } } \\mathbb { I }$ . ",
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+        "image_caption": [
+            "Figure 1: In this work, we evaluate a variety of agent training methods (Section 2) in zero-shot coordination with agents (Section 4). We then run a human-agent collaborative study designed to elicit human preferences over agents (Section 5) "
+        ],
+        "image_footnote": [],
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+        "text": "Our approach starts with the intuition that the key to producing robust agent collaborators is exposure to diverse training partners. We find that a surprisingly simple strategy is effective in generating sufficient diversity. We train $N$ self-play agents varying only their random seed for neural network initialization. Periodically during training, we save agent “checkpoints” representing their strategy at that point in time. Then, we train an agent partner as the best-response to both the fully-trained agents and their past checkpoints. The different checkpoints simulate different skill levels, and the different random seeds simulate breaking symmetries in different ways. We refer to this agent training procedure as Fictitious $\\mathbf { C o }$ -Play (FCP) for its relationship to fictitious self-play [7, 27, 28, 69]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate FCP in a fully-observable two-player common-payoff collaborative cooking simulator. Based on the game Overcooked $\\boldsymbol { \\left[ \\left[ 2 5 \\right] \\right] }$ , it has recently been proposed as a coordination challenge for AI [12, 50, 70]. State-of-the-art performance in producing agents capable of generalization to novel humans was achieved in $[ \\mathbb { 1 2 } ]$ via behavioral cloning (BC) of human data. More precisely, BC was used to produce models that can stand in as human proxies during training in simulation, a method we call behavioral cloning play (BCP). We demonstrate that FCP outperforms BCP in generalizing to both novel agent and human partners, and that humans express a significant preference for partnering with FCP over BCP. Our method avoids the cost and potential privacy concerns of collecting human data for training, while achieving better outcomes for humans at test time. ",
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+    {
+        "type": "text",
+        "text": "We summarize the novel contributions of this paper as follows: ",
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+    {
+        "type": "text",
+        "text": "1. We propose Fictitious Co-Play (FCP) to train agents capable of zero-shot coordination with humans (Section 2.1).   \n2. We demonstrate that FCP agents generalize better than SP, PP, and BCP in zero-shot coordination with a variety of held-out agents (Section 4.2).   \n3. We propose a rigorous human-agent interaction study with behavioral analysis and participant feedback (Section $\\boxed { 5 . 1 }$ .   \n4. We demonstrate that FCP significantly outperforms the BCP state-of-the-art, both in task score and in human partner preference (Section $5 . 2 )$ . ",
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+        "type": "text",
+        "text": "2 Methods ",
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+        "type": "text",
+        "text": "2.1 Fictitious Co-Play (FCP) ",
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+        "type": "text",
+        "text": "Diverse training conditions have been shown to make agents more robust, from environmental variations (i.e. domain randomization $\\textcircled { 1 5 4 } , \\textcircled { 5 6 } , \\textcircled { 6 7 } \\textcircled { 1 }$ to heterogeneity in training partners $\\left[ \\left[ 6 9 \\right] \\right]$ . We seek to train agents that are robust partners for humans in common-payoff games, and so extend this line of work to that setting. ",
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+    {
+        "type": "text",
+        "text": "One important challenge in collaborating with novel partners is dealing with symmetries $\\textcircled { \\left| 3 1 \\right| }$ . For example, two agents A and B facing each other may move past each other by A going left and B going right, or vice versa. Both are valid solutions, but a good agent partner will adaptively switch between these conventions if a human clearly prefers one over the other. A second important challenge is dealing with variations in skill level. Good agent partners should be able to assist both highly-skilled partners, as well as partners who are still learning. ",
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+        "image_caption": [
+            "Figure 2: The four agent training methods we evaluate in this work. Self-play (SP) where an agent learns with itself, population-play (PP) where a population of agents are co-trained together, and behavioral cloning play (BCP) where data from human games is used to create a behaviorally cloned agent with which an RL agent is then trained. In our method, Fictitious Co-Play (FCP), $N$ self-play agents are trained independently and checkpointed throughout training. An agent is then trained to best respond to the entire population of SP agents and their checkpoints. "
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+        "text": "Fictitious co-play (FCP) is a simple two-stage approach for training agents that overcomes both of these challenges (Figure $\\bigstar$ right). In the first stage, we train a diverse pool of partners. To allow the pool to represent different symmetry breaking conventions, we train $N$ partner agents in self-play. Since these partners are trained independently, they can arrive at different arbitrary conventions for breaking symmetries. To allow the pool to represent different skill levels, we use multiple checkpoints of each self-play partner throughout training. The final checkpoint represents a fully-trained “skillful” partner, while earlier checkpoints represent less skilled partners. Notably, by using multiple checkpoints per partner, this additional diversity in skill incurs no extra training cost. ",
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+        "type": "text",
+        "text": "In the second stage, we train an FCP agent as the best response to the pool of diverse partners created in the first stage. Importantly, the partner parameters are frozen and thus FCP must learn to adapt to partners, rather than expect partners to adapt to it. In this way, FCP agents are prepared to follow the lead of human partners, and learn a general policy across a range of strategies and skills. We call our method “fictitious” co-play for its relationship to fictitious self-play in which competitive agents are trained with past checkpoints (in that case, to avoid strategy cycling) [7, 27, 28, 39, 69]. ",
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+        "text": "2.2 Baselines and ablations ",
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+        "text": "We compare FCP agents to the three baseline training methods listed below, each varying only in their set of training partners, with the RL algorithm and architecture consistent across all agents: ",
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+        "type": "text",
+        "text": "1. Self-play (SP), where agents learn solely through interaction with themselves.   \n2. Population-play (PP), where a population of agents are co-trained through random pairings.   \n3. Behavioral cloning play (BCP), where an agent is trained with a BC model of a human [12]. ",
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+        "type": "text",
+        "text": "We also evaluate three variations on FCP to better understand the conditions for its success: ",
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+        "text": "1. To test the importance of including past checkpoints in training, we evaluate an ablation of FCP in which agents are trained only with the converged checkpoints of their partners $\\mathrm { F C P } _ { - T }$ for “FCP minus time”). 2. To test whether FCP would benefit from additional diversity in its partner population, we evaluate an augmentation of FCP in which the population of SP partners varies not just in random seed, but also in architecture $\\operatorname { F C P } _ { + A }$ for “FCP plus architectural variation”). 3. To test whether architectural variation can serve as a full replacement for playing with past checkpoints, we evaluate the combination of both modifications $( \\mathrm { F C P } _ { - T , + A } )$ . ",
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+        "type": "text",
+        "text": "2.3 Environment ",
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+        "type": "text",
+        "text": "Following prior work on zero-shot coordination in human-agent interaction, we study the Overcooked environment (see Figure 3) [12, 13, 38, 50, 70]. We draw particular inspiration from the environment in Carroll et al. [12]. For full details, see Appendix A. ",
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+    {
+        "type": "text",
+        "text": "In this environment, players are placed into a gridworld kitchen as chefs and tasked with delivering as many cooked dishes of tomato soup as possible within an episode. This involves a series of sequential high-level actions to which both players can contribute: collecting tomatoes, depositing them into cooking pots, letting the tomatoes cook into soup, collecting a dish, getting the soup, and delivering it. Upon a successful delivery, both players are rewarded equally. ",
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+        "type": "text",
+        "text": "To effectively complete the task, players must learn to navigate the kitchen and interact with objects in the correct order, all while maintaining awareness of their partner’s behavior to coordinate with them. This environment therefore presents the challenges of both movement and strategic coordination. ",
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+        "type": "text",
+        "text": "Each player observes an egocentric RGB view of the world, and at every step can perform one of six actions: stand still, move {up, down, left, right}, interact. The behavior ofLÈ¡áyҏįŘį\u000eºµò‹µÒ¡y® interact varies based on the cell which the player is facing (e.g. place tomato on counter). ",
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+    {
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+        "img_path": "images/48e8af75423e0fa5e5c23e2088c49716c4c6044ee3579f26cdd9911dbb4b4925.jpg",
+        "image_caption": [
+            "Figure 3: The Overcooked environment: a two-player common-payoff game in which players must coordinate to cook and deliver soup. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "img_path": "images/2b3253fcaac0677dbb15cc1ed5aa0663e561ff52605ba15bd61605f96c716872.jpg",
+        "image_caption": [
+            "Figure 4: Layouts: the kitchens which agents and humans play in, each emphasizing different coordination strategies. Highlighted in bold are the terms used to refer to each in the rest of this paper. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "2.4 Implementation details ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Here we highlight several key implementation details for our training methods. For full details, including the architectures, hyperparameters, and compute used, please see Appendix B. ",
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+        "type": "text",
+        "text": "For our reinforcement learning agents, we use the V-MPO $\\pmb { \\Vert 6 5 \\Vert }$ algorithm along with a ResNet [26] plus LSTM $\\mathbb { \\left| \\bigstar \\bigstar \\right\\| }$ architecture which we found led to optimal behavior across all layouts. Agents are trained using a distributed set of environments running in parallel $\\textcircled { 1 1 7 }$ , each sampling two agents from the training population to play together every episode. ",
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+        "type": "text",
+        "text": "Both PP and FCP are trained with a population size of $N = 3 2$ agents which are sampled uniformly. For FCP, we use 3 checkpoints for each agent, therefore incurring no additional training burden: (1) at initialization (i.e. a low-skilled agent), (2) at the end of training (i.e. a fully-trained expert agent), and (3) at the middle of training, defined as when the agent reaches $50 \\%$ of its final reward (i.e. an average-skilled agent). When varying architecture for the training partners of the $\\operatorname { F C P } _ { + A }$ and $\\mathrm { F C P } _ { - T , + A }$ variants, we vary whether the partners use memory (i.e. LSTM vs not) and the width of their policy and value networks (i.e. 16 vs 256). In total, we train 8 agents for each of the 4 combinations, leaving the total population size of $N = 3 2$ unchanged, ensuring a fair comparison. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "To train agents via behavioral cloning $\\left[ \\left[ 5 8 \\right] \\right]$ , we use the open-source Acme $\\pmb { \\mathbb { B } } 0 \\|$ to learn a policy from human gameplay data. Specifically, we collected 5 human-human trajectories of length 1200 time steps for each of the 5 layouts, resulting in 60k total environment steps. We divide this data in half and train two BC agents: (1) a partner for training a BCP agent, and (2) a “human proxy” partner for agent-agent evaluation. Following Carroll et al. $\\mathbb { \\overline { { \\lVert \\lambda \\rVert } } }$ , we use a set of feature-based observations for the agents (as opposed to RGB) and generate comparable results: performance is higher on 3 layouts (asymmetric, cramped, and ring) but poorer on the other 2 (circuit and forced). ",
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+    {
+        "type": "text",
+        "text": "3 Related work ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Ad-hoc team play There is a large and diverse body of literature on ad-hoc team-play $ { \\mathbb { B } } ,  { \\mathbb { G } } 6  { \\mathbb { I } }$ , also known as zero-shot coordination $[ \\overbrace { 3 \\mathrm { 1 } } ]$ . Prior work based in game-theoretic settings has suggested the benefits of planning $\\mathbb { [ [  \\mathrm { { 7 1 } ] }  }$ , online learning $\\mathbb { \\left[ \\left. 5 1 \\right] \\right. }$ , and novel solution concepts $\\bar { \\mathbb { D } }$ , to name a few examples. More recently, multi-agent deep reinforcement learning has provided the tools to scale to more complex gridworld or continuous control settings, leading to work on hierarchical social planning $\\widehat { \\left\\| 3 6 \\right\\| }$ , adapting to existing social conventions $\\checkmark$ , trajectory diversity $\\lVert \\boldsymbol { \\mathsf { A } } \\boldsymbol { \\mathsf { S } } \\rVert$ , and theory of mind [14]. Ad-hoc team-play among novel agent partners is also an object of active study in the emergent communication literature [10, 11, 43]. This prior work has tended to focus on generalization to held-out agent partners as a proxy for human co-players. ",
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+    {
+        "type": "text",
+        "text": "Collaborative play with novel humans has been evaluated more actively in the context of training agent assistants; see for instance [57, 68]. To our knowledge, our FCP agents represent the stateof-the-art in coordinating with novel human partners on an equal footing of capabilities in a rich gridworld environment, as measured by the challenge tasks in Carroll et al. [12]. ",
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+    {
+        "type": "text",
+        "text": "Diversity in multi-agent reinforcement learning In multi-agent reinforcement learning, agents that train with behaviorally diverse populations of game partners tend to demonstrate stronger performance than their self-play counterparts. For example, across a range of multi-agent games, generalization to held-out populations can be improved by training larger and more diverse populations [13, 42, 50]. In mixed-motive settings, cooperation among agents can be encouraged through social diversity, such as in player preferences and rewards [3, 47, 49]. Similarly, competitiveness can be optimized through selective matchmaking between increasingly diverse agents $\\pm \\boxed { 1 2 4 } \\boxed { 3 9 } \\boxed { 6 9 }$ . ",
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+    {
+        "type": "text",
+        "text": "Despite the increased focus on improving multi-agent performance, evaluation has typically been constrained to agent-agent settings. High-performing agents have infrequently been evaluated with humans, particularly in non-competitive domains $\\dot { \\left. \\overline { { \\dot { \\left. \\dot { \\theta } \\right\\| } } } \\right. }$ . We add to this growing literature, showing that training with diversity is a powerful approach for effective human-agent collaboration. ",
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+    {
+        "type": "text",
+        "text": "Human-agent interaction In recent years, increased attention has been directed toward designing machine learning agents capable of collaborating with humans [41, 57, 68, 72] (see also $\\textcircled { 1 1 6 } \\textcircled { }$ for a broader review on Cooperative AI). Tylkin et al. $\\lVert \\overline { { 6 8 } } \\rVert$ is particularly notable in also demonstrating that partially trained agents can be useful learning targets for human helpers, although in a different domain (cooperative Atari). Our method, FCP, can be seen as extending theirs by training with multiple “skill levels” and random seeds, rather than just one, which we demonstrate to be crucial to our agents’ performance (Tables 1 and 2 and Figure 7b). ",
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+        "text": "A key preceding entry in this research area is Carroll et al. $[ \\mathbb { 1 2 } ]$ , who similarly investigated humanagent coordination in Overcooked. We use their method (BCP) as a baseline throughout our experiments (Section $\\boxed { 2 . 2 }$ . Relative to BCP, our approach removes the need for the expensive step of human data collection for agent training. Furthermore, through our novel human-agent experimental design, we go beyond objective performance metrics to compare the subjective preferences that agents generate. For a detailed comparison of methods and results, see Appendix E. ",
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+    {
+        "type": "text",
+        "text": "4 Zero-shot coordination with agents ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we evaluate our FCP agent, its ablations, and the baselines with held-out agents. ",
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+    {
+        "type": "text",
+        "text": "4.1 Evaluation method: collaborative evaluation with agent partners ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our primary concern in this work is generalization to novel human partners (as investigated in Section $\\textcircled{5}$ . However, just as collecting human-human data for behavioral cloning is expensive, so too is evaluating agents with humans. Consequently, we instead use generalization to held-out agent partners as a cheap proxy of performance with humans. This is then used to guide our model selection process, allowing us to be more targeted with the agents we select for our human-agent evaluations. ",
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+    {
+        "type": "text",
+        "text": "We evaluate with three held-out populations: ",
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+    {
+        "type": "text",
+        "text": "1. A BC model trained on human data, $H _ { \\mathrm { p r o x y } }$ , intended as a proxy of generalization to humans, as done by Carroll et al. $[ \\overbrace { | 1 2 | }$ . 2. A set of self-play agents varying in seed, architecture, and training time (specifically, heldout seeds of the $N = 3 2$ partners trained for the $\\mathrm { F C P } _ { + A }$ agent; see Section 2.4). These are intended to test generalization to a diverse yet still skillful population. 3. Randomly initialized agents intended to test generalization to low-skill partners. ",
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+        "type": "text",
+        "text": "For all results, we report the average number of deliveries made by both players within an episode, aggregated across the 5 different layouts from Figure $\\boxed { 4 }$ (with the per-layout results reported in Appendix $\\underline { { \\overline { { ( \\mathrm { C . 2 } ) } } } }$ . We estimate mean and standard deviation across 5 random seeds. For each seed, we evaluate the agent with all members of the held-out population for 10 episodes per agent-partner pair. ",
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+    {
+        "type": "text",
+        "text": "4.2 Results ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Finding 1: FCP significantly outperforms all baselines ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To begin, we compare our FCP agent and the baselines when partnered with the three held-out populations introduced above. As can be seen in Figure $\\boxed { 5 }$ FCP significantly outperforms all baselines when partnered with all three held-out populations. Notably, it performs better than BCP with $H _ { \\mathrm { p r o x y } }$ even though BCP trains with such a model and FCP does not. Similar to Carroll et al. $\\mathbb { \\lVert 1 2 \\rVert }$ , we find that BCP significantly outscores SP. ",
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+    {
+        "type": "text",
+        "text": "When paired with a randomly initialized partner which behaves suboptimally, we see an even greater difference between FCP and the baselines. Given that FCP is trained with non-held-out versions of such agents, it may not be surprising that it does so well with partners that behave poorly. However, what is surprising is how brittle the other training methods are. This suggests that they may not perform well with humans who are not highly skilled players, which we will see in Section 5. ",
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+    {
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+        "img_path": "images/8083dcac415b6ef9624e1aa4f428a0a33c3e9d47ce8903df3798f3e80ef0427c.jpg",
+        "image_caption": [
+            "Figure 5: Agent-agent collaborative evaluation: Performance of each agent when partnered with each of the held-out populations (Section $4 . 1 )$ in episodes of length $T = 5 4 0$ . Importantly, FCP scores higher than all baselines with a variety of test partners. Error bars represent standard deviation over five random training seeds. Plots aggregate data across kitchen layouts; results calculated by individual layout can be found in Appendix C.2. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finding 2: Training with past checkpoints is the most beneficial variation for performance Next, we investigate how the different training partner variations influence FCP’s performance. In particular, we separately ablate the past checkpoints $( T )$ and architecture $( A )$ variations, evaluating them with the same partners as in Figure 5. The results of this evaluation are presented in Table 1. Comparing the FCP and $\\mathrm { F C P } _ { - T }$ columns, we see that removing past checkpoints from training significantly reduces performance. Comparing the FCP and $\\operatorname { F C P } _ { + A }$ columns, we see that adding architectural variation to the training population offers no improvement over training with past ",
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+    {
+        "type": "table",
+        "img_path": "images/56b2734b6d3a9e5266ca9b0dd8d3ba9f6d27df29190a11ac23347bb9e9fb3c37.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Partner</td><td>FCP</td><td>FCP-T</td><td>FCP+A</td><td>FCP-T,+A</td></tr><tr><td>Hproxy</td><td>10.6± 0.5</td><td>4.7± 0.4</td><td>9.9±0.6</td><td>7.0±0.8</td></tr><tr><td>Diverse SP</td><td>11.2 ± 0.1</td><td>6.9 ± 0.1</td><td>11.1 ± 0.4</td><td>8.6 ± 0.4</td></tr><tr><td>Random</td><td>8.6± 0.2</td><td>1.0 ± 0.1</td><td>8.4±0.4</td><td>3.2 ± 0.5</td></tr></table>",
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+        "type": "text",
+        "text": "Table 1: Ablation results: Performance of each variation of FCP – training with past partner checkpoints $T$ for time) and adding partner variation in architecture $( A )$ . Scores are mean deliveries with standard deviation over 5 random seeds. Notably, we find that the inclusion of past checkpoints is essential for strong performance $( \\mathrm { F C P } > \\mathrm { F C P } _ { - T }$ ), and additionally including architectural variation does not improve performance $( \\mathrm { F C P } \\approx \\mathrm { F C P } _ { + A . }$ ). However, architectural variation is better than no variation, improving performance when past checkpoints are not available $( \\mathrm { F C P } _ { - T , + A } > \\mathrm { F C P } _ { - T , }$ ). ",
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+        "type": "text",
+        "text": "checkpoints. However, comparing the $\\mathrm { F C P } _ { - T }$ and $\\mathrm { F C P } _ { - T , + A }$ columns, we see that without training with past checkpoints, architectural variation in the population does improve performance. ",
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+        "type": "text",
+        "text": "5 Zero-shot coordination with humans ",
+        "text_level": 1,
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+        "text": "Ultimately, our goal is to develop agents capable of coordinating with novel human partners. In this section, we run an online study to evaluate our FCP agent and the baseline agents in collaborative play with human partners. ",
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+            "Figure 6: Human-agent collaborative study: For our human-agent collaboration study, we recruited participants online to play games with FCP and baseline agents. Participants played a randomized sequence of episodes with different agent partners and kitchen layouts. After every two episodes, participants reported the direction and strength of their preference between their last two partners. "
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+        "type": "text",
+        "text": "5.1 Evaluation method: collaborative evaluation with human participants ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To test how effectively FCP’s performance generalizes to human partners, we recruited participants from Prolific $\\mathbb { 1 1 8 } , \\lvert 5 5 \\rvert$ for an online collaboration study $N = 1 1 4$ ; $3 7 . 7 \\%$ female, $5 9 . 6 \\%$ male, $1 . 8 \\%$ nonbinary; median age between 25–34 years). We used a within-participant design for the study: each participant played with a full cohort of agents (i.e. generated through every training method). This design allowed us to evaluate both objective performance as well as subjective preferences. ",
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+        "type": "text",
+        "text": "Participants first read game instructions and played a short tutorial episode guiding them through the dish preparation sequence (see Appendix $\\underline { { \\overline { { \\mathbb { D . 1 . 1 } } } } }$ for instruction text and study screenshots). Participants then played 20 episodes with a randomized sequence of agent partners and kitchen layouts. Episodes lasted $T = 3 0 0$ steps (1 minute) each. After every two episodes, participants reported their preference over the agent partners from those episodes on a five-point Likert-type scale. After playing all 20 episodes, participants completed a debrief questionnaire collecting standard demographic information and open-ended feedback on the study. Our statistical analysis below primarily relies upon the repeated-measures analysis of variance (ANOVA) method. See Appendix D for additional details of our study design and analysis, including independent ethical review. ",
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+        "type": "text",
+        "text": "5.2 Results ",
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+        "type": "text",
+        "text": "Finding 1: FCP coordinates best with humans, achieving the highest score across maps To begin, we compare the objective team performance supported by our FCP and baseline agents. The strong FCP performance observed in agent-agent play generalizes to human-agent collaboration: ",
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+        "type": "text",
+        "text": "the FCP-human teams significantly outperform all other agent-human teams, achieving the highest average scores across maps, every $p < 0 . 0 0 1$ (Figure $\\lvert \\overline { { 7 \\mathrm { a } } } \\rvert$ , while performing as well as or better than the other teams on each individual map (see Appendix $\\mathbf { D } . 3 )$ . Echoing the results from our agent-agent ablation experiments (Table $^ { 1 ) }$ , the inclusion of past checkpoints in training proves critical to FCP’s strong performance, $p < 0 . 0 { \\overline { { 0 1 } } }$ (Figure $\\textcircled { 7 6 }$ . Similar to Carroll et al. $\\mathbb { \\lVert \\rVert }$ , we find that BCP outscores SP when collaborating with human players, $p < 0 . 0 0 1$ . ",
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+        "type": "text",
+        "text": "Finding 2: Participants prefer FCP over all baselines ",
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+        "type": "text",
+        "text": "FCP’s strong collaborative performance carries over to our participants’ subjective partner preferences. Participants expressed a significant preference for FCP partners over all other agents, including BCP, with every $p < 0 . 0 5$ (Figure $\\bar { 7 \\mathrm { c } } )$ . Notably, while human-BCP and human-PP teams did not significantly differ in their completed deliveries, participants reported significantly preferring BCP over PP, $p = 0 . 0 0 3$ , highlighting the informativeness of our subjective analysis. ",
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+        "image_caption": [
+            "Figure 7: Human-agent collaborative evaluation: Evaluation and preference metrics from humanagent play in episodes of length $T = 3 0 0$ . Error bars represents $9 5 \\%$ confidence intervals, calculated over episodes. Plots aggregate data across kitchen layouts; results calculated by individual layout can be found in Appendix D.3. "
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+        "type": "text",
+        "text": "5.3 Exploratory behavioral analysis ",
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+        "type": "text",
+        "text": "To better understand how the human-agent scores and preferences may have arisen, here we analyze the resulting action trajectories of each human and agent player in our experiment. ",
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+        "img_path": "images/65959c0ab23a18c0f8de72bc451d199e7d9d9cc2698e7ca2ac4f320e26dac829.jpg",
+        "image_caption": [],
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+    {
+        "type": "text",
+        "text": "Figure 8: Behavioral analysis: (a) FCP is able to move most frequently $3 5 \\%$ of the time), corresponding to the best movement coordination with human partners. (b) FCP exhibits the most equal preferences over cooking pots (0.11 difference), aligning with human preferences. Values are calculated as the absolute difference in preferences between the two pots; 1 indicates that the player only uses one of the two available pots, while 0 indicates that the player uses both pots equally. ",
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+    {
+        "type": "text",
+        "text": "Finding 1: FCP exhibits the best movement coordination with humans ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "First, we investigate how much each player moves in an episode (Figure $\\textcircled { 8 \\mathrm { a } }$ , where moving in a higher fraction of timesteps may suggest fewer collisions and thus better coordination with a partner. Notably, we observe two results: (1) humans rarely move, a behavior which is out-of-distribution for typical training methods (e.g. SP, PP) but is seen in the training distribution for BCP and FCP. ",
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+    {
+        "type": "text",
+        "text": "(2) FCP moves the most on all layouts other than Forced, suggesting it is better at coordinating its movement strategy with its partner. This result was also reported by human participants, for example: “I noticed that some of my partners seemed to know they needed to move around me, while others seemed to get ‘stuck’ until I moved out of their way” (see Appendix D for more examples). ",
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+    {
+        "type": "text",
+        "text": "Finding 2: FCP’s preferences over cooking pots aligns best with that of humans ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Next, we investigate whether there was a preference for a specific cooking pot in the layouts which included two cooking pots (Figure $\\textcircled { 8 6 }$ . To do this, we calculate the difference in the number of times each pot was used by each player, where a high value indicates a strong preference for one pot and a low value indicates more equal preference for the two pots. ",
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+    {
+        "type": "text",
+        "text": "As can be seen in the FCP column, our agent typically has the most aligned preferences with that of humans (0.11 for FCP to 0.14 for humans). Behaviorally speaking, this means that our agent prefers one cooking pot over the other $5 5 . 5 \\%$ of the time (i.e. a 0.11 point difference). In contrast, all other agents have a strong preference for a single pot. This is a non-adaptive strategy which generalizes poorly to typical human behavior of using both pots, leading to worse performance. ",
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+    {
+        "type": "text",
+        "text": "6 Discussion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Summary In this work, we investigated the challenging problem of zero-shot collaboration with humans without using human data in the training pipeline. To accomplish this, we introduced Fictitious Co-Play (FCP) – a surprisingly simple yet effective method based on creating a diverse set of training partners. We found that FCP agents scored significantly higher than all baselines when partnered with both novel agent and human partners. Furthermore, through a rigorous human-agent experimental design, we also found that humans reported a strong subjective preference to partnering with FCP agents over all baselines. ",
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+    {
+        "type": "text",
+        "text": "Limitations and future work Our method currently relies on the manual process of initially training and selecting a diverse set of partners. This is not only time consuming, but also prone to researcher biases that may negatively influence the behavior of the created agents. Additionally, while we found FCP with a partner population size of $N = 3 2$ sufficient here, for more complex games, FCP may require an unrealistically large partner population size to represent sufficiently diverse strategies. To address these concerns, methods for automatically generating partner diversity for common-payoff games may be important. Possibilities include adaptive population matchmaking as been used in competitive zero-sum games $\\mathbb { \\lVert 6 9 \\rVert }$ , as well as auxiliary objectives that explicitly encourage behavioral diversity [19, 45, 46]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our method requires a known and fixed reward function. We also focus on one domain in order to compare with prior work which has argued that human-in-the-loop training is necessary. Consequently, the resulting agents are only designed to adaptively collaborate on a single task, and not to infer human preferences in general $\\textcircled { 1 1 } \\textcircled { 3 3 } \\textcircled { 5 9 }$ . Moreover, if a task’s reward function is poorly aligned with how humans approach the task, our method may well produce subpar partners, as would any method without access to human data. Thus, additional domains and tasks should be studied to better understand how our method generalizes. Targeted experiments to test specific forms of generalization may be especially helpful in this regard $\\overline { { \\| 3 8 \\| } }$ , as could approaches that procedurally generate environment layouts requiring diverse solutions $\\lVert 2 2 \\rVert$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finally, it may be possible to produce even stronger agent assistants by combining the strengths of FCP (i.e. diversity) and BCP (i.e. human-like play). Indeed, Knott et al. $\\pmb { \\Vert 3 8 \\Vert }$ recently demonstrated that modifying BCP to train with multiple BC partners produces more robust collaboration with held-out agents, a finding that would be interesting to test with human partners. ",
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+    {
+        "type": "text",
+        "text": "Societal impact A challenge for this line of work is ensuring agent behavior is aligned with human values (i.e. the AI value alignment problem [23, 59]). Our method has no guarantees that the resulting policy aligns with the preferences, intentions, or welfare of its potential partners. It likewise does not exclude the possibility that the target being optimized for is harmful (e.g. if the agent’s partner expresses preferences or intentions to harm others). This could therefore produce negative societal effects either if training leads to poor alignment or if agents are optimized for harmful metrics. ",
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+    {
+        "type": "text",
+        "text": "One potential strategy for mitigating these risks is the use of human preference data [15]. Such data could be used to fine-tune and filter trained agents before deployment, encouraging better alignment with human values. A key question in this line of research is how human preference data should be aggregated—or selected, in the case of expert preferences—when our aim is to create socially aligned agents (i.e. agents that are sufficiently aligned for everyone). Relatedly, targeted research on human beliefs and perceptions of AI $\\pm 8 \\jmath$ , and how they steer human-agent interaction, would help inform agent design for positive societal impact. For instance, developers could incorporate specific priors into agents to reinforce tendencies for fair outcomes $\\pm \\mathbb { Z } 0 . \\pm \\mathbb { B } 2 \\mathbb { I }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Conclusion We proposed a method which is both effective at collaborating with humans and simple to implement. We also presented a rigorous and general methodology for evaluating with humans and eliciting their preferences. Together, these establish a strong foundation for future research on the important challenge of human-agent collaboration for benefiting society. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank Mary Cassin for creating the game sprite art; Rohin Shah, Thore Graepel, and Iason Gabriel for feedback on the draft; Lucy Campbell-Gillingham, Tina Zhu, and Saffron Huang for support in evaluating agents with humans; and Max Kleiman-Weiner, Natasha Jaques, Marc Lanctot, Mike Bowling, and Dan Roberts for useful discussions. ",
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+    {
+        "type": "text",
+        "text": "Funding disclosure ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This work was funded solely by DeepMind. The authors declare no competing interests. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "References ",
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+        "bbox": [
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+        "type": "text",
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parse/train/BklfR3EYDH/BklfR3EYDH.md

@@ -0,0 +1,401 @@
+# KEYFRAMING THE FUTURE: DISCOVERING TEMPORAL HIERARCHY WITH KEYFRAME-INPAINTER PREDICTION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+To flexibly and efficiently reason about temporal sequences, abstract representations that compactly represent the important information in the sequence are needed. One way of constructing such representations is by focusing on the important events in a sequence. In this paper, we propose a model that learns both to discover such key events (or keyframes) as well as to represent the sequence in terms of them. We do so using a hierarchical Keyframe-Inpainter (KEYIN) model that first generates keyframes and their temporal placement and then inpaints the sequences between keyframes. We propose a fully differentiable formulation for efficiently learning the keyframe placement. We show that KEYIN finds informative keyframes in several datasets with diverse dynamics. When evaluated on a planning task, KEYIN outperforms other recent proposals for learning hierarchical representations.
+
+# 1 INTRODUCTION
+
+When thinking about the future, humans focus their thoughts on the important things that may happen (When will the plane depart?) without fretting about the minor details that fill each intervening moment (What is the last word I will say to the taxi driver?). Because the vast majority of elements in a temporal sequence contains redundant information, a temporal abstraction can make reasoning and planning both easier and more efficient. How can we build such an abstraction? Consider the example of a lead animator who wants to show what happens in the next scene of a cartoon. Before worrying about every low-level detail, the animator first sketches out the story by keyframing, drawing the moments in time when the important events occur. The scene can then be easily finished by other animators who fill in the rest of the sequence from the story laid out by the keyframes. In this paper, we argue that learning to discover such informative keyframes from raw sequences is an efficient and powerful way to learn to reason about the future.
+
+Our goal is to learn such an abstraction for future image prediction. In contrast, much of the work on future image prediction has focused on frame-by-frame synthesis (Oh et al. (2015); Finn et al. (2016)). This strategy puts an equal emphasis on each frame, irrespective of the redundant content it may contain or its usefulness for reasoning relative to the other predicted frames. Other recent work has considered predictions that “jump” more than one step into the future, but these approaches either used fixed-offset jumps (Buesing et al., 2018) or used heuristics to select the predicted frames (Neitz et al., 2018; Jayaraman et al., 2019; Gregor et al., 2019). In this work, we propose a method that selects the keyframes that are most informative about the full sequence, so as to allow us to reason about the sequence holistically while only using a small subset of the frames. We do so by ensuring that the full sequence can be recovered from the keyframes with an inpainting strategy, similar to how a supporting animator finishes the story keyframed by the lead.
+
+One possible application for a model that discovers informative keyframes is in long-horizon planning. Recently, predictive models have been employed for model-based planning and control (Ebert et al. (2018)). However, they reason about every single future time step, limiting their applicability to short horizon tasks. In contrast, we show that a model that reasons about the future using a small set of informative keyframes enables visual predictive planning for horizons much greater than previously possible by using keyframes as subgoals in a hierarchical planning framework.
+
+![](images/614250f5f00c823ae4387cda2a004a949a64510d762852882ef1b18a5bd49cb6.jpg)  
+Figure 1: Keyframing the future. Instead of predicting one frame after the other, we propose to represent the sequence with the keyframes that depict the interesting moments of the sequence. The remaining frames can be inpainted given the keyframes.
+
+To discover informative frames in raw sequence data, we formulate a hierarchical probabilistic model in which a sequence is represented by a subset of its frames (see Fig. 1). In this two-stage model, a keyframing module represents the keyframes as well as their temporal placement with stochastic latent variables. The images that occur at the timepoints between keyframes are then inferred by an inpainting module. We parametrize this model with a neural network and formulate a variational lower bound on the sequence log-likelihood. Optimizing the resulting objective leads to a model that discovers informative future keyframes that can be easily inpainted to predict the full future sequence.
+
+Our contributions are as follows. We formulate a hierarchical approach for the discovery of informative keyframes using joint keyframing and inpainting (KEYIN). We propose a soft objective that allows us to train the model in a fully differentiable way. We first analyze our model on a simple dataset with stochastic dynamics in a controlled setting and show that it can reliably recover the underlying keyframe structure on visual data. We then show that our model discovers hierarchical temporal structure on more complex datasets of demonstrations: an egocentric gridworld environment and a simulated robotic pushing dataset, which is challenging for current approaches to visual planning. We demonstrate that the hierarchy discovered by KEYIN is useful for planning, and that the resulting approach outperforms other proposed hierarchical and non-hierarchical planning schemes on the pushing task. Specifically, we show that keyframes predicted by KEYIN can serve as useful subgoals that can be reached by a low-level planner, enabling long-horizon, hierarchical control.
+
+# 2 RELATED WORK
+
+Hierarchical temporal structure. Hierarchical neural models for efficiently modeling sequences were proposed in Liu et al. (2015); Buesing et al. (2018). These approaches were further extended to predict with an adaptive step size so as to leverage the natural hierarchical structure in language data (Chung et al., 2016; Kádár et al., 2018). However, these models rely on autoregressive techniques for text generation and applying them to structured data, such as videos, might be impractical.
+
+The video processing community has used keyframe representations as early as 1991 in the MPEG codec (Gall, 1991). Wu et al. (2018) adapted this algorithm in the context of neural compression; however, these approaches use constant offsets between keyframes and thus do not fully reflect the temporal structure of the data. Recently, several neural methods were proposed to leverage such temporal structure. Neitz et al. (2018) and Jayaraman et al. (2019) propose models that find and predict the least uncertain “bottleneck” frames. Gregor et al. (2019) construct a representation that can be used to predict any number of frames into the future. In contrast, we propose an approach for hierarchical video representation that discovers the keyframes that best describe a certain sequence.
+
+In parallel to our work, Kipf et al. (2019) propose a related method for video segmentation via generative modeling. Kipf et al. (2019) focus on using the discovered task boundaries for training hierarchical RL agents, while we show that our model can be used to perform efficient hierarchical planning by representing the sequence with only a small set of keyframes. Also concurrently, Kim et al. (2019) propose a similar method to KEYIN for learning temporal abstractions. While Kim et al. (2019) focuses on learning hierarchical state-space models, we propose a model that operates directly in the observation space and performs joint keyframing and inpainting.
+
+Video modeling. Early approaches to probabilistic video modeling include autoregressive models that factorize the distribution by considering pixels sequentially (Kalchbrenner et al., 2017; Reed et al., 2017). To reason about the images in the video holistically, latent variable approaches were developed based on variational inference (Chung et al., 2015; Rezende et al., 2014; Kingma & Welling, 2014), including (Babaeizadeh et al., 2018; Denton & Fergus, 2018; Lee et al., 2018) and large-scale models such as (Castrejon et al., 2019; Villegas et al., 2019). Kumar et al. (2019) is a recently proposed approach that uses exact inference based on normalizing flows (Dinh et al., 2014; Rezende & Mohamed, 2015). We build on existing video modeling approaches and show how they can be used to learn temporal abstractions with a novel keyframe-based generative model.
+
+Visual planning and model predictive control. We build on recent work that explored applications of learned visual predictive models to planning and control. Several groups (Oh et al., 2015; Finn et al., 2016; Chiappa et al., 2017) have proposed models that predict the consequences of actions taken by an agent given its control output. Recent work (Byravan et al., 2017; Hafner et al., 2018; Ebert et al., 2018) has shown that visual model predictive control based on such models can be applied to a variety of different settings. In this work, we show that the hierarchical representation of a sequence in terms of keyframes improves planning performance in the hierarchical planning setting.
+
+# 3 KEYFRAMING THE FUTURE
+
+Our goal is to develop a model that generates sequences by first predicting key observations and the time steps when they occur and then filling in the remaining observations in between. To achieve this goal, in the following we (i) define a probabilistic model for joint keyframing and inpainting, and (ii) show how a maximum likelihood objective leads to the discovery of keyframe structure.
+
+# 3.1 A PROBABILISTIC MODEL FOR JOINT KEYFRAMING AND INPAINTING
+
+We first describe a probabilistic model for joint keyframing and inpainting of a sequence $I _ { 1 : T }$ . The model consists of two parts: the keyframe predictor and the sequence inpainter (see Fig. 2).
+
+![](images/98bcd37d48a6589ff12be0ddb6202dc874f4766da02360f6c63243bee596b2e9.jpg)  
+Figure 2: A probabilistic model for jointly keyframing and inpainting a future sequence. First, a sequence of keyframes $K ^ { 1 : N }$ is generated, as well as corresponding temporal indices $\tau ^ { 1 : N }$ , defining the structure of the underlying sequence. In the second stage, for each pair of keyframes $K ^ { n }$ and $K ^ { n + 1 }$ , the frames $I _ { \tau ^ { n } : \tau ^ { n + 1 } - 1 }$ are inpainted.
+
+The keyframe predictor takes in $C$ conditioning frames $I _ { c o }$ and produces $N$ keyframes $K ^ { 1 : N }$ as well as the corresponding time indices τ 1:N :
+
+$$
+p ( K ^ { 1 : N } , \tau ^ { 1 : N } | I _ { c o } ) = \prod _ { n } p ( K ^ { n } , \tau ^ { n } | K ^ { 1 : n - 1 } , \tau ^ { 1 : n - 1 } , I _ { c o } ) .
+$$
+
+From each pair of keyframes, the sequence inpainter generates the sequence of frames in between:
+
+$$
+p ( I _ { \tau ^ { n } : \tau ^ { n + 1 } - 1 } | K ^ { n } , K ^ { n + 1 } , \tau ^ { n + 1 } - \tau ^ { n } ) = \prod _ { n } p ( I _ { t } | K ^ { n } , K ^ { n + 1 } , I _ { \tau ^ { n } : t - 1 } , \tau ^ { n + 1 } - \tau ^ { n } ) ,
+$$
+
+which completes the generation of the full sequence. The inpainter additionally observes the number of frames it needs to generate $\tau ^ { n + 1 } - \tau ^ { n }$ . The temporal spacing of the most informative keyframes is data-dependent: shorter keyframe intervals might be required in cases of rapidly fluctuating motion, while longer intervals can be sufficient for steadier motion. Our model handles this by predicting the keyframe indices $\tau$ and inpainting $\tau ^ { n + 1 } - \tau ^ { n }$ frames between each pair of keyframes. We parametrize the prediction of $\tau ^ { n }$ in relative terms by predicting offsets $\delta ^ { n }$ : $\tau ^ { n } \stackrel { - } { = } \tau ^ { n - 1 } \stackrel { . } { + } \delta ^ { n }$ .
+
+# 3.2 KEYFRAME DISCOVERY
+
+To produce a complex multimodal distribution over $K$ we use a per-keyframe latent variable $z$ with prior distribution $p ( z )$ and approximate posterior $q ( z | I , I _ { c o } )$ .1 We construct a variational lower bound
+
+on the likelihood of both $I$ and $K$ as follows2:
+
+$$
+\begin{array} { r l } & { \displaystyle \ln p ( I , K | I _ { c o } ) \geq \mathbb { E } _ { q ( z | I , I _ { c o } ) } \left[ \sum _ { n = 1 } ^ { N } \underbrace { \ln \mathbb { E } _ { p ( \tau ^ { n } , \tau ^ { n + 1 } | z ^ { 1 : n } , I _ { c o } ) } \left[ p ( I _ { \tau ^ { n } : \tau ^ { n + 1 } } | K ^ { n , n + 1 } , \tau ^ { n + 1 } - \tau ^ { n } ) \right] } _ { \mathrm { i n p a i n i t i g } } \right. } \\ & { \quad \quad \quad \left. + \underbrace { \ln p ( K | z , I _ { c o } ) } _ { \mathrm { k e y f r a m i n g } } \right] - \underbrace { D _ { \mathrm { K L } } \left( q ( z | I , I _ { c o } ) | | p ( z ) \right) } _ { \mathrm { r e g u l a r i z a t i o n } } . } \end{array}
+$$
+
+In practice, we use a weight $\beta$ on the KL-divergence term, as is common in amortized variational inference (Higgins et al., 2017; Alemi et al., 2018; Denton & Fergus, 2018).
+
+If a simple model is used for inpainting, most of the representational power of the model has to come from the keyframe predictor. We use a relatively powerful latent variable model for the keyframe predictor and a simpler Gaussian distribution produced with a neural network for inpainting. Because of this structure, the keyframe predictor has to predict keyframes that describe the underlying sequence well enough to allow a simpler inpainting process to maximize the likelihood. We will show that pairing a more flexible keyframe predictor with a simpler inpainter allows our model to discover semantically meaningful keyframes in video data.
+
+# 4 CONTINUOUS RELAXATION BY LINEAR INTERPOLATION IN TIME
+
+Our model can dynamically predict the keyframe placement $\tau ^ { n }$ . However, learning a distribution over the discrete variable $\tau ^ { n }$ is challenging due to the expensive evaluation of the expectation over $p ( \tau ^ { n } | z ^ { 1 : n } , I _ { c o } )$ in the objective in Eq. 3. To be able to evaluate this term efficiently and in a differentiable manner while still learning the keyframe placement, we propose a continuous relaxation of the objective. The placement distribution $\cdot$ defines a probability for each predicted frame to match to a certain frame in the ground truth sequence. Instead of sampling from this distribution to pick a target frame we produce a soft target for each predicted frame by computing the expected target frame, i.e. the weighted sum of all frames in the true sequence, each multiplied with the probability of matching to the predicted frame. When the
+
+![](images/ace5c4f21ee1c810c8b8897f991d0acb4eb2138160c16d1893de517a49bcb006.jpg)  
+Figure 3: Soft keyframe loss in the relaxed formulation. For each predicted keyframe ${ \hat { K } } ^ { n }$ we compute a target image ${ \tilde { K } } ^ { n }$ as the sum of the ground truth images weighted with the corresponding distribution over index $\tau ^ { n }$ . Finally, we compute the reconstruction loss between the estimated image ${ \hat { K } } ^ { n }$ and the soft target ${ \tilde { K } } ^ { n }$ .
+
+entropy of $\tau ^ { n }$ converges to zero, the continuous relaxation objective is equivalent to the original, discrete objective. 3
+
+Keyframe targets. To produce a keyframe target, ${ \tilde { K } } ^ { n }$ , we linearly interpolate between the ground truth images according to the predicted distribution over the keyframe’s temporal placement $\tau ^ { n }$ : $\begin{array} { r } { \tilde { K } ^ { n } = et { } { ' } \sum _ { t } \tau _ { t } ^ { n } I _ { t } } \end{array}$ , where $\tau _ { t } ^ { n }$ is the probability that the $n ^ { t h }$ keyframe occurs at timestep $t$ . This process is depicted in Fig. 3.
+
+We parametrize temporal placement prediction in terms of offsets $\delta$ with a maximum offset of $J$ Because of this, the maximum possible length of the predicted sequence is $N J$ . It is desirable for $J$ to be large enough to be able to capture the distribution of keyframes in the data, but this may lead to the generation of sequences longer than the target $N J > T$ . To correctly compute the value of the relaxed objective in this case, we discard predicted frames at times $> T$ and normalize the placement probability output by the network so that it sums to one over the first $T$ steps. Specifically, for each keyframe we compute this probability as $c ^ { n }$ : $\begin{array} { r } { c ^ { n } = \sum _ { t \leq T } \tau _ { t } ^ { n } } \end{array}$ . The loss corresponding to the last two terms of Eq. (3) then becomes:
+
+![](images/a46c93e9771bc10fe7f96464599f2f60152d74fdb03c092e11f95962ae24de86.jpg)  
+Figure 4: Sequences generated by KEYIN and a method with constant temporal keyframe offset (Jumpy) on Brownian Motion data. Generation is conditioned on the first five frames. The first half of the sequence is shown. Movement direction changes are marked red in the ground truth sequence and predicted keyframes are marked blue. We see that KEYIN can correctly reconstruct the motion as it selects an informative set of keyframes. The sequence generated by the Jumpy method does not reproduce the direction changes since they cannot be inferred from the selected keyframes.
+
+$$
+\mathcal { L } _ { k e y } = \frac { \sum _ { n } c ^ { n } \left( | | \hat { K } ^ { n } - \tilde { K } ^ { n } | | ^ { 2 } + \beta D _ { \mathrm { K L } } \left( q ( z ^ { n } | I _ { - C + 1 : T } , z ^ { 1 : n - 1 } ) | | p ( z ^ { n } ) \right) \right) } { \sum _ { n } c ^ { n } } .
+$$
+
+Inpainting targets. To complete our relaxed objective, for each ground truth frame, we produce a target image composed from the inpainted frames.4 We note that as offsets $\delta$ have a maximum range of $\cdot$ , and in general have non-zero probability on each timestep, the inpainting network needs to produce $J$ frames $\cdot$ between each pair of keyframes $\cdot$ . As in the previous section, the targets for ground truth images are given as an interpolation between generated images weighted by the probability of the predicted frame $\hat { I } _ { j } ^ { n }$ being matched to ground truth frame $I _ { t }$ : $\begin{array} { r } { \tilde { I } _ { t } \ = \ ( \sum _ { n , j } m _ { j , t } ^ { n } \hat { I } _ { j } ^ { n } ) / \sum _ { n , j } m _ { j , t } ^ { n } } \end{array}$ . Here, $m _ { j , t } ^ { n }$ is the probability that the $j$ -th predicted image in segment $n$ has an offset of $t$ from the beginning of the predicted sequence, which can be computed from $\tau ^ { n }$ . To obtain a probability distribution over produced frames, we normalize the result with $\textstyle \sum _ { n , j } m _ { j , t } ^ { n }$ . The full loss for our model is:
+
+$$
+\mathcal { L } _ { t o t a l } = \mathcal { L } _ { k e y } + \beta _ { I } \sum _ { t } | | I _ { t } - \tilde { I } _ { t } | | ^ { 2 } .
+$$
+
+# 5 DEEP VIDEO KEYFRAMING
+
+We show how to instantiate KEYIN with deep neural networks and train it on high-dimensional observations, such as images. We further describe an effective training procedure for KEYIN.
+
+# 5.1 ARCHITECTURE
+
+We use a common encoder-recurrent-decoder architecture (Denton & Fergus (2018); Hafner et al. (2018)). Video frames are first processed with a convolutional encoder module to produce image embeddings $\iota _ { t } = \mathbf { C } \mathbf { N } \mathbf { N } _ { e n c } ( I _ { t } )$ . Inferred frame embeddings $\hat { \iota }$ are decoded with a convolutional decoder $\hat { I } _ { j } ^ { n } = \mathbf { C } \mathbf { N } \mathbf { N } _ { d e c } ( \hat { \iota } _ { j } ^ { n } )$ . The keyframe predictor $p ( K ^ { 1 : N } , \tau ^ { 1 : N } | z ^ { 1 : N } , I _ { c o } )$ is parametrized with a Long Short-Term Memory network (LSTM, Hochreiter & Schmidhuber (1997)). To condition the keyframe predictor on past frames, we initialize its state with the final state of another LSTM that processes the conditioning frames. Similarly, we parametrize the sequence inpainter $p ( I _ { \tau ^ { n } : \tau ^ { n + 1 } } | \dot { K } ^ { n } , K ^ { n + 1 } , \tau ^ { n + 1 } - \tau ^ { n } )$ with an LSTM. We condition the inpainting on both keyframe embeddings, $\hat { \kappa } ^ { n - 1 }$ and $\hat { \kappa } ^ { n }$ , as well as the temporal offset between the two, $\delta ^ { n }$ , by passing these inputs through a multi-layer perceptron that produces the initial state of the inpainting LSTM.
+
+![](images/d02697610133d19680c65f2a746aba88f635d4895a91a5752add180aba804112.jpg)  
+Figure 5: Example generations by KEYIN on (top) Pushing and (bottom) Gridworld data. The generation is conditioned on a single ground truth frame. Twelve of the 30 predicted frames are shown. We observe that for each transition between pushes and each action of the Gridworld agent our network predicts a keyframe either exactly at the timestep of the event or one timestep apart. Note, although agent position is randomized, objects not visible in the first image can be predicted in Gridworld because the maze is fixed across episodes.
+
+We use a Gaussian distribution with identity variance as the output distribution for both the keyframe predictor and the inpainting model and a multinomial distribution for $\delta ^ { n }$ . We parametrize the inference $q ( z ^ { 1 : N } | I _ { - C + 1 : T } )$ with an LSTM with attention over the entire input sequence. The inference distribution is a diagonal covariance Gaussian, and the prior $p ( z ^ { 1 : N } )$ is a unit Gaussian. Further details of the inference procedure are given in Sec. B and Fig. 8 of the Appendix.
+
+# 5.2 TRAINING PROCEDURE
+
+We train our model in two stages. First, we train the sequence inpainter to inpaint between ground truth frames sampled with random offsets, thus learning interpolation strategies for a variety of different inputs. In the second stage, we train the keyframe predictor using the loss from Eq. 5 by feeding the predicted keyframe embeddings to the inpainter. In this stage, the weights of the inpainter are frozen and are only used to backpropagate errors to the rest of the model. We found that this simple two-stage procedure improves optimization of the model.
+
+We use L1 reconstruction losses to train the keyframe predictor. We found that this and adding a reconstruction loss on the predicted embeddings of the keyframes, weighted with a factor $\beta _ { \kappa }$ , improved the ability of the model to produce informative keyframes. Target embeddings are computed using the same soft relaxation used for the target keyframes. More details of the loss computation are given in Sec. E and Algorithm 1 of the Appendix.
+
+# 6 EXPERIMENTS
+
+We evaluate the quality of KEYIN’s representation for future sequences by addressing the following questions: (i) Can it discover and predict informative keyframes? (ii) Can it model complex data distributions? (iii) Is the discovered hierarchy useful for long-horizon hierarchical planning?
+
+Datasets. We evaluate our model on three datasets containing structured long-term behavior. The Structured Brownian motion (SBM) dataset consists of binary image sequences of size $3 2 \times 3 2$ pixels in which a ball randomly changes directions after periods of straight movement of six to eight frames.
+
+The Gridworld Dataset consists of $2 0 \mathrm { k }$ sequences of an agent traversing a maze with different objects. The agent sequentially navigates to objects and interacts with them following a task sketch.We use the same maze for all episodes and randomize the initial position of the agent and the task sketch. We use $6 4 \times 6 4$ pixel image observations and further increase visual complexity by constraining the field of view to a $5 \times 5$ -cells egocentric window.
+
+The Pushing Dataset consists of $5 0 \mathrm { k }$ sequences of a robot arm pushing a puck towards a goal on the opposite side of a wall. Each sequence consists of six consecutive pushes. We vary start and target position of the puck, as well as the placement of the wall. The demonstrations were generated with the MuJoCo simulator (Todorov et al., 2012) at a resolution of $6 4 \times 6 4$ pixels. For more details on the data generation process, see Sec.D of the Appendix.
+
+Further details about the experimental setup are given in Sec. C of the Appendix.
+
+# 6.1 KEYFRAME DISCOVERY
+
+To evaluate KEYIN’s ability to discover keyframes, we train KEYIN on all three datasets with $N = 6$ , which can be interpreted as selecting the $N$ most informative frames from a sequence. We show qualitative examples of keyframe discovery for the SBM dataset in Fig. 4 and for the Gridworld and Pushing datasets in Fig. 5.
+
+Table 1: F1 accuracy score for keyframe discovery on all three datasets. Higher is better.   
+
+<table><tr><td>METHOD</td><td>BROWNIAN</td><td>PUSH</td><td>GRIDWORLD</td></tr><tr><td>RANDOM</td><td>0.15</td><td>0.18</td><td>0.12</td></tr><tr><td>STATIC</td><td>0.21</td><td>0.18</td><td>0.25</td></tr><tr><td>SURPRISE</td><td>0.73</td><td>0.10</td><td>0.32</td></tr><tr><td>KEYIN (OURS)</td><td>0.94</td><td>0.43</td><td>0.42</td></tr></table>
+
+On all datasets the model discovers meaningful keyframes which mark direction changes of the ball, transitions between pushes or interactions with objects, adapting its keyframe prediction patterns to the data. Consequently, the inpainter network is able to produce frames of high visual quality. Misplaced keyframes yield blurry interpolations, as can be seen for the jumpy prediction in Fig. 4. This suggests that keyframes found by KEYIN describe the overall sequences better.
+
+To show that KEYIN discovers informative keyframes, we compare keyframe predictions against an alternative approach that measures the surprise associated with observing a frame given the previous frames. This approach selects keyframes as the $N$ frames with the largest peaks in “surprise” as measured by the KL-divergence $\mathsf { \bar { D } } _ { \mathrm { K L } } [ q ( z _ { t } | I _ { 1 : t } ) | | p ( z _ { t } ) ]$ between the prior and the posterior of a stochastic predictor based on Denton & Fergus (2018) (see Sec. F and Algorithm 2 of the Appendix for details). We provide comparisons to alternative formulations of surprise in Appendix Sec. F, Tab. 3.
+
+For quantitative analysis, we define approximate ground truth keyframes to be the points of direction change for the SBM dataset, the moments when the robot lifts its arm to transitions between pushes, or when the agent interacts with objects in the gridworld. We report F1 scores that capture both the precision and recall of keyframe discovery.
+
+![](images/fa92d6d39fd0214b0a578cc36ccbeccdcc42928d6b7dbf03d009d7a9147cb1e2.jpg)  
+Figure 6: Distribution of trajectories sampled from KEYIN. Each black line denotes one of 100 trajectories of the manipulated object. The obstacle is shown in blue and the initial position in pink. We see that our model covers both modes of the distribution, producing both trajectories that go to the right and to the left of the obstacle.
+
+We additionally compare to random keyframe placement, and a learned but static baseline that is the same for all sequences. The evaluation in Tab. 1 shows that KEYIN discovers better keyframes than alternative methods. The difference is especially large on the more complex Pushing and Gridworld datasets. The surprise-based method does not reason about which frames are most helpful to reconstruct the entire trajectory and thus is unable to discover the correct structure on the more complex datasets. In addition to the F1 scores, we report temporal distance between predicted and annotated keyframes in Appendix, Tab. 4, also indicating that KEYIN is better able to discover the temporal structure in both datasets.
+
+# 6.2 KEYFRAME-BASED VIDEO MODELING
+
+Even though the focus of this work is on discovering temporal structure via keyframing and not on improving video prediction quality, we verify that KEYIN can represent complex data distributions in terms of discovered keyframes and attains high diversity and visual quality. We show sample generations from our model on the Pushing and Gridworld datasets on the supplementary website5.
+
+We see that KeyIn is able to faithfully model complex distributions of video sequences. We further visualize multiple sampled Pushing sequences from our model conditioned on the same start position in Fig. 6, showing that KEYIN is able to cover both modes of the demonstration distribution. We further show that KEYIN compares favorably to prior work on video prediction metrics on sequence modeling in Tab. 5 of the Appendix, and outperforms prior approaches in terms of keyframe modeling in Appendix, Tab. 6.
+
+# 6.3 ROBUSTNESS OF KEYFRAME DETECTION
+
+In the previous sections, we showed that when the sequence can indeed be summarized with $N$ keyframes, KEYIN predicts the keyframes that correspond to our notion of salient frames. However, what happens if we train KEYIN to select a larger or a smaller amount of keyframes?
+
+To evaluate this, we measure KEYIN recall with extra and precision with fewer available keyframes. We note that high precision is unachievable in the first case and high recall is unachievable in the second case, since these problems are misspecified. As these numbers are not
+
+Table 2: Keyframe discovery for varied number of predicted keyframes. The data has approximately 6 keyframes. Uninterpretable entries are omitted for clarity: see the text for details.   
+
+<table><tr><td colspan="2"># KEYFRAMES</td><td>4</td><td>6</td><td>8</td></tr><tr><td>BROWNIAN</td><td>PRECISION RECALL</td><td>0.92 -</td><td>0.92 0.96</td><td>- 0.90</td></tr><tr><td>PUSH</td><td>PRECISION RECALL</td><td>0.30</td><td>0.38</td><td>-</td></tr><tr><td rowspan="2">GRIDWORLD</td><td>PRECISION</td><td>1 0.37</td><td>0.48 0.43</td><td>0.46 -</td></tr><tr><td>RECALL</td><td>1</td><td>0.41</td><td>0.40</td></tr></table>
+
+informative, we do not report them. In Tab. 2, we see that KEYIN is able to find informative keyframes even when $N$ does not exactly match the structure of the data. We further qualitatively show that KEYIN selects a superset or a subset of the original keyframes respectively in Sec. G. This underlines that our method’s ability to discover keyframe structure is robust to the choice of the number of predicted keyframes.
+
+As a first step towards analyzing the robustness of KEYIN under more realistic conditions we report keyframe discovery when trained and tested on sequences with additive Gaussian noise, a noise characteristic commonly found in real-world camera sensors. We find that KEYIN is still able to discover the temporal structure on both the Pushing and the Gridworld dataset. For qualitative and quantitative results, see Appendix Fig. 11 and Tab. 7.
+
+# 6.4 HIERARCHICAL KEYFRAME-BASED PLANNING
+
+We have seen that KEYIN can find frames that correspond to an intuitive notion of keyframes. This demonstrates that the keyframes discovered by KEYIN do indeed capture an abstraction that compactly describes the sequence. In light of this, we hypothesize that an informative set of keyframes contains sufficient information about a sequence to effectively follow the trajectory it shows. To test this, we use the inferred keyframes as subgoals for hierarchical planning in the pushing environment. During task execution, we first plan a sequence of keyframes that reaches the target using our learned keyframe predictor. Specifically, we generate keyframe trajectories from our model by sampling latent variables $z$ from the prior and using them to roll out the keyframe prediction model. We optimize for a sequence of latent variables $z$ that results in a keyframe trajectory which reaches the goal using the Cross-Entropy Method (CEM, Rubinstein & Kroese (2004)). We then execute the plan by using the keyframes as subgoals for a low-level planner. This planner reaches each subgoal via model predictive control using ground truth dynamics, again employing CEM for optimization of the action trajectory. This planning procedure is illustrated in Fig. 7 (left). For more details, see Sec. I and Algs. 3 and 4 of the Appendix.
+
+We find that KEYIN is able to plan coherent subgoal paths towards the final goal that often lead to successful task execution (executions are shown on the supplementary website6). To quantitatively evaluate the keyframes discovered, we compare to alternative subgoal selection schemes: fixed time offset (Jumpy, similar to Buesing et al. (2018)), a method that determines points of peak surprise (Surprise, see Sec. 6.1), and a bottleneck-based subgoal predictor (time-agnostic prediction or TAP, Jayaraman et al. (2019)). We additionally compare to an approach that plans directly towards the final goal using the low-level planner (Flat). We evaluate all methods with the shortest path between the target and the actual position of the object after the plan is executed. All compared methods use the same low-level planner as we only want to measure the quality of the predicted subgoals.
+
+![](images/88ca63f94046a9eb68c76a5247393d213017a4658751186c1e11bef2569d19f1.jpg)  
+Figure 7: Hierarchical planning on the Pushing dataset. Left: We use the model to produce keyframes that represent the sequence between the current observation image and the goal. A low-level planner based on model predictive control produces the actions, $a _ { t }$ , executed to reach each keyframe, until the final goal is reached. Right: Planning performance on a Pushing task. The hierarchy discovered by KEYIN outperforms comparable planning approaches.
+
+<table><tr><td>METHOD</td><td>POSITION ERROR</td><td>SUCCESS RATE</td></tr><tr><td>INTITIAL</td><td>1.32 ± 0.06</td><td></td></tr><tr><td>RANDOM</td><td>1.32 ± 0.07</td><td>-</td></tr><tr><td>FLAT</td><td>0.90 ±0.14</td><td>15.0%</td></tr><tr><td>TAP</td><td>0.80 ±0.16</td><td>23.3%</td></tr><tr><td>SURPRISE</td><td>0.64±0.28</td><td>50.8%</td></tr><tr><td>JUMPY</td><td>0.62 ±0.33</td><td>58.8%</td></tr><tr><td>KEYIN (OURS)</td><td>0.50 ±0.26</td><td>64.2%</td></tr></table>
+
+As shown in Fig. 7 (right), our method outperforms all prior approaches. TAP shows only a moderate increase in performance over the Flat planner, which we attribute to the fact that it fails to predict good subgoals and often simply predicts the final image as the bottleneck. This is likely due to the relatively large stochasticity of our dataset and the absence of the clear bottlenecks that TAP is designed to find. Our method outperforms the planners that use Jumpy and Surprise subgoals. This further confirms that KEYIN is able to produce keyframes that are informative about the underlying trajectory, such that planning toward these keyframes makes it easier to follow the trajectory.
+
+# 7 DISCUSSION
+
+We presented KEYIN, a method for representing a sequence by its informative keyframes by jointly keyframing and inpainting. KEYIN first generates the keyframes of a sequence and their temporal placement and then produces the full sequence by inpainting between keyframes. We showed that KEYIN discovers informative keyframes on several datasets with stochastic dynamics. Furthermore, by using the keyframes for planning, we showed our method outperforms several other hierarchical planning schemes. Our method opens several avenues for future work. First, an improved training procedure that allows end-to-end training is desirable. Second, more powerful hierarchical planning approaches can be designed using the keyframe representation to scale to long-term real-world tasks. Finally, the proposed keyframing method can be applied to a variety of applications, including video summarization, video understanding, and multi-stage hierarchical video prediction.
+
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+![](images/4f6210e7a6ddf4eeba22999d399e0f64326987225db7f68d6afe570c1fd7629b.jpg)  
+Figure 8: Structure of the keyframe inference network. This diagram depicts the procedure to infer the embedding of the $n$ -th keyframe, $\kappa ^ { n }$ , given the previously inferred keyframe embedding $\kappa ^ { n - 1 }$ and the future images. The initial state of $\mathrm { L S T M _ { \mathrm { k e y } } }$ is produced by $\mathrm { L S T M _ { \mathrm { c o n d } } }$ (not shown), which takes the embedding of past images as input. This ensures that the See the text for more details.
+
+A VIDEOS
+
+We include video results on the supplementary website at https://sites.google.com/ view/keyin. The website includes inference samples, prior samples, and exectued trajectories for all methods.
+
+# B ARCHITECTURE DETAILS
+
+We found simple attention over LSTM outputs to be an effective inference procedure. Our approximate inference network $\mathrm { L S T M } _ { i n f }$ outputs $( \kappa _ { t } ^ { i n f } , \zeta _ { t } ) _ { t \leq T }$ , where $\kappa ^ { i n f }$ is an embedding used to compute an attention weight and the $\zeta _ { t }$ are values to be attended over. We compute the posterior distribution over $z ^ { t }$ using a key-value attention mechanism (Bahdanau et al., 2015; Luong et al., 2015):
+
+$$
+\begin{array} { c } { \displaystyle { a _ { n , t } = \exp ( d ( \hat { \kappa } ^ { n - 1 } , \kappa _ { t } ^ { i n f } ) ) } } \\ { \displaystyle { \mu ^ { n } , \sigma ^ { n } = ( \sum _ { t } a _ { n , t } \zeta _ { t } ) / \sum _ { t } a _ { n , t } . } } \end{array}
+$$
+
+The distance metric, $d$ , is the standard inner product. The architecture used for keyframe inference, including the attention mechanism, is depicted in Supplemental Fig. 8.
+
+# C EXPERIMENTAL SETUP
+
+We set the prediction horizon to $T = 3 0$ frames and predict $N = 6$ segments with $J = 1 0$ frames each for the SBM dataset and 6 frames each for the Pushing dataset. We pre-train the interpolator on segments of two to eight frames for Structured Brownian motion data, and two to six frames for Pushing data. The weight on the KL-divergence term for the interpolator VAE is 1e−3. For training the keyframe predictor, we set $\beta _ { K } = 0$ , $\beta _ { \kappa } = 1 , \beta = 5 \mathrm { e } { - 2 }$ . The hyperparameters were hand-tuned. We activate the generated images with a sigmoid and use BCE losses on each color channel to avoid saturation. The convolutional encoder and decoder both have three layers for the Structured Brownian motion dataset and four layers for the Pushing dataset. We use a simple two-layer LSTM with a 256-dimensional state in each layer for all recurrent modules. Each LSTM has a linear projection layer before and after it that projects the observations to and from the correct dimensions. We use the Adam optimizer (Kingma & Ba, 2015) with $\beta _ { 1 } = 0 . 9$ and $\beta _ { 2 } = 0 . 9 9 9$ , batch size of 30, and a learning rate of 2e−4. For more details please refer to the appendix. Each network was trained on a single high-end NVIDIA GPU. We trained the interpolator for 100K iterations, and the keyframe predictor for 200K iterations. The toal training time took about a day.
+
+In the Pushing environment, we use a held-out test set of 120 of sequences. The Structured Brownian Motion dataset is generated automatically and is potentially infinite. We used 1000 testing samples on the Structured Brownian Motion generated using a different random seed.
+
+# D DATA COLLECTION IN THE MUJOCO ENVIRONMENT
+
+The data collection for our pushing dataset was performed in an environment simulated in MuJoCo Todorov et al. (2012). In the environment, a robot arm initialized at the center of the table pushes an object to a goal position located at the other side of a wall-shaped obstacle.
+
+The demonstrations followed a rule-based algorithm that first samples subgoals between the initial position of the object and the goal and then runs a deterministic pushing procedure to the subgoals in order. The ground truth keyframes of the demonstrations were defined by frames at which subgoals were completed.
+
+We subsampled demonstration videos by a factor of two when saving them to the dataset, dropping every other frame in the trajectory and averaging actions of every two consecutive frames. For all datasets we generated for this environment following a rule-based algorithm, we only kept successful demonstrations and dropped the ones that fail to push the object to the goal position within a predefined horizon.
+
+# E DETAILS OF THE LOSS COMPUTATION ALGORITHM
+
+We describe the details of the continuous relaxation loss computation in Algorithm 1.
+
+Note that we efficiently implement computing of the cumulative distributions $\tau$ as a convolution, which allows us to vectorize much of the computation. Computational complexity of the proposed implementation scales linearly with the number of keyframes $N$ , and number of allowed frames per segment $J$ and number of ground truth frames $T$ . The final complexity is $\mathcal { O } ( N T J )$ , which we find in practice to be negligible compared to the time needed for the forward and backward pass.
+
+# F SURPRISE BASELINE
+
+Standard stochastic video prediction methods do not attempt to estimate keyframes, as they are designed to densely estimate future videos frame-by-frame. Accordingly, they cannot be used directly as baselines for keyframe prediction methods, such as KEYIN. However, Denton & Fergus (2018) observe that the variance of the learned prior of a stochastic video prediction model tends to spike before an uncertain event happens. We exploit this observation to find the points of high uncertainty for our strong Surprise baseline. We use the KL divergence between the prior and the approximate posterior $\mathrm { K L } [ q ( \boldsymbol { z } _ { t } | I _ { 1 : t } ) | | p ( \boldsymbol { z } _ { t } ) ]$ to measure the surprise. This quantity can be interpreted as the number of bits needed to encode the latent variable describing the next state, it will be larger if the next state is more stochastic.
+
+We train a stochastic video prediction network with a fixed prior (SVG-FP, Denton & Fergus (2018)) with the same architectures of encoder, decoder, and LSTM as our model. We found that selecting the peaks of suprise works the best for finding true keyframes. The procedure we use to select the keyframes is described in Algorithm 2. In order to find the keyframes in a sequence sampled from the prior, we run the inference network on the generated sequence.
+
+Algorithm 1 Continuous relaxation loss computation
+
+Parameters: Number of ground truth frames $T$ , Number of keyframes $N$
+
+Input: Ground truth frames $I _ { 1 : T }$ , Generated frames $\hat { I } _ { i } ^ { t }$ , generated offset distributions $\delta ^ { n }$ Convert the distributions of interframe offsets $\delta ^ { n }$ to keyframe timesteps $\tau ^ { n }$ . For the first keyframe, $\tau ^ { 1 } = \delta ^ { 1 }$ .
+
+for $t = 2 \dots M$ do
+
+Compute further $\tau ^ { n }$ with chain rule. This can be efficiently computed via convolution:
+
+$$
+\tau ^ { n } = \tau ^ { n - 1 } * \delta ^ { n } , \mathrm { i . e . } \tau _ { t } ^ { n } = \sum _ { j } \tau _ { n - j + 1 } ^ { n - 1 } \delta _ { j } ^ { n } .
+$$
+
+# end for
+
+Compute probabilities of keyframes being within the predicted sequence: $\begin{array} { r } { c ^ { n } = \sum _ { t \leq T } \tau _ { t } ^ { n } } \end{array}$
+
+Compute soft keyframe targets: $\begin{array} { r } { \tilde { K } ^ { n } = \sum _ { t } \tau _ { t } ^ { n } I _ { t } } \end{array}$ .   
+Compute the keyframe loss: $\textstyle \bigl ( \sum _ { n } c ^ { n } | | \hat { K } ^ { n } - \tilde { K } ^ { n } | | ^ { 2 } \bigr ) / \sum _ { n } c ^ { n }$ . Get probabilities of segments ending after particular frames: $\begin{array} { r } { e _ { j } ^ { n } = \sum _ { j > i } \delta _ { j } ^ { n } } \end{array}$ Get distributions of individual frames timesteps: $m _ { j , t } ^ { n } \propto \tau _ { t - j + 1 } ^ { n - 1 } e _ { j } ^ { n }$ .   
+Compute soft individual frames: $\begin{array} { r } { \tilde { I } _ { t } = \sum _ { t , i } m _ { j , t } ^ { n } \hat { I } _ { i } ^ { t } } \end{array}$   
+Compute the sequence loss: $\begin{array} { r } { \sum _ { t } | | I _ { t } - \tilde { I } _ { t } | | ^ { 2 } } \end{array}$ .
+
+# Algorithm 2 Selecting keyframes via Surprise
+
+<table><tr><td>Parameters:Number of ground truth frames N,Desired number of keyframes M Input: Input sequence I1:T, Stochastic Video Prediction model SVG(.) add M - |S| maximum surprise points to S.</td></tr><tr><td>Run the inference network over the sequence: q(z1:T|I1:T) = SVG(I1:T).</td></tr><tr><td>Get the surprise measure: St = KL[q(zt|I1:t)llp(zt)]. Find the set of peak surprise points S where: St &gt; St+1 ∧ St &lt; St-1· if |S|&lt;M then</td></tr></table>
+
+# G ABLATION OF THE NUMBER OF PREDICTED KEYFRAMES
+
+We show qualitative results of training KEYIN with $N = 4$ , 6 (optimal number), and 8 on the SBM dataset in Fig. 9. We observe that if we train KEYIN to select a smaller or a larger number of keyframes than needed, it learns to predict a subset or a superset of the true keyframes, respectively. This property follows from the structure of the model, which encourages the model to predict the keyframes that allow the full sequence to be inpainted. When too few keyframes are available, the model will be unable to put keyframes at all important times, but those it picks must still be good for inpainting. When more keyframes are available than necessary, the model can place the additional keyframes at any time, as only a subset of the keyframes are needed to ensure good inpainting.
+
+<table><tr><td rowspan=1 colspan=1>····</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>+</td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>+</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>+</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1>：</td></tr><tr><td rowspan=1 colspan=2>4 Keyframes</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=2>6Keyframes</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=2>8Keyframes</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr></table>
+
+Figure 9: Qualitative keyframe discovery on the Structured Brownian Motion dataset for varying number of predicted keyframes. Top: Ground truth sequence, keyframes with bold white frame. Bottom: KEYIN keyframe predictions at their predicted temporal placement. Even if the number of predicted keyframes does not match the true number of keyframes KEYIN correctly discovers the keyframes and their temporal placement.
+
+Table 3: We compare different formulations for a surprise-based keyframe detection method: (1) detecting maxima of the KL divergence between prior and posterior in a stochastic prediction model, (2) detecting maxima in the lower bound on data likelihood $\log p$ (ELBO) of a stochastic prediction model, (3) the formulation proposed in Denton & Fergus (2018) that detects maxima of the variance of a learned prior distribution.   
+
+<table><tr><td rowspan="2">DATASET METHOD</td><td colspan="3">PUSH</td><td colspan="3">GRIDWORLD</td></tr><tr><td>F1个</td><td>mind</td><td>mindD √</td><td>F1个</td><td>mind</td><td>mindED √</td></tr><tr><td>KL-SUPRISE</td><td>0.25</td><td>1.44</td><td>2.05</td><td>0.32</td><td>1.42</td><td>1.86</td></tr><tr><td>log p-SURPRISE</td><td>0.24</td><td>1.42</td><td>2.08</td><td>0.31</td><td>1.45</td><td>1.83</td></tr><tr><td>DENTON&amp; FERGUS(2018)</td><td>0.17</td><td>1.73</td><td>2.01</td><td>0.35</td><td>1.10</td><td>1.53</td></tr></table>
+
+Table 4: In addition to the F1 scores we report the minimal temporal distance to the next keyframe as an additional metric that is more graceful with respect to "close misses". Specifically, we report the distance to the next annotated keyframe averaged across predicted keyframes, $\operatorname* { m i n } d _ { \mathrm { K F } } ^ { \mathrm { t r u e } }$ , and, inversely, the distance to the next predicted keyframe for each annotated keyframe, min $d _ { \mathrm { K F } } ^ { \mathrm { p r e d } }$ . For both datasets the distance metrics support the F1 results: KEYIN discovers keyframes that are better aligned with the annotated keyframes than the baselines.   
+
+<table><tr><td rowspan="2">DATASET METHOD</td><td colspan="3">PUSH</td><td colspan="3">GRIDWORLD</td></tr><tr><td>F1↑</td><td>mind ←</td><td>min dKFD</td><td>F1↑</td><td></td><td>4</td></tr><tr><td>STATIC</td><td>0.18</td><td>1.67</td><td>1.25</td><td>0.25</td><td>1.22</td><td>1.07</td></tr><tr><td>SURPRISE</td><td>0.25</td><td>1.44</td><td>2.05</td><td>0.32</td><td>1.42</td><td>1.86</td></tr><tr><td>KEYIN (OURS)</td><td>0.43</td><td>1.25</td><td>1.86</td><td>0.42</td><td>1.03</td><td>0.99</td></tr></table>
+
+# H VIDEO MODELING PERFORMANCE
+
+We further report quantitative results on standard video prediction metrics, Structural Similarity Index (SSIM) and Peak Signal-to-Noise Ratio (PSNR), in Tab. 5. KEYIN is able to match performance of two comparable prior approaches, showing that the keyframe-based modeling is able to represent complex data distributions.
+
+# I PLANNING ALGORITHM
+
+<table><tr><td>Algorithm 3 Planning in the subgoal space.</td></tr><tr><td>Input:Keyframe model KEYIN(.,.),cost function c Input: Start and target images Il and Itarget Set the sampling distribution to the prior: μi=0,Oi=I fori=1...Hdo</td></tr><tr><td>Sample L sequences of latent variables: z1:N~N(μi,Oi)</td></tr><tr><td>Produce L subgoal plans: K1:N = KEYIN(I1, z1:N) Compute cost between generated and true targets: c(KN,Itarget)</td></tr><tr><td>Choose L&#x27;best plans,refit sampling distribution: μi+1,Oi+1= fit(z&#x27;)</td></tr><tr><td>end for Return: Best subgoal plan K1:N</td></tr></table>
+
+![](images/b9666548c2a81c5c54da582412076a2f048882aaa33d51aa50af32bdc8a378ce.jpg)  
+Figure 10: A training sequence and two samples from our model on the Structured Brownian motion dataset. Each image shows an entire trajectory. Our model first samples the keyframes (shown in red), and then deterministically predicts the rest of the sequence. The image resolution was enhanced for viewability.
+
+Table 5: SSIM and PSNR scores on pushing and gridworld dataset. Higher is better.   
+
+<table><tr><td rowspan="2">DATASET METHOD</td><td colspan="2">PUSH</td><td colspan="2">GRIDWORLD</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td>DENTON &amp; FERGUS (2018)</td><td>33.3 ± 0.1</td><td>0.956 ± 0.001</td><td>29.4±0.1</td><td>0.812 ±0.01</td></tr><tr><td>JUMPY</td><td>33.7 ± 0.1</td><td>0.960 ± 0.001</td><td>29.9 ± 0.1</td><td>0.831 ±0.001</td></tr><tr><td>KEYIN (OURS)</td><td>33.4 ± 0.7</td><td>0.959 ± 0.001</td><td>29.3 ±0.1</td><td>0.820 ±0.001</td></tr></table>
+
+To apply the KEYIN model for planning, we use the approach for visual planning outlined in Algorithm 1. At the initial timestep, we use the cross-entropy method (CEM) Rubinstein & Kroese (2004) to select subgoals for the task. To do so, we sample $\tilde { M }$ latent sequences $z _ { \mathrm { 0 } }$ from the prior $\mathcal { N } ( 0 , I )$ and use the keyframe model to retrieve $\tilde { M }$ corresponding keyframe sequences, each with $L$ frames. We define the cost of an image trajectory as the distance between the target image and the final image of each keyframe sequence defined under a domain-specific distance function (see below). In the update step of the CEM algorithm, we rank the trajectories based on their cost and fit a diagonal Gaussian distribution to the latents $z ^ { \prime }$ that generated the $\tilde { M } ^ { \prime } = r \tilde { M }$ best sequences, where $r$ is the elite ratio. We repeat the procedure above for a total of $N$ iterations.
+
+![](images/cdeeac6511e96f013cf49c40378ef5b5dbce4df497f23a83bd31925eef259d91.jpg)
+
+![](images/554ea6f9608590974566c99e6a65c3511e79fedab8610b872b3964e2784c24b0.jpg)  
+Figure 11: Example keyframe detections on noisy sequences. Red frames mark annotated keyframes. Top: Pushing dataset. Bottom: Gridworld dataset. Each triplet depicts top: ground truth sequence with additive Gaussian noise, middle: predicted keyframes at the predicted time steps, bottom: predicted full sequence. KEYIN is reliably able to detect keyframes and reconstruct the full sequence.
+
+Table 6: Keyframe SSIM and PSNR scores on pushing and gridworld dataset. Higher is better.   
+
+<table><tr><td rowspan="2">DATASET METHOD</td><td colspan="2">PUSH</td><td colspan="2">GRIDWORLD</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td>JUMPY</td><td>28.25 ± 0.11</td><td>0.904 ± 0.001</td><td>16.3 ± 0.17</td><td>0.632 ± 0.001</td></tr><tr><td>KEYIN (OURS)</td><td>29.5 ± 0.16</td><td>0.911 ± 0.001</td><td>18.4 ± 0.17</td><td>0.636 ± 0.001</td></tr></table>
+
+Table 7: F1 score and distance to closest annotated / predicted keyframe when trained and tested on sequences with additive Gaussian noise. KEYIN is able to reliably find keyframes on both datasets even when trained and tested on noisy sequences. Even though the F1 score is lower on the Pushing dataset, the distances indicate that the discovered keyframes are well aligned with the annotated keyframes even under noise.   
+
+<table><tr><td>DATASET</td><td colspan="3">PUSH</td><td colspan="3">GRIDWORLD</td></tr><tr><td>METHOD</td><td>F1↑</td><td>mind ↓</td><td>mind ↓</td><td>F1↑</td><td>mind</td><td>mind</td></tr><tr><td>KEYIN,NO-NOISE</td><td>0.43</td><td>1.25</td><td>1.86</td><td>0.42</td><td>1.03</td><td>0.99</td></tr><tr><td>KEYIN, GAUSS-NOISE</td><td>0.25</td><td>1.21</td><td>1.34</td><td>0.43</td><td>1.00</td><td>0.96</td></tr></table>
+
+We define the cost between two frames used during planning as the Euclidean distance between the center pixels of the object in both frames. We recover the center pixel via color-based segmentation of the object. While this cost function is designed for the particular planning environment we are testing on, our algorithm can be easily extended to use alternative, more domain-agnostic cost formulations that have been proposed in the literature Finn & Levine (2017); Ebert et al. (2017; 2018).
+
+After subgoals are selected, we use a CEM based planner to produce rollout trajectories. Similar to the subgoal generation procedure, at each time step, we initially sample $M$ action sequences $\mathbf { \delta } \mathbf { \em u } _ { 0 }$ from the prior $\mathcal { N } ( 0 , I )$ and use the ground truth dynamics of the simulator to retrieve $M$ corresponding image sequences, each with $l$ frames7. We define the cost of an image trajectory as the distance between the target image and the final image of each trajectory. In the update step, we rank the trajectories based on their cost and fit a diagonal Gaussian distribution to the actions $\mathbf { { \boldsymbol { u } } } ^ { \prime }$ that generated the $M ^ { \prime } = r M$ best sequences. After sampling a new set of actions ${ \pmb u } _ { n + 1 }$ from the fitted Gaussian distributions we repeat the procedure above for a total of $N$ iterations.
+
+Finally, we execute the first action in the action sequence corresponding to the best rollout of the final CEM iteration. The action at the next time step is chosen using the same procedure with the next observation as input and reinitialized action distributions. The algorithm terminates when the specified maximal number of planning steps $T _ { \mathrm { m a x } }$ has been executed or the distance to the goal is below a set threshold.
+
+Table 8: Hyperparameters for the visual planning experiments.   
+
+<table><tr><td>planning Parameters</td><td></td></tr><tr><td>Max. planning timesteps (Tmax) Max. per subgoal timesteps (Ts,max) Keyframe prediction horizon (L)</td><td>60 10 6</td></tr><tr><td># keyframe sequences (M)</td><td>200</td></tr><tr><td>planning horizon (l) # planning sequences (M)</td><td>8</td></tr><tr><td>Elite fraction (r = M&#x27;/M)</td><td>200</td></tr><tr><td># refit iterations (N)</td><td>0.05 3</td></tr><tr><td>max.action (amax) dswitch</td><td>1.0 5</td></tr></table>
+
+We switch between planned subgoals if (i) the subgoal is reached, i.e. the distance to the subgoal is below a threshold $d _ { \mathrm { s w i t c h } }$ measured in pixels, or (ii) the current subgoal was not reached for $T _ { s , \mathrm { m a x } }$ execution steps. We use the true goal image as an additional, final subgoal.
+
+<table><tr><td>Algorithm 4 Full two-stage planning algorithm</td></tr><tr><td>Input: Keyframe model K1:L = LSTMkey(I, z1:L)</td></tr><tr><td>Input: Video prediction model It:t+l = LSTMinter(I1:t-1, U2:t+l)</td></tr><tr><td>Input: Subgoal index update heuristics ixt+1 = f(ixt,It,K1:L)</td></tr><tr><td>Input: Start and goal images I1 and Igoal:</td></tr><tr><td>Initialize latents from prior: zo ~ N(O, I). for i=1...Nit do</td></tr><tr><td>Rollout keyframe model for L steps,obtain M future keyframe sequences K1:L.</td></tr><tr><td>Compute distance between final and goal image: c = dist(KL, Igoal).</td></tr><tr><td>Choose M&#x27; best sequences, refit Gaussian distribution: μi+1,oi+1 key key = fit(K).</td></tr><tr><td>key)</td></tr><tr><td>end for Feed best sequence of latents into keyframe model to obtain subgoals: K1L</td></tr><tr><td>LSTMkey(I1, zNit,0).</td></tr><tr><td>Set current subgoal to ix1 = 1.</td></tr><tr><td>for t = 1...Tplan do Perform subgoal update ixt = f(ixt-1,It-1,K:L).</td></tr><tr><td></td></tr><tr><td>Initialize latents from prior: uo ~ N(O,I). fori=O...Nit do</td></tr><tr><td>Rollout prediction model for l steps, obtain M future sequences It:t+l·</td></tr><tr><td></td></tr><tr><td>Compute distance between final and subgoal image: c = dist(It+t, Kixt).</td></tr><tr><td>Choose M&#x27; best sequences, refit Gaussian distribution: μi+1, Oi+1 = fit(ui). Sample new latents from updated distribution: Ui+1 ~ N(μi+1, Oi+1).</td></tr><tr><td>end for Execute uNit,0 and observe next image It.</td></tr></table>
+
+![](images/ae708062f0e0467d47f4b6afa75ad2af322443a7d67b9af0566171cf46339f50.jpg)  
+Figure 13: Sample planning task executions from the test set. From a start state depicted on the left, the robot arm successfully pushes the object into the goal position (semi-transparent object) guided by the KEYIN subgoals. The right side of the figure shows intermediate frames of the execution trajectories.

parse/train/Byg1v1HKDB/Byg1v1HKDB.md

@@ -0,0 +1,419 @@
+# ABDUCTIVE COMMONSENSE REASONING
+
+Chandra Bhagavatula♦, Ronan Le Bras♦, Chaitanya Malaviya♦, Keisuke Sakaguchi♦, Ari Holtzman♦, Hannah Rashkin♦, Doug Downey♦, Scott Wen-tau $\mathbf { Y i h ^ { \alpha } }$ , Yejin Choi♦♥
+
+♦Allen Institute for AI, Seattle, WA, USA, ♣Facebook AI, Seattle, WA, USA   
+♥Paul G. Allen School of Computer Science & Engineering, WA, USA   
+{chandrab,ronanlb,chaitanyam,keisukes}@allenai.org   
+{arih,hannahr,dougd}@allenai.org   
+{yejin}@cs.washington.edu   
+{scottyih}@fb.com∗
+
+# ABSTRACT
+
+Abductive reasoning is inference to the most plausible explanation. For example, if Jenny finds her house in a mess when she returns from work, and remembers that she left a window open, she can hypothesize that a thief broke into her house and caused the mess, as the most plausible explanation. While abduction has long been considered to be at the core of how people interpret and read between the lines in natural language (Hobbs et al., 1988), there has been relatively little research in support of abductive natural language inference and generation.
+
+We present the first study that investigates the viability of language-based abductive reasoning. We introduce a challenge dataset, ART, that consists of over 20k commonsense narrative contexts and $2 0 0 \mathrm { k }$ explanations. Based on this dataset, we conceptualize two new tasks – (i) Abductive NLI: a multiple-choice question answering task for choosing the more likely explanation, and (ii) Abductive NLG: a conditional generation task for explaining given observations in natural language. On Abductive NLI, the best model achieves $6 8 . 9 \%$ accuracy, well below human performance of $9 1 . 4 \%$ . On Abductive NLG, the current best language generators struggle even more, as they lack reasoning capabilities that are trivial for humans. Our analysis leads to new insights into the types of reasoning that deep pre-trained language models fail to perform—despite their strong performance on the related but more narrowly defined task of entailment NLI—pointing to interesting avenues for future research.
+
+# 1 INTRODUCTION
+
+The brain is an abduction machine, continuously trying to prove abductively that the observables in its environment constitute a coherent situation. – Jerry Hobbs, ACL 2013 Lifetime Achievement Award1
+
+Abductive reasoning is inference to the most plausible explanation for incomplete observations (Peirce, 1965a). Figure 1 illustrates an example. Given the incomplete observations about the world that $O _ { 1 }$ : “Jenny cleaned her house and went to work, leaving the window just a crack open.” and sometime later $O _ { 2 }$ : “When Jenny returned home, she saw her house was a mess.”, we can hypothesize different potential explanations and reason about which is the most likely. We can readily rule out $H _ { 3 }$ since it fails to justify the observation $O _ { 2 }$ . While $H _ { 1 }$ and $H _ { 2 }$ are both plausible, the most likely explanation based on commonsense is $H _ { 1 }$ as $H _ { 2 }$ is somewhat implausible given $O _ { 1 }$ .
+
+One crucial observation Peirce makes about abductive reasoning is that abduction is “the only logical operation which introduces any new ideas”, which contrasts with other types of inference such as entailment, that focuses on inferring only such information that is already provided in the premise.
+
+![](images/67ca285bb25d55bf804e5545459afcebac377e1d64228ee3fd79e44a38e2ca1c.jpg)  
+Figure 1: Example of Abductive Reasoning. Given observations $O _ { 1 }$ and $O _ { 2 }$ , the $\alpha \mathbf { N L I }$ task is to select the most plausible explanatory hypothesis. Since the number of hypotheses is massive in any given situation, we make a simplifying assumption in our ART dataset to only choose between a pair of explanations.
+
+Abductive reasoning has long been considered to be at the core of understanding narratives (Hobbs et al., 1988), reading between the lines (Norvig, 1987; Charniak & Shimony, 1990), reasoning about everyday situations (Peirce, 1965b; Andersen, 1973), and counterfactual reasoning (Pearl, 2002; Pearl & Mackenzie, 2018). Despite the broad recognition of its importance, however, the study of abductive reasoning in narrative text has very rarely appeared in the NLP literature, in large part because most previous work on abductive reasoning has focused on formal logic, which has proven to be too rigid to generalize to the full complexity of natural language.
+
+In this paper, we present the first study to investigate the viability of language-based abductive reasoning. This shift from logic-based to language-based reasoning draws inspirations from a significant body of work on language-based entailment (Bowman et al., 2015; Williams et al., 2018b), language-based logic (Lakoff, 1970; MacCartney & Manning, 2007), and language-based commonsense reasoning (Mostafazadeh et al., 2016; Zellers et al., 2018). In particular, we investigate the use of natural language as the representation medium, and probe deep neural models on language-based abductive reasoning.
+
+More concretely, we propose Abductive Natural Language Inference $\mathbf { \alpha } _ { \mathrm { ( \alpha } } \mathrm { \alpha } _ { \mathrm { ( \alpha } } \mathrm { \alpha } _ { \mathrm { ) } }$ and Abductive Natural Language Generation (αNLG) as two novel reasoning tasks in narrative contexts.2 We formulate $\alpha \mathbf { N L I }$ as a multiple-choice task to support easy and reliable automatic evaluation: given a context, the task is to choose the more likely explanation from a given pair of hypotheses choices. We also introduce a new challenge dataset, ART, that consists of 20K narratives accompanied by over 200K explanatory hypothesis.34 We then establish comprehensive baseline performance based on state-of-the-art NLI and language models. The best baseline for $\alpha \mathbf { N L I }$ based on BERT achieves $6 8 . 9 \%$ accuracy, with a considerable gap compared to human performance of $9 1 . 4 \% ( \ S 5 . 2 )$ . The best generative model, based on GPT2, performs well below human performance on the $\alpha \mathbf { N L G }$ task (§5.2). Our analysis leads to insights into the types of reasoning that deep pre-trained language models fail to perform — despite their strong performance on the closely related but different task of entailment NLI — pointing to future research directions.
+
+# 2 TASK DEFINITION
+
+Abductive Natural Language Inference We formulate $\alpha \mathbf { N L I }$ as multiple choice problems consisting of a pair of observations as context and a pair of hypothesis choices. Each instance in ART is defined as follows:
+
+• $O _ { 1 }$ : The observation at time $t _ { 1 }$ .
+
+• $O _ { 2 }$ : The observation at time $t _ { 2 } > t _ { 1 }$ .   
+• $h ^ { + }$ : A plausible hypothesis that explains the two observations $O _ { 1 }$ and $O _ { 2 }$ .   
+• $h ^ { - }$ : An implausible (or less plausible) hypothesis for observations $O _ { 1 }$ and $O _ { 2 }$ .
+
+Given the observations and a pair of hypotheses, the $\alpha$ NLI task is to select the most plausible explanation (hypothesis).
+
+Abductive Natural Language Generation αNLG is the task of generating a valid hypothesis $h ^ { + }$ given the two observations $O _ { 1 }$ and $O _ { 2 }$ . Formally, the task requires to maximize $P ( h ^ { + } | O _ { 1 } , O _ { 2 } )$ .
+
+# 3 MODELS FOR ABDUCTIVE COMMONSENSE REASONING
+
+# 3.1 ABDUCTIVE NATURAL LANGUAGE INFERENCE
+
+A Probabilistic Framework for $\alpha \mathbf { N L I }$ : A distinct feature of the $\alpha \mathbf { N L I }$ task is that it requires jointly considering all available observations and their commonsense implications, to identify the correct hypothesis. Formally, the $\alpha \mathbf { N L I }$ task is to select the hypothesis $h ^ { * }$ that is most probable given the observations.
+
+$$
+h ^ { * } = \arg \operatorname* { m a x } _ { h ^ { i } } P ( H = h ^ { i } | O _ { 1 } , O _ { 2 } )
+$$
+
+Rewriting the objective using Bayes Rule conditioned on $O _ { 1 }$ , we have:
+
+$$
+P ( h ^ { i } | O _ { 1 } , O _ { 2 } ) \propto P ( O _ { 2 } | h ^ { i } , O _ { 1 } ) P ( h ^ { i } | O _ { 1 } )
+$$
+
+We formulate a set of probabilistic models for $\alpha \mathbf { N L I }$ that make various independence assumptions on Equation 2 – starting from a simple baseline that ignores the observations entirely, and building up to a fully joint model. These models are depicted as Bayesian Networks in Figure 2.
+
+![](images/ef656f8cd949aa01dc7b3e52cbcaba4694d9fe72e2f513166ac1e02d18d9a34b.jpg)  
+Figure 2: Illustration of the graphical models described in the probabilistic framework. The “Fully Connected” model can, in theory, combine information from both available observations.
+
+Hypothesis Only: Our simplest model makes the strong assumption that the hypothesis is entirely independent of both observations, i.e. $( H \perp O _ { 1 } , O _ { 2 } )$ , in which case we simply aim to maximize the marginal $P ( H )$ .
+
+First (or Second) Observation Only: Our next two models make weaker assumptions: that the hypothesis depends on only one of the first $O _ { 1 }$ or second $O _ { 2 }$ observation.
+
+Linear Chain: Our next model uses both observations, but considers each observation’s influence on the hypothesis independently, i.e. it does not combine information across the observations. Formally, the model assumes that the three variables $\langle O _ { 1 } , H , O _ { 2 } \rangle$ form a linear Markov chain, where the second observation is conditionally independent of the first, given the hypothesis (i.e. $( O _ { 1 } \perp O _ { 2 } | H ) \rangle$ ). Under this assumption, we aim to maximize a somewhat simpler objective than Equation 2:
+
+$$
+h ^ { * } = \arg \operatorname* { m a x } _ { h ^ { i } } P ( O _ { 2 } | h ^ { i } ) P ( h ^ { i } | O _ { 1 } ) \mathrm { w h e r e } ( O _ { 1 } \perp O _ { 2 } | H )
+$$
+
+Fully Connected: Finally, our most sophisticated model jointly models all three random variables as in Equation 2, and can in principle combine information across both observations to choose the correct hypothesis.
+
+![](images/5e1e3a50c96750a4875733f4dc0c4780335ee75484838b84aced992eaaa60e51.jpg)  
+Figure 3: Overview of an αNLG model that integrates commonsense representations obtained from COMeT (Bosselut et al., 2019) with GPT2. Each observation is input to the COMeT model to obtain nine embeddings, each associated with one commonsense inference type.
+
+To help illustrate the subtle distinction between how the Linear Chain and Fully Connected models consider both observations, consider the following example. Let observation $O _ { 1 }$ : “Carl went to the store desperately searching for flour tortillas for a recipe.” and $O _ { 2 }$ : “Carl left the store very frustrated.”. Then consider two distinct hypotheses, an incorrect $h ^ { 1 }$ : “The cashier was rude” and the correct $h ^ { 2 }$ : “The store had corn tortillas, but not flour ones.”. For this example, a Linear Chain model could arrive at the wrong answer, because it reasons about the observations separately—taking $O _ { 1 }$ in isolation, both $h ^ { 1 }$ and $h ^ { 2 }$ seem plausible next events, albeit each a priori unlikely. And for $O _ { 2 }$ in isolation—i.e. in the absence of $O _ { 1 }$ , as for a randomly drawn shopper—the $h ^ { 1 }$ explanation of a rude cashier seems a much more plausible explanation of Carl’s frustration than are the details of the store’s tortilla selection. Combining these two separate factors leads the Linear Chain to select $h ^ { 1 }$ as the more plausible explanation. It is only by reasoning about Carl’s goal in $O _ { 1 }$ jointly with his frustration in $O _ { 2 }$ , as in the Fully Connected model, that we arrive at the correct answer $h ^ { 2 }$ as the more plausible explanation.
+
+In our experiments, we encode the different independence assumptions in the best performing neural network model. For the hypothesis-only and single observation models, we can enforce the independencies by simply restricting the inputs of the model to only the relevant variables. On the other hand, the Linear Chain model takes all three variables as input, but we restrict the form of the model to enforce the conditional independence. Specifically, we learn a discriminative classifier:
+
+$$
+P _ { \mathrm { L i n e a r } \mathrm { C h a i n } } ( h | O _ { 1 } , O _ { 2 } ) \propto e ^ { \phi ( O _ { 1 } , h ) + \phi ^ { \prime } ( h , O _ { 2 } ) }
+$$
+
+where $\phi$ and $\phi ^ { \prime }$ are neural networks that produce scalar values.
+
+# 3.2 ABDUCTIVE NATURAL LANGUAGE GENERATION
+
+Given $h ^ { + } = \lbrace w _ { 1 } ^ { h } \dots w _ { l } ^ { h } \rbrace$ , $O _ { 1 } { = } \{ w _ { 1 } ^ { o 1 } \dots w _ { m } ^ { o 1 } \}$ and $O _ { 2 } { = } \{ w _ { 1 } ^ { o 2 } \dots w _ { n } ^ { o 2 } \}$ as sequences of tokens, the $\alpha \mathbf { N L G }$ l task can be modeled as $P ( h ^ { + } | O _ { 1 } , O _ { 2 } ) = \prod P ( w _ { i } ^ { h } | w _ { < i } ^ { h } , w _ { 1 } ^ { o 1 } . . . w _ { m } ^ { o 1 } , w _ { 1 } ^ { o 2 } . . . w _ { n } ^ { o 2 } )$ Option$\kappa$ then be trained to minimize the negative log-likelihood over instances in ART:
+
+$$
+\mathcal { L } = - \sum _ { i = 1 } ^ { N } \log P ( w _ { i } ^ { h } | w _ { < i } ^ { h } , w _ { 1 } ^ { o 1 } \dots w _ { m } ^ { o 1 } , w _ { 1 } ^ { o 2 } \dots w _ { n } ^ { o 2 } , \mathcal { K } )
+$$
+
+# 4 ART DATASET: ABDUCTIVE REASONING IN NARRATIVE TEXT
+
+ART is the first large-scale benchmark dataset for studying abductive reasoning in narrative texts. It consists of $\sim 2 0 \mathrm { K }$ narrative contexts (pairs of observations $\langle O _ { 1 } , O _ { 2 } \rangle ,$ ) with over 200K explanatory hypotheses. Table 6 in the Appendix summarizes corpus-level statistics of the ART dataset.5 Figure 4 shows some illustrative examples from $\mathcal { A R T }$ (dev split). The best model based on BERT fails to correctly predict the first two dev examples.
+
+![](images/eb7406a89dda86ffad67f70804dfdcede43b15cd13cabdf197ae2587f7c2d3b9.jpg)  
+Figure 4: Examples from $\mathcal { A R T }$ (dev split). The best model based on BERT fails to correctly predict the first two examples.
+
+Collecting Observations: The pairs $O _ { 1 }$ , $O _ { 2 }$ in $\mathcal { A R T }$ are drawn from the ROCStories dataset (Mostafazadeh et al., 2016). ROCStories is a large collection of short, manually curated fivesentence stories. It was designed to have a clear beginning and ending for each story, which naturally map to the first $( O _ { 1 } )$ and second $( O _ { 2 } )$ observations in ART.
+
+Collecting Hypotheses Options: We crowdsourced the plausible and implausible hypotheses options on Amazon Mechanical Turk (AMT) in two separate tasks6:
+
+1. Plausible Hypothesis Options: We presented $O _ { 1 }$ and $O _ { 2 }$ as narrative context to crowdworkers who were prompted to fill in “What happened in-between?” in natural language. The design of the task motivates the use of abductive reasoning to hypothesize likely explanations for the two given observations.   
+2. Implausible Hypothesis Options: In this task, we presented workers with observations $O _ { 1 }$ , $O _ { 2 }$ and one plausible hypothesis option $h ^ { + } \in \mathcal { H } ^ { + }$ collected from the previous task. Crowdworkers were instructed to make minimal edits (up to 5 words) to a given $\bar { h ^ { + } }$ to create implausible hypothesis variations for each plausible hypothesis.
+
+A significant challenge in creating datasets is avoiding annotation artifacts – unintentional patterns in the data that leak information about the target label – that several recent studies (Gururangan et al., 2018; Poliak et al., 2018; Tsuchiya, 2018) have reported on crowdsourced datasets . To tackle this challenge, we collect multiple plausible and implausible hypotheses for each $\langle O _ { 1 } , O _ { 2 } \rangle$ pair (as described above) and then apply an adversarial filtering algorithm to retain one challenging pair of hypotheses that are hard to distinguish between. We describe our algorithm in detail in Appendix A.5. While our final dataset uses BERT as the adversary, preliminary experiments that used GPT as an adversary resulted in similar drops in performance of all models, including all BERT variants. We compare the results of the two adversaries in Table 1.
+
+# 5 EXPERIMENTS AND RESULTS
+
+We now present our evaluation of finetuned state-of-the-art pre-trained language models on the ART dataset, and several other baseline systems for both $\alpha \mathbf { N L I }$ and $\alpha \mathbf { N L G }$ . Since $\alpha \mathbf { N L I }$ is framed as a binary classification problem, we choose accuracy as our primary metric. For αNLG, we report performance on automated metrics such as BLEU (Papineni et al., 2002), CIDEr (Vedantam et al., 2015), METEOR (Banerjee & Lavie, 2005) and also report human evaluation results.
+
+# 5.1 ABDUCTIVE NATURAL LANGUAGE INFERENCE
+
+Despite strong performance on several other NLP benchmark datasets, the best baseline model based on BERT achieves an accuracy of just $6 8 . 9 \%$ on ART compared to human performance of $9 1 . 4 \%$ . The large gap between human performance and that of the best system provides significant scope for development of more sophisticated abductive reasoning models. Our experiments show that introducing the additional independence assumptions described in Section 3.1 over the fully connected model tends to degrade system performance (see Table 1) in general.
+
+Human Performance We compute human performance using AMT. Each instance (two observations and two hypothesis choices) is shown to three workers who were prompted to choose the more plausible hypothesis choice.7 We compute majority vote on the labels assigned which leads to a human accuracy of $9 1 . 4 \%$ on the ART test set.
+
+Table 1: Performance of baselines and finetuned-LM approaches on the test set of ART. Test accuracy is reported as the mean of five models trained with random seeds, with the standard deviation in parenthesis.   
+
+<table><tr><td>Model</td><td>GPTAF Acc. (%)</td><td>ART Acc. (%)</td></tr><tr><td>Random (2-way choice)</td><td>50.1</td><td>50.4</td></tr><tr><td>Majority (from dev set)</td><td>50.1</td><td>50.8</td></tr><tr><td>Infersent (Conneau et al., 2017)</td><td>50.9</td><td>50.8</td></tr><tr><td>ESIM+ELMo (Chen et al.,2017)</td><td>58.2</td><td>58.8</td></tr><tr><td>Finetuning Pre-trained LMs</td><td></td><td></td></tr><tr><td>GPT-ft</td><td>52.6 (0.9)</td><td>63.1 (0.5)</td></tr><tr><td>BERT-ft [h Only]</td><td>55.9 (0.7)</td><td>59.5 (0.2)</td></tr><tr><td>BERT-ft [O Only]</td><td>63.9 (0.8)</td><td>63.5 (0.7)</td></tr><tr><td>BERT-ft [O2 Only]</td><td>68.1 (0.6)</td><td>66.6 (0.2)</td></tr><tr><td>BERT-ft[Linear Chain]</td><td>65.3 (1.4)</td><td>68.9 (0.5)</td></tr><tr><td>BERT-ft [Fully Connected]</td><td>72.0 (0.5)</td><td>68.6 (0.5)</td></tr><tr><td>Human Performance</td><td>=</td><td>91.4</td></tr></table>
+
+Baselines We include baselines that rely on simple features to verify that ART is not trivially solvable due to noticeable annotation artifacts, observed in several crowdsourced datasets. The accuracies of all simple baselines are close to chance-performance on the task – indicating that the dataset is free of simple annotation artifacts.
+
+A model for the related but distinct task of entailment NLI (e.g. SNLI) forms a natural baseline for $\alpha \mathbf { N L I }$ . We re-train the ESIM $+$ ELMo (Chen et al., 2017; Peters et al., 2018) model as its performance on entailment NLI $( 8 8 . 9 \% )$ is close to state-of-the-art models (excluding pre-trained language models). This model only achieves an accuracy of $5 8 . 8 \%$ highlighting that performing well on $\mathcal { A R T }$ requires models to go far beyond the linguistic notion of entailment.
+
+Pre-trained Language Models BERT (Devlin et al., 2018) and GPT (Radford, 2018) have recently been shown to achieve state-of-the-art results on several NLP benchmarks (Wang et al., 2018). We finetune both BERT-Large and GPT as suggested in previous work and we present each instance in their natural narrative order. BERT-ft (fully connected) is the best performing model achieving $6 8 . 9 \%$ accuracy, compared to GPT’s $6 3 . 1 \%$ .8 Our AF approach was able to reduce BERT performance from over $8 8 \%$ by 20 points.
+
+![](images/7ca97c54b3a9c08fd6f31f67434432ad67641bbe2a13f0dc9a635be06e7e6445.jpg)  
+Figure 5: BERT learning curve on the dev set of ART. For each point on the $\mathbf { X }$ -axis, we fine-tune BERT with five random seeds. Human performance is $9 1 . 4 \%$ .
+
+Learning Curve and Dataset Size While there is enough scope for considerably scaling up the dataset based on ROCStories, the learning curve in Figure 5 shows that the performance of the best
+
+model plateaus after ${ \sim } 1 0$ , 000 instances. In addition, there is still a wide gap $( \sim 2 3 \%$ ) between the performance of the best model and human performance.
+
+Table 2: Performance of generative models on the test set of ART. All models except GPT2-Fixed are finetuned on ART.   
+
+<table><tr><td>Model</td><td>BLEU</td><td>METEOR</td><td>ROUGE</td><td>CIDEr</td><td>BERT-Score</td><td>Human</td></tr><tr><td>GPT2-Fixed</td><td>0.0</td><td>9.29</td><td>9.99</td><td>3.34</td><td>36.69</td><td>-</td></tr><tr><td>O1-Oz-Only</td><td>2.23</td><td>16.71</td><td>22.83</td><td>33.54</td><td>48.74</td><td>42.26</td></tr><tr><td>COMeT-Txt+GPT2</td><td>2.29</td><td>16.73</td><td>22.51</td><td>31.99</td><td>48.46</td><td>38.28</td></tr><tr><td>COMeT-Emb+GPT2</td><td>3.03</td><td>17.66</td><td>22.93</td><td>32.00</td><td>48.52</td><td>44.56</td></tr><tr><td>Human-written Hypotheses</td><td>8.25</td><td>26.71</td><td>30.40</td><td>53.56</td><td>53.30</td><td>96.03</td></tr></table>
+
+GPT Adversary Table 1 also includes results of our experiments where GPT was used as the adversary. Notably, in this case, adversarially filtering the dataset brings down GPT performance under $53 \%$ . On the other hand, the best BERT model, that encodes the fully connected bayesian network performs significantly better than the BERT model that encodes the linear chain assumptions $- 7 2 \%$ compared to $65 \%$ . Therefore, we use the BERT fully connected model as the adversary in ART. The gap between the linear chain and fully connected BERT models diminishes when BERT is used as an adversary – in spite of being a more powerful model – which indicates that adversarial filtering disproportionately impacts the model used as the adversary. However, the dataset also becomes more difficult for the other models that were not used as adversaries. For example, before any filtering, BERT scores $8 8 \%$ and OpenGPT gets $8 0 \%$ , which is much higher than either model achieves in Table 1 when the other model is used for filtering. This result is a reasonable indicator, albeit not a guarantee, that ART will remain challenging for new models released in the future.
+
+# 5.2 ABDUCTIVE NATURAL LANGUAGE GENERATION
+
+Generative Language Models As described in Equation 4, we train GPT2 conditioned on the tokens of the two observations $O _ { 1 }$ and $O _ { 2 }$ . Both observations are enclosed with field-specific tags. ATOMIC (Sap et al., 2019), a repository of inferential $i f .$ -then knowledge is a natural source of background commonsense required to reason about narrative contexts in ART. Yet, there is no straightforward way to include such knowledge into a neural model as ATOMIC’s nodes are not canonicalized and are represented as short phrases of text. Thus, we rely on COMeT – a transformer model trained on ATOMIC that generates nine commonsense inferences of events in natural language.9 Specifically, we experiment with two ways of integrating information from COMeT in GPT2: (i) as textual phrases, and (ii) as embeddings.
+
+Figure 3 shows how we integrate COMeT representations. Concretely, after the input tokens are embedded by the word-embedding layer, we append eighteen (corresponding to nine relations for each observation) embeddings to the sequence before passing through the layers of the Transformer architecture. This allows the model to learn each token’s representation while attending to the COMeT embeddings – effectively integrating background commonsense knowledge into a language model.10
+
+Discussion Table 2 reports results on the αNLG task. Among automatic metrics, we report BLEU4 (Papineni et al., 2002), METEOR (Banerjee & Lavie, 2005), ROUGE (Lin, 2004), CIDEr (Vedantam et al., 2015) and BERT-Score (Zhang et al., 2019) (with the bert-base-uncased model). We establish human performance through crowdsourcing on AMT. Crowdworkers are shown pairs of observations and a generated hypothesis and asked to label whether the hypothesis explains the given observations. The last column reports the human evaluation score. The last row reports the score of a held-out human-written hypothesis and serves as a ceiling for model performance. Human-written hypotheses are found to be correct for $96 \%$ of instances, while our best generative models, even when enhanced with background commonsense knowledge, only achieve $45 \%$ – indicating that the αNLG generation task is especially challenging for current state-of-the-art text generators.
+
+# 6 ANALYSIS
+
+# 6.1 αNLI
+
+Commonsense reasoning categories We investigate the categories of commonsense-based abductive reasoning that are challenging for current systems and the ones where the best model over-performs. While there have been previous attempts to categorize commonsense knowledge required for entailment (LoBue & Yates, 2011; Clark et al., 2007), crowdsourcing this task at scale with high fidelity and high agreement across annotators remains challenging. Instead, we aim to probe the model with soft categories identified by matching lists of category-specific keywords to the hypothesis choices.
+
+Table 3: BERT’s performance and human evaluation on categories for 1,000 instances from the test set, based on commonsense reasoning domains (Numerical, Spatial, Emotional). The number in parenthesis indicates the size of the category.   
+
+<table><tr><td>Category</td><td>Human Accuracy</td><td>BERT Accuracy</td><td>△</td></tr><tr><td>All (1, 000)</td><td>91.4</td><td>68.8</td><td>22.6</td></tr><tr><td>Numerical (44)</td><td>88.6</td><td>56.8</td><td>21.8</td></tr><tr><td>Spatial (130)</td><td>91.5</td><td>65.4</td><td>26.1</td></tr><tr><td>Emotional (84)</td><td>86.9</td><td>72.6</td><td>14.3</td></tr></table>
+
+Table 3 shows the accuracy of the best model (BERT-ft) across various categories of commonsense knowledge. BERT-ft significantly underperforms on instances involving Numerical $( 5 6 . 8 \% )$ and Spatial $( 6 5 . 4 \% )$ commonsense. These two categories include reasoning about numerical quantities and the spatial location of agents and objects, and highlight some of the limitations of the language models. In contrast, it significantly overperforms on the Emotional category $( 7 2 . 6 \% )$ where the hypotheses exhibit strong textual cues about emotions and sentiments.
+
+Implausible transitions A model for an instance of the ART dataset should discard implausible hypotheses in the context of the two given observations. In narrative contexts, there are three main reasons for an implausible hypothesis to be labeled as such:
+
+<table><tr><td>Story Transition</td><td>%of Dataset</td><td>BERT-ft Fully Connected Acc. (%)</td><td>BERT-ft Linear Chain Acc. (%)</td></tr><tr><td>O1h-</td><td>32.5</td><td>73.6</td><td>71.6</td></tr><tr><td>hO2</td><td>45.3</td><td>69.0</td><td>70.5</td></tr><tr><td>Plausible</td><td>22.2</td><td>62.5</td><td>58.5</td></tr><tr><td>All (1,000)</td><td>100.0</td><td>69.1</td><td>68.2</td></tr></table>
+
+1. $O _ { 1 } \not \to h ^ { - }$ : $h ^ { - }$ is unlikely to follow after the first observation $O _ { 1 }$ . 2. $h ^ { - } \hat { \rho } O _ { 2 }$ : $h ^ { - }$ is plausible after $O _ { 1 }$ but unlikely to precede the second observation $O _ { 2 }$ . 3. Plausible: $\langle O _ { 1 } , h ^ { - } , O _ { 2 } \rangle$ is a coherent narrative and forms a plausible alternative, but it is less plausible than $\langle O _ { 1 } , h ^ { + } , O _ { 2 } \rangle$ .
+
+Table 4: Fraction of dataset for which a particular transition in the story is broken for the negative hypothesis, for 1,000 random instances from the test set.
+
+We analyze the prevalence of each of these reasons in ART. We design a crowdsourcing task in which we show the implausible option along with the narrative context $\langle O _ { 1 } , O _ { 2 } \rangle$ and get labels for which transition $( O _ { 1 } \not \to h ^ { - }$ , $h ^ { - } \hat { \rho } O _ { 2 }$ or neither) in the narrative chain is broken. Table 4 shows the proportion of each category from a subset of $1 , 0 0 0$ instances from the test set. While $h ^ { - } \not \to O _ { 2 }$ accounts for almost half of the implausible transitions in ART, all three categories are substantially present in the dataset. BERT performance on each of these categories indicates that the model finds it particularly hard when the narrative created by the incorrect hypothesis is plausible, but less plausible than the correct hypothesis. On that subset of the test set, the fully connected model performs better than the linear chain model where it is important to consider both observations jointly to arrive at the more likely hypothesis.
+
+# 6.2 αNLG
+
+Figure 6 shows some examples of generations from the trained models compared to human-written generations. The example on the left is an example of an instance that only humans could get correct, while for the one on the right, COMeT-Emb $^ +$ GPT2also generates the correct explanation for the observations.
+
+![](images/648efe55a19523c64a9b5f6e2b2f9b60e1a301c7a4114c3496cd6a7e9fd18912.jpg)  
+Figure 6: Examples of generated hypotheses from different models and human-written hypothesis for 2 instances from ART.
+
+# 7 TRANSFER LEARNING FROM ART
+
+ART contains a large number of questions for the novel abductive reasoning task. In addition to serving as a benchmark, we investigate if ART can be used as a resource to boost performance on other commonsense tasks. We apply transfer learning by first training a model on ART, and subsequently training on four target datasets – WinoGrande Sakaguchi et al. (2020), WSC Levesque et al. (2011), DPR Rahman & $\mathrm { N g }$ (2012) and HellaSwag Zellers et al. (2019). We show that compared to a model that is only trained on the target dataset, a model that is sequentially trained on ART first and then on the target dataset can perform better. In particular, pre-training on ART consistently improves performance on related datasets when they have relatively few training examples.
+
+On the other hand, for target datasets with large amounts of training data, pre-training on ART does not provide a significant improvement.
+
+# 8 RELATED WORK
+
+Table 5: Transfer Learning from ART   
+
+<table><tr><td>Dataset</td><td>BERT-ft(D)</td><td>BERT-ft(AR→ BERT-ft(D)</td></tr><tr><td>WinoGrande Sakaguchi et al. (2020)</td><td>65.8%</td><td>67.2%</td></tr><tr><td>WSC Levesque et al. (2011)</td><td>70.0%</td><td>74.0%</td></tr><tr><td>DPR Rahman &amp; Ng (2012)</td><td>72.5%</td><td>86.0%</td></tr><tr><td>Hellaswag Zellers et al. (2019)</td><td>46.7%</td><td>46.1%</td></tr></table>
+
+Cloze-Style Task vs. Abductive Reasoning Since abduction is fundamentally concerned with plausible chains of cause-and-effect, our work draws inspiration from previous works that deal with narratives such as script learning
+
+(Schank & Abelson, 1975) and the narrative cloze test (Chambers & Jurafsky, 2009; Jans et al., 2012; Pichotta & Mooney, 2014; Rudinger et al., 2015). Rather than learning prototypical scripts or narrative chains, we instead reason about the most plausible events conditioned on observations. We make use of the ROCStories dataset (Mostafazadeh et al., 2016), which was specifically designed for the narrative cloze task. But, instead of reasoning about plausible event sequences, our task requires reasoning about plausible explanations for narrative omissions.
+
+Entailment vs. Abductive Reasoning The formulation of $\alpha \mathbf { N L I }$ is closely related to entailment NLI, but there are two critical distinctions that make abductive reasoning uniquely challenging. First, abduction requires reasoning about commonsense implications of observations (e.g., if we observe that the “grass is wet”, a likely hypothesis is that “it rained earlier”) which go beyond the linguistic notion of entailment (also noted by Josephson (2000)). Second, abduction requires non-monotonic reasoning about a set of commonsense implications collectively, to check the potential contradictions against multiple observations and to compare the level of plausibility of different hypotheses. This makes abductive reasoning distinctly challenging compared to other forms of reasoning such as induction and deduction (Shank, 1998). Perhaps more importantly, abduction is closely related to the kind of reasoning humans perform in everyday situations, where information is incomplete and definite inferences cannot be made.
+
+Generative Language Modeling Recent advancements in the development of large-scale pretrained language models (Radford, 2018; Devlin et al., 2018; Radford et al., 2019) have improved the quality and coherence of generated language. Although these models have shown to generate reasonably coherent text when condition on a sequence of text, our experiments highlight the limitations of these models to 1) generate language non-monotonically and 2) adhere to commonsense knowledge. We attempt to overcome these limitations with the incorporation of a generative commonsense model during hypothesis generation.
+
+Related Datasets Our new resource ART complements ongoing efforts in building resources for natural language inference (Dagan et al., 2006; MacCartney & Manning, 2009; Bowman et al., 2015; Williams et al., 2018a; Camburu et al., 2018). Existing datasets have mostly focused on textual entailment in a deductive reasoning set-up (Bowman et al., 2015; Williams et al., 2018a) and making inferences about plausible events (Maslan et al., 2015; Zhang et al., 2017). In their typical setting, these datasets require a system to deduce the logically entailed consequences of a given premise. In contrast, the nature of abduction requires the use of commonsense reasoning capabilities, with less focus on lexical entailment. While abductive reasoning has been applied to entailment datasets (Raina et al., 2005), they have been applied in a logical theorem-proving framework as an intermediate step to perform textual entailment – a fundamentally different task than αNLI.
+
+# 9 CONCLUSION
+
+We present the first study that investigates the viability of language-based abductive reasoning. We conceptualize and introduce Abductive Natural Language Inference (αNLI) – a novel task focused on abductive reasoning in narrative contexts. The task is formulated as a multiple-choice questionanswering problem. We also introduce Abductive Natural Language Generation $( \alpha \mathbf { N L G } ) - \mathbf { a }$ novel task that requires machines to generate plausible hypotheses for given observations. To support these tasks, we create and introduce a new challenge dataset, ART, which consists of 20,000 commonsense narratives accompanied with over 200,000 explanatory hypotheses. In our experiments, we establish comprehensive baseline performance on this new task based on state-of-the-art NLI and language models, which leads to $6 8 . 9 \%$ accuracy with a considerable gap with human performance $( 9 1 . 4 \% )$ . The $\alpha \mathbf { N L G }$ task is significantly harder – while humans can write a valid explanation $96 \%$ of times, the best generator models can only achieve $45 \%$ . Our analysis leads to new insights into the types of reasoning that deep pre-trained language models fail to perform – despite their strong performance on the closely related but different task of entailment NLI – pointing to interesting avenues for future research. We hope that ART will serve as a challenging benchmark for future research in languagebased abductive reasoning and the $\alpha \mathbf { N L I }$ and αNLG tasks will encourage representation learning that enables complex reasoning capabilities in AI systems.
+
+# ACKNOWLEDGMENTS
+
+We thank the anonymous reviewers for their insightful feedback. This research was supported in part by NSF (IIS-1524371), the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1256082, DARPA CwC through ARO (W911NF15-1- 0543), DARPA MCS program through NIWC Pacific (N66001-19-2-4031), and the Allen Institute for AI. Computations on beaker.org were supported in part by credits from Google Cloud.
+
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+
+# A APPENDICES
+
+# A.1 DATA COLLECTION DETAILS
+
+We describe the crowdsourcing details of our data collection method.
+
+Task 1 - Plausible Hypothesis Options In this task, participants were presented an incomplete three-part story, which consisted of the first observation $( O _ { 1 } )$ and the second observation $( O _ { 2 } )$ o f the story. They were then asked to complete the story by writing a probable middle sentence that explains why the second observation should follow after the first one. We instructed participants to make sure that the plausible middle sentence (1) is short (fewer than 10 words) and (2) simple as if narrating to a child, (3) avoids introducing any extraneous information, and (4) uses names instead of pronouns (e.g., he/she) wherever possible.
+
+All participants were required to meet the following qualification requirements: (1) their location is in the US, (2) HIT approval rate is greater than $9 5 ( \% )$ , and (3) Number of HITs approved is greater than 5,000. The reward of this task was set to be $\$ 0.07$ per question ( $\$ 14$ hour in average), and each HIT was assigned to five different workers (i.e., 5-way redundancy).
+
+Task 2 - Implausible Hypothesis Options In this task, participants were presented a three-part story, which consisted of the first observation $( O _ { 1 } )$ , a middle sentence $( h ^ { + } )$ collected in Task 1, and the second observation $( O _ { 2 } )$ of the story. They were then asked to rewrite the middle sentence $( h ^ { + } )$ with minimal changes, so that the story becomes unlikely, implausible or inconsistent $( h ^ { - } )$ . We asked participants to add or remove at most four words to $h ^ { + }$ , while ensuring that the new middle sentence is grammatical. In addition, we asked them to stick to the context in the given story. For example, if the story talks about “doctors”, they are welcome to talk about “health” or “diagnosis”, but not mention “aliens”. Finally, we also asked workers to verify if the given middle $( h ^ { + } )$ makes a plausible story, in order to confirm the plausibility of $h ^ { + }$ collected in Task 1.
+
+With respect to this task’s qualification, participants were required to fulfill the following requirements: (1) their location is the US or Canada, (2) HIT approval rate is greater than or equal to $9 9 ( \% )$ , and (3) number of HITs approved is greater than or equal to $1 0 , 0 0 0$ . Participants were paid $\$ 0.1$ per question (\$14/hour in average), and each HIT was assigned to three different participants (i.e., 3-way redundancy).
+
+Task $\mathbf { 3 } - \alpha \mathbf { N L I }$ Human Performance Human performance was evaluated by asking participants to answer the αNLI questions. Given a narrative context $\langle O _ { 1 } , O _ { 2 } \rangle$ and two hypotheses, they were asked to choose the more plausible hypothesis. They were also allowed to choose “None of the above“ when neither hypothesis was deemed plausible.
+
+We asked each question to seven participants with the following qualification requirements: (1) their location is either in the US, UK, or Canada, (2) HIT approval rate is greater than $9 8 ( \% )$ , (3) Number of HITs approved is greater than 10, 000. The reward was set to $\$ 0.05$ per HIT. We took the majority vote among the seven participants for every question to compute human performance.
+
+# A.2 ART DATA STATISTICS
+
+Table 6 shows some statistics of the ART dataset.
+
+A.3 FINE-TUNING BERT
+
+We fine-tuned the BERT model using a grid search with the following set of hyper-parameters:
+
+• batch size: $\{ 3 , 4 , 8 \}$ • number of epochs: $\{ 3 , 4 , 1 0 \}$ • learning rate: {1e-5, 2e-5, 3e-5, 5e-5}
+
+The warmup proportion was set to 0.2, and cross-entropy was used for computing the loss. The best performance was obtained with a batch size of 4, learning rate of 5e-5, and number of epochs equal to 10. Table 7 describes the input format for GPT and BERT (and its variants).
+
+Table 6: Some statistics summarizing the ART dataset. The train set includes all plausible and implausible hypotheses collected via crowdsourcing, while the dev and test sets include the hypotheses selected through the Adversarial Filtering algorithm.   
+
+<table><tr><td></td><td>Train</td><td>Dev</td><td>Test</td></tr><tr><td>Total unique occurrences</td><td></td><td></td><td></td></tr><tr><td>Contexts (O1, O2)</td><td>17,801</td><td>1,532</td><td>3,059</td></tr><tr><td>Plausible hyp. h+</td><td>72,046</td><td>1,532</td><td>3,059</td></tr><tr><td>Implausible hyp. h-</td><td>166,820</td><td>1,532</td><td>3,059</td></tr><tr><td>Avg. size per context</td><td></td><td></td><td></td></tr><tr><td>Plausible hyp. h+</td><td>4.05</td><td>1</td><td>1</td></tr><tr><td>Implausible hyp.h-</td><td>9.37</td><td>1</td><td>1</td></tr><tr><td>Avg. word length</td><td></td><td></td><td></td></tr><tr><td>Plausible hyp. h+</td><td>8.34</td><td>8.62</td><td>8.54</td></tr><tr><td>Implausible hyp.h-</td><td>8.28</td><td>8.55</td><td>8.53</td></tr><tr><td>First observation Oi</td><td>8.09</td><td>8.07</td><td>8.17</td></tr><tr><td>Second observation O2</td><td>9.29</td><td>9.3</td><td>9.31</td></tr></table>
+
+# A.4 BASELINES
+
+The SVM classifier is trained on simple features like word length, overlap and sentiment features to select one of the two hypothesis choices. The bag-of-words baseline computes the average of GloVe (Pennington et al., 2014) embeddings for words in each sentence to form sentence embeddings. The sentence embeddings in a story (two observations and a hypothesis option) are concatenated and passed through fully-connected layers to produce a score for each hypothesis. The accuracies of both baselines are close to $50 \%$ (SVM: 50.6; BOW: 50.5).
+
+Specifically, we train an SVM classifier and a bag-of-words model using GLoVE embeddings. Both models achieve accuracies close to $50 \%$ . An Infersent (Conneau et al., 2017) baseline that uses sentences embedded by max-pooling over Bi-LSTM token representations achieves only $5 0 . 8 \%$ accuracy.
+
+Table 7: Input formats for GPT and BERT fine-tuning.   
+
+<table><tr><td>Model</td><td colspan="5">Input Format</td></tr><tr><td>GPT</td><td colspan="5">[START]O+h[SEP]O2[SEP]</td></tr><tr><td>BERT-ft [Hypothesis Only]</td><td>[CLS]h[SEP]</td><td></td><td></td><td></td><td></td></tr><tr><td>BERT-ft [First Observation Only]</td><td>[CLS]</td><td>O1</td><td></td><td>[SEP]h[SEP]</td><td></td></tr><tr><td>BERT-ft [Second Observation Only]</td><td>[CLS]</td><td>h</td><td>[SEP]</td><td>O2[SEP]</td><td></td></tr><tr><td>BERT-ft [Linear Chain]</td><td>[CLS]</td><td>O1</td><td>[SEP]</td><td></td><td>h[SEP]；[CLS]h[SEP]O2[SEP]</td></tr><tr><td>BERT-ft [Fully Connected]</td><td>[CLS]</td><td></td><td></td><td>O+ O2[SEP]h[SEP]</td><td></td></tr></table>
+
+# A.5 ADVERSARIAL FILTERING OF HYPOTHESES CHOICES
+
+Given an observation pair and sets of plausible and implausible hypotheses $\langle O _ { 1 } , O _ { 2 } , \mathcal { H } ^ { + } , \mathcal { H } ^ { - } \rangle$ , our adversarial filtering algorithm selects one plausible and one implausible hypothesis $\langle O _ { 1 } , O _ { 2 } , h ^ { + }$ , $h ^ { - } \ \rangle$ such that $h ^ { + }$ and $h ^ { - }$ are hard to distinguish between. We make three key improvements over the previously proposed Adversarial Filtering (AF) approach in Zellers et al. (2018). First, Instead of a single positive sample, we exploit a pool $\mathcal { H } ^ { + }$ of positive samples to choose from (i.e. plausible hypotheses). Second, Instead of machine generated distractors, the pool $\varkappa ^ { - }$ of negative samples (i.e. implausible hypotheses) is human-generated. Thus, the distractors share stylistic features of the positive samples as well as that of the context (i.e. observations $O _ { 1 }$ and $O _ { 2 }$ ) – making the negative samples harder to distinguish from positive samples. Finally, We use BERT (Devlin et al., 2018) as the adversary and introduce a temperature parameter that controls the maximum number of instances that can be modified in each iteration of AF. In later iterations, fewer instances get modified resulting in a smoother convergence of the AF algorithm (described in more detail below).
+
+Algorithm 1 provides a formal description of our approach. In each iteration $i$ , we train an adversarial model $M _ { i }$ on a random subset $\tau _ { i }$ of the data and update the validation set $\nu _ { i }$ to make it more challenging for $M _ { i }$ . For a pair $( h _ { k } ^ { + } , h _ { k } ^ { - } )$ of plausible and implausible hypotheses for an instance $k$ , we denote $\delta = \Delta _ { M _ { i } } ( h _ { k } ^ { + } , h _ { k } ^ { - } )$ the difference in the model evaluation of $h _ { k } ^ { + }$ and $h _ { k } ^ { - }$ . A positive value of $\delta$ indicates that the model $M _ { i }$ favors the plausible hypothesis $h _ { k } ^ { + }$ over the implausible one $h _ { k } ^ { - }$ . With probability $t _ { i }$ , we update instance $k$ that $M _ { i }$ gets correct with a pair $( h ^ { + } , h ^ { - } ) \in \mathcal { H } _ { k } ^ { + } \times \mathcal { H } _ { k } ^ { - }$ of hypotheses that reduces the value of $\delta$ , where $\mathcal { H } _ { k } ^ { + }$ (resp. $\mathcal { H } _ { k } ^ { - }$ ) is the pool of plausible (resp. implausible) hypotheses for instance $k$ .
+
+We ran AF for 50 iterations and the temperature $t _ { i }$ follows a sigmoid function, parameterized by the iteration number, between $t _ { s } = 1 . 0$ and $t _ { e } = 0 . 2$ . Our final dataset, ART, is generated using BERT as the adversary in Algorithm 1.
+
+# Algorithm 1: Dual Adversarial Filtering
+
+input : dataset $\mathcal { D } _ { 0 }$ , plausible & implausible hypothesis sets $( \mathcal { H } ^ { + } , \mathcal { H } ^ { - } )$ , number of iterations $n$ ,   
+initial & final temperatures $( t _ { s } , t _ { e } )$   
+output: dataset $\mathcal { D } _ { n }$   
+1 for iteration $i : 0 . . n - 1$ do   
+2 $\begin{array} { r } { t _ { i } = t _ { e } + \frac { t _ { s } - t _ { e } } { 1 + e ^ { 0 . 3 ( i - \frac { 3 n } { 4 } ) } } } \end{array}$   
+3 Randomly partition $\mathcal { D } _ { i }$ into $( \mathcal { T } _ { i } , \mathcal { V } _ { i } )$ .   
+4 Train model $M _ { i }$ on $\mathcal { T } _ { i }$ .   
+5 $\mathcal { S } _ { i } = \varnothing$ , the selected hypotheses for $\nu _ { i }$ .   
+6 for $( h _ { k } ^ { + } , h _ { k } ^ { - } ) \in \mathcal { V } _ { i }$ do   
+7 Pick $r$ uniformly at random in $[ 0 , 1 ]$ .   
+8 if $r > t _ { i }$ or $\Delta _ { M _ { i } } ( h _ { k } ^ { + } , h _ { k } ^ { - } ) < 0$ then   
+9 Add $( h _ { k } ^ { + } , h _ { k } ^ { - } )$ to $s _ { i }$ .   
+10 else   
+11 Pick $( h ^ { + } , h ^ { - } ) \in \mathcal { H } _ { k } ^ { + } \times \mathcal { H } _ { k } ^ { - }$ s.t. ${ \Delta } _ { M _ { i } } ( h ^ { + } , h ^ { - } ) < \Delta _ { M _ { i } } ( h _ { k } ^ { + } , h _ { k } ^ { - } )$   
+12 Add $( h ^ { + } , h ^ { - } )$ to $s _ { i }$ .   
+13 end   
+14 end   
+15 $\mathcal { D } _ { i + 1 } = \mathcal { T } _ { i } \cup \mathcal { S } _ { i }$   
+16 end
+
+# A.6 ATOMIC RELATIONS
+
+ATOMIC (Sap et al., 2019) represents commonsense knowledge as a graph with events are nodes and the following nine relations as edges:
+
+1. xIntent: Why does X cause an event?   
+2. xNeed: What does X need to do before the event?   
+3. xAttr: How would X be described?   
+4. xEffect: What effects does the event have on X?   
+5. xWant: What would X likely want to do after the event?   
+6. xReaction: How does X feel after the event?   
+7. oReact: How do others’ feel after the event?   
+8. oWant: What would others likely want to do after the event?   
+9. oEffect: What effects does the event have on others?
+
+<table><tr><td>Model</td><td>Input Format</td></tr><tr><td>GPT2-Fixed</td><td>w1...wnw² ...w Because,</td></tr><tr><td>O1-Oz-Only</td><td>{o1)wi...wn&lt;/o1)(o2)w² ...wn&lt;/o2)(h)</td></tr><tr><td>COMeT-Txt+GPT2</td><td>(pi&gt;T1...T(pg&gt;&lt;p²)T²...T²(p²)(o1)wi ...wn&lt;/o1)(o2)w² ...w²&lt;/02)(h)</td></tr><tr><td>COMeT-Emb+GPT2</td><td>c1...c§;c²...c²{(o1)w1 ...wn&lt;/o1)(02)w² ...w²&lt;/o2)(h)</td></tr></table>
+
+Table 8: Input format used to training and generated text from various GPT2 based models. $c _ { i } ^ { j }$ refers to the COMeTembeddings obtained using a separate transformer model for relation $i$ and observation $j$ . Similarly, $T _ { i } ^ { j }$ is the textual phrase for relation $i$ , observation $j$ . Where appropriate, field specific start and end-tags are added to the sequence of inputs.
+
+# A.7 GENERATION MODELS INPUT FORMAT
+
+Table 8 describes the format of input to each variation of the generative model evaluated.

parse/train/gDcaUj4Myhn/gDcaUj4Myhn_content_list.json

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+    {
+        "type": "text",
+        "text": "Laplace Redux – Effortless Bayesian Deep Learning ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Erik Daxberger⇤,c,m ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Agustinus Kristiadi⇤,t Matthias Bauerd ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Runa Eschenhagen⇤,t ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Alexander Immer⇤,e,p Philipp Hennigt,m ",
+        "text_level": 1,
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+        ],
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+    {
+        "type": "text",
+        "text": "c University of Cambridge mMPI for Intelligent Systems, Tübingen t University of Tübingen   \ne Department of Computer Science, ETH Zurich   \npMax Planck ETH Center for Learning Systems dDeepMind, London ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Bayesian formulations of deep learning have been shown to have compelling theoretical properties and offer practical functional benefits, such as improved predictive uncertainty quantification and model selection. The Laplace approximation (LA) is a classic, and arguably the simplest family of approximations for the intractable posteriors of deep neural networks. Yet, despite its simplicity, the LA is not as popular as alternatives like variational Bayes or deep ensembles. This may be due to assumptions that the LA is expensive due to the involved Hessian computation, that it is difficult to implement, or that it yields inferior results. In this work we show that these are misconceptions: we (i) review the range of variants of the LA including versions with minimal cost overhead; (ii) introduce laplace, an easy-to-use software library for PyTorch offering user-friendly access to all major flavors of the LA; and (iii) demonstrate through extensive experiments that the LA is competitive with more popular alternatives in terms of performance, while excelling in terms of computational cost. We hope that this work will serve as a catalyst to a wider adoption of the LA in practical deep learning, including in domains where Bayesian approaches are not typically considered at the moment. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "laplace library: https://github.com/AlexImmer/Laplace Experiments: https://github.com/runame/laplace-redux ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            678,
+            310,
+            695
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Despite their successes, modern neural networks (NNs) still suffer from several shortcomings that limit their applicability in some settings. These include (i) poor calibration and overconfidence, especially when the data distribution shifts between training and testing [1], (ii) catastrophic forgetting of previously learned tasks when continuously trained on new tasks [2], and (iii) the difficulty of selecting suitable NN architectures and hyperparameters [3]. Bayesian modeling [4, 5] provides a principled and unified approach to tackle these issues by (i) equipping models with robust uncertainty estimates [6], (ii) enabling models to learn continually by capturing past information [7], and (iii) allowing for automated model selection by optimally trading off data fit and model complexity [8]. ",
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+    {
+        "type": "text",
+        "text": "Even though this provides compelling motivation for using Bayesian neural networks (BNNs) [9], they have not gained much traction in practice. Common criticisms include that BNNs are difficult to implement, finicky to tune, expensive to train, and hard to scale to modern models and datasets. For instance, popular variational Bayesian methods [10–12, etc.] require considerable changes to the training procedure and model architecture. Also, their optimization process is slower and typically more unstable unless carefully tuned [13]. Other methods, such as deep ensembles [14], Monte Carlo dropout [6], and SWAG [15] promise to bring uncertainty quantification to standard NNs in simple manners. But these methods either require a significant cost increase compared to a single network, have limited empirical performance, or an unsatisfying Bayesian interpretation. ",
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+    {
+        "type": "image",
+        "img_path": "images/2cd8979d494e70cbf9f9fc80b1e30136132d33c723973cb95205c950f704e8c9.jpg",
+        "image_caption": [
+            "Figure 1: Probabilistic predictions with the Laplace approximation in three steps. (a) We find a MAP estimate (yellow star) via standard training (background contours $=$ log-posterior landscape on the two-dimensional PCA subspace of the SGD trajectory [30]). (b) We locally approximate the posterior landscape by fitting a Gaussian centered at the MAP estimate (yellow contours), with covariance matrix equal to the negative inverse Hessian of the loss at the MAP—this is the Laplace approximation (LA). (c) We use the LA to make predictions with predictive uncertainty estimates— here, the black curve is the predictive mean, and the shading covers the $9 5 \\%$ confidence interval. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we argue that the Laplace approximation (LA) is a simple and cost-efficient, yet competitive approximation method for inference in Bayesian deep learning. First proposed in this context by MacKay [16], the LA dates back to the 18th century [17]. It locally approximates the posterior with a Gaussian distribution centered at a local maximum, with covariance matrix corresponding to the local curvature. Two key advantages of the LA are that the local maximum is readily available from standard maximum a posteriori (MAP) training of NNs, and that curvature estimates can be easily and efficiently obtained thanks to recent advances in second-order optimization, both in terms of more efficient approximations to the Hessian [18–20] and easy-to-use software libraries [21]. Together, they make the LA practical and readily applicable to many already-trained NNs—the LA essentially enables practitioners to turn their high-performing point-estimate NNs into BNNs easily and quickly, without loss of predictive performance. Furthermore, the LA to the marginal likelihood may even be used for Bayesian model selection or NN training [8, 22]. Figure 1 provides an intuition of the LA—we first fit a point estimate of the model and then estimate a Gaussian distribution around that. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Yet, despite recent progress in scaling and improving the LA for deep learning [23–29], it is far less widespread than other methods. This is likely due to misconceptions, like that the LA is hard to implement due to the Hessian computation, that it must necessarily perform worse than the competitors due to its local nature, or quite simply that it is old and too simple. Here, we show that these are indeed misconceptions. Moreover, we argue that the LA deserves a wider adoption in both practical and research-oriented deep learning. To this end, our work makes the following contributions: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. We first survey recent advances and present the key components of scalable and practical Laplace approximations in deep learning (Section 2). 2. We then introduce laplace, an easy-to-use PyTorch-based library for “turning a NN into a BNN” via the LA (Section 3). laplace implements a wide range of different LA variants. 3. Lastly, using laplace, we show in an extensive empirical study that the LA is competitive to alternative approaches, especially considering how simple and cheap it is (Section 4). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 The Laplace Approximation in Deep Learning ",
+        "text_level": 1,
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "The LA can be used in two different ways to benefit deep learning: Firstly, we can use the LA to approximate the model’s posterior distribution (see Eq. (5) below) to enable probabilistic predictions (as also illustrated in Fig. 1). Secondly, we can use the LA to approximate the model evidence (see Eq. (6)) to enable model selection (e.g. hyperparameter tuning). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The canonical form of (supervised) deep learning is that of empirical risk minimization. Given, e.g., an i.i.d. classification dataset $\\mathcal { D } : = \\{ ( x _ { n } \\in \\mathbb { R } ^ { M } , y _ { n } \\in \\mathbb { R } ^ { C } ) \\} _ { n = 1 } ^ { N }$ , the weights $\\boldsymbol { \\theta } \\in \\mathbb { R } ^ { D }$ of an $L$ -layer NN $f _ { \\theta } : \\mathbb { R } ^ { M }  \\mathbb { R } ^ { C }$ are trained to minimize the (regularized) empirical risk, which typically decomposes into a sum over empirical loss terms $\\ell ( x _ { n } , y _ { n } ; \\theta )$ and a regularizer $r ( \\theta )$ , ",
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+    {
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+        "img_path": "images/d8d8fbb1383388e0b45dfe3111e7b2046612a2480ef8d51349cd8e9dad761618.jpg",
+        "text": "$$\n\\begin{array} { r } { \\theta _ { \\mathrm { M A P } } = \\arg \\operatorname* { m i n } _ { \\theta \\in \\mathbb { R } ^ { D } } \\mathcal { L } ( \\mathcal { D } ; \\theta ) = \\arg \\operatorname* { m i n } _ { \\theta \\in \\mathbb { R } ^ { D } } \\left( r ( \\theta ) + \\sum _ { n = 1 } ^ { N } \\ell ( x _ { n } , y _ { n } ; \\theta ) \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "From the Bayesian viewpoint, these terms can be identified with i.i.d. log- likelihoods and a log-prior, respectively and, thus, $\\theta _ { \\mathrm { M A P } }$ is indeed a maximum a-posteriori (MAP) estimate: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/c87a87be07ebbfa9d45ff11bc0d90be177b92b2922d0d9c056cc50e13dfde0b4.jpg",
+        "text": "$$\n\\ell ( x _ { n } , y _ { n } ; \\theta ) = - \\log p ( y _ { n } | f _ { \\theta } ( x _ { n } ) ) \\qquad { \\mathrm { a n d } } \\qquad r ( \\theta ) = - \\log p ( \\theta )\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "For example, the widely used weight regularizer $\\begin{array} { r } { r ( \\theta ) = \\frac { 1 } { 2 } \\gamma ^ { - 2 } \\lVert \\theta \\rVert ^ { 2 } } \\end{array}$ (a.k.a. weight decay) corresponds to a centered Gaussian prior $p ( \\theta ) = \\mathcal { N } ( \\theta ; 0 , \\gamma ^ { 2 } I )$ , and the cross-entropy loss amounts to a categorical likelihood. Hence, the exponential of the negative training loss $\\mathrm { e x p } \\big ( { - \\infty } ( \\mathcal { D } ; \\theta ) \\big )$ amounts to an unnormalized posterior. By normalizing it, we obtain ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f066fc7f3047e4b5f2de0f66bb38b888af4f0112a44778c4251a2882c56621cc.jpg",
+        "text": "$$\n\\begin{array} { r } { p ( \\theta | \\mathcal D ) = \\frac 1 Z p ( \\mathcal D | \\theta ) p ( \\theta ) = \\frac 1 Z \\exp ( - \\mathcal L ( \\mathcal D ; \\theta ) ) , \\qquad Z : = \\int p ( \\mathcal D | \\theta ) p ( \\theta ) d \\theta } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            294,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "with an intractable normalizing constant $Z$ . Laplace approximations [17] use a second-order expansion of $\\mathcal { L }$ around $\\theta _ { \\mathrm { M A P } }$ to construct a Gaussian approximation to $p ( \\theta \\mid \\mathcal { D } )$ . I.e. we consider: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c93cab2fb82d6ce6723a48668d4d0461cecfd210ce9a9a0a154c7ca41fbea7a8.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } ( \\mathcal { D } ; \\theta ) \\approx \\mathcal { L } ( \\mathcal { D } ; \\theta _ { \\mathrm { M A P } } ) + \\frac { 1 } { 2 } { ( \\theta - \\theta _ { \\mathrm { M A P } } ) ^ { \\top } } \\left( \\nabla _ { \\theta } ^ { 2 } \\mathcal { L } ( \\mathcal { D } ; \\theta ) | _ { \\theta _ { \\mathrm { M A P } } } \\right) ( \\theta - \\theta _ { \\mathrm { M A P } } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            349,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the first-order term vanishes at $\\theta _ { \\mathrm { M A P } }$ . Then we can identify the Laplace approximation as ",
+        "bbox": [
+            176,
+            371,
+            794,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b24bf48d5c92361129a350a3a5eb397df8a2b3864602a290fe3966c56bfb7e29.jpg",
+        "text": "$$\np ( \\theta | \\mathcal { D } ) \\approx \\mathcal { N } ( \\theta ; \\theta _ { \\mathrm { M A P } } , \\Sigma ) \\qquad \\mathrm { w i t h } \\qquad \\Sigma : = - \\left( \\nabla _ { \\theta } ^ { 2 } \\mathcal { L } ( \\mathcal { D } ; \\theta ) | _ { \\theta _ { \\mathrm { M a P } } } \\right) ^ { - 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The normalizing constant $Z$ (which is typically referred to as the marginal likelihood or evidence) is useful for model selection and can also be approximated as ",
+        "bbox": [
+            176,
+            429,
+            823,
+            457
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4bd6a7171354234d59c33e26b1e7b8ba517a623f4364a61d1c708e90e55a77e7.jpg",
+        "text": "$$\nZ \\approx \\exp ( - \\mathcal { L } ( \\mathcal { D } ; \\theta _ { \\mathrm { M A P } } ) ) ( 2 \\pi ) ^ { \\bar { D } / 2 } ( \\operatorname* { d e t } { \\varSigma } ) ^ { 1 / 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "See Appendix A for more details. Thus, to obtain the approximate posterior, we first need to find the argmax $\\theta _ { \\mathrm { M A P } }$ of the log-posterior function, i.e. do “standard” deep learning with regularized empirical risk minimization. The only additional step is to compute the inverse of the Hessian matrix at $\\theta _ { \\mathrm { M A P } }$ (see Figure 1(b)). The LA can therefore be constructed post-hoc to a pre-trained network, even one downloaded off-the-shelf. As we discuss below, the Hessian computation can be offloaded to recently advanced automatic differentiation libraries [21]. LAs are widely used to approximate the posterior distribution in logistic regression [31], Gaussian process classification [32, 33], and also for Bayesian neural networks (BNNs), both shallow [34] and deep [23]. The latter is the focus of this work. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Generally, any prior with twice differentiable log-density can be used. Due to the popularity of the weight decay regularizer, we assume that the prior is a zero-mean Gaussian $p ( \\theta ) \\overset { \\cdot } { = } \\mathcal { N } ( \\theta ; 0 , \\gamma ^ { 2 } I )$ unless stated otherwise.2 The Hessian $\\nabla _ { \\theta } ^ { 2 } \\mathcal { L } ( \\mathcal { D } ; \\theta ) | _ { \\theta _ { \\mathrm { M A P } } }$ then depends both on the (simple) log-prior $/$ regularizer and the (complicated) log-likelihood / empirical risk: ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b5eb6452a13d4f47b1b48152897952f07bd40d6610d37f555b445de4e27bc48a.jpg",
+        "text": "$$\n\\begin{array} { r } { \\nabla _ { \\theta } ^ { 2 } \\mathcal { L } ( \\mathcal { D } ; \\theta ) | _ { \\theta _ { \\mathrm { M A P } } } = - \\gamma ^ { - 2 } I - \\sum _ { n = 1 } ^ { N } \\nabla _ { \\theta } ^ { 2 } \\log p ( y _ { n } | f _ { \\theta } ( x _ { n } ) ) | _ { \\theta _ { \\mathrm { M A P } } } . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "A naive implementation of the Hessian is infeasible because the second term in Eq. (7) scales quadratically with the number of network parameters, which can be in the millions or even billions [35, 36]. In recent years, several works have addressed scalability, as well as other factors that affect approximation quality and predictive performance of the LA. In the following, we identify, review, and discuss four key components that allow LAs to scale and perform well on modern deep architectures. See Fig. 2 for an overview and Appendix B for a more detailed version of the review and discussion. ",
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+    {
+        "type": "text",
+        "text": "Four Components of Scalable Laplace Approximations for Deep Neural Networks ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "$\\textcircled{1}$ Inference over all Weights or Subsets of Weights ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In most cases, it is possible to treat all weights probabilistically when using appropriate approximations of the Hessian, as we discuss below in $\\textcircled{2}$ . Another simple way to scale the LA to large NNs (without Hessian approximations) is the subnetwork $\\pmb { L A }$ [27], which only treats a subset of the model parameters probabilistically with the LA and leaves the remaining parameters at their MAP-estimated values. An important special case of this applies the LA to only the last linear layer of an $L$ -layer NN, while fixing the feature extractor defined by the first $L - 1$ layers at its MAP estimate [37, 28]. This last-layer $\\pmb { L A }$ is cost-effective yet compelling both theoretically and in practice [28]. ",
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+        "img_path": "images/00a601d4d4902eb04ba88847f92e868cc249cf34c759b9bac557dcb04925b14b.jpg",
+        "image_caption": [
+            "Figure 2: Four key components to scale and apply the LA to a neural network $f _ { \\theta }$ (with randomlyinitialized or pre-trained weights $\\theta$ ), with corresponding laplace code. $\\textcircled{1}$ We first choose which part of the model we want to perform inference over with the LA. $\\textcircled{2}$ We then select how to to approximate the Hessian. $\\textcircled{3}$ We can then perform model selection using the evidence: (a) If we started with an untrained model $f _ { \\theta }$ , we can jointly train the model and use the evidence to tune hyperparameters online. (b) If we started with a pre-trained model, we can use the evidence to tune the hyperparameters post-hoc. Here, shades represent the loss landscape, while contours represent LA log-posteriors—faded contours represent intermediate iterates during hyperparameter tuning to obtain the final log-posterior (thick yellow contours). $\\textcircled{4}$ Finally, to make predictions for a new input $x _ { * }$ , we have several options for computing/approximating the predictive distribution $p ( \\boldsymbol { y } | f _ { \\boldsymbol { \\theta } } ( x _ { * } ) , \\mathcal { D } )$ . "
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+        "type": "text",
+        "text": "$\\textcircled{2}$ Hessian Approximations and Their Factorizations ",
+        "text_level": 1,
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+        "text": "One advance in second-order optimization that the LA can benefit from are positive semi-definite approximations to the (potentially indefinite) Hessian of the log-likelihoods of NNs in the second term of Eq. (7) [38]. The Fisher information matrix [39], abbreviated as the Fisher and defined by ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/4f6b0f1fd7192418349cd1f9fbf3f84932d350485057cff3134c652e0292ca58.jpg",
+        "text": "$$\n\\begin{array} { r } { F : = \\sum _ { n = 1 } ^ { N } \\mathbb { E } _ { \\widehat { \\mathcal { I } } \\sim p ( y \\mid f _ { \\theta } ( x _ { n } ) ) } \\left[ \\left( \\nabla _ { \\theta } \\log p ( \\widehat { y } \\mid f _ { \\theta } ( x _ { n } ) ) | _ { \\theta _ { \\mathrm { M a p } } } \\right) ( \\nabla _ { \\theta } \\log p ( \\widehat { y } \\mid f _ { \\theta } ( x _ { n } ) ) | _ { \\theta _ { \\mathrm { M a p } } } ) ^ { \\top } \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As $F$ and $G$ are still quadratically large, we typically need further factorization assumptions. The most lightweight is a diagonal factorization which ignores off-diagonal elements [42, 43]. More expressive alternatives are block-diagonal factorizations such as Kronecker-factored approximate curvature (KFAC) [18–20], which factorizes each within-layer Fisher4 as a Kronecker product of two smaller matrices. KFAC has been successfully applied to the LA [23, 24] and can be improved by low-rank approximations of the KFAC factors [29] by leveraging their eigendecompositions [44]. Finally, recent work has studied/enabled low-rank approximations of the Hessian/Fisher [45–47]. ",
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+    {
+        "type": "text",
+        "text": "$\\textcircled{3}$ Hyperparameter Tuning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As with all approximate inference methods, the performance of the LA depends on the (hyper)parameters of the prior and likelihood. For instance, it is typically beneficial to tune the prior variance $\\gamma ^ { 2 }$ used for inference [23, 28, 27, 26, 22]. Commonly, this is done through cross-validation, e.g. by maximizing the validation log-likelihood [23, 48] or, additionally, using out-of-distribution data [28, 49]. When using the LA, however, marginal likelihood maximization (a.k.a. empirical Bayes or the evidence framework [34, 50]) constitutes a more principled alternative to tune these hyperparameters, and requires no validation data. Immer et al. [22] showed that marginal likelihood maximization with LA can work in deep learning and even be performed in an online manner jointly with the MAP estimation. Note that such approach is not necessarily feasible for other approximate inference methods because most do not provide an estimate of the marginal likelihood. Other recent approaches for hyperparameter tuning for the LA include Bayesian optimization [51] or the addition of dedicated, trainable hidden units for the sole purpose of uncertainty tuning [49]. ",
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+    {
+        "type": "text",
+        "text": "$\\textcircled{4}$ Approximate Predictive Distribution ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To predict using a posterior (approximation) $p ( \\theta \\mid \\mathcal { D } )$ , we need to compute $p ( y \\mid f ( x _ { * } ) , \\mathcal { D } ) \\ =$ $\\begin{array} { r } { \\int p \\dot { ( \\boldsymbol { y } \\vert } f _ { \\boldsymbol { \\theta } } ( x _ { * } ) ) \\dot { p ( \\boldsymbol { \\theta } \\vert } \\dot { \\mathcal { D } } ) d \\boldsymbol { \\theta } } \\end{array}$ for any test point $x _ { * } \\in \\mathbb { R } ^ { n }$ , which is intractable in general. The sim| ⇤ D       (✓s)Ss=1 from p(✓ | D): p(y | f (x⇤), D) ⇡ S\u00001 PSs=1 p(y | f✓s (x⇤)). However, for LAs with GGN [26] attribute this to the inconsistency between Hessian approximation and the predictive and suggest to use a linearized predictive instead, which can also be useful for theoretic analyses [28]. For the last-layer LA, the Hessian coincides with the GGN and the linearized predictive is exact. ",
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+    {
+        "type": "text",
+        "text": "The predictive of a linearized neural network with a LA approximation to the posterior $p ( \\boldsymbol { \\theta } | \\mathcal { D } ) \\approx$ $\\mathcal { N } ( \\theta ; \\theta _ { \\mathrm { M A P } } , \\mathcal { \\Sigma } )$ results in a Gaussian distribution on neural network outputs $f _ { * } : = f ( x _ { * } )$ and therefore enables simple approximations or even a closed-form solution. The distribution on the outputs is given by $\\bar { p } ( f _ { * } | \\bar { x } _ { * } , \\mathcal { D } ) \\approx \\mathcal { N } ( f _ { * } ; f _ { \\theta _ { \\mathrm { M A P } } } ( x _ { * } ) , J ( x _ { * } ) ^ { \\intercal } \\varSigma J ( x _ { * } ) )$ and is typically significantly lowerdimensional (number of outputs $C$ instead of parameters $D$ ). It can also be inferred entirely in function space as a Gaussian process [25, 26]. Given the distribution on outputs $f _ { * }$ , the predictive distribution can be obtained by integration against the likelihood: $\\begin{array} { r } { p ( y | x _ { * } , \\mathcal { D } ) = \\int p ( y | \\bar { f } _ { * } ) p ( f _ { * } | x _ { * } , \\mathcal { D } ) d \\theta } \\end{array}$ . In the case of regression with a Gaussian likelihood with variance $\\sigma ^ { 2 }$ , the solution can even be obtained analytically: $\\bar { p } ( y | x _ { * } , \\mathcal { D } ) \\approx \\mathcal { N } ( y ; f _ { \\theta _ { \\mathrm { M A P } } } ( x _ { * } ) , J ( x _ { * } ) ^ { \\intercal } \\varSigma J ( x _ { * } ) + \\sigma ^ { 2 } I )$ . For non-Gaussian likelihoods, e.g. in classification, a further approximation is needed. Again, the simplest approximation to this is Monte Carlo integration. In the binary case, we can employ the probit approximation [31, 16] which approximates the logistic function with the probit function. In the multi-class case, we can use its generalization, the extended probit approximation [52]. Finally, first proposed for non-BNN applications [53, 54], the Laplace bridge approximates the softmax-Gaussian integral via a Dirichlet distribution [55]. The key advantage is that it yields a distribution of the integral solutions. ",
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+        "type": "text",
+        "text": "3 laplace: A Toolkit for Deep Laplace Approximations ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Implementing the LA is non-trivial, as it requires efficient computation and storage of the Hessian. While this is not fundamentally difficult, there exists no complete, easy-to-use, and standardized implementation of various LA flavors—instead, it is common for deep learning researchers to repeatedly re-implement the LA and Hessian computation with varying efficiency [56–58, etc.]. An efficient implementation typically requires hundreds of lines of code, making it hard to quickly prototype ",
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+        "type": "text",
+        "text": "Listing 1: Fit diagonal LA over all weights of a pre-trained classification model, do post-hoc tuning of the prior precision hyperparameter using cross-validation, and make a prediction for input $x$ with the probit approximation. ",
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+        "type": "text",
+        "text": "with the LA. To address this, we introduce laplace: a simple, easy-to-use, extensible library for scalable LAs of deep NNs in PyTorch [59]. laplace enables all sensible combinations of the four components discussed in Section 2—see Fig. 2 for details. Listings 1 and 2 show code examples. ",
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+        "type": "text",
+        "text": "The core of laplace consists of efficient implementations of the LA’s key quantities: (i) posterior (i.e. Hessian computation and storage), (ii) marginal likelihood, and (iii) posterior predictive. For (i), to take advantage of advances in automatic differentiation, we outsource the Hessian computation to state-of-the-art, optimized second-order optimization libraries: BackPACK [21] and ASDL [60]. Moreover, we design laplace in a modular manner that makes it easy to add new backends and approximations in the future. For (ii), we follow Immer et al. [22] in our implementation of the LA’s marginal likelihood—it is thus both efficient and differentiable and allows the user to implement both online and post-hoc marginal likelihood tuning, cf. Listing 2. Note that laplace also supports standard cross-validation for hyperparameter tuning [23, 28], as shown in Listing 1. Finally, for (iii), laplace supports all approximations to the posterior predictive distribution discussed in Section 2—it thus provides the user with flexibility in making predictions, depending on the computational budget. ",
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+        "text": "Default behavior To abstract away from a large number of options available (Section 2), we provide the following default choices based on our extensive experiments (Section 4); they should be applicable and perform decently in the majority of use cases: we assume a pre-trained network and treat only the last-layer weights probabilistically (last-layer LA), use the KFAC factorization of the GGN and tune the hyperparameters post-hoc using empirical Bayes. To make predictions, we use the closed-form Gaussian predictive distribution for regression and the (extended) probit approximation for classification. Of course, the user can pick custom choices (Listings 1 and 2). ",
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+        "type": "text",
+        "text": "Limitations Because laplace employs external libraries (BackPACK [21] and ASDL [60]) as backends, it inherits the available choices of Hessian factorizations from these libraries. For instance, the LA variant proposed by Lee et al. [29] can currently not be implemented via laplace, because neither backend supports eigenvalue-corrected KFAC [44] (yet). ",
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+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+        "text": "We benchmark various LAs implemented via laplace. Section 4.1 addresses the question of “which are the best design choices for the LA”, in light of Figure 2. Section 4.2 shows that the LA is competitive to strong Bayesian baselines in in-distribution, dataset-shift, and out-of-distribution (OOD) settings. We then showcase some applications of the LA in downstream tasks. Section 4.3 demonstrates the applicability of the (last-layer) LA on various data modalities and NN architectures (including transformers [61])—settings where other Bayesian methods are challenging to implement. Section 4.4 shows how the LA can be used as an easy-to-use yet strong baseline in continual learning. In all results, arrows behind metric names denote if lower (#) or higher (\") values are better. ",
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+        "type": "image",
+        "img_path": "images/aca1294dce9f89ed6bda1908df9957f01fe321679803ea5b42ba687564ecf9c4.jpg",
+        "image_caption": [
+            "Figure 3: In- vs. out-of-distribution (ID and OOD, resp.) performance on CIFAR-10 of different LA configurations (dots), each being a combination of settings for 1) subset-of-weights, 2) covariance structure, 3) hyperparameter tuning, and 4) predictive approximation (see Appendix C.1 for details). “DA” stands for “data augmentation”. Post-hoc performs better with DA and a strong pre-trained network, while online performs better without DA where optimal hyperparameters are unknown. "
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+    {
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+        "table_caption": [
+            "Table 1: OOD detection performance averaged over all test sets (see Appendix C.2 for details). Confidence is defined as the max. of the predictive probability vector [62] (e.g. Confidence $( [ 0 . 7 , 0 . 2 , 0 . 1 ] ) ~ = ~ 0 . 7 )$ . LA and especially $\\mathrm { L A ^ { * } }$ reduce the overconfidence of MAP and achieve better results than the VB, CSGHMC (HMC), and SWAG (SWG) baselines. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">Confidence ↓</td><td colspan=\"2\">AUROC个</td></tr><tr><td>Methods</td><td>MNIST</td><td>CIFAR-10</td><td>MNIST</td><td>CIFAR-10</td></tr><tr><td>MAP</td><td>75.0±0.4</td><td>76.1±1.2</td><td>96.5±0.1</td><td>92.1±0.5</td></tr><tr><td>DE</td><td>65.7±0.3</td><td>65.4±0.4</td><td>97.5±0.0</td><td>94.0±0.1</td></tr><tr><td>VB</td><td>73.2±0.8</td><td>58.8±0.7</td><td>95.8±0.2</td><td>88.7±0.3</td></tr><tr><td>HMC</td><td>69.2±1.7</td><td>69.4±0.6</td><td>96.1±0.2</td><td>90.6±0.2</td></tr><tr><td>SWG</td><td>75.8±0.3</td><td>68.1±2.3</td><td>96.5±0.1</td><td>91.3±0.8</td></tr><tr><td>LA</td><td>67.5±0.4</td><td>69.0±1.3</td><td>96.2±0.2</td><td>92.2±0.5</td></tr><tr><td>LA*</td><td>56.1±0.5</td><td>55.7±1.2</td><td>96.4±0.2</td><td>92.4±0.5</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.1 Choosing the Right Laplace Approximation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In Section 2 we presented multiple options for each component of the design space of the LA, resulting in a large number of possible combinations, all of which are supported by laplace. Here, we try to reduce this complexity and make suggestions for sensible default choices that cover common application scenarios. To this end, we performed a comprehensive comparison between most variants; we measured in- and out-of-distribution performance on standard image classification benchmarks (MNIST, FashionMNIST, CIFAR-10) but also considered the computational complexity of each variant. We provide details of the comparison and a list of the considered variants in Appendix C.1 and summarize the main arguments and take-aways in the following. ",
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+    {
+        "type": "text",
+        "text": "Hyperparameter tuning and parameter inference. We can apply the LA purely post-hoc (only tune hyperparameters of a pre-trained network) or online (tune hyperparameters and train the network jointly, as e.g. suggested by Immer et al. [22]). We find that the online LA only works reliably when it is applied to all weights of the network. In contrast, applying the LA post-hoc only on the last layer instead of all weights typically yields better performance due to less underfitting, and is significantly cheaper. For problems where a pre-trained network or optimal hyperparameters are available, e.g. for well-studied data sets, we, therefore, suggest using the post-hoc variant on the last layer. This LA has the benefit that it has minimal overhead over a standard neural network forward pass (cf. Fig. 5) while performing on par or better than state-of-the-art approaches (cf. Fig. 4). When hyperparameters are unknown or no validation data is available, we suggest training the neural network online by optimizing the marginal likelihood, following Immer et al. [22] (cf Section 4.4). Figure 3 illustrates this on CIFAR-10: for CIFAR-10 with data augmentation, strong pre-trained networks and hyperparameters are available and the post-hoc methods directly profit from that while the online methods merely reach the same performance. On the less studied CIFAR-10 without data augmentation, the online method can improve the performance over the post-hoc methods. ",
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+    {
+        "type": "text",
+        "text": "Covariance approximation and structure. Generally, we find that a more expressive covariance approximation improves performance, as would be expected. However, a full covariance is in most cases intractable for full networks or networks with large last layers. The KFAC structured covariance provides a good trade-off between expressiveness and speed. Diagonal approximations perform significantly worse than KFAC and are therefore not suggested. Independent of the structure, we find that the empirical Fisher (EF) approximations perform better on out-of-distribution detection tasks while GGN approximations tend to perform better on in-distribution metrics. ",
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+    {
+        "type": "text",
+        "text": "Predictive distribution. Considering in- and out-of-distribution (OOD) performance as well as cost, the probit provides the best approximation to the predictive for the last-layer LA. MC integration can sometimes be superior for OOD detection but at an increased computational cost. The Laplace bridge has the same cost as the probit approximation but typically provides inferior results in our experiments. When using the LA online to optimize hyperparameters, we find that the resulting MAP predictive provides good performance in-distribution, but a probit or MC predictive improves OOD performance. ",
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+    {
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+        "img_path": "images/85013079cfb36603abfbc81a41bd09b8e1e0a7f28e6c050a63c2f8fc175e90fb.jpg",
+        "image_caption": [
+            "Figure 4: Assessing model calibration (a) on in-distribution data and $^ { ( \\mathbf { b } , \\mathbf { c } ) }$ under distribution shift, for the MNIST (top row) and CIFAR-10 (bottom row) datasets. For (b,c), we use the Rotated-MNIST (top) and Corrupted-CIFAR-10 (bottom) benchmarks [63, 64]. In (a), we report accuracy and, to measure calibration, negative log-likelihood (NLL) and expected calibration error (ECE)—all evaluated on the standard test sets. In (b) and (c), we plot shift intensities against NLL and ECE, respectively. For Rotated-MNIST (top), shift intensities denote degrees of rotation of the images, while for CorruptedCIFAR-10 (bottom), they denote the amount of image distortion (see [63, 64] for details). (a) On in-distribution data, LA is the best-calibrated method in terms of ECE, while also retaining the accuracy of MAP (unlike VB and CSGHMC). (b,c) On corrupted data, all Bayesian methods improve upon MAP significantly. Even though post-hoc, all LAs achieve competitive results, even to DE. In particular, $\\mathrm { L A ^ { * } }$ achieves the best results, at the expense of slightly worse in-distribution calibration— this trade-off between in- and out-of-distribution performance has been observed previously [65]. "
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+        "type": "text",
+        "text": "Overall recommendation. Following the experimental evidence, the default in laplace is a posthoc KFAC last-layer LA with a GGN approximation to the Hessian. This default is applicable to all architectures that have a fully-connected last layer and can be easily applied to pre-trained networks. For problems where trained networks are unavailable or hyperparameters are unknown, the online KFAC LA with a GGN or empirical Fisher provides a good baseline with minimal effort. ",
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+        "type": "text",
+        "text": "4.2 Predictive Uncertainty Quantification ",
+        "text_level": 1,
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+        "text": "We consider two flavors of LAs: the default flavor of laplace (LA) and the most robust one in terms of distribution shift found in Section 4.1 (LA\\*—last-layer, with a full empirical Fisher Hessian approximation, and the probit approximation). We compare them with the MAP network (MAP) and various popular and strong Bayesian baselines: Deep Ensemble [DE, 14], mean-field variational Bayes [VB, 11, 12] with the flipout estimator [66], cyclical stochastic-gradient Hamiltonian Monte Carlo [CSGHMC / HMC, 67], and SWAG [SWG, 15]. For each baseline, we use the hyperparameters recommended in the original paper—see Appendix A for details. First, Fig. 4 shows that LA and $\\mathbf { L A ^ { * } }$ are, respectively, competitive with and superior to the baselines in trading-off between in-distribution calibration and dataset-shift robustness. Second, Table 1 shows that LA and $\\mathbf { L A ^ { * } }$ achieve better results on out-of-distribution (OOD) detection than even VB, CSGHMC, and SWG. ",
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+        "type": "text",
+        "text": "The LA shines even more when we consider its (time and memory) cost relative to the other, more complex baselines. In Fig. 5 we show the wall-clock times of each method relative to MAP’s for training and prediction. As expected, DE, VB, and CSGHMC are slow to train and in making predictions: they are between two to five times more expensive than MAP. Meanwhile, despite being post-hoc, SWG is almost twice as expensive as MAP during training due to the need for sampling and updating its batch normalization statistics. Moreover, with 30 samples, as recommended by its authors [15], it is very expensive at prediction time—more than ten times more expensive than MAP. ",
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+        "image_caption": [
+            "Figure 6: Assessing real-world distribution shift robustness on five datasets from the WILDS benchmark [68], covering different data modalities, model architectures, and output types. Camelyon17: Tissue slide image tumor classification across hospitals (DenseNet-121 [69]). FMoW: Satellite image land use classification across regions/years (DenseNet-121). CivilCommments: Online comment toxicity classification across demographics (DistilBERT [70]). Amazon: Product review sentiment classification across users (DistilBERT). PovertyMap: Satellite image asset wealth regression across countries (ResNet-18 [35]). We plot means $\\pm$ standard errors of the NLL (top) and ECE (for classification) or regression calibration error [71] (bottom). The in-distribution (left panels) and OOD (right panels) dataset splits correspond to different domains (e.g. hospitals for Camelyon17). LA is much better calibrated than MAP, and competitive with temp. scaling and DE, especially on the OOD splits. "
+        ],
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+        "type": "text",
+        "text": "Meanwhile, LA (and LA\\*) is the cheapest of all methods considered: it only incurs a negligible overhead on top of the costs of MAP. This is similar for the memory consumption (see Table 5 in Appendix C.5). This shows that the LA is significantly more memory- and compute-efficient than all the other methods, adding minimal overhead over MAP inference and prediction. This makes the LA particularly attractive for practitioners, especially in low-resource environments. Together with Fig. 4 and Table 1, this justifies our default flavor in laplace, and importantly, shows that Bayesian deep learning does not have to be expensive. ",
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+        "image_caption": [
+            "Figure 5: Wall-clock time costs relative to MAP. LA introduces negligible overhead over MAP, while all other baselines are significantly more expensive. "
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+        "text": "4.3 Realistic Distribution Shift ",
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+        "type": "text",
+        "text": "So far, our experiments focused on comparably simple benchmarks, allowing us to comprehensively assess different LA variants and compare to more involved Bayesian methods such as VB, MCMC, and SWAG. In more realistic settings, however, where we want to improve the uncertainty of complex and costly-to-train models, such as transformers [61], these methods would likely be difficult to get to work well and expensive to run. However, one might often have access to a pre-trained model, allowing for the cheap use of post-hoc methods such as the LA. To demonstrate this, we show how laplace can improve the distribution shift robustness of complex pre-trained models in large-scale settings. To this end, we use WILDS [68], a recently proposed benchmark of realistic distribution shifts encompassing a variety of real-world datasets across different data modalities and application domains. While the WILDS models employ complex (e.g. convolutional or transformer) architectures as feature extractors, they all feed into a linear output layer, allowing us to conveniently and cheaply apply the last-layer LA. As baselines, we consider: 1) the pre-trained MAP models [68], 2) post-hoc temperature scaling of the MAP models (for classification tasks) [1], and 3) deep ensembles [14].5 More details on the experimental setup are provided in Appendix C.3. Fig. 6 shows the results on five different WILDS datasets (see caption for details). Overall, Laplace is significantly better calibrated than MAP, and competitive with temperature scaling and ensembles, especially on the OOD splits. ",
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+        "type": "text",
+        "text": "4.4 Further Applications ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Beyond predictive uncertainty quantification, the LA is useful in wide range of applications such as Bayesian optimization [37], bandits [72], active learning [34, 73], and continual learning [24]. The laplace library conveniently facilitates these applications. As an example, we demonstrate the performance of the LA on the standard continual learning benchmark with the PermutedMNIST dataset, consisting of ten tasks each containing pixel-permuted MNIST images [74]. Figure 7 shows how the all-layer diagonal and Kronecker-factored LAs can overcome catastrophic forgetting. In this experiment, we update the LAs after each task as suggested by Ritter et al. [24] and improve upon their result by tuning the prior precision through marginal likelihood optimization during training, following Immer et al. [22] (details in Appendix C.4). Using this scheme, the performance after 10 tasks is at around $9 6 \\%$ accuracy, outperforming other Bayesian approaches for continual learning [7, 75, 76]. Concretely, we show that the KFAC LA, while much simpler when applied via laplace, can achieve better performance to a recent VB baseline [VOGN, 13]. Our library thus provides an easy and quick way of constructing a strong baseline for this application. ",
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+        "img_path": "images/c57807865bc426e868ac3d967fdb140129f548f82f96c23b66f4a6de203120ea.jpg",
+        "image_caption": [
+            "Figure 7: Continual learning results on Permuted-MNIST. MAP fails catastrophically as more tasks are added. The Bayesian approaches substantially outperform MAP, with LA-KFAC performing the best, closely followed by VOGN. "
+        ],
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+    {
+        "type": "text",
+        "text": "5 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The LA is fundamentally a local approximation that covers a single mode of the posterior; similarly, other Gaussian approximations such as mean-field variational inference [11–13] or SWAG [15] also only capture local information. SWAG uses the first and second empirical moment of SGD iterates to form a diagonal plus low-rank Gaussian approximation but requires storing many NN copies and applying a (costly) heuristic related to batch normalization at test time. In contrast, the LA directly uses curvature information of the loss around the MAP and can be applied post-hoc to pre-trained NNs. ",
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+    {
+        "type": "text",
+        "text": "In contrast to local Gaussian approximations, (stochastic-gradient) MCMC methods [77, 78, 67, 79, 80, etc.] and deep ensembles [14] can explore several modes. Nevertheless, prior works—also validated in our experiments in Section 4—indicate that using a single mode might not be as limiting in practice as one might think. Wilson and Izmailov [81] conjecture that this is due to the complex, nonlinear connection between the parameter space and the function (output) space of NNs. Moreover, while unbiased compared to its simpler alternatives, MCMC methods are notoriously expensive in practice and, thus, often require further approximations such as distillation [82, 83]. Finally, note that both the LA as well as SWAG can be extended to ensembles of modes in a post-hoc manner [84, 81]. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we argued that the Laplace approximation is a simple yet competitive and versatile method for Bayesian deep learning that deserves wider adoption. To this end, we reviewed many recent advances to and variants of the Laplace approximation, including versions with minimal cost overhead that can be applied post-hoc to pre-trained off-the-shelf models. In a comprehensive evaluation we demonstrated that the Laplace approximation is on par with other approaches that approximate the intractable network posterior, but at typically much lower computational cost. A particularly simple variant that only treats some weights probabilistically can even be used in the context of pre-trained transformer models to improve predictive uncertainty. As an efficient implementation is not straightforward, we introduced laplace, a modular and extensible software library for PyTorch offering user-friendly access to all major flavors of the Laplace approximation. In this way, Laplace approximations provide drop-in Bayesian functionality for most types of deep neural networks. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We thank Kazuki Osawa for providing early access to his automatic second-order differentiation (ASDL) library for PyTorch and Alex Botev for feedback on the manuscript. We also thank the anonymous reviewers for their helpful suggestions for our paper. ",
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+    {
+        "type": "text",
+        "text": "E.D. acknowledges funding from the EPSRC and Qualcomm. A.I. gratefully acknowledges funding by the Max Planck ETH Center for Learning Systems (CLS). R.E., A.K. and P.H. gratefully acknowledge financial support by the European Research Council through ERC StG Action 757275 / PANAMA; the DFG Cluster of Excellence “Machine Learning - New Perspectives for Science”, EXC 2064/1, project number 390727645; the German Federal Ministry of Education and Research (BMBF) through the Tübingen AI Center (FKZ: 01IS18039A); and funds from the Ministry of Science, Research and Arts of the State of Baden-Württemberg. A.K. is grateful to the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for support. ",
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+    {
+        "type": "text",
+        "text": "References ",
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+        "type": "text",
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Bayesian-Torch: Bayesian Neural Network Layers for Uncertainty Estimation. https://github.com/IntelLabs/bayesian-torch, 2020.   \n[98] Florian Wenzel, Jasper Snoek, Dustin Tran, and Rodolphe Jenatton. Hyperparameter Ensembles for Robustness and Uncertainty Quantification. In NeurIPS, 2020.   \n[99] Ferenc Huszár. Note on the quadratic penalties in elastic weight consolidation. Proceedings of the National Academy of Sciences, page 201717042, 2018. ",
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parse/train/hQGRD1Zael7/hQGRD1Zael7.md

@@ -0,0 +1,397 @@
+# The Regularizing Effect of Different Output Layer Designs in Deep Neural Networks
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Deep neural networks are prone to overfitting, especially on small datasets. Com  
+2 mon regularizers such as dropout or dropconnect reduce overfitting, but are complex   
+3 and prone to hyperparameter choices, thus prolonging development cycles in prac  
+4 tice. In this paper, we propose simple but effective design changes to the output   
+5 layer - namely randomization, sparsity, activation scaling, and ensembling - that   
+6 lead to improved regularization. These designs are motivated by experiments   
+7 showing that standard fully-connected output layers tend to rely on individual   
+8 input neurons, which in turn do not cover the variance of the data. We call these   
+9 two related phenomena neuron dependency and expressivity, propose different   
+10 ways to measure them, and optimize the presented output layers for them. In our   
+11 experiments, we compare these layer types for image classification and semantic   
+12 segmentation across architectures, datasets, and application settings. We report sig  
+13 nificantly and consistently improved performance of up to $10 \%$ points in accuracy   
+14 over standard output layers while reducing the number of trainable parameters by   
+15 up to $90 \%$ . It is demonstrated that neither training of output layers is required, nor   
+16 are output layers themselves crucial components of deep networks.
+
+# 17 1 Introduction
+
+18 Neural networks are powerful feature extractors that have become the standard approach for a myriad   
+19 of tasks. New architectures are continuously introduced and set records on benchmark datasets   
+20 (e.g. [24, 15, 44]). These networks differ in layer composition, depth/width or use specific concepts   
+21 such as residual connections [15] or self-attention [34]. With growing capacity, their performance on   
+22 large datasets tends to increase [44]. However, model complexity is also associated with overfitting,   
+23 especially for small datasets where fine details of the training data are easily memorized [52, 2, 53].   
+24 Rather than defining another, possibly more complex architecture, we analyze what often remains   
+25 unconsidered: the output layer. In image classification, networks usually end with a fully-connected   
+26 (fc) layer that combines extracted features for the final output [43, 15, 17, 44]. As we will show, this   
+27 layer is prone to overfitting since high dependencies on individual, possibly memorized features can   
+28 arise. The same neurons are subsequently not able to generalize across examples. We call these two   
+29 related phenomena neuron dependency and expressivity and illustrate a simplified example in Fig. 2.   
+30 Both problems can be improved by simple but effective changes to the output layer that require only   
+31 few lines of code and achieve better generalization (i.e., better results on the test set [25], see e.g.   
+32 Fig. 1). Those changes rely on four principles: activation scaling, fixed randomization, sparsity and   
+33 in-layer ensembling (see Fig. 5). This work analyzes all layers in terms of their capability to reduce   
+34 dependencies and/or increase expressivity. Then, the connection to network performance is shown   
+35 through a comprehensive empirical study across datasets, architectures and application settings in   
+36 image classification and segmentation. Furthermore, we investigate how stronger regularization   
+37 can be induced by applying the identified principles to other parts of a network while reducing the   
+38 computational footprint. In contrast to common practice, we find neither training of output layers   
+39 to be necessary, nor that output layers are crucial components of deep networks. In summary, our   
+40 contributions are:
+
+![](images/8e43eeee593d328662090cd17f785e27e200555070f9bdf398e263d153622b69.jpg)  
+Figure 1: Effect of different output layer designs on cross-entropy loss (left) and accuracy (right) in a ResNet-50 for the STL-10 dataset. Best viewed in color.
+
+![](images/f3006ae6c8378d574fca6155922ed230bfe6cb0a5b920eee8e8a32624e25d218.jpg)  
+Figure 2: Schematic of neuron dependency/expressivity in fc output layers. The left side of each subfigure represents penultimate layer activations (A1-2), the right shows output neurons for each class (O1-2). Filled/blank circles indicate high/low activation, up-/downward facing arrows signal positive/negative weights. Higher activations of O1 lead to correct predictions in this example. Model 1 depends on neuron A1 to be activated to give high prediction scores to O1. This is the case for a training instance in a). If Model 1 is applied to an unseen input pattern of same class in b), higher scores are erroneously given to O2 since A1 remains inactive and A2 slightly favors O2. Model 1 fails to generalize as it depends on A1, which is not expressive enough to cover the variance of the target class. Instead, Model 2 shown in c) exhibits neurons with low dependency and high expressivity, where A1 generalizes to unseen patterns, while the activation of A2 can be regarded as backup. Note that this example is simplified and educational. See Sect. 3.3 for measurements.
+
+• Introducing neuron dependency and expressivity as two factors contributing to overfitting and proposing ways to measure these factors • Showing improved regularization of 5 different output layer designs up to $10 \%$ in absolute accuracy compared to standard fc layers and other common regularizers • Empirical results showing that the proposed layers have improved dependency and expressivity, computational efficiency, wide applicability to both small and large datasets, extensibility to other parts of the network, and robustness in the choice of hyperparameters
+
+# 48 2 Related Work
+
+49 Regularization in deep learning is approached in various ways. Widely used methods are, e.g.,   
+50 normalization [19, 3], weight decay [32], data and adversarial augmentation [40, 1], early stopping [7],   
+51 boosting [38], multitask learning [6], dropout [42], dropconnect [50], and Gaussian noise layers [10].   
+52 To the best of our knowledge, this is the first work that evaluates regularization with respect to   
+53 different output layer designs. Similar to dropout/dropconnect, output layers can be categorized as   
+54 affecting the architecture according to the regularization taxonomy described in [25]. Unlike other   
+55 regularizers, our methods are either hyperparameter-free or robust to their choice and can be applied   
+56 to any deep net, including pre-trained ones that are less affected by overfitting (see Sect. 5.4 and 5.7).   
+57 Related to fixed randomization are the output layers used in [16, 39, 14], which show comparable   
+58 performance to trained layers. One can also preallocate output layer weights with a defined struc  
+59 ture [31, 16]. Besides output layers, weight fixing is for example applied to the first layer in the   
+60 Extreme Learning Machine [18], or to different weight dimensions in [36]. In contrast, we omit   
+61 hand-crafted weights, show improved regularization and relate to neuron dependencies. Further, we   
+62 show that fixing or scaling the last conv block next to the output layer has a strong regularizing effect.   
+63 Sparsity is common in deep learning, e.g. the ReLU activation [13] or a $L _ { 1 }$ penalty term in the loss   
+64 function [46]. Sparsity has also been applied to the channels of Convolutional Neural Networks   
+65 (CNNs) [8, 29]. Others induce sparsity by pruning connections before training under the lottery   
+66 ticket hypothesis [11, 30], with the goal of reducing the number of parameters while not sacrificing   
+67 performance [27, 45, 51]. Different to them, we show that (extreme) sparsity is not merely useful to   
+68 improve computational efficiency, but to improve performance when applied to the output layer.   
+69 The Network in Network (NIN) [28] and All-CNN [41] both use global average pooling (GAP)   
+70 followed by softmax, which replaces the fc output layer with an identify transform to simplify the   
+71 network. This is further analyzed in [33]. We show its connection to neuron dependency/expressivity   
+72 and achieve comparable or better performance on various datasets. Further, we observe that previous   
+73 works do not leverage the full capacity of the last layer in modern networks, which enables the   
+74 construction of computationally efficient in-layer ensembles that further boost performance in small   
+75 and large datasets.
+
+# 76 3 Neuron dependency and expressivity
+
+# 3.1 Setting and notation
+
+We consider neural networks consisting of an encoder $f _ { e n c } : \pmb { \mathsf { X } }  \pmb { a }$ followed by an output layer $f _ { o u t } : \pmb { a }  \hat { \pmb y }$ . In this paper, the encoder is a CNN, taking as input an image $\pmb { \chi } \doteq \mathbb { R } ^ { C \times H ^ { \star } W }$ with $C , W$ and $H$ being input channels, width and height, respectively; and transforming it to a feature vector $\pmb { a } \in \mathbb { R } ^ { 1 \times N }$ . Commonly in CNNs, 2D representations resulting from the final conv layer are aggregated by GAP [28] where $N$ corresponds to the number of pooled conv channels. The output layer transforms the embedding to output $\hat { \pmb y } \in \mathbb { R } ^ { K }$ holding the probabilities of $K$ classes. The output layer is parameterized by a weight matrix $\pmb { W } \in \mathbb { R } ^ { N \times K }$ , and is commonly initialized as $W ^ { r a n d o m } \sim \mathcal { U } ( - \overset { \cdot } { \sqrt { 1 / N } } , \overset { } { \sqrt { 1 / N } } )$ [26]. Both $\hat { y }$ and target $\textbf {  { y } }$ are used to compute the cross-entropy loss $\begin{array} { r } { \ell = - \sum _ { i } ^ { K } y _ { i } \log ( \hat { y } _ { i } ) } \end{array}$ . We use the terms features/channels/nodes or neurons interchangeably meaning activations $\textbf { \em a }$ . When required, we refer to individual instances with a superscript, e.g. $( \mathbf { X } ^ { ( i ) } , \pmb { y } ^ { ( i ) } ) \in \mathcal { D }$ , with $\mathcal { D }$ being a dataset. Corresponding subsets are denoted as $\mathcal { D } _ { t r a i n }$ and $\mathcal { D } _ { t e s t }$ .
+
+# 3.2 Concepts
+
+During training, CNNs learn a set of visual patterns that are combined for a classification decision. However, if patterns remain undetected, e.g. due to noise in the image or inherent but unseen variance in the data, their activation values can become small and thus reduce the output values for the target class. When a network is overfitting, it learns malignant image-specific patterns by heart [52]. Such a network may depend on the activation of individual nodes, which in turn fail to generalize to patterns that are salient to a class. We call these two related phenomena neuron dependency and expressivity.
+
+97 Neuron dependency: How much does a model depend on a single neuron? In a network   
+98 with high neuron dependencies, output scores and thus performance drop significantly when certain   
+99 neurons remain inactive. In contrast, a network with low neuron dependencies distributes activations   
+00 across many neurons, so that a single inactive node does not have much influence on the classification.   
+101 Neuron expressivity: How much class-specific variance does a neuron cover? Neurons with low   
+02 expressivity focus on unimportant details that do not characterize the properties of a class. In contrast,   
+03 a neuron with high expressivity generalizes by activating to various patterns pertinent to a given class.
+
+104 An example of neuron dependency/expressivity for a simplified fc output layer is illustrated in Fig. 2.
+
+![](images/6ba3f7354c2569c371b9a496a44617a877179672678813c1f8a0b9f0fd35d4af.jpg)  
+Figure 3: The effect of dataset size and activation scale on neuron dependency in a ResNet-50 trained on subsets of CIFAR-100, evaluated on the test set. Left: Small training sets lead to high neuron dependencies. Center: Scaling activations results in larger absolute logits. Right: Larger scales lead to higher dependencies. Best viewed in color.
+
+![](images/536b1049d52311a96dfd6c592448af58dbc321fabc96aa3ba43a984aee61f334.jpg)  
+Figure 4: Neuron dependency (left) and expressivity (right) in a ResNet-50 with 2048 penultimate layer channels trained on CIFAR-100 for different output layer designs, showing the change in accuracy on the test set. Best viewed in color.
+
+# 105 3.3 Measuring dependency and expressivity
+
+106 We introduce two ways of measuring dependency/expressivity: instance-based and class-based. The   
+107 former is used to determine the dependency on the most important node for the predicted class given an   
+108 instance. Importance scores for node $n$ and output class $\hat { k }$ are computed with Gradient $\odot .$ Activation [4],   
+109 a global attribution method where we leverage the partial derivative of the softmax values:
+
+$$
+a _ { n } \frac { \partial \hat { y } _ { \hat { k } } } { \partial a _ { n } } .
+$$
+
+110 Instance-based dependency is then measured as avg. reduction in output probabilities when ablating   
+111 the most important feature w.r.t. the output class of any instance. This is illustrated for various   
+112 training set sizes of CIFAR-100 [23] in Fig. 3 (left). With less data, fc output layers tend to depend   
+113 more on single nodes. This is in contrast to class-based measures, which enable quantifying both   
+114 dependency/expressivity and use various features jointly. Importances are determined for each class   
+115 $k$ and over all test instances:
+
+$$
+\sum _ { i = 1 } ^ { \left| \mathcal { D } _ { t e s t } \right| } a _ { n } ^ { ( i ) } \frac { \partial \hat { y } _ { k } ^ { ( i ) } } { \partial a _ { n } ^ { ( i ) } } .
+$$
+
+116 Class-based dependency is then measured as drop in accuracy when ablating a given number of most   
+117 important neurons per class. Measuring expressivity reverses this - the most important neurons per   
+118 class are retained, all others are ablated. This is illustrated for both dependency/expressivity in Fig. 4.   
+119 We see that standard (i.e. trained) fc output layers tend to depend on single channels to achieve high   
+120 performance, but these very channels hold only limited class information.
+
+# 21 4 Output Layer Types
+
+122 We describe several simple output layer variants that require minimal changes to standard networks,   
+123 decrease neuron dependency and/or increase neuron expressivity. All types are illustrated in Fig. 5.
+
+![](images/0bb6eaba53161f189e9e2cd5c71c027bc083eae6e5a9d0722ef82f50cdf90dc5.jpg)  
+Figure 5: A visual comparison of various output layer types. Red/blue represent variable/fixed.
+
+# 124 4.1 Standard output layers
+
+The ubiquitous approach to compute class scores is to learn the parameters of a weight matrix $W ^ { t r a i n e \bar { d } }$ , s.t. $\pmb { \hat { y } } = \sigma _ { S M } ( \pmb { a } \pmb { W } ^ { t r a i n e d } )$ with $\sigma _ { S M } ( \cdot )$ being softmax. Each feature is considered in the computation of each class score. As shown in Sect. 3, trained fc output layers can lead to high neuron dependencies, where the deletion of a single neuron might cause significant loss in performance, and low neuron expressivity, where multiple features are required for adequate predictions.
+
+# 4.2 Scaled output layers
+
+The reduction of an activation, e.g. due to changing light conditions, has a large influence on the output scores. This is simulated in Fig. 3 (center) by multiplying features during training with a scalar $\alpha > 0$ , so that $\pmb { \hat { y } } = \sigma _ { S M } ( \alpha \pmb { a } \pmb { W } ^ { s \bar { c } a l e d } )$ . Note that the variances of the output logit distributions increase with $\alpha$ , resulting in larger differences (or smaller entropies) after softmax normalization. This results in greater dependencies of the model on individual neurons, as shown in Fig. 3 (right). However, if $\alpha$ is chosen small, the activations of individual neurons become insufficient for class discrimination with high confidence. The model is therefore forced to learn multiple class-specific features for each instance, which increases the expressivity of the neurons and also reduces their dependencies to some extent, as shown in Fig. 4. If not specified otherwise, we use $\alpha = 0 . 1$ .
+
+# 40 4.3 Random fixed layers
+
+This setting uses $W ^ { r a n d o m }$ during training/inference, and its classification performance was first analyzed in [16]. The encoder learns to extract patterns that adjust to predetermined weights. Unlike activation scaling, the parameters are bounded and fixed to a small value range. For any class, the chosen uniform initialization is expected to assign similar weight values to multiple neurons, making them learn similar features. We suppose that the enforced similarity reduces dependency shown in Fig. 3 and 4 (both left), while small initialization values increase expressivity as in Sect. 4.2, shown in Fig. 4 (right).
+
+# 8 4.4 Sparse fixed layers
+
+In sparse output layers, class nodes use predetermined sets of channels, some of which might be shared across classes. First, a set of cutting indices $\mathcal { T } _ { k }$ is randomly sampled for each class $k$ , where sparsity is determined by the proportion $q$ of class-specific connections to cut, so that $\left| \mathcal { T } _ { k } \right| = \left\lfloor q \bar { N } \right\rfloor$ with $0 < q < 1$ . Then, starting from a fixed random initialization as in Sect. 4.3, weights connecting to a given class are ablated so that: $W _ { i , k } ^ { s p a r s e } = 0 \forall i \in \mathcal { T } _ { k }$ . Hyperparameter $q$ trades off dependency/expressivity. Larger values induce more sparsity, leading to greater dependencies to the remaining nodes, but forcing them to activate across instances, making them expressive. We set $q = 0 . 9$ in the experiments to show that high sparsity benefits generalization.
+
+# 4.5 1-to-1 correspondence layers
+
+158 The most extreme type of sparsity in an output layer is one with a single connection between a   
+159 feature and a class. If these connections correspond to an identity transform, the activations of the   
+160 penultimate layer are equivalent to the class logits - in practice, the output layer can hence be omitted.   
+161 This was analyzed in [33, 28] and showed comparable results to a standard output layer. Formally, we   
+162 have $\pmb { \hat { y } } = \sigma _ { S M } \big ( \pmb { a } \pmb { W } ^ { 1 t o 1 } \big )$ with $\pmb { a } \in \mathbb { R } ^ { 1 \times K }$ and $W ^ { 1 t o 1 } \in \mathbb { R } ^ { K \times K }$ , where ${ \cal W } ^ { 1 t \dot { o } 1 } = \dot { d } i a g ( 1 , 1 , \dot { \bf \Phi } , \dot { \bf \Phi } , 1 )$   
+163 In this layer, both the model’s dependency on individual neurons as well as each neuron’s expressivity   
+164 are maximal. If a single neuron is ablated, the output logits for the class this neuron is connected to is   
+165 reduced to zero. However, individual neurons learn to cover the whole variance of a given class in the   
+166 training set, which is one conjecture for their performance. Note that as mentioned in [33], we have   
+167 the constraint $N = K$ , which might be restrictive for small networks and large numbers of classes.
+
+# 168 4.6 Ensemble layers
+
+169 Is there a way to optimize for both low neuron dependency and high expressivity? Of the approaches   
+170 discussed, 1-to-1 correspondence layers have the highest expressivity. Starting from this layer, a   
+171 simple approach to reduce neuron dependency is to use the capacity of the penultimate layer and   
+172 create multiple heads $h = 1 \ldots H$ with $N = K H$ , each head being a 1-to-1 correspondence layer.   
+173 Each head’s output is effectively computed as $\hat { \pmb { y } } ^ { h } = \sigma _ { S M } ( \alpha \pmb { a } ^ { h } )$ with $\mathbf { \Omega } _ { a } \mathbf { \Omega } _ { h } ^ { h }$ being the activation part of   
+174 head $h$ . As in Sect 4.2, we introduce a scalar $\alpha$ , which controls the magnitude of feature activations.   
+175 176 For consisaveraged: $\begin{array} { r } { \frac { 1 } { H } \dot { \sum _ { h = 1 } ^ { H } } \ell ( \hat { \pmb y } ^ { h } , \pmb y ) } \end{array}$ this approach as . Similarly, logit $W ^ { h e a d s }$ . The loss is computed for each head andraged over heads for inference. Due to the   
+177 induced redundancy, the performance only drops considerably after removing class-related neurons   
+178 from all heads. In our experiments, we set $H$ to its maximum given any setting (architecture/dataset).   
+179 Note that hyperparameter $\alpha$ in ensemble layers is the only one which is tuned to individual settings.
+
+# 180 5 Experiments
+
+181 We aim to show that the presented output layers from Sect. 4 outperform standard output layers and   
+182 common regularization methods in various settings. Details about training, compute resources, code,   
+183 datasets as well as additional experiments on dependency/expressivity are included in the appendix.
+
+# 184 5.1 Small-scale and fine-grained classification
+
+All layer types are first applied to small-scale and/or fine-grained classification, both of which are challenging and require regularization. Datasets include STL-10 (500 img/class) [9], CUB-200 ${ \sim } 3 0$ img/class) [49], Cars-196 ${ \sim } 4 0$ img/class) [22] and Food-101 (750 img/class) [5]. Table 1 shows results for the two popular backbones ResNet-50 [15] and DenseNet-169 [17], exchanging the output layer accordingly. In 53/56 settings, we see improved results over standard layers. Of these, 48 and 36 are significant with $p < 0 . 1$ and $p < 0 . 0 0 1$ , respectively. Although there is no clear best method, it is worth noting that sparse and ensemble layers as enhancements of both random and 1-to-1 layers are significantly better $\mathit { p } < 0 . 0 0 1 )$ ) in 7/8 settings, respectively. As expected, smaller performance differences are exhibited in Food-101, which is a considerably larger dataset, thus requiring less regularization. Among the worse settings, only 1 is significant $( p < 0 . 1 )$ for Food-101 since it involves strong regularization to multiple layers. These regularizers are discussed seperately in Sect. 5.5.
+
+# 197 5.2 Large-scale classification and transfer learning
+
+198 Machine learning models are subject to the bias-variance tradeoff [12], in which induced biases of   
+199 the presented output layers might be too strong to fit the training data. We therefore want to shed   
+200 light on how these layers behave in large-scale and transfer learning settings, where overfitting is   
+201 less problematic. Datasets include CIFAR-100 (C100, 5000 img/class) [23], ImageNet (IN, ${ \sim } 1 2 0 0$   
+202 img/class) from ILSVRC2012 [37] reported on the validation set, as well as CUB/Cars/Food with   
+203 models being pre-trained on IN. Table 2 shows the results for the ResNet-50 backbone. In C100,   
+204 we see consistent improvements with at least $p < 0 . 1$ . On the other datasets, results are mostly   
+205 comparable corroborating widespread applicability. It is worth mentioning that $W ^ { 1 t o 1 }$ and $W$ ensemble   
+206 perform consistently better, and $W ^ { e n s e m b l e }$ significantly $( p < 0 . 1 )$ in multiple cases. With growing   
+207 dataset sizes, both layers expose a strong constraint on the class neurons to fit an increasing number   
+208 of examples. We believe this to be responsible for progressively separating the signal from the noise,   
+209 leading to better generalization. On the other hand, neuron dependency is reduced in larger datasets   
+210 (see Fig. 3 left) diminishing the effect of $W ^ { s c a l e }$ and $W ^ { r a n d o \bar { m } }$ . Moreover, $W ^ { r a n d o m }$ and $W ^ { s p a r s e }$   
+211 can be affected by predetermined feature-class weights that do not have to match features learned   
+212 during pre-training, which might require larger adjustments to the weights of the last conv layer.
+
+Table 1: Classification accuracy for different output layer designs in small-scale and fine-grained classification without pre-training. Exponent repeats describe probability values $( * { } ; \ p \ < \ 0 . 1$ , $^ { * * }$ : $p < 0 . 0 1$ , $^ { \ast \ast \ast }$ : $p \ < \ 0 . 0 0 1 $ indicating statistical significance based on a one-tailed normal approximation interval test comparing accuracy of the proposed layer designs to a baseline fc layer $( \dot { W } ^ { t r a i n e d } )$ . Symbols $^ *$ and $\dagger$ denote better/worse performance than baseline, respectively. Bold denotes best performance.   
+
+<table><tr><td colspan="2"></td><td>STL-10</td><td>CUB-200</td><td>Cars-196</td><td>Food-101</td></tr><tr><td rowspan="7">RrsSee50</td><td>Wtrained (baseline)</td><td>81.36</td><td>57.18</td><td>81.20</td><td>83.70</td></tr><tr><td>Wscaled</td><td>83.33*</td><td>63.46***</td><td>87.07***</td><td>85.46***</td></tr><tr><td>W scaled block</td><td>86.42***</td><td>66.74***</td><td>87.53***</td><td>85.03**</td></tr><tr><td>Wrandom</td><td>86.08***</td><td>60.91**</td><td>83.02*</td><td>84.20</td></tr><tr><td>Wrandom block</td><td>86.59***</td><td>67.21***</td><td>84.07***</td><td>84.04</td></tr><tr><td>Wsparse</td><td>87.23***</td><td>66.27***</td><td>85.47***</td><td>85.45***</td></tr><tr><td>W1to1</td><td>84.78***</td><td>58.56</td><td>80.51</td><td>84.41*</td></tr><tr><td rowspan="6">Grr-nseese1g</td><td>Wensemble</td><td>87.94***</td><td>62.98***</td><td>85.76***</td><td>85.36***</td></tr><tr><td>Wtrained (baseline) W scaled</td><td>81.88 86.53***</td><td>55.33</td><td>80.82</td><td>84.31</td></tr><tr><td>W scaled block</td><td>85.89***</td><td>63.31*** 65.57***</td><td>85.85***</td><td>85.05*</td></tr><tr><td>Wrandom</td><td>86.11***</td><td>61.24***</td><td>85.35***</td><td>85.44**</td></tr><tr><td>Wrandom block</td><td>86.64***</td><td>65.99***</td><td>83.52**</td><td>84.90*</td></tr><tr><td>Wsparse</td><td></td><td></td><td>82.93*</td><td>83.25t</td></tr><tr><td>Wltol</td><td></td><td>86.58***</td><td>62.75***</td><td>85.79***</td><td>84.63</td></tr><tr><td>Wensemble</td><td></td><td>86.06***</td><td>55.37</td><td>83.75**</td><td>84.11</td></tr><tr><td></td><td></td><td>87.00***</td><td>64.15***</td><td>85.09***</td><td>84.91*</td></tr></table>
+
+Table 2: Classification results for different output layer designs in large-scale image recognition and transfer learning. $^ +$ denotes fine-tuning from ImageNet. See Table 1 for other symbols.   
+
+<table><tr><td></td><td>C100</td><td>IN-top1</td><td>IN-top5</td><td>CUB-200+</td><td>Cars-196+</td><td>Food-101+</td></tr><tr><td>Wtrained</td><td>77.75</td><td>76.36</td><td>93.12</td><td>80.91</td><td>91.73</td><td>87.32</td></tr><tr><td>Wscaled</td><td>79.65*</td><td>76.08</td><td>92.84</td><td>78.68t</td><td>90.91t</td><td>87.21</td></tr><tr><td>Wrandom</td><td>78.91*</td><td>76.08</td><td>93.15</td><td>80.89</td><td>91.72</td><td>87.29</td></tr><tr><td>Wsparse</td><td>79.46*</td><td>75.32tt</td><td>92.36ttt</td><td>80.38</td><td>92.07</td><td>87.31</td></tr><tr><td>W1to1</td><td>79.07*</td><td>76.53</td><td>93.32</td><td>81.79</td><td>91.87</td><td>87.31</td></tr><tr><td>Wensemble</td><td>80.38***</td><td>76.62</td><td>93.46*</td><td>82.22*</td><td>92.77*</td><td>87.76</td></tr></table>
+
+# 213 5.3 Use Case: Medical imaging
+
+Output layer design is critical in fields such as medical imaging, which presents special challenges to regularization: Datasets tend to be small, imbalanced, abnormalities might fill only a few pixels of the image, and appearances between classes are often similar. In addition, transfer learning with IN weights is either inaccessible due to architectural differences (e.g. image segmentation, 3D Magnetic Resonance Imaging) or less effective due to large domain differences. This is first illustrated on the APTOS Kaggle challenge dataset (3662 images, 193-1805 img/class) [20], with the goal of detecting diabetic retinopathy severities in retinal fundus images. We use the public training dataset to train a multi-class classifier and perform 5-fold cross-validation. Table 3 shows the results. We consistently get better performance with regularization and reduce the gap to a pre-trained network. Furthermore, an additional experiment in the appendix indicates that the standard output layer is biased towards the prevalent class, which is inherently remedied through randomization.
+
+225 We provide further evidence that the proposed layer designs positively affect tasks other than   
+226 classification. We learn a U-Net [35] for binary semantic slice-based segmentation of Computed   
+227 Tomography scans of livers comparing a standard 1x1 conv output layer with 64 parameters to both a   
+228 fixed randomized and an ensemble layer. Due to the limited number of parameters, we omit $W ^ { s c a l e }$   
+229 and $W ^ { s p a r s e }$ here. Different to classification, the output of a U-Net itself can be interpreted as a 1-to-1   
+230 layer. One can still build an ensemble by treating each output channel as a head. Both $W ^ { r a n d o m }$ and   
+231 W ensemble $H = 1 0$ ) are then applied to the CHAOS [21] and SLIVER [47] datasets. For CHAOS,
+
+Table 3: Quadratic weighted kappa and accuracy (with significance) for different output layers in ResNet-50 for the APTOS dataset. $^ +$ denotes fine-tuning from IN. See Table 1 for other symbols.   
+
+<table><tr><td></td><td>Kappa</td><td>Acc.</td></tr><tr><td>Wtrained</td><td>0.816</td><td>77.44</td></tr><tr><td>W scaled</td><td>0.818</td><td>78.38</td></tr><tr><td>Wrandom</td><td>0.848</td><td>79.32*</td></tr><tr><td>Wsparse</td><td>0.840</td><td>79.60*</td></tr><tr><td>Wltol</td><td>0.856</td><td>80.08*</td></tr><tr><td>Wensemble</td><td>0.866</td><td>80.78**</td></tr><tr><td>Wtrained+</td><td>0.909</td><td>85.02</td></tr><tr><td>W sparse+</td><td>0.910</td><td>85.17</td></tr><tr><td>Wensemble+</td><td>0.912</td><td>85.56</td></tr></table>
+
+Table 4: Jaccard coefficients in segmentation   
+
+<table><tr><td></td><td>CHAOS</td><td>SLIVER</td></tr><tr><td>Wtrained</td><td>0.77</td><td>0.83</td></tr><tr><td>Wrandom</td><td>0.80</td><td>0.85</td></tr><tr><td>Wensemble</td><td>0.78</td><td>0.86</td></tr></table>
+
+Table 5: Regularization comparison   
+
+<table><tr><td></td><td>STL</td><td>CUB</td><td>CUB+</td><td>Cars</td></tr><tr><td>Dropout [42]</td><td>82.73</td><td>63.20</td><td>80.26</td><td>83.98</td></tr><tr><td>Dropconn. [50]</td><td>86.15</td><td>61.48</td><td>80.41</td><td>85.06</td></tr><tr><td>Add. Noise [10]</td><td>82.51</td><td>52.74</td><td>80.91</td><td>76.77</td></tr><tr><td>Wtrained</td><td>81.36</td><td>57.18</td><td>80.91</td><td>81.20</td></tr><tr><td>Wsparse</td><td>87.23</td><td>66.27</td><td>80.38</td><td>85.47</td></tr><tr><td>Wensemble</td><td>87.94</td><td>62.98</td><td>82.22</td><td>85.76</td></tr></table>
+
+we train on 15 randomly selected patients (2155 slices) and evaluate on the remaining 5 (719 slices). We then test for generalization by training on all 20 patients from CHAOS and evaluating on the external SLIVER dataset consisting of 20 patients (4159 slices). Table 4 shows improved results in both settings.
+
+# 5.4 Other regularization techniques
+
+Table 5 compares our most competitive methods to other popular regularizers when applied to a standard output layer. Nodes/connections in dropout/dropconnect are both removed with $p = 0 . 7$ , and the noise layer adds a Gaussian with $\mu = 0$ and $\sigma = 0 . 1$ before applying the fc layer. In all cases, our variants perform better. Whereas noise does not benefit training here, dropout/-connect is supporting regularization. However, both of the latter methods come with two main disadvantages. First, they add complexity by changing states in each iteration and having different behavior during training and inference. Second, hyperparameter tuning is necessary, while our layers are either hyperparameter-free or stable to them. See the ablation study in Sect. 5.7 for evidence.
+
+# 5.5 Beyond output layers - block scaling and randomization
+
+Activation scaling and randomization are techniques applicable to any layer and increase regularization further. This is demonstrated for ResNet-50 and DenseNet-169 in Table 1. Both architectures consist of multiple blocks, each holding groups of conv layer, batch normalization (BN) and activation function. For $W ^ { r a n d o m }$ block, all layers of the last block and the output layer are kept in their initialized state during training. Similarly, in $W ^ { s c a l e d }$ block, activations of all layer groups in the last block are scaled during training. In ResNet-50, block scaling outperforms output layer scaling in $3 / 4$ datasets by up to $3 \%$ points, and block randomization increases performance in 3/4 datasets by up to $6 \%$ points compared to output layer randomization. In DenseNet-169, block scaling outperforms output layer scaling in 2/4 datasets by up to $2 \%$ points, and block randomization increases performance in 2/4 datasets by up to $4 \%$ points compared to output layer randomization. Only in Food-101 and DenseNet, block randomization performs significantly worse than baseline because regularization is too strong leading to underfitting (tr $a i n l o s s = 0 . 4 7$ compared to 0.02 in $W ^ { r a n d o m }$ ).
+
+# 5.6 Computational efficiency
+
+The design of the head of deep CNNs has a great impact on computational efficiency. Standard output layers alone can contain a large amount of parameters, as CNNs typically hold more channels as they get deeper and the number of classes can become large. In ImageNet and a ResNet-50, for example, the output layer alone generates over 2 million parameters, which are saved in $W ^ { 1 t o 1 }$ and $W ^ { e n s e m b l e }$ . This problem compounds when using multiple fc layers. In a VGG-16, for instance, 3 fc layers are employed after the last conv layer. As Table 6 shows, omitting all fc layers saves up to $90 \%$ in parameters, a considerable amount of memory, and time for a forward/backward pass while
+
+Table 6: Computational efficiency comparison in CUB-200 highlighting that the number of trainable parameters can often be reduced while accuracy is improved. $^ +$ denotes fine-tuning from ImageNet.   
+
+<table><tr><td>Architecture</td><td>#Params in M.</td><td>Mem.[GB]</td><td>GFLOPS</td><td>Time [ms/it.]</td><td>Accuracy</td></tr><tr><td>VGG16 Wtrained+</td><td>135.1</td><td>7.5</td><td>31.1</td><td>114</td><td>78.68</td></tr><tr><td>VGG16 W1to1+</td><td>13.5</td><td>5.9</td><td>30.4</td><td>106</td><td>79.27</td></tr><tr><td>VGG16 Wensemble+</td><td>14.7</td><td>5.9</td><td>30.8</td><td>106</td><td>81.15**</td></tr><tr><td>Res50 Wtrained</td><td>23.9</td><td>5.1</td><td>8.2</td><td>71</td><td>57.18</td></tr><tr><td> Res50 Wrandom block</td><td>8.5</td><td>5.0</td><td>8.2</td><td>67</td><td>67.21***</td></tr></table>
+
+![](images/fe7727447bdb0dffda41c5653723499e8d22299b6bfed4abf9900ce522c38049.jpg)  
+Figure 6: Ablation study showing stability and consistency of our output layer designs
+
+266 increasing accuracy. If a ResNet-50 is used, randomization of the last conv block next to the output   
+267 layer yields savings of about $65 \%$ in trainable parameters while increasing accuracy by $10 \%$ points.
+
+# 268 5.7 Ablation study
+
+269 Note that $W ^ { r a n d o m }$ and $W ^ { 1 t o 1 }$ are hyperparameter-free compared to other regularizers such as   
+270 dropout/-connect, thus saving the cost of tuning them. Although other layer variants possess hyperpa  
+271 rameters, we show in Fig. 6 for the ResNet backbone that they are stable (no large jumps in vicinity)   
+272 and consistent (tend to monotonicity w.r.t. performance). In $W ^ { s p a r s e }$ , the maximum accuracy for   
+273 both datasets is at $q = 0 . 9 9$ (20 nodes per class) and drops only slightly for $q = 0 . 9 9 5$ . Similarly,   
+274 downscaling in $W ^ { s c a l e }$ improves performance at a small cost if the optimum is not hit. Also, more   
+275 heads in $W ^ { \bar { e } n s e m b l e }$ tend to increase performance. What is the result of adding more heads than   
+276 given by the constraint $N = K H ?$ If $N < K H$ , which is the case for CUB-200 and $H > 1 0$ , we   
+277 add an additional 1x1 conv layer, BN and ReLU with $K H$ nodes to adjust for the missing channels.   
+278 Although this leads to a considerable increase in parameters ( $. N K H$ for the conv layer), it helps with   
+279 generalization, contradicting the common belief that overparameterization leads to overfitting [48].   
+280 In contrast, dropout is not stable or consistent. With a dropout rate of 0.9, the network fails to train   
+281 in both datasets. Furthermore, the optimum for CUB lies at 0.8, the same hyperparameter choice in   
+282 STL would result in worse performance than baseline.
+
+# 283 6 Conclusion
+
+284 In this work, we introduced neuron dependency and expressivity as factors contributing to overfitting.   
+285 Then, different output layers were defined to optimize both and showed improved regularization in   
+286 various settings while being efficient and robust to hyperparameters. Although these layers are simple,   
+287 they have high practical relevance due to the importance of regularization and the ubiquity of output   
+288 layers in deep nets. In addition to their application, they may also be useful as primitives in future   
+289 (automatically created) architectures. Although improving regularization, we note that optimizing   
+290 for neuron dependencies/expressivity does not solve overfitting. For example, an unknown or noisy   
+291 instance may result in reduced activations in the majority of nodes in the penultimate layer. Finally,   
+292 we speculate that overfitting may not just be a function of the number of parameters in the encoder.   
+293 Instead, it might be more important how the extracted features are combined in the output layer.
+
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+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Sect. 3, 5 and the appendix.   
+(b) Did you describe the limitations of your work? [Yes] See Sect. 4.5 $N = K$ constraint), Sect. 5.2 (some layers are not optimal for large-scale datasets and fine-tuning), Sect. 5.5 (strong regularization may lead to underfitting), and Sect. 6 (neuron dependency/expressivity are factors of overfitting, but this does not constitute all factors).   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] Focus is on architecture and technical, no particular application affecting society is targeted. Note that medical ML applications should be thoroughly evaluated, e.g., for bias and generalizability, before being used in practice.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] Only empirical results are included.   
+(b) Did you include complete proofs of all theoretical results? [N/A] Only empirical results are included.
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+3. If you ran experiments...
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+(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] Code implementing the shown layer types and dataset descriptions are given in the supplemental material/appendix.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We report the corresponding splits if not defined by the datasets themselves. Hyperparameters and information about the training are partly provided in the main text (e.g. architecture type, layer hyperparameters) and are detailed further in the appendix.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We indicate statistical significance for our classification results.   
+(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] We have not tracked CO2 due to missing awareness but some details about the setup are provided in the appendix. We plan to track CO2 in subsequent works.
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+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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+(a) If your work uses existing assets, did you cite the creators? [Yes] See Sect. 5. This is done after first mentioning the corresponding datasets in the main text.   
+(b) Did you mention the license of the assets? [Yes] We mention asset licenses in the appendix.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We provide code as supplemental material.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] The datasets in Sect. 5.3 are open source from past challenges and contain de-identified data to the best of our knowledge. Further details on data collection are provided in the corresponding references.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] All datasets are commonly used for benchmarks and/or do not contain obviously offensive content. However, ImageNet likely contains images showing persons.
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+5. If you used crowdsourcing or conducted research with human subjects...
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+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] No crowdsourcing or research with human subjects has been conducted.   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] See above.   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] See above.