Discriminator-Weighted Offline Imitation Learning from Suboptimal Demonstrations

1 INTRODUCTION . The recent success of reinforcement learning ( RL ) in many domains showcases the great potential of applying this family of learning methods to real-world applications . A key prerequisite for RL is to design a reward function that specifies what kind of agent behavior is preferred . However , in many real-world applications , designing a reward function is prohibitively difficult ( Ng et al. , 1999 ; Irpan , 2018 ) . By contrast , imitation learning ( IL ) provides a much easier way to leverage the reward function implicitly from the collected demonstrations and has achieved great success in many sequential decision making problems ( Pomerleau , 1989 ; Ng et al. , 2000 ; Ho & Ermon , 2016 ) . However , popular IL methods such as behavioral cloning ( BC ) and generative adversarial imitation learning ( GAIL ) ( Ho & Ermon , 2016 ) , assume the expert demonstration is optimal . Unfortunately , it is often difficult to obtain optimal demonstrations for many real-world tasks , because human experts often make mistakes due to various reasons , such as the difficulty of the task , partial observability of the environment , or the presence of distraction . Given such noisy expert demonstrations , which contain records of both optimal and non-optimal behaviors , BC and GAIL all fail to imitate the optimal policy ( Wu et al. , 2019b ; Ma , 2020 ) . Current methods that deal with suboptimal demonstrations either require additional labels , which can be done explicitly by annotating each demonstration with confidence scores by human experts ( Wu et al. , 2019b ) , or implicitly by ranking noisy demonstrations according to their relative performance through interacting with the environment ( Brown et al. , 2019 ; 2020 ) . However , human annotation and environment interaction are laborious and expensive in real-world settings , such as in medicine , healthcare , and industrial processes . In this work , we investigate a pure offline learning setting where the agent has access to neither the expert nor the environment for additional information . The agent , instead , has only access to a small pre-collected dataset of state-action-next state transition triples sampled from the expert and a large batch offline dataset sampled from one or multiple behavior policies that could be highly sub-optimal . Prior works that address the problem are based on variants of BC or inverse RL , Sasaki & Yamashina ( 2021 ) reuse another policy learned by BC as the weight of original BC objective , however , this requires that expert data occupy the majority proportion of the offline dataset , otherwise the policy will be misguided to imitate the suboptimal data . Zolna et al . ( 2020a ) first learns a reward function that prioritizes expert data over others and then performs offline RL based on this reward function . This algorithm is extremely expensive to run , requiring solving offline RL in an inner loop , which itself is a challenging problem and prone to training instability ( Kumar et al. , 2019 ) and hyperparameter sensitivity ( Wu et al. , 2019a ) . In this paper , we propose an offline imitation learning algorithm to learn from demonstrations that ( perhaps ) contain a large proportion of suboptimal data without additional steps of reward learning and offline RL training . Built upon the task of behavioral cloning ( BC ) , we introduce an additional task to learn a discriminator to distinguish expert and non-expert data , we propose a cooperation strategy to boost the performance of both tasks . This results in a new policy learning objective and surprisingly , we find its equivalence to a generalized BC objective , where the outputs of the discriminator serve as the weights of the BC loss function . We thus term our resulting algorithm Discriminator-Weighted Behavioral Cloning ( DWBC ) . Experimental results show that DWBC achieves higher returns and faster training speed compared to baseline algorithms , under different scenarios . 2 PRELIMINARY . 2.1 PROBLEM SETTING . We consider the standard fully observed Markov Decision Process ( MDP ) setting ( Sutton et al. , 1998 ) , M = { S , A , P , r , γ , d0 } , where S is the state space , A is the action space , P : S × A → ∆ ( S ) is the MDP ’ s transition probability , r is the reward function , γ ∈ [ 0 , 1 ) is the discount factor for future reward and d0 is the initial distribution . A policy π : S → ∆ ( A ) maps from state to distribution over actions . We denote dπ ∈ ∆ ( S × A ) as the discounted state-action distribution of π under transition kernel P , that is , dπ = ( 1− γ ) ∑∞ t=0 γ tdπt , where d π t ∈ ∆ ( S × A ) is the distribution of ( s ( t ) , a ( t ) ) under π at step t. Following the standard IL setting , the ground truth reward function r is unknown . Instead , we have the demonstrations by the expert specified by πe : S → ∆ ( A ) ( potentially stochastic and not necessarily optimal ) . Concretely , we have an expert dataset in the form of i.i.d tuples De = { si , ai , s′i } ne i=1 where ( s , a ) is sampled from distribution d πe and s′ is sampled from P ( s , a ) . In our problem setting , we also have an offline static dataset consisting of i.i.d tuples Do = { si , ai , s′i } no i=1 s.t . ( s , a ) ∼ ρ ( s , a ) , s′ ∼ P ( s , a ) , where ρ ∈ ∆ ( S × A ) is an offline state-action distribution resulting from some other behavior policies . Note that these behavior policies could be much worse than the expert πe . Our goal is to only leverage the offline batch data Db = De ∪ Do to learn an optimal policy π with regard to optimizing the ground truth reward r , without any interaction with the environment or expert . 2.2 A GENERALIZED BEHAVIORAL CLONING OBJECTIVE . In order to discard low-quality demonstrations and only clone the best behavior available , previous works ( Sasaki & Yamashina , 2021 ; Zolna et al. , 2020a ) consider a generalized behavioral cloning objective to imitate demonstrations unequally , that is , min π E ( s , a ) ∼Db [ − log π ( a|s ) · f ( s , a ) ] ( 1 ) where f : S ×A → [ 0 , 1 ] denotes an arbitrary weight function . • If f ( s , a ) = 1 for ∀ ( s , a ) ∈ S ×A , the objective ( 1 ) corresponds to the vanilla BC objective . • If f ( s , a ) = π′ ( a|s ) , where π′ is an old policy which was previously optimized with Db , the objective ( 1 ) corresponds to the objective of Behavioral Cloning from Noisy Demonstrations ( Sasaki & Yamashina , 2021 ) . Since ∑ a π ′ ( a|s ) = 1 for ∀s ∈ S is satisfied , π′ ( a|s ) can be interpreted as the weights for the weighted action sampling . • If f ( s , a ) = 1 [ Aπ ( s , a ) ] , where 1 is the indicator function which creates a boolean mask that elim- inates samples which are thought to be worse than the current policy , the objective ( 1 ) corresponds to the objective of Offline Reinforced Imitation Learning ( Zolna et al. , 2020a ) . The objective ( 1 ) can also be deemed as the objective of Soft Q Imitation Learning ( Reddy et al. , 2019 ) with f ( s , a ) = 1 for ( s , a ) ∈ De and f ( s , a ) = 0 for ( s , a ) ∈ Do in online IL literature ; or the objective of off-policy actor-critic ( Off-PAC ) algorithm ( Degris et al. , 2012 ) with f ( s , a ) = Qπ ( s , a ) · π ( a|s ) /πb ( a|s ) in online RL literature . 3 DISCRIMINATOR-WEIGHTED BEHAVIORAL CLONING . We now continue to describe our approach for offline imitation learning from demonstrations that ( perhaps ) contain large-proportional suboptimal data without additional steps of reward learning and offline RL training . Built upon the task of BC , we introduce an additional task to learn a discriminator to distinguish expert and non-expert data , we propose a cooperation strategy to boost the performance of both tasks , this will result in a new generalized BC objective . We then provide the interpretation of weights in our generalized BC objective , this gives the intuition about why our method can work . 3.1 LEARN THE POLICY AND DISCRIMINATOR SEPARATELY . It is obvious that we can avoid the negative impact of suboptimal demonstrations presented in Do by only imitating De , which can be written as min π E ( s , a ) ∼De [ − log π ( a|s ) ] ( 2 ) We call the task of learning a policy using objective ( 2 ) as the BC task . The drawback of BC task is that the learned policy may not be able to generalize due to the potential limited size and state coverage of De . It does not fully utilize the information from Do . If we can select those high-reward transitions from Do and combine them with De , we are expected to get a better policy . Now let ’ s consider another different task , which aims to learn a discriminator by contrasting expert and non-expert transitions , given by min d E ( s , a ) ∼De [ − log d ( s , a ) ] + E ( s , a ) ∼Do [ − log ( 1− d ( s , a ) ) ] ( 3 ) We call this the discriminating task , which is similar to how the discriminator is trained in GAIL ( Ho & Ermon , 2016 ) and GAN ( Goodfellow et al. , 2014a ) , except that the second term is sampled from a fixed dataset instead of new samples obtained from interacting with the environment . However , optimizing objective ( 3 ) will make the discriminator learned to assign 1 to all transitions from De and 0 to all transitions from Do . This limiting behavior is unsatisfactory because Do can contain some successful ( high-reward ) transitions . This bears similarity to the positive-unlabeled classification problem ( Elkan & Noto , 2008 ) , where both positive and negative samples exist in the unlabeled data . To solve this problem , we adopt the approach from positive-unlabeled ( PU ) learning ( Du Plessis et al. , 2015 ; Xu & Denil , 2019 ; Zolna et al. , 2020b ) . The main idea is to re-weight different losses for positive and unlabeled data , in order to obtain an estimate of model loss on negative samples that is not directly available . Applying PU learning to objective ( 3 ) yields the following objective : min d η E ( s , a ) ∼De [ − log d ( s , a ) ] + E ( s , a ) ∼Do [ − log ( 1− d ( s , a ) ) ] − η E ( s , a ) ∼De [ − log ( 1− d ( s , a ) ) ] ( 4 ) where η is a hyperparameter , corresponds to the proportion of positive samples to unlabeled samples . Intuitively , the second term in ( 4 ) could make d ( s , a ) of state-action pairs from De become 0 if similar state-action pairs are included in Do , and the third term in ( 4 ) balances the impact of the second term , i.e. , avoids d ( s , a ) of state-action pairs from De becoming 0 . To summarize , the BC task aims to imitate the expert behavior from De , but ignores the valuable information in Do ; the discriminating task aims to contrast expert and non-expert transitions from De and Do , by only using state-action information as input . Both tasks lack enough information to improve their own performance , which , however , can be obtained from the other task . It then seems natural to find a scheme to incorporate the policy into the training of the discriminator and effectively use the discriminator to help the training of the policy .

Summary: This paper proposes an offline imitation learning framework that incorporates both optimal and suboptimal datasets to learn decision-making tasks, without requiring any reward annotations. To leverage high reward transitions from the suboptimal dataset, the authors formulate a discriminator that optimizes a positive-unlabeled learning objective, where positive samples come from the optimal dataset and unlabeled samples come from the suboptimal dataset. This discriminator is trained in an adversarial fashion along with the policy, resulting in a behavior cloning objective where samples from the optimal and suboptimal datasets are weighted differently according to the discriminator’s predictions. Experiments demonstrate that on a set of simulated locomotion domains, the proposed algorithm can leverage the suboptimal dataset to learn more performant policies compared to vanilla behavior cloning objectives and prior offline IL/RL baselines.