As robots and other autonomous agents enter our homes, schools, workplaces, and hospitals, it is important that these agents can safely learn from and adapt to a variety of human preferences and goals. One common way to learn preferences and goals is via imitation learning, in which an autonomous agent learns how to perform a task by observing demonstrations of the task. While there exists a large body of literature on high-confidence off-policy evaluation in the reinforcement learning (RL) setting, there has been much less work on high-confidence policy evaluation in the imitation learning setting where reward samples are unavailable.
Prior work on high-confidence policy evaluation for imitation learning has used Bayesian inverse reinforcement learning (IRL) ramachandran2007bayesian to allow an agent to reason about reward uncertainty and policy robustness brown2018efficient ; brown2018risk . However, Bayesian IRL is typically intractable for complex problems due to the need to repeatedly solve an MDP in the inner loop. This high computational cost precludes robust safety and uncertainty analysis for imitation learning in complex high-dimensional problems.
. We next propose a novel algorithm, Bayesian Reward Extrapolation (B-REX), that uses a pairwise ranking likelihood to significantly reduce the computational complexity of generating samples from the posterior distribution over reward functions when performing Bayesian IRL. We demonstrate that B-REX can leverage neural network function approximation and successor featuresbarreto2017successor to efficiently perform deep Bayesian reward inference given preferences over demonstrations that consist of raw visual observations. Finally, we demonstrate that samples obtained from B-REX can be used to solve the high-confidence off-policy evaluation problem for imitation learning in high-dimensional tasks. We evaluate our method on imitation learning for Atari games and demonstrate that we can efficiently compute high-confidence bounds on the worst-case performance of a policy and that these bounds are beneficial when comparing different evaluation policies and may provide a useful tool for detecting reward hacking amodei2016concrete .
2 Related work
Imitation learning is the problem of learning a policy from demonstrations of desired behavior. Imitation learning can roughly be divided into techniques that use behavioral cloning and techniques that use inverse reinforcement learning. Behavioral cloning methods pomerleau1991efficient
seek to solve the imitation learning problem via supervised learning where the goal is to learn a mapping from states to actions that mimics the demonstrator. While computationally efficient, these methods can suffer from compounding errorsross2011reduction . Methods such as DAgger ross2011reduction and DART laskey2017dart avoid this problem by collecting additional state-action pairs from a demonstrator in an online fashion.
Inverse reinforcement learning methods ng2000algorithms typically seek to solve the imitation learning problem by first estimating a reward function that makes the demonstrations appear near optimal and then performing reinforcement learning sutton1998introduction on the inferred reward function to learn a policy that can generalize to states not seen in the demonstrations. Classical approaches typically repeatedly alternate between reward estimation and full policy optimization. Bayesian IRL ramachandran2007bayesian generates samples from the posterior distribution over rewards, whereas other methods seek a single estimate of the reward that matches the demonstrator’s state occupancy abbeel2004apprenticeship , often while also seeking to maximize the entropy of the resulting policy ziebart2008maximum
. Modern, deep learning approaches to inverse reinforcement learning are typically based on a maximum entropy frameworkfinn2016guided or an occupancy matching framework ho2016generative and are related to Generative Adversarial Networks goodfellow2014generative ; finn2016connection . These methods scale to complex control problems by iterating between reward learning and policy learning steps. Recently, Brown et al. proposed to use preferences over suboptimal demonstrations to efficiently learn a reward function via supervised learning without requiring fully or partially solving an MDP browngoo2019trex ; brown2019drex . The reward function is then used to optimize a potentially better-than-demonstrator policy. However, despite recent successes of deep IRL, existing methods typically return a point estimate of the reward function, precluding the rich uncertainty and robustness analysis possible with a full Bayesian approach. One of our contributions is to propose the algorithm B-REX, first deep Bayesian IRL algorithm that can scale to complex control problems with visual observations.
Another contribution of this paper is an application of B-REX to safe imitation learning brown2018efficient . While there has been much recent interest and progress in imitation learning arora2018survey , less attention has been given to problems related to safe imitation learning. Zhang and Cho propose SafeDAgger safedagger a variant of DAgger that predicts in which states the novice policy will have a large action difference from the expert policy. Control is given the the expert policy only if the predicted action difference of the novice is above some hand-tuned parameter, . Other work has focused making generative adversarial imitation learning ho2016generative more robust and risk-sensitive. Lacotte et al. lacotte2019risk propose an imitation learning algorithm that seeks to match the tail risk of the expert as well as find a policy that is indistinguishable from the demonstrations. Brown and Niekum brown2018efficient propose a Bayesian sampling approach to provide explicit high-confidence safety bounds in the imitation learning setting. Their method uses samples from the posterior distribution to compute sample efficient probabilistic upper bounds on the policy loss of any evaluation policy. Brown et al. brown2018risk
extend this work by proposing an active learning algorithm that uses these high-confidence performance bounds for risk-aware policy improvement via active queries. Our work presented in this paper extends and generalizes the work of Brown and Niekumbrown2018efficient by demonstrating, for the first time, that high-confidence performance bounds can be obtained for imitation learning problems where demonstrations consist of high-dimensional visual observations.
Safety has been extensively studied within the reinforcement learning community (see Garcia et al. garcia2015comprehensive for a survey). These approaches usually either seek safe exploration strategies or seek to optimize an objective other than expected return. Recently, objectives based on measures of risk such as VaR and Conditional VaR have been shown to provide tractable and useful risk-sensitive measures of performance for MDPs tamar2015optimizing ; chow2015risk . Other related work on safe reinforcement learning has focused finding robust solutions to MDPs using Bayesian ambiguity sets petrik2019beyond and on obtaining high-confidence off-policy bounds on the performance of an evaluation policy thomas2015high ; hanna2017bootstrapping . Recently, it has been shown that high-confidence off policy evaluation is possible when samples of the true reward are available but the behavior policy is unknown hanna2019importance . Our work complements existing work on high-confidence off policy evaluation by formulating and providing a deep learning solution to the problem of high-confidence off-policy evaluation in the imitation learning setting, where samples of rewards are not observed and the demonstrator’s policy (the behavioral policy) is unknown.
We model the environment as a Markov Decision Process (MDP) consisting of states , actions , transition dynamics , reward function , initial state distribution , and discount factor . A policy
is a mapping from states to a probability distribution over actions. We denote the value of a policyunder reward function as and denote the value of executing policy starting at state as . Given a reward function , we denote the Q-value of a state-action pair as . We use the notation and .
3.2 Bayesian Inverse Reinforcement Learning
In inverse reinforcement learning, the environment is modeled as an MDPR where the reward function is internal to the demonstrator and is unknown and unobserved by the learner. The goal of inverse reinforcement learning (IRL) is to infer the latent reward function of the demonstrator given demonstrations consisting of state-action pairs from the demonstrator’s policy. Bayesian IRL models the demonstrator as a Boltzman rational agent that follows the softmax policy
where is the reward function of the demonstrator, and is the inverse temperature parameter that represents the confidence that the demonstrator is acting optimally. Given the assumption of Boltzman rationality, the likelihood of a set of demonstrations , given a specific reward function hypothesis , can be written as
Bayesian IRL ramachandran2007bayesian generates samples from the posterior distribution
via Markov Chain Monte Carlo (MCMC) sampling. This requires repeatedly solving forin order to compute the likelihood of each new proposal. Thus, Bayesian IRL methods are typically only used for low-dimensional problems with reward functions that are often linear combinations of a small number of hand-crafted features brown2018efficient ; biyik2019asking . One of our contributions is to propose an efficient deep Bayesian reward learning algorithm that leverages preferences to allow Bayesian IRL to be scaled to high-dimensional visual control problems.
4 High Confidence Off-Policy Evaluation for Imitation Learning
Before detailing B-REX, we first formalize the problem of high-confidence off-policy evaluation for imitation learning. We assume an MDPR, an evaluation policy , a set of demonstrations, , confidence level , and performance statistic , where denotes the space of all reward functions and is the space of all policies.
The High-Confidence Off-Policy Evaluation problem for Imitation Learning (HCOPE-IL) is to find a high-confidence lower bound such that , where denotes the demonstrator’s true reward function, and denotes the space of all demonstration sets . HCOPE-IL takes as input an evaluation policy , a set of demonstrations , and a performance statistic, , which evaluates a policy under a reward function. The goal of HCOPE-IL is to return a high-confidence lower bound on the performance statistic .
Note that this problem setting is significantly more challenging than the standard high-confidence off-policy evaluation problem in reinforcement learning, which we denote as HCOPE-RL. In HCOPE-RL the behavior policy is typically known and the demonstrations from the behavior policy contain ground-truth reward samples thomas2015high . In HCOPE-IL, the behavior policy is the demonstrator’s policy , which is unknown. Furthermore, in HCOPE-IL the demonstration data from contains only state-action pairs; samples of the true reward signal are not available. In the following sections we describe how to use preferences to scale Bayesian IRL to high-dimensional visual control tasks as a way to efficiently solve the HCOPE-IL problem for complex, visual imitation learning tasks.
5 Deep Bayesian Reward Extrapolation
Prior work brown2018efficient ; brown2018risk has investigated HCOPE-IL for simple problem domains where repeatedly solving for optimal Q-values is possible. However, for high-dimensional tasks such as learning control policies from pixel observations, even solving a single MDP can be challenging and sampling from becomes intractable. We now describe one of the main contribution of this paper: scaling Bayesian IRL to high-dimensional visual imitation learning problems.
Our first insight towards solving this problem is that the main bottleneck for standard Bayesian IRL ramachandran2007bayesian is computing the softmax likelihood function:
which requires solving for optimal Q-values. Thus, to make Bayesian IRL scale to high-dimensional visual domains, it is necessary to either efficiently solve for optimal Q-values or to formulate a new likelihood. Value-based reinforcement learning focuses on solving for optimal Q-values quickly; however, even for low-resolution visual control tasks such as Atari, RL algorithms can several hours or even days to train mnih2015human ; hessel2018rainbow
. Because MCMC is sequential in nature, evaluating large numbers of proposal steps is infeasible given the current state-of-the-art in RL. Methods such as transfer learning could reduce the time needed to calculatefor a new proposed reward ; however, transfer learning is not guaranteed to speed up reinforcement learning on the new task taylor2009transfer and transfer learning methods that avoid performing reinforcement learning only provide loose bounds on policy performance barreto2017successor
, making it difficult to compute accurate likelihood ratios needed for Bayesian inferenceramachandran2007bayesian . Thus, we focus on reformulating the likelihood function to speed up Bayesian IRL.
An ideal likelihood function would require little computation and minimal interaction with the environment. One promising candidate is to leverage recent work on learning from ranked demonstrations christiano2017deep ; browngoo2019trex ; brown2019drex . Given ranked demonstrations, Brown et al. browngoo2019trex proposed the algorithm Trajectory-ranked Reward Extrapolation (T-REX) that performs efficient reward inference by transforming reward function learning into a classification problem using a standard pairwise ranking loss. T-REX removes the need to repeatedly sample from or solve an MDP in the inner loop, allowing IRL to scale to visual imitation learning domains such as Atari. However, T-REX only solves for the maximum likelihood estimate of the reward function. One of our contributions is to show that a similar approach based on a pairwise preference likelihood can allow for efficient sampling from the posterior distribution over reward functions.
We assume that we have a set of trajectories along with a set of pairwise preferences over trajectories . Note that we do not require a total-ordering over trajectories. These preferences may come from a human demonstrator or could be automatically generated by watching a learner improve at a task browngoo2019trex or via noise injection brown2019drex . Some trajectory pairs may not have preference information and some trajectories maybe equally preferred, i.e. and may both be in set . The benefit of pairwise preferences over trajectories is that we can now leverage a pair-wise ranking loss to compute the likelihood of a given a parameterized reward function hypothesis . We use the standard Bradley-Terry model bradley1952rank , alternatively called the Luce’s choice axiom luce2012individual , to obtain the following pairwise ranking likelihood function:
where is the predicted return of trajectory under the reward function , and is the inverse temperature parameter that models the confidence in the preference labels.
Note that using the likelihood function defined in Equation (4) does not require solving an MDP. In fact, it does not require any rollouts or access to the MDP. All that is required is that we first calculate the return of each trajectory under . We then compare the relative predicted returns to the preference labels to determine the likelihood of the demonstrations under the reward hypothesis . Given this preference-based likelihood function we can perform preference-based Bayesian reward learning using standard MCMC.
5.1 Optimizations via Successor Features
B-REX uses a deep network to represent the reward function . However, MCMC proposal generation and mixing time can be slow if there are many demonstrations and if the network is especially large. To make B-REX more efficient and practical, we propose to limit the proposal to only change the last layer of weights in when generating MCMC proposals—we will discuss pretraining in a later section. We freeze all but the last layer of weights and use the activations of the penultimate layer as our reward features . This allows us to represent the reward at a state as a linear combination of features . There are two advantages to this formulation: (1) the proposal dimension for MCMC is significantly reduced, allowing for faster convergence; (2) we can efficiently compute the expected value of a policy via a single dot product, and (3) the computation required to calculated the proposal likelihood is significantly reduced.
Given , the value function of a policy can be written as
where we assume a finite horizon MDP with horizon and where are the successor features barreto2017successor of . Given any evaluation policy , we can compute the successor feature once to obtain, . We can then compute the expected value of as for any reward function weights, .
Using a linear combination of features also allows us to efficiently compute the pairwise ranking losses in the likelihood function. Consider the likelihood function in Equation (4). A naive computation of the likelihood would require forward passes through the deep neural network per proposal evaluation, where is the number of pairwise preferences over demonstration trajectories and is the length of the trajectories. Given that we would like to potentially generate thousands of samples from the posterior distribution over reward functions, this can significantly slow down MCMC. However, we can reduce this computational cost by noting that
Thus, we can precompute and cache for . The likelihood can then be quickly evaluated as
This results in only dot products per proposal, resulting in a significant computational savings when generating long MCMC chains over deep neural networks.
When we refer to B-REX in the remainder of this paper we will refer to the optimized version described in this section. See Algorithm 1 in the Appendix for full pseudo-code. We found that generating 100,000 reward hypothesis for Atari imitation learning tasks takes approximately 5 minutes on a Dell Inspiron 5577 personal laptop with an Intel i7-7700 processor and an NVIDIA GTX 1050 GPU. In comparison, using standard Bayesian IRL to generate one sample from the posterior takes 10+ hours of training for a parallelized PPO reinforcement learning agent schulman2017proximal ; baselines .
5.2 Pretraining the Reward Function Network
Precompute the successor features assumes that we already know a good . But how do we train from raw visual features? One way is to pretrain using T-REX browngoo2019trex to find the weight parameters that result in a maximum likelihood estimate given the rankings. Then we can freeze all but the last layer of weights and perform MCMC. Another option is to train the network using an auxiliary loss. Possible candidate auxiliary losses are (1) an inverse dynamics model that uses embeddings and to predict the corresponding action torabi2018behavioral ; hanna2017grounded
, (2) a variational pixel-to-pixel autoencoder whereis the learned latent encoding makhzani2017pixelgan ; doersch2016tutorial , (3) a cross-entropy loss to learn an embedding
that can be used to classify how many timesteps apart are two randomly chosen framesimitationyoutube , and (4) a forward dynamics model that predicts from and oh2015action ; thananjeyan2019extending .
5.3 HCOPE-IL via B-REX
We now discuss how to use B-REX to find solutions to the high-confidence off-policy evaluation for imitation learning (HCOPE-IL) problem (see Section 4) when learning from raw visual demonstrations. Given samples from the distribution , where , we can compute the posterior distribution over any performance statistic
as follows. For each sampled weight vectorproduced by B-REX, we compute . This results in a sample from the posterior distribution , the posterior distribution over performance statistic conditioned on and . We then compute a confidence lower bound, , by finding the
-quantile offor . In our experiments we focus on bounding the expected value of the evaluation policy, i.e., . To compute a confidence bound on , we take full advantage of the successor feature representation to efficiently calculate the posterior distribution over policy returns given preferences and demonstrations via a simple matrix vector product, , where each row of is a sample, , from the MCMC chain and is the evaluation policy. We then sort the elements of this vector and select the -quantile. This gives us a confidence lower bound on and corresponds to calculating the -Value at Risk (VaR) over brown2018efficient ; jorion1997value ; tamar2015optimizing .
6 Experimental Results
6.1 Imitation Learning via B-REX
We first tested the efficacy of B-REX to see if it can be used to find a reward function that leads to good policies via reinforcement learning. We enforce constraints on the weight vectors by normalizing the output of the weight vector proposal such that and use a Gaussian proposal function centered on
with standard deviation. Thus, given the current sample , the proposal is defined as , where normalize projects the sample back to the surface of the L1-unit ball. We used models pretrained from pairwise preferences using T-REX to obtain browngoo2019trex .111Pretrained networks are available at https://github.com/hiwonjoon/ICML2019-TREX/tree/master/atari/learned_models/icml_learned_rewards This results in a 65 dimensional features vector . We ran MCMC for 100,000 steps with and with a uniform prior. Due to our proposed optimizations this only required a few minutes of computation time. We then took the MAP and mean reward estimates and optimized a policy using Proximal Policy Optimization schulman2017proximal .
|Ranked Demonstrations||B-REX Mean||B-REX MAP||T-REX|
We tested our approach on five Atari games from the Arcade Learning Environment bellemare2013arcade . Because we are concerned with imitation learning, we mask game scores and life information and the imitation learning agent does not receive the ground-truth reward signal. All that is available are pairwise preferences on state trajectories. Table 1 shows results of performing RL on the mean and MAP rewards found using B-REX. We ran all algorithms using the same demonstrations and preference labels. We used the same 12 suboptimal demonstrations used by Brown et al. and give each algorithm all pairwise preference labels based on the ground-truth returns. T-REX uses a sigmoid to normalize rewards before passing them to the RL algorithm; however, we obtained better performance for B-REX by feeding the unnormalized predicted reward into PPO for policy optimization.
Table 1 shows that, similar to T-REX, B-REX is able to utilize preferences to outperform the demonstrator. B-REX is competitive with T-REX, achieving better average scores on 3 out of 5 games. Additionally, we found that using the MAP reward from the posterior was superior to optimizing for the mean reward on 4 out of 5 games. We also found that B-REX can successfully use pairwise preferences over suboptimal demonstrations to learn a better-than-demonstrator policy. When optimizing for the mean reward, B-REX is able to obtain an average performance that surpasses the performance of the best demonstration in every game except for Beam Rider. When optimizing for the MAP reward, B-REX is obtains an average performance that surpasses the best demonstration on all games, except for Seaquest.
6.2 High-Confidence Lower Bounds on Policy Performance
Next we ran an experiment to validate whether the posterior distribution generated by B-REX can be used for accurately bounding the expected return of the evaluation policy under the unknown reward function . We estimated using 30 Monte Carlo rollouts. We first evaluated four different evaluation policies, , created by partially training a PPO agent on the ground-truth reward function. We ran B-REX to generate 100,000 samples from . Figure 1 shows predicted and ground truth distributions for the four different evaluation policies: A–D. We found that the predicted distributions (100,000 MCMC samples) have roughly similar shape to the ground truth distribution (30 rollouts). Note we do not know the scale of the true reward . Thus, the results from B-REX are most useful when comparing the relative performance of several different evaluation policies brown2018efficient . We see that the modes predicted by B-REX match the ordering of the modes of policies A–D under the true reward function.
|Policy||Mean Chain||0.05-VaR Chain||Traj. Length||GT Avg. Return||GT Min. Return|
Table 2 shows the numerical results for evaluating policies under . We show results for partially trained policies A-D as well as well as policies trained on the MAP reward, the mean reward, and a No-Op policy. We found that the ground-truth returns for the checkpoints were highly correlated with the mean reward found under the chain and the 0.05-VaR (5th percentile policy return) under the chain. However, we also noticed that the trajectory length was also highly correlated with the ground-truth reward. If the reward function learned via IRL gives a small positive reward at every timestep, then long polices that do the wrong thing may look good under the posterior. To test this we evaluated a No-Op policy that seeks to hack the learned reward function by never releasing the ball in Breakout. We ran the No-Op policy until the Atari emulator timed out after 99,994 no-ops.
The bottom row of Table 2 shows that while the No-Op policy has a high expected return over the chain, looking at the 0.05-VaR shows that the No-Op policy has high risk under the distribution, much lower than even policy A which on average only scores 2.0 points. This finding validates the results by Brown and Niekum brown2018efficient that demonstrated the value of using a probabilistic worst-case bound for evaluating the performance of policies when the true reward function is unknown. Our results demonstrate that reasoning about probabilistic worst-case performance may be one potential way to detect policies that have overfit to certain features in the demonstrations that are correlated with the intent of the demonstrations, but do not lead to desired behavior, the so-called reward hacking problem amodei2016concrete . See the Appendix for results for all games. We found that for some of the games, the learned posterior is not as useful for accurately ranking policies. We hypothesize that this may be because the pretrained features are overfit to the rankings. In the future we hope to improve these results by using additional auxiliary losses when pretraining the reward features (see Section 5.2).
Bayesian reasoning is a powerful tool when dealing with uncertainty and risk; however, existing Bayesian inverse reinforcement learning algorithms require solving an MDP in the inner loop, rendering them intractable for complex problems where solving an MDP may take several hours or even days. In this paper we propose a novel deep learning algorithm, Bayesian Reward Extrapolation (B-REX), that leverages preference labels over demonstrations to make Bayesian IRL tractable for high-dimensional visual imitation learning tasks. B-REX can sample tens of thousands of reward functions from the posterior in a matter of minutes using a consumer laptop. We tested our approach on five Atari imitation learning tasks and demonstrated B-REX is competitive with state-of-the-art imitation learning methods. Using the posterior samples produced by B-REX, we demonstrated for the first time that it is computationally feasible to compute high-confidence performance bounds for arbitrary evaluation policies given demonstrations of visual imitation learning tasks. Our proposed bounds can allow accurate comparison of different evaluation policies and provide a potential way to detect reward hacking. In the future we are interested in using high-confidence bounds on policy performance to implement safe and robust policy improvement in the imitation learning setting. Given a starting policy we want to optimize a policy such that it maximizes some safety threshold. One possible way to improve the policy would be to use an evolutionary strategy where the fitness is simply the lower bound on the performance metric calculated over the posterior distribution of reward functions. We also plan to experiment with different architectures and different pretraining schemes for learning reward features automatically from raw visual features.
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Appendix A Preference-based Bayesian IRL
Pseudo-code for the optimized version of Bayesian Reward Extrapolation (B-REX) is shown in Algorithm 1.
Appendix B Mixing plots
The mixing plots for three randomly chosen features for Breakout are shown in Figure 2 and appear to be rapidly mixing.
Appendix C Plots of return distributions for MCMC chain and ground-truth rewards
Appendix D High-Confidence Lower Bounds (Full results)
|Policy||Mean Chain||0.05-VaR Chain||Traj. Length||GT Avg. Return||GT Min. Return|
|Policy||Mean Chain||0.05-VaR Chain||Traj. Length||GT Avg. Return||GT Min. Return|
|Policy||Mean Chain||0.05-VaR Chain||Traj. Length||GT Avg. Return||GT Min. Return|
|Policy||Mean Chain||0.05-VaR Chain||Traj. Length||GT Avg. Return||GT Min. Return|