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Walking the Values in Bayesian Inverse Reinforcement Learning

Ondrej Bajgar, Alessandro Abate, Konstantinos Gatsis, Michael A. Osborne

TL;DR

This work tackles the computational bottleneck of Bayesian inverse reinforcement learning by reorienting inference to the Q-value space with ValueWalk, an MCMC method that uses gradient information via Hamiltonian dynamics to sample from the posterior over Q-values and rewards. By deriving a Q-space prior from the reward prior and the environment, and employing a Boltzmann likelihood over actions, ValueWalk achieves a fully Bayesian treatment in continuous and discrete settings, including unknown transitions. Empirical results in gridworlds and classic control tasks show improved sampling efficiency, better uncertainty calibration, and competitive imitation performance relative to prior Bayesian IRL and imitation-learning baselines, with model-based variants highlighting the value of environment dynamics. The approach provides a principled, uncertainty-aware foundation for safe and robust imitation learning and offers a useful benchmark for future variational or scalable IRL methods.

Abstract

The goal of Bayesian inverse reinforcement learning (IRL) is recovering a posterior distribution over reward functions using a set of demonstrations from an expert optimizing for a reward unknown to the learner. The resulting posterior over rewards can then be used to synthesize an apprentice policy that performs well on the same or a similar task. A key challenge in Bayesian IRL is bridging the computational gap between the hypothesis space of possible rewards and the likelihood, often defined in terms of Q values: vanilla Bayesian IRL needs to solve the costly forward planning problem - going from rewards to the Q values - at every step of the algorithm, which may need to be done thousands of times. We propose to solve this by a simple change: instead of focusing on primarily sampling in the space of rewards, we can focus on primarily working in the space of Q-values, since the computation required to go from Q-values to reward is radically cheaper. Furthermore, this reversion of the computation makes it easy to compute the gradient allowing efficient sampling using Hamiltonian Monte Carlo. We propose ValueWalk - a new Markov chain Monte Carlo method based on this insight - and illustrate its advantages on several tasks.

Walking the Values in Bayesian Inverse Reinforcement Learning

TL;DR

This work tackles the computational bottleneck of Bayesian inverse reinforcement learning by reorienting inference to the Q-value space with ValueWalk, an MCMC method that uses gradient information via Hamiltonian dynamics to sample from the posterior over Q-values and rewards. By deriving a Q-space prior from the reward prior and the environment, and employing a Boltzmann likelihood over actions, ValueWalk achieves a fully Bayesian treatment in continuous and discrete settings, including unknown transitions. Empirical results in gridworlds and classic control tasks show improved sampling efficiency, better uncertainty calibration, and competitive imitation performance relative to prior Bayesian IRL and imitation-learning baselines, with model-based variants highlighting the value of environment dynamics. The approach provides a principled, uncertainty-aware foundation for safe and robust imitation learning and offers a useful benchmark for future variational or scalable IRL methods.

Abstract

The goal of Bayesian inverse reinforcement learning (IRL) is recovering a posterior distribution over reward functions using a set of demonstrations from an expert optimizing for a reward unknown to the learner. The resulting posterior over rewards can then be used to synthesize an apprentice policy that performs well on the same or a similar task. A key challenge in Bayesian IRL is bridging the computational gap between the hypothesis space of possible rewards and the likelihood, often defined in terms of Q values: vanilla Bayesian IRL needs to solve the costly forward planning problem - going from rewards to the Q values - at every step of the algorithm, which may need to be done thousands of times. We propose to solve this by a simple change: instead of focusing on primarily sampling in the space of rewards, we can focus on primarily working in the space of Q-values, since the computation required to go from Q-values to reward is radically cheaper. Furthermore, this reversion of the computation makes it easy to compute the gradient allowing efficient sampling using Hamiltonian Monte Carlo. We propose ValueWalk - a new Markov chain Monte Carlo method based on this insight - and illustrate its advantages on several tasks.
Paper Structure (23 sections, 1 theorem, 13 equations, 4 figures, 1 table, 3 algorithms)

This paper contains 23 sections, 1 theorem, 13 equations, 4 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Bayesian inverse reinforcement learning
  4. Markov-chain Monte Carlo (MCMC)
  5. Related Work
  6. Method
  7. Finite state and action spaces
  8. Continuous state representations
  9. Continuous actions
  10. Experiments
  11. Gridworld
  12. Results
  13. Classic Control Environments
  14. Results
  15. Discussion
  16. ...and 8 more sections

Key Result

Theorem 1

Assume that the transition kernel $q_Q$ satisfies the detailed balance condition with respect to the posterior over Q values defined in Algorithm 1. Then the associated implicit Markov chain over rewards also satisfies the detailed balance condition with respect to the posterior $p_R(R|D)$.

Figures (4)

  • Figure 1: Left: Illustrative 3x3 gridworld. The agent always starts in the top left corner. The top right corner yields a reward of 10 and is terminal. The top centre tile represents an unsafe state that should be avoided and yields a reward of -30. Centre: Histograms of the samples from the posterior over rewards recovered by our ValueWalk algorithm corresponding to the 9 states of the gridworld. The red line indicates the mean. Right: Density functions of the posterior over rewards recovered by AVRIL and its model-based version, MB-AVRIL. Note the much narrower range of the reward axis relative to the histograms.
  • Figure 2: The test performance of an apprentice agent for ValueWalk and 3 baseline methods for different numbers of demonstration trajectories. The ValueWalk apprentice agent takes the action that maximizes the median of the posterior Q-value samples. The line shows mean performance across 5 runs with different sets of expert demonstrations; the shaded region shows mean$\pm$std.
  • Figure 3: The log likelihood on a hold-out set of 100 test demonstrations and the entropy of the action predictions produced by ValueWalk and AVRIL. The plot shows the mean and the 90% confidence interval on the value of the mean calculated using the bootstrap.
  • Figure 4: 2-D histograms representing the joint posteriors of the rewards associated with the 9 states of the gridworld (enumerated left-to-right, top-to-bottom, so state 3 is the goal state in the top right corner.

Theorems & Definitions (2)

  • Theorem 1
  • proof