Inverse reinforcement learning (IRL) recovers the loss function of a forward learner from its observed responses. Adaptive IRL aims to reconstruct the loss function of a forward learner by passively observing its gradients as it performs reinforcement learning (RL). This paper proposes a novel passive Langevin-based algorithm that achieves adaptive IRL. The key difficulty in adaptive IRL is that the required gradients in the passive algorithm are counterfactual, that is, they are conditioned on events of probability zero under the forward learner's trajectory. Therefore, naive Monte Carlo estimators are prohibitively inefficient, and kernel smoothing, though common, suffers from slow convergence. We overcome this by employing Malliavin calculus to efficiently estimate the required counterfactual gradients. We reformulate the counterfactual conditioning as a ratio of unconditioned expectations involving Malliavin quantities, thus recovering standard estimation rates. We derive the necessary Malliavin derivatives and their adjoint Skorohod integral formulations for a general Langevin structure, and provide a concrete algorithmic approach which exploits these for counterfactual gradient estimation.
Figures & tables
Symbol
Meaning
L
unknown loss of the forward learner
Xt , Xk
forward Langevin state (continuous / discrete)
α
query point; state of the inverse Langevin chain
∇^L(α)
counterfactual estimate of ∇L at α
β
inverse temperature; π∞∝e−βL
ε , η
forward / inverse step sizes
TABLE I: Notation
Fig. 1: Estimated gradient L′(α) versus the true gradient L′(α) over a grid of target locations α , using Monte Carlo estimation with sample size N=5000 .
Fig. 2: Sequential gradient estimates L′(α) versus the true gradient L′(α) across iterations of the outer passive Langevin chain ( 29 ). Occasional outliers appear, yet the overall procedure recovers the target Gibbs distribution and hidden loss function accurately (see Figures 3 and 4 )
Fig. 3: Empirical histogram of the stationary distribution of the Langevin chain {αk} ( 2000 MCMC steps after a 300 -step burn-in), compared with the target Gibbs density πβ(α)∝e−βL(α) .
Fig. 4: Adaptive IRL for reconstructing the loss function L(⋅) from MCMC samples of exp(−βL(⋅) . Specifically, L(⋅)=−β1log(π^(⋅)) , where π^(⋅) is the empirical MCMC distribution in Fig. 3 .
Reinforcement learning studies how agents learn sequential decisions from rewards and interaction, balancing exploration, performance, safety, and adaptation.
In the forward reinforcement-learning problem, the reward is fixed and known; the learner is asked to find a good policy or value function. Here we turn the question around. Given offline data generated by an expert, can we recover the reward the expert was optimizing? This is the inverse reinforcement learning problem, and remarkably, two communities, structural econometricians studying dynamic discrete choice (DDC) and machine learners studying entropy-regularized IRL, have been working on exactly the same probabilistic model under different names. We begin by proving their equivalence. We then develop the classical identification result of Magnac and Thesmar and the classical computational paradigms that grew out of it: Rust's nested fixed-point algorithm, the conditional-choice-probability approach of Hotz and Miller, and the two temporal-difference approaches of Adusumilli and Eckardt: linear semi-gradient TD and approximate value iteration. Each route has its limits: dimensionality, transition-kernel estimation, the deadly triad, or projected fixed-point bias. We then walk through the modern ML/IRL strand: adversarial IRL, occupancy matching, IQ-Learn, and offline ML-IRL, deriving each method's actual objective and stating precisely what it does and does not identify. We close with the empirical-risk-minimization framework of Kang et al., which yields a gradient-based estimator for offline IRL/DDC.
Enoch Hyunwook Kang
University of Washington, Foster School of Business
Inverse reinforcement learning (IRL) is typically formulated as maximizing entropy subject to matching the distribution of expert trajectories. Classical (dual-ascent) IRL guarantees monotonic performance improvement but requires fully solving an RL problem each iteration to compute dual gradients. More recent adversarial methods avoid this cost at the expense of stability and monotonic dual improvement, by directly optimizing the primal problem and using a discriminator to provide rewards. In this work, we bridge the gap between these approaches by enabling monotonic improvement of the reward function and policy without having to fully solve an RL problem at every iteration. Our key theoretical insight is that a trust-region-optimal policy for a reward function update can be globally optimal for a smaller update in the same direction. This smaller update allows us to explicitly optimize the dual objective while only relying on a local search around the current policy. In doing so, our approach avoids the training instabilities of adversarial methods, offers monotonic performance improvement, and learns a reward function in the traditional sense of IRL--one that can be globally optimized to match expert demonstrations. Our proposed algorithm, Trust Region Inverse Reinforcement Learning (TRIRL), outperforms state-of-the-art imitation learning methods across multiple challenging tasks by a factor of 2.4x in terms of aggregate inter-quartile mean, while recovering reward functions that generalize to system dynamics shifts.
Anish Diwan, Davide Tateo, Christopher E. Mower +3
Technical University of Darmstadt · Robotics Institute Germany (RIG) · Lund University +3
We establish novel structural and statistical results for entropy-regularized min-max inverse reinforcement learning (Min-Max-IRL) with linear reward classes in finite-horizon MDPs with Borel state and action spaces. On the structural side, we show that maximum likelihood estimation (MLE) and Min-Max-IRL are equivalent at the population level, and at the empirical level under deterministic dynamics. On the statistical side, exploiting pseudo-self-concordance of the Min-Max-IRL loss, we prove that both the trajectory-level KL divergence and the squared parameter error in the Hessian norm decay at the fast rate O(n−1), where n is the number of expert trajectories. Our guarantees apply under misspecification and require no exploration assumptions. We further extend reward-identifiability results to general Borel spaces and derive novel results on the derivatives of the soft-optimal value function with respect to reward parameters.