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Apprenticeship Learning using Inverse Reinforcement Learning and Gradient Methods
Gergely Neu, Csaba Szepesvari
TL;DR
The paper addresses learning an expert policy when the expert is optimal for an unknown reward in an MDP, while parameter-to-policy mappings are nonsmooth and redundant. It proposes gradient-based reward tuning with natural gradients and evaluates the approach in two artificial domains, reporting greater reliability and efficiency than some previous methods. Its scope remains limited by reward non-uniqueness, repeated MDP solving, and assumptions about observed states or sufficiently rich features.
Problem
Inverse reinforcement learning seeks an expert-matching policy from behavior despite non-unique reward solutions and redundant policy parameterizations.
Method
The paper combines supervised-style policy-deviation loss minimization with reward tuning, repeated MDP solution, and natural gradients in policy space.
Results
The proposed method was more reliable and efficient than some previous methods across two artificial domains.
Takeaways & Limitations
Unifying direct and indirect apprenticeship learning can combine policy matching with model-based behavior and may support sample efficiency and vaguely known features.
Takeaways & Limitations
The method requires solving MDPs many times and assumes observed states or sufficiently rich features; generalization to large state spaces remains open.
Abstract
from arXiv · showhide
In this paper we propose a novel gradient algorithm to learn a policy from an expert's observed behavior assuming that the expert behaves optimally with respect to some unknown reward function of a Markovian Decision Problem. The algorithm's aim is to find a reward function such that the resulting optimal policy matches well the expert's observed behavior. The main difficulty is that the mapping from the parameters to policies is both nonsmooth and highly redundant. Resorting to subdifferentials solves the first difficulty, while the second one is over- come by computing natural gradients. We tested the proposed method in two artificial domains and found it to be more reliable and efficient than some previous methods.
1 INTRODUCTION
Apprenticeship learning estimates an expert’s policy from behavioral samples, using either direct policy imitation or indirect inverse reinforcement learning. The paper proposes a gradient method that combines supervised policy matching with reward-based MDP optimization.
- Apprenticeship learning aims to estimate an expert’s policy from samples of the expert’s behavior.
- Direct and indirect approaches: Direct methods learn a state-to-action policy with supervised learning by minimizing a loss measuring deviation from the expert.
- Direct and indirect approaches: Sparse samples in states avoided by the expert can make direct methods struggle to learn a good policy there.
- Direct and indirect approaches: Indirect inverse reinforcement learning assumes the expert acts optimally in a known or learnable Markovian decision problem with an unknown reward function.
- The proposed gradient algorithm penalizes deviations from the expert’s policy while tuning a reward function and solving the resulting MDP.
2 BACKGROUND
The paper fixes notation for finite discounted infinite-horizon MDPs, stationary stochastic policies, and their value and action-value functions. Optimal policies maximize expected discounted reward and can be characterized through Bellman optimality and greediness with respect to Q∗.
- MDPs: A finite discounted infinite-horizon MDP is defined by states, actions, discount factor, transition probabilities, and a reward function.
- Policies: A stationary stochastic policy assigns each state a probability distribution over actions, while a deterministic policy selects one action per state.
- Value functions: The value function gives a policy’s expected discounted reward from each state, and the action-value function gives the corresponding quantity for a state-action pair.
- Optimality: An optimal policy maximizes expected total discounted reward, with V∗(x) and Q∗(x,a) defined as suprema over policies.
- Optimality: Policies greedy with respect to Q∗ are optimal, and all stationary optimal policies can be obtained this way.
3 APPRENTICESHIP LEARNING
The paper combines supervised policy matching with inverse reinforcement learning: reward parameters are tuned so the resulting near-optimal policy matches the expert. Natural gradients address redundant parameterizations, while smooth policy mappings and subdifferentials handle the optimization geometry and nonsmoothness.
- 3 APPRENTICESHIP LEARNING: Apprenticeship learning seeks to recover an expert policy from observed state-action traces, either directly through supervised policy learning or indirectly through inverse reinforcement learning.Direct methods can suffer in sparsely sampled regions of the state space, whereas IRL assumes the expert acts optimally for an unknown reward.
- 3 APPRENTICESHIP LEARNING: Direct policy estimates may be undefined for many unvisited states when the state space is large, motivating reward-based generalization beyond the expert’s sampled states.The empirical expert policy is defined arbitrarily for states not visited by the expert.
- 3 APPRENTICESHIP LEARNING: The proposed method tunes a parametric reward so its corresponding near-optimal policy matches the expert’s empirical policy.The loss measures policy deviation, while the policy is obtained by solving the MDP induced by the tuned reward.
- 3.1 NATURAL GRADIENTS: The parameter-to-policy mapping is potentially redundant, because multiple reward parameterizations can produce the same loss or policy; natural gradients address this redundancy.Natural-gradient trajectories are invariant under equivalent smooth reparameterizations and can alleviate plateaus.
- 3.1 NATURAL GRADIENTS: Natural gradients choose parameter-space directions that correspond to steepest descent in policy space rather than merely steepest descent in the parameters.The construction uses the geometry induced by the mapping h(θ)=πθ and a pseudo-metric based on local policy changes.
- 3.1 NATURAL GRADIENTS: The paper’s theorem identifies a natural-gradient direction by multiplying the ordinary gradient by the Moore-Penrose inverse of the induced pseudo-metric.Under the stated differentiability and local-Lipschitz assumptions, G†_θ∇J̃(θ) belongs to the natural-gradient set.
4 CALCULATING THE GRADIENT
The gradient calculation differentiates the policy map through a smooth action-selection rule, despite the nonsmooth optimal action-value function. Lipschitz continuity and subdifferential arguments establish almost-everywhere differentiability and fixed-point equations for the required derivatives.
- 4 CALCULATING THE GRADIENT: The policy derivative is obtained by applying the chain rule to the loss and differentiating the map from reward parameters to action-value-induced policies.The experiments use a smooth Boltzmann mapping from action-value functions to policies.
- 4 CALCULATING THE GRADIENT: Boltzmann action selection uses β>0 to control how closely the smooth policy mapping approaches greedy action selection.The parameter β determines the degree of concentration toward greedy actions.
- 4 CALCULATING THE GRADIENT: Subdifferentials handle the nondifferentiability of the optimal action-value function, whose derivatives can be computed almost everywhere through Bellman-like fixed-point equations.The paper introduces Fréchet subdifferentials and establishes a fixed-point characterization for the derivative components.
- 4 CALCULATING THE GRADIENT: If reward derivatives are uniformly bounded, the optimal action-value function is uniformly Lipschitz in the reward parameters.The Lipschitz bound follows from contraction-based value iteration and has the form R/(1−γ).
- 4 CALCULATING THE GRADIENT: The derivative fixed point is constructed using policy-specific operators that act componentwise and contract, allowing convergence from derivatives of successive value-iteration approximations.The limiting derivative is a subdifferential of the optimal action-value function, and almost-everywhere differentiability then identifies it with the derivative.
5 COMPUTER EXPERIMENTS
Experiments in grid worlds and sailing evaluated efficiency, robustness, feature sensitivity, and comparison with prior methods. Natural gradients were generally robust to feature transformations and perturbations, while performance varied with sample size.
- Experimental setup: The study compared plain-gradient descent, natural-gradient descent, RPROP, and Abbeel–Ng max-margin and projection methods in two artificial domains.The domains were grid worlds and sailing; the reward functions were linear in unknown parameters.
- Grid world: Grid-world experiments used 10×10 environments, five features, value iteration, and usually 10 optimal-policy trajectories of 100 steps.Actions moved in four compass directions with 70% success, and learning ran for up to 100 iterations.
- Grid world: For small sample sizes plain gradient performed best, while natural gradient eventually became the winner; max margin and RPROP also caught up at the end.Figure 1 averages 10 runs with 1/10 s.e. error bars, and its y-axis is logarithmic.
- Grid world: Under linearly transformed features, max margin suffered badly, natural gradient and RPROP were little affected, and plain gradient eventually converged to good solutions.The true reward remained in the span of the transformed features; Figure 2 averages 25 runs with 1/10 s.e. error bars.
- Grid world: With perturbed features, natural gradients and RPROP remained robust, whereas plain gradients and max margin suffered large losses.Perturbations added uniform random noise to the features, testing performance without assuming the reward-containing subspace was known.
6 RELATED WORK
The related-work discussion argues that feature-expectation matching can fail when feature scaling is unknown, even when the matching error is small. This exposes a limitation in prior performance guarantees and motivates the paper’s alternative approach.
- Feature-scaling sensitivity: Abbeel and Ng’s guarantee depends on the scaling used for the features matching the true scaling.The paper identifies this as a hidden assumption underlying their linearly parameterized reward formalism.
- Comparison with prior work: The analysis concludes that feature scaling must be known for the earlier guarantee to reliably reflect performance under the original reward.The claimed ε-vicinity guarantee is therefore sensitive to whether algorithmic and true feature scales coincide.
- Feature-scaling sensitivity: A small Euclidean feature-expectation error does not ensure near-expert performance under mismatched feature scaling.The paper’s two-feature example has ∥φπ − φE∥2 ≤ ε, yet the performance ratio can tend to −∞ as λ1/λ2 grows.
- Comparison with prior work: The prior algorithm measures performance against the expert’s unknown reward function rather than directly comparing the expert’s and returned policies.Ratliff et al. are described as using similar ideas and the same performance-measurement choice.
7 CONCLUSIONS
The conclusion presents the method as a unified direct–indirect apprenticeship-learning procedure that combines cost optimization with model-based policy selection. It reports stable experiments while identifying repeated MDP solving and broader state representations as important boundaries for further work.
- Conclusions: The proposed procedure optimizes a cost function while selecting the policy through a model, potentially avoiding problems of direct policy matching.The authors report stable behavior in their experiments but say the method’s limitations require further study.
- Conclusions: Repeatedly solving MDPs is a significant barrier because solving an MDP is itself challenging.The conclusion suggests two-time-scale algorithms as one possible way to reduce this burden.
- Future directions and limitations: The present work assumes that states are observed; insufficiently rich observed features would make the method fail.For large state spaces, function approximation is needed, but generalization to such settings remains an open question.
- Future directions and limitations: The authors identify infinite MDPs and replacing the parametric framework with a non-parametric one as further directions.They note technical difficulties for infinite MDPs but expect the methods could still be generalized.