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Multi-Agent Adversarial Inverse Reinforcement Learning
Lantao Yu, Jiaming Song, Stefano Ermon
TL;DR
Multi-agent reward design is difficult because reward mis-specification causes undesired behavior and multi-agent rationality is more complex. MA-AIRL addresses this with a new equilibrium concept, maximum pseudolikelihood estimation, and adversarial reward learning. It recovers rewards highly correlated with ground truth and imitates expert behavior in high-dimensional Markov games with unknown dynamics.
Problem
Reward design is difficult in multi-agent reinforcement learning, while extending inverse reinforcement learning requires handling more complex notions of rational behavior.
Method
MA-AIRL uses logistic stochastic best response equilibrium, maximum pseudolikelihood estimation, and adversarial reward learning within a multi-agent MaxEnt IRL framework.
Results
MA-AIRL imitates expert behaviors in high-dimensional complex environments and learns reward functions highly correlated with ground-truth rewards.
Takeaways & Limitations
MA-AIRL provides an effective and scalable approach to multi-agent IRL for Markov games with high-dimensional state-action spaces and unknown dynamics.
Takeaways & Limitations
The partition function is intractable to compute when state-action spaces are large or continuous and environment dynamics are unknown.
Abstract
from arXiv · showhide
Reinforcement learning agents are prone to undesired behaviors due to reward mis-specification. Finding a set of reward functions to properly guide agent behaviors is particularly challenging in multi-agent scenarios. Inverse reinforcement learning provides a framework to automatically acquire suitable reward functions from expert demonstrations. Its extension to multi-agent settings, however, is difficult due to the more complex notions of rational behaviors. In this paper, we propose MA-AIRL, a new framework for multi-agent inverse reinforcement learning, which is effective and scalable for Markov games with high-dimensional state-action space and unknown dynamics. We derive our algorithm based on a new solution concept and maximum pseudolikelihood estimation within an adversarial reward learning framework. In the experiments, we demonstrate that MA-AIRL can recover reward functions that are highly correlated with ground truth ones, and significantly outperforms prior methods in terms of policy imitation.
1. Introduction
Reward mis-specification makes reward design difficult, especially when multiple agents have different goals and representations. The paper proposes MA-AIRL to infer multi-agent rewards from demonstrations using a new equilibrium concept and adversarial learning.
- Reward mis-specification can produce undesired behaviors, while hand-tuning rewards becomes harder as agents have different goals and state-action representations.
- Inverse reinforcement learning infers reward functions from expert demonstrations, offering a more succinct and transferable task representation than direct policy imitation.
- Multi-agent IRL must replace single-agent optimality with an equilibrium concept because each agent’s optimal policy depends on the others’ policies.
- Nash and correlated-equilibrium approaches may be unable to model irrational or computationally bounded experts because they assume equilibrium behavior.
- MA-AIRL combines logistic stochastic best response equilibrium, maximum pseudolikelihood estimation, and adversarial reward learning for scalable multi-agent IRL.
2. Preliminaries
This section introduces Markov games, equilibrium concepts, and inverse reinforcement learning foundations. It then motivates adversarial reward learning by describing scalability limits of maximum-entropy IRL and reward ambiguity.
- Markov games: Markov games generalize Markov decision processes to N interacting agents with joint actions, stochastic transitions, and agent-specific rewards.Each agent selects actions through a stochastic policy, potentially using coordination signals.
- Equilibrium concepts: Correlated equilibrium permits joint strategies where no agent benefits from unilateral policy changes, while Nash equilibrium additionally requires independent actions.Many decentralized adaptive strategies have been shown to converge to correlated equilibrium rather than the more restrictive Nash equilibrium.
- Equilibrium concepts: Logistic quantal response equilibrium models bounded rationality by assigning higher action probabilities to actions with higher expected returns.It is presented as a stochastic generalization of Nash and correlated equilibrium.
- Inverse reinforcement learning: Expert demonstrations provide the supervision for imitation learning and inverse reinforcement learning when the ground-truth reward and additional expert interactions are unavailable.Imitation learning directly matches demonstrated behavior, whereas inverse reinforcement learning infers reward functions that induce expert policies.
- Inverse reinforcement learning: Maximum-entropy IRL resolves some reward ambiguity but requires an intractable trajectory partition function in large or continuous spaces with unknown dynamics.Adversarial IRL uses sampling-based approximation for efficiency, while reward estimators and potential shaping functions can recover the true reward and value function up to a constant under suitable assumptions.
3. Method
The method characterizes multi-agent expert behavior with a stochastic best-response equilibrium and learns reward functions through pseudolikelihood-based adversarial training. It targets Markov games with unknown dynamics while addressing reward ambiguity and intractable joint likelihoods.
- 3.1. Logistic Stochastic Best Response Equilibrium: LSBRE characterizes rational joint policies by repeatedly applying entropy-regularized stochastic best responses while holding other agents’ actions fixed.Its rationality parameter λ controls action randomness: low values produce near-uniform choices, whereas high values approach greedy best responses.
- 3.1. Logistic Stochastic Best Response Equilibrium: The LSBRE construction extends from normal-form games to Markov games through time-dependent conditional policies and stationary joint distributions.The procedure recursively constructs policies across time and uses stationary distributions of Markov chains to define joint behavior.
- 3.1. Logistic Stochastic Best Response Equilibrium: When agent conditionals are incompatible, the resulting pseudo Gibbs sampler can still yield nearly consistent stationary conditionals with sufficiently large data.Compatible conditionals correspond to a Gibbs sampler; incompatible ones motivate the pseudo Gibbs formulation.
- 3.2. Maximum Pseudolikelihood: Maximum pseudolikelihood replaces intractable joint likelihood optimization by a product of conditional likelihoods while retaining an asymptotically consistent estimator.The estimated parameters converge to the maximizer of the joint likelihood under the stated demonstration assumptions.
- 3.3. Multi-Agent Adversarial IRL: MA-AIRL estimates partition functions with adaptive importance samplers and trains discriminators to recover reward-related functions.The sampler reduces importance-sampling variance by matching the reward-induced trajectory distribution, while the discriminator estimates the reward function.
- 3.3. Multi-Agent Adversarial IRL: At optimality, the discriminator function approximates the expert-policy advantage and the generator approximates the expert policy.This connects the adversarial components to the policy and reward quantities required by the LSBRE-based formulation.
- 3.4. Solving Reward Ambiguity in Multi-Agent IRL: Potential-based reward shaping leaves optimal policies invariant, so demonstrations alone cannot identify a unique ground-truth reward without additional assumptions.The same reward ambiguity extends to multi-agent settings, where common shaping can preserve the set of Nash equilibria.
4. Related Work
Multi-agent imitation and inverse reinforcement learning remain less developed than their single-agent counterparts, with prior work often restricting reward structures. MA-AIRL differs from MA-GAIL by using LSBRE, MaxEnt RL, and pseudolikelihood to target joint-trajectory likelihood.
- Existing Multi-Agent Methods: Prior multi-agent inverse reinforcement learning methods often assume cooperative games, two-player zero-sum games, or rewards that are linear combinations of prespecified features.These assumptions limit the reward structures handled by much existing work.
- Comparison with MA-GAIL: MA-GAIL extends GAIL to general Markov games, whereas MA-AIRL is derived from MaxEnt RL and LSBRE.The two methods therefore use different solution concepts and derivations despite both relying on adversarial training.
- Comparison with MA-GAIL: MA-AIRL connects joint maximum-likelihood estimation to conditional likelihoods through pseudolikelihood before solving the resulting objectives with adversarial reward learning.This contrasts with MA-GAIL’s Nash-equilibrium formulation and specific Lagrange-multiplier motivation.
5. Experiments
The experiments evaluate MA-AIRL on simulated cooperative and competitive particle environments without task-type knowledge, measuring policy imitation and reward recovery against MA-GAIL and ground-truth rewards. MA-AIRL consistently achieves better received rewards and substantially stronger reward correlations, with correlation remaining high during training.
- Experimental setup: Experiments use cooperative navigation, cooperative communication, and competitive keep-away particle environments to assess policy imitation and reward recovery.The learning algorithms use decentralized forms without prior knowledge of whether interactions are cooperative or competitive.
- Experimental setup: Policies are trained from demonstrations generated by experts whose supervision comes from ground-truth rewards, while learned policies and rewards are evaluated against those ground truths.The experiments compare MA-AIRL with prior methods using expected returns and statistical correlations.
- Policy imitation: MA-AIRL consistently performs better than MA-GAIL in received reward across all considered cooperative and competitive environments.Cooperative performance uses shared ground-truth rewards; competitive evaluation compares the learned policy’s return with its opponent’s return.
- Reward recovery: MA-AIRL achieves much higher PCC and SCC than MA-GAIL in cooperative and competitive reward-recovery evaluations.PCC measures linear correlation, while SCC measures dependence between reward rankings.
- Reward recovery: MA-AIRL maintains high reward correlation throughout training, whereas MA-GAIL’s initially high correlation dramatically decreases as training continues.Figure 1 tracks PCC over training epochs in cooperative navigation.
6. Discussion and Future Work
The paper presents MA-AIRL as a scalable multi-agent MaxEnt IRL framework for complex Markov games. Its experiments show expert imitation and reward recovery, while future work targets reward regularization and task-structure priors.
- Discussion: MA-AIRL is presented as effective and scalable for Markov games with high-dimensional state-action spaces and unknown dynamics.The framework is described as the first multi-agent MaxEnt IRL framework with this scope.
- Discussion: The method uses LSBRE and maximum pseudolikelihood estimation to obtain a tractable multi-agent IRL procedure.These components support the framework’s derivation and computational tractability.
- Discussion: Experiments demonstrate expert-behavior imitation and reward functions highly correlated with ground-truth rewards.The reported conclusions cover both policy and reward recovery.
- Future work: Future work includes reward regularization to mitigate overfitting and leveraging prior knowledge of task structure.These are identified as directions for extending the method.
A.1. Trajectory Distribution Induced by Logistic Stochastic Best Response Equilibrium −i(at
This appendix derives the trajectory distribution induced by LSBRE and connects it to an optimization problem. Dynamic programming then constructs the optimal conditional policies.
- Trajectory distribution: The joint trajectory distribution is formed from agents’ conditional policies using the chain rule.The resulting conditionals are the distributions used to produce LSBRE.
- Trajectory distribution: The desired distribution is represented through an optimization problem whose optimal solution corresponds to LSBRE conditional policies.The derivation relates the target distribution to the conditional-policy formulation.
- Trajectory distribution: The optimization objective is equivalent to the target objective because the desired distribution’s partition function is constant with respect to the optimized policies.This equivalence enables optimization over the policies directly.
- Dynamic-programming solution: Dynamic programming solves the objective by first handling a base case and then deriving optimal policies for successive conditional optimization problems.The appendix presents these steps through the displayed equations and their associated optimal policies.
A.2. Maximum Pseudolikelihood Estimation for LSBRE
This appendix analyzes maximum pseudolikelihood estimation for LSBRE using expert demonstrations from a parameterized game. It shows that the true parameters satisfy the limiting first-order conditions.
- Setup: The analysis considers a normal-form game with N players, parameterized reward functions, and demonstrations generated by the true parameter value.The proof is stated for normal-form games, with extension to Markov games by induction.
- Pseudolikelihood objective: The pseudolikelihood objective is defined from the empirical demonstration distribution and conditional partition functions.The objective is the quantity maximized for estimation.
- Consistency argument: Derivatives of the pseudolikelihood objective are analyzed as the sample size tends to infinity.The limiting expression is obtained from the derivative equations.
- Consistency argument: At the true parameter value, the limiting gradients are zero, establishing the key first-order condition for consistency.The appendix attributes the result to the asymptotic consistency property of maximum pseudolikelihood estimation.