Policy Gradient

Cooperative equilibria are fragile. When agents learn alongside each other rather than in a fixed environment, the process of learning destabilizes the cooperation they are trying to sustain: every gradient step an agent takes shifts the distribution of actions its partner will play, turning a cooperative partner into a source of stochastic noise precisely where the cooperation decision is most sensitive. We study how this co-learning noise propagates through the structure of coordination games, and find that the cooperative equilibrium, even when strongly Pareto-dominant, is exponentially unstable under standard risk-neutral learning, collapsing irreversibly once partner noise crosses the game's critical cooperation threshold. The natural response to apply distributional robustness to hedge against partner uncertainty makes things strictly worse: risk-averse return objectives penalize the high-variance cooperative action relative to defection, widening the instability region rather than shrinking it, a paradox that reveals a fundamental mismatch between the domains where robustness is applied and instability originates. We resolve this by showing that robustness should target the policy gradient update variance induced by partner uncertainty, not the return distribution. This distinction yields an algorithm whose gradient updates are modulated by an online measure of partner unpredictability, provably expanding the cooperation basin in symmetric coordination games. To unify stability, sample complexity, and welfare consequences of this approach, we introduce the Price of Paranoia as the structural dual of the Price of Anarchy. Together with a novel Cooperation Window, it precisely characterizes how much welfare learning algorithms can recover under partner noise, pinning down the optimal degree of robustness as a closed-form balance between equilibrium stability and sample efficiency.

We develop a unified treatment of credit assignment for RL training in multi-agent LLM systems. We show that observed reward alone cannot distinguish an agent that determines it from one that never affects it, and that standard shared-reward training performs exact gradient ascent on each agent's private utility rather than system performance. Moreover, we prove no single scalar per agent can consistently account for joint performance once agents interact. We thus develop the unique background-dependent notion of marginal contribution satisfying natural consistency requirements. From it we derive gradient-correct marginal contribution training signals, identify them from filtered feedback, and optimally allocate a budget of exact counterfactual evaluations against learned-signal error. Instantiated in GRPO, our signal improves routed GSM8K accuracy over winner-take-all training at no extra generation cost.

Multi-agent games in dynamic nonlinear settings are challenging due to the time-varying interactions among the agents and the non-stationarity of the (potential) Nash equilibria. In this paper we consider model-free games, where agent transitions and costs are observed without knowledge of the transition and cost functions that generate them. We propose a novel distributed policy structure that follows the communication constraints in multi-team games, with multiple agents per team, and learned through policy gradients. Our formulation is inspired by the structure of distributed policies in linear quadratic games, which take the form of time-varying linear feedback gains. In the nonlinear case, we model the policies as nonlinear feedback gains, parameterized by self-attention layers to account for the time-varying multi-agent communication topology. We demonstrate that our approach achieves strong performance in several settings, including distributed linear and nonlinear regulation, and simulated and real multi-robot pursuit-and-evasion games.

Regularization is a cornerstone of modern reinforcement learning. Regularized policy iteration (RPI) provides a fundamental scheme for solving regularized Markov decision processes (RMDPs), and the widely used soft actor-critic algorithm arises as a special case when the regularizer is Shannon entropy. Despite its empirical success, the theoretical underpinnings of RPI remain unclear. In this paper, we address this gap by proving that RPI is formally equivalent to the standard Newton-Raphson method applied to the Bellman equation smoothed by strongly convex regularizers. This equivalence enables a unified convergence analysis of existing methods and supports the development of accelerated algorithms. We show that RPI enjoys local quadratic convergence; notably, for Shannon entropy, the guarantee is dimension-free. We further study RPI with inexact policy evaluation, establishing its equivalence to an inexact Newton method in which each Newton step is solved via truncated iterations, and derive an asymptotic linear convergence rate of $γ^{M}$, where $M$ denotes the number of operator steps used in policy evaluation. Finally, motivated by higher-order Newton schemes, we propose a new algorithm for RMDPs that achieves third-order local convergence. Numerical experiments corroborate our theory and demonstrate the practical advantages of the proposed algorithm. Overall, our results advance the theoretical understanding of regularization in reinforcement learning and suggest new directions for efficient algorithm design.