Batuhan Yardim

Batuhan Yardim, Semih Cayci, Matthieu Geist et al.

Mean-field games have been used as a theoretical tool to obtain an approximate Nash equilibrium for symmetric and anonymous $N$-player games. However, limiting applicability, existing theoretical results assume variations of a "population generative model", which allows arbitrary modifications of the population distribution by the learning algorithm. Moreover, learning algorithms typically work on abstract simulators with population instead of the $N$-player game. Instead, we show that $N$ agents running policy mirror ascent converge to the Nash equilibrium of the regularized game within $\widetilde{\mathcal{O}}(\varepsilon^{-2})$ samples from a single sample trajectory without a population generative model, up to a standard $\mathcal{O}(\frac{1}{\sqrt{N}})$ error due to the mean field. Taking a divergent approach from the literature, instead of working with the best-response map we first show that a policy mirror ascent map can be used to construct a contractive operator having the Nash equilibrium as its fixed point. We analyze single-path TD learning for $N$-agent games, proving sample complexity guarantees by only using a sample path from the $N$-agent simulator without a population generative model. Furthermore, we demonstrate that our methodology allows for independent learning by $N$ agents with finite sample guarantees.

Antonio Terpin, Nicolas Lanzetti, Batuhan Yardim et al.

Policy Optimization (PO) algorithms have been proven particularly suited to handle the high-dimensionality of real-world continuous control tasks. In this context, Trust Region Policy Optimization methods represent a popular approach to stabilize the policy updates. These usually rely on the Kullback-Leibler (KL) divergence to limit the change in the policy. The Wasserstein distance represents a natural alternative, in place of the KL divergence, to define trust regions or to regularize the objective function. However, state-of-the-art works either resort to its approximations or do not provide an algorithm for continuous state-action spaces, reducing the applicability of the method. In this paper, we explore optimal transport discrepancies (which include the Wasserstein distance) to define trust regions, and we propose a novel algorithm - Optimal Transport Trust Region Policy Optimization (OT-TRPO) - for continuous state-action spaces. We circumvent the infinite-dimensional optimization problem for PO by providing a one-dimensional dual reformulation for which strong duality holds. We then analytically derive the optimal policy update given the solution of the dual problem. This way, we bypass the computation of optimal transport costs and of optimal transport maps, which we implicitly characterize by solving the dual formulation. Finally, we provide an experimental evaluation of our approach across various control tasks. Our results show that optimal transport discrepancies can offer an advantage over state-of-the-art approaches.

Batuhan Yardim, Niao He

Mean-field games (MFG) have become significant tools for solving large-scale multi-agent reinforcement learning problems under symmetry. However, the assumption of exact symmetry limits the applicability of MFGs, as real-world scenarios often feature inherent heterogeneity. Furthermore, most works on MFG assume access to a known MFG model, which might not be readily available for real-world finite-agent games. In this work, we broaden the applicability of MFGs by providing a methodology to extend any finite-player, possibly asymmetric, game to an "induced MFG". First, we prove that $N$-player dynamic games can be symmetrized and smoothly extended to the infinite-player continuum via explicit Kirszbraun extensions. Next, we propose the notion of $α,β$-symmetric games, a new class of dynamic population games that incorporate approximate permutation invariance. For $α,β$-symmetric games, we establish explicit approximation bounds, demonstrating that a Nash policy of the induced MFG is an approximate Nash of the $N$-player dynamic game. We show that TD learning converges up to a small bias using trajectories of the $N$-player game with finite-sample guarantees, permitting symmetrized learning without building an explicit MFG model. Finally, for certain games satisfying monotonicity, we prove a sample complexity of $\widetilde{\mathcal{O}}(\varepsilon^{-6})$ for the $N$-agent game to learn an $\varepsilon$-Nash up to symmetrization bias. Our theory is supported by evaluations on MARL benchmarks with thousands of agents.

Nathan Corecco, Batuhan Yardim, Vinzenz Thoma et al.

Designing incentives for a multi-agent system to induce a desirable Nash equilibrium is both a crucial and challenging problem appearing in many decision-making domains, especially for a large number of agents $N$. Under the exchangeability assumption, we formalize this incentive design (ID) problem as a parameterized mean-field game (PMFG), aiming to reduce complexity via an infinite-population limit. We first show that when dynamics and rewards are Lipschitz, the finite-$N$ ID objective is approximated by the PMFG at rate $\mathscr{O}(\frac{1}{\sqrt{N}})$. Moreover, beyond the Lipschitz-continuous setting, we prove the same $\mathscr{O}(\frac{1}{\sqrt{N}})$ decay for the important special case of sequential auctions, despite discontinuities in dynamics, through a tailored auction-specific analysis. Built on our novel approximation results, we further introduce our Adjoint Mean-Field Incentive Design (AMID) algorithm, which uses explicit differentiation of iterated equilibrium operators to compute gradients efficiently. By uniting approximation bounds with optimization guarantees, AMID delivers a powerful, scalable algorithmic tool for many-agent (large $N$) ID. Across diverse auction settings, the proposed AMID method substantially increases revenue over first-price formats and outperforms existing benchmark methods.

Jiawei Huang, Batuhan Yardim, Niao He

In this paper, we study the fundamental statistical efficiency of Reinforcement Learning in Mean-Field Control (MFC) and Mean-Field Game (MFG) with general model-based function approximation. We introduce a new concept called Mean-Field Model-Based Eluder Dimension (MF-MBED), which characterizes the inherent complexity of mean-field model classes. We show that a rich family of Mean-Field RL problems exhibits low MF-MBED. Additionally, we propose algorithms based on maximal likelihood estimation, which can return an $ε$-optimal policy for MFC or an $ε$-Nash Equilibrium policy for MFG. The overall sample complexity depends only polynomially on MF-MBED, which is potentially much lower than the size of state-action space. Compared with previous works, our results only require the minimal assumptions including realizability and Lipschitz continuity.