Proof of Theorem 4.7 We first prove a Lemma. Theorem A.2. (Theorem 1 in [36]) Let ϵ = max Theorem 4.7 In a two-player game, suppose that According to Theorem A.2, we have J ( π, µ) J ( π, α) E CQL [20] puts regularization on the learning of Q function to penalize out-of-distribution actions. The CSP algorithm is illustrated in Algorithm 1. The proxy model is trained adversarially against our agent, therefore, we set the proxy's reward function to be the negative of our agent's reward. We show experiment details of the Maze example in this section.

When one agent interacts with a multi-agent environment, it is challenging to deal with various opponents unseen before.

In zero-sum games, an NE strategy tends to be overly conservative confronted with opponents of limited rationality, because it does not actively exploit their weaknesses. From another perspective, best responding to an estimated opponent model is vulnerable to estimation errors and lacks safety guarantees. Inspired by the recent success of real-time search algorithms in developing superhuman AI, we investigate the dilemma of safety and opponent exploitation and present a novel real-time search framework, called Safe Exploitation Search (SES), which continuously interpolates between the two extremes of online strategy refinement. We provide SES with a theoretically upper-bounded exploitability and a lower-bounded evaluation performance. Additionally, SES enables computationally efficient online adaptation to a possibly updating opponent model, while previous safe exploitation methods have to recompute for the whole game. Empirical results show that SES significantly outperforms NE baselines and previous algorithms while keeping exploitability low at the same time.

There also exist many application scenarios involving multiagent interactions, commonly known as multiagent systems (MAS).

Despite the significant advancements of self-play fine-tuning (SPIN), which can transform a weak large language model (LLM) into a strong one through competitive interactions between models of varying capabilities, it still faces challenges in the Text-to-SQL task. SPIN does not generate new information, and the large number of correct SQL queries produced by the opponent model during self-play reduces the main model's ability to generate accurate SQL queries. To address this challenge, we propose a new self-play fine-tuning method tailored for the Text-to-SQL task, called SPFT-SQL. Prior to self-play, we introduce a verification-based iterative fine-tuning approach, which synthesizes high-quality fine-tuning data iteratively based on the database schema and validation feedback to enhance model performance, while building a model base with varying capabilities. During the self-play fine-tuning phase, we propose an error-driven loss method that incentivizes incorrect outputs from the opponent model, enabling the main model to distinguish between correct SQL and erroneous SQL generated by the opponent model, thereby improving its ability to generate correct SQL. Extensive experiments and in-depth analyses on six open-source LLMs and five widely used benchmarks demonstrate that our approach outperforms existing state-of-the-art (SOTA) methods.

The proof is similar to the one above for Lemma A.1: Proof.

The Control as Inference (CAI) framework has successfully transformed single-agent reinforcement learning (RL) by reframing control tasks as probabilistic inference problems. However, the extension of CAI to multi-agent, general-sum stochastic games (SGs) remains underexplored, particularly in decentralized settings where agents operate independently without centralized coordination. In this paper, we propose a novel variational inference framework tailored to decentralized multi-agent systems. Our framework addresses the challenges posed by non-stationarity and unaligned agent objectives, proving that the resulting policies form an $\epsilon$-Nash equilibrium. Additionally, we demonstrate theoretical convergence guarantees for the proposed decentralized algorithms. Leveraging this framework, we instantiate multiple algorithms to solve for Nash equilibrium, mean-field Nash equilibrium, and correlated equilibrium, with rigorous theoretical convergence analysis.