Solving the Nash equilibrium in normal-form games with large-scale strategy spaces presents significant challenges. Open-ended learning frameworks, such as PSRO and its variants, have emerged as effective solutions. However, these methods often lack an efficient metric for evaluating strategy improvement, which limits their effectiveness in approximating equilibria. In this paper, we introduce a novel evaluative metric called Advantage, which possesses desirable properties inherently connected to the Nash equilibrium, ensuring that each strategy update approaches equilibrium. Building upon this, we propose the Advantage Policy Space Response Oracle (A-PSRO), an innovative unified open-ended learning framework applicable to both zero-sum and general-sum games. A-PSRO leverages the Advantage as a refined evaluation metric, leading to a consistent learning objective for agents in normal-form games. Experiments showcase that A-PSRO significantly reduces exploitability in zero-sum games and improves rewards in general-sum games, outperforming existing algorithms and validating its practical effectiveness.

Achieving generalization in neural approaches across different scales and distributions remains a significant challenge for routing problems. A key obstacle is that neural networks often fail to learn robust principles for identifying universal patterns and deriving optimal solutions from diverse instances. In this paper, we first uncover Purity Law, a fundamental structural principle for optimal solutions of routing problems, defining that edge prevalence grows exponentially with the sparsity of surrounding vertices. Statistically and theoretically validated across diverse instances, Purity Law reveals a consistent bias toward local sparsity in global optima. Building on this insight, we propose Purity Policy Optimization (PUPO), a novel training paradigm that explicitly aligns characteristics of neural solutions with Purity Law during the solution construction process to enhance generalization. Extensive experiments demonstrate that PUPO can be seamlessly integrated with popular neural solvers, significantly enhancing their generalization performance without incurring additional computational overhead during inference.

The capabilities of large language models (LLMs) significantly depend on training data drawn from various domains. Optimizing domain-specific mixture ratios can be modeled as a bi-level optimization problem, which we simplify into a single-level penalized form and solve with twin networks: a proxy model trained on primary data and a dynamically updated reference model trained with additional data. Our proposed method, Twin Networks for bi-level DatA mixturE optiMization (TANDEM), measures the data efficacy through the difference between the twin models and up-weights domains that benefit more from the additional data. TANDEM provides theoretical guarantees and wider applicability, compared to prior approaches. Furthermore, our bi-level perspective suggests new settings to study domain reweighting such as data-restricted scenarios and supervised fine-tuning, where optimized mixture ratios significantly improve the performance. Extensive experiments validate TANDEM's effectiveness in all scenarios.

Low-Rank Adaptation (LoRA) offers a parameter-efficient paradigm for tuning large models. While recent spectral initialization methods improve convergence and performance over the naive “Noise \& Zeros” scheme, their extra computational and storage overhead undermines efficiency. In this paper, we establish update magnitude as the fundamental driver of LoRA performance and propose LoRAM, a magnitude-driven “Basis \& Basis” initialization scheme that matches spectral methods without their inefficiencies. Our key contributions are threefold: (i) Magnitude of weight updates determines convergence. We prove low-rank structures intrinsically bound update magnitudes, unifying hyperparameter tuning in learning rate, scaling factor, and initialization as mechanisms to optimize magnitude regulation. (ii) Spectral initialization succeeds via magnitude amplification. We demystify that the presumed knowledge-driven benefit of spectral component essentially arises from the boost in the weight update magnitude. (iii) A novel and compact initialization strategy, LoRAM, scales deterministic orthogonal bases using pretrained weight magnitudes to simulate spectral gains. Extensive experiments show that LoRAM serves as a strong baseline, retaining the full efficiency of LoRA while matching or outperforming spectral initialization across benchmarks.

Establishing robust policies is essential to counter attacks or disturbances affecting deep reinforcement learning (DRL) agents. Recent studies explore state-adversarial robustness and suggest the potential lack of an optimal robust policy (ORP), posing challenges in setting strict robustness constraints. This work further investigates ORP: At first, we introduce a consistency assumption of policy (CAP) stating that optimal actions in the Markov decision process remain consistent with minor perturbations, supported by empirical and theoretical evidence. Building upon CAP, we crucially prove the existence of a deterministic and stationary ORP that aligns with the Bellman optimal policy. Furthermore, we illustrate the necessity of $L^{\infty}$-norm when minimizing Bellman error to attain ORP. This finding clarifies the vulnerability of prior DRL algorithms that target the Bellman optimal policy with $L^{1}$-norm and motivates us to train a Consistent Adversarial Robust Deep Q-Network (CAR-DQN) by minimizing a surrogate of Bellman Infinity-error. The top-tier performance of CAR-DQN across various benchmarks validates its practical effectiveness and reinforces the soundness of our theoretical analysis.

Abstract Multiagent reinforcement learning (MARL) has been used extensively in the game environment. One of the main challenges in MARL is that the environment of the agent system is dynamic, and the other agents are also updating their strategies. Therefore, modeling the opponents’ learning process and adopting specific strategies to shape learning is an effective way to obtain better training results. Previous studies such as DRON, LOLA and SOS approximated the opponent’s learning process and gave effective applications. However, these studies modeled only transient changes in opponent strategies and lacked stability in the improvement of equilibrium efficiency. In this article, we design the MOL (modeling opponent learning) method based on the Stackelberg game. We use best response theory to approximate the opponents’ preferences for different actions and explore stable equilibrium with higher rewards. We find that MOL achieves better results in several games with classical structures (the Prisoner’s Dilemma, Stackelberg Leader game and Stag Hunt with 3 players), and in randomly generated bimatrix games. MOL performs well in competitive games played against different opponents and converges to stable points that score above the Nash equilibrium in repeated game environments. The results may provide a reference for the definition of equilibrium in multiagent reinforcement learning systems, and contribute to the design of learning objectives in MARL to avoid local disadvantageous equilibrium and improve general efficiency.