Adversarial training is often modeled as a two-player zero-sum game, relying on strong assumptions that limit its practical guidance. In this paper, we instead analyze the interactions between training samples and show that even $\textbf{the fundamental objective—minimizing training loss—may not converge}$. To address this, we propose AT-Field, an adversarial training framework guided by sample-wise game-theoretic relationships. Specifically, we prove that training samples across different batches can form a none-potential game, where gradient descent induces cyclic behaviors, preventing convergence. By strategically searching and grouping these samples within the same batch, AT-Field transforms none-potential games into exact potential games, which are more effectively optimized using gradient-based methods. Experiments demonstrate that AT-Field integrates seamlessly with existing adversarial training techniques, enhancing both accuracy and robustness.