Black-box On-Policy Distillation (OPD) seeks to improve a student from its own generations when the teacher provides sampled responses but not token probabilities. Adversarial distillation offers one route: it learns a discriminator over prompt-matched teacher and student responses and uses its score as the policy reward. However, sampling discriminator negatives from the latest student at each step couples the learned reward to a negative distribution that changes after every policy update. We address this moving-target problem with persistent-negative adversarial distillation, a live-pool method that replaces a fraction of each discriminator batch with historical, prompt-matched teacher--student comparisons. Under matched discriminator compute, historical comparisons train the discriminator, while GRPO remains on-policy with fresh student responses. Our analysis identifies the Bayes-optimal reward as a teacher-to-negative log-density ratio and, under explicit assumptions, shows how persistent negatives anchor the discriminator and reduce reward-estimation MSE relative to fresh-negative training. Across two student families, three judges, and four judged-chat benchmarks, persistent-negative adversarial distillation consistently improves performance over current methods at matched discriminator compute. It also yields smoother fresh-policy discriminator trajectories, with fewer below-chance dips. These findings identify the discriminator's negative distribution as an important design axis in black-box on-policy distillation.

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