KL for a KL: On-Policy Distillation with Control Variate Baseline

On-Policy Distillation (OPD) has emerged as a dominant post-training paradigm for large language models, especially for reasoning domains like mathematics. However, OPD remains unstable in practice due to the high gradient variance of its single-sample Monte Carlo estimator, and recipes for stable training are still immature. We propose vOPD (On-Policy Distillation with a control variate baseline), which casts OPD as policy-gradient RL and stabilizes it by introducing a control variate baseline—canonically a value function—from the RL literature. We show that the OPD value function admits a closed form as the per-token negative reverse KL divergence between the student and the teacher, available directly from the already-computed forward pass with no additional critic or inference. Existing stabilization methods either compute the full token-level reverse KL over the entire vocabulary, adding significant overhead, or restrict it to a top-k support, biasing the objective. vOPD instead preserves the lightweight single-sample estimator, subtracting the value function as a detached baseline to keep the gradient unbiased while reducing variance. Furthermore, we show that a top-k approximation of the baseline further lowers cost without compromising performance. Across mathematical and scientific reasoning, vOPD consistently outperforms vanilla OPD and matches the most expensive full-vocabulary baseline, offering an efficient stabilization of On-Policy Distillation through principled RL variance reduction.