Title: On Kernel RL and Optimistic Closure
Primary area: reinforcement learning
Keywords: RL, kernel, theory

Abstract: We study episodic reinforcement learning with a kernel (RKHS) structure on state-action pairs. Previous optimistic analyses in this case either pay a data-dependent covering-number penalty that can grow with time and undermine no-regret guarantees, or it assumes a strong “optimistic closure” condition requiring all optimistic proxies to lie in a fixed state-RKHS ball. We take a different approach that removes the covering-number dependence without invoking optimistic closure. Our analysis builds a uniform confidence bound, derived via conditional mean embeddings, that holds simultaneously for all proxy value functions within a bounded state-RKHS class. We introduce \textbf{KOVI-Proj}, an optimistic value-iteration scheme that explicitly projects the optimistic proxy back into the state-RKHS ball at every step, ensuring that the uniform bound applies throughout the learning process. Under a restricted Bellman-embedding assumption (bounded conditional mean embeddings), KOVI-Proj enjoys a high-probability no-regret guarantee whose rate is governed by the task horizon and the kernel’s information gain. When the optimal value function lies in the chosen state-RKHS ball (realizability), the regret is sublinear; in the agnostic case, an explicit approximation term reflects the best RKHS approximation error. Overall, this work provides a new pathway to no-regret kernel RL that is strictly weaker than optimistic closure and avoids covering-number penalties.

TL;DR: kernals, RL, closure