A Closed-Form Upper Bound for Admissible Learning-Rate Steps in Belief-Space Dynamics

Follow-the-regularized-leader framework has shown effectiveness and flexibility in online learning problems, where the choice of learning rates are known to be crucial. Recently, adaptive learning rates defined in terms of the arm-selection probabilities, obtained by solving convex optimization, have achieved improved best-of-both-worlds (BOBW) guarantees in various bandit problems. In contrast, BOBW guarantees for its computationally efficient alternative, follow-the-perturbed-leader (FTPL), remain relatively limited since its optimization-free nature ironically makes the design of adaptive, probability-dependent learning rates non-trivial. To address this challenge, we propose an adaptive learning rate for FTPL by introducing surrogate probability functions that can be computed only from the available quantities, without requiring the exact probabilities. Based on these learning rates with surrogate functions, we provide the BOBW guarantee for FTPL with Pareto perturbations for any shape parameter $α>1$, generalizing prior results restricted to specific choices of $α=2$. We further show the BOBW guarantees for FTPL with adaptive learning rates in the bandit problem with expert advices. Our approach preserves the computational simplicity of FTPL while enabling probability-dependent adaptivity, and the surrogate-based methodology may be of independent interest in other algorithmic frameworks beyond FTPL and learning rate designs.

Independent evaluation can reject harmful policy updates yet also prevent useful continual learning. We argue that update admission must be assessed through both error control and retained learning opportunities at a stated interaction budget. We identify a concrete failure: a range-based confidence gate cannot certify unchanged old-task behavior within otherwise substantial budgets. A standard paired-binomial construction reduces this burden when outcome disagreements are rare. We also specify certified historical-reference promotion and a round-level missed-opportunity metric. In a constructed one-step pushing diagnostic with 32 seeds, fresh paired checks admit 31.6% of a common update stream at 2,000 episodes per stage, versus zero for the range-based gate; unconditional replay nevertheless learns better in closed-loop runs. A separate learned-dynamics stress test distinguishes model bias from feedback-selection error. The contribution is an admission-audit protocol with analytical and synthetic evidence; physical-robot and VLA validation remain open.

Adaptive stochastic convex optimization (SCO) methods face a fundamental ``price of adaptivity'' barrier: under the standard set of assumptions, they cannot efficiently adapt to large uncertainty in both the initial distance to optimality and the Lipschitz constant. We circumvent this barrier by requiring a small amount of additional structure common to many learning problems. Specifically, we assume that the objective decomposes into a model and a loss function, enabling us to intervene by modifying the model's output before it passes to the loss function. Under this assumption, we design a method that clips the learned model output in tail events where it deviates too much from the output of a fixed reference model. Our method matches the optimal bounds for known-parameter SCO up to logarithmic factors in the uncertainty in the distance and Lipschitz parameters, thus efficiently adapting to large uncertainty in both.