Policy-Controlled Generalized Share: A General Framework with a Transformer Instantiation for Strictly Online Switching-Oracle Tracking

In non-stationary online learning, dynamic regret has attracted increasing attention as a measure of how well an online learner performs against a time-varying comparator sequence. Despite considerable advances, attaining optimal bounds for strongly convex and exp-concave losses often involves intricate analysis. In this paper, we present a simple framework that reduces dynamic regret minimization to switching regret minimization. As a result, we can derive dynamic regret bounds by using off-the-shelf algorithms with switching regret guarantees. The key idea of our reduction is to construct, for any comparator sequence, an auxiliary random sequence that is unbiased at each round, with the controlled variance and a manageable number of switches. Combining this construction with suitable surrogate losses, we can decompose dynamic regret into the expected switching regret against the random sequence and its controlled variance. Theoretically, for strongly convex and exp-concave losses, we establish the $\widetilde{O}(T^{1/3}P_T^{2/3})$ dynamic regret bounds, where $T$ denotes the time horizon and $P_T$ denotes the path-length of the comparator sequence. Moreover, for general convex losses, the same reduction also recovers the $O(\sqrt{T(1+P_T)})$ dynamic regret bound. Notably, all our findings match the minimax optimal results for these three types of losses, highlighting the versatility of our proposed framework.

We study online convex optimization with one unbiased stochastic subgradient per round and noise having a finite $p$-th central moment, where $p\in(1,2]$ is unknown. For a bounded convex domain of diameter $D$, subgradients bounded by $G$, noise scale $σ$, and comparator path length $P_T$, let $Λ_T=1+P_T/D$. A single algorithm, using none of $G,σ,p,P_T$, attains expected dynamic regret $O_p\left(\min\{GD\sqrt{TΛ_T}+σDT^{1/p}Λ_T^{(p-1)/p},\,GDT\}\right)$ against every fixed comparator sequence. Restarted AdaGrad experts produce the noise-path exponent $(p-1)/p$, and a prior favoring longer restart intervals removes horizon-dependent logarithmic overhead. We give an explicit bound uniform in $p$; its logarithm-free form has noise coefficient $O(1+\log(p/(p-1)))$, while the static-regret constant is universal. The analysis requires only marginal noise moments and permits dependent errors. Complete pathwise proofs retain both the expert-loss range and the gradient energies preceding comparator movement. Matching lower bounds hold on every bounded convex domain of positive diameter, under the same gradient-only information model. Together with a path-budget-tuned upper bound, they characterize the minimax rate with universal constants, including its linear-regret saturation.

We study dynamic regret in online convex optimization with an indicator switching cost: a fixed penalty incurred whenever two consecutive decisions differ. This captures startup overheads such as server activation, model deployment, and cache updates, and on a bounded domain it recovers norm-based movement costs as a special case. Existing guarantees for indicator costs handle only static comparators. We show that a direct extension of these techniques to dynamic regret provably fails, motivating a different approach. We propose a meta-learning framework: a set of randomized lazy FTRL base learners restarted at dyadic time scales, aggregated by a movement-aware master that mixes their proposal densities and samples actions via maximal coupling of consecutive mixtures. The resulting algorithm satisfies, in expectation, $\mathcal{R}^{\mathbf{1}}_T \le \tilde{\mathcal{O}}(\min\{\sqrt{T(S_T{+}1)},T^{2/3}(P_T+1)^{1/3}\})$, where $\mathcal{R}^{\mathbf{1}}_T$ is the dynamic regret plus the cumulative indicator switching cost, $S_T$ counts comparator switches, and $P_T$ is the comparator path length. The bound holds simultaneously for all sequences and requires no prior knowledge of $S_T$ or $P_T$: it is minimax-optimal (up to logarithmic factors) for tracking piecewise-constant comparators, and also captures frequently moving comparators with small total path length.