Adaptivity and Non-stationarity: Problem-dependent Dynamic Regret for Online Convex Optimization
Peng Zhao, Yu-Jie Zhang, Lijun Zhang, Zhi-Hua Zhou
TL;DR
This work advances online convex optimization in non-stationary environments by deriving problem-dependent dynamic regret bounds that replace time-based dependence with problem-specific quantities like the gradient variation $V_T$ and the comparator sequence loss $F_T$. It introduces two online ensemble algorithms, Sword (multi-gradient feedback) and Sword++ (one-gradient feedback), which achieve a unified bound of $O(\sqrt{(1+P_T+\min\{V_T,F_T\})(1+P_T)})$, thus adapting to easy problem instances while preserving worst-case rates. The core methodology combines Optimistic Online Mirror Descent with a collaborative two-layer meta-base framework, augmented by decision-deviation correction terms that enable effective cooperation between layers and enable a single gradient per iteration in the one-gradient setting. The paper provides a comprehensive theoretical treatment, including bounds for meta- and base-regret, a general online-ensemble guarantee, and implications to small-loss and worst-case dynamic regret, along with experimental validation on synthetic and real datasets. Overall, the results offer practical, adaptive guarantees for non-stationary online learning with efficient gradient usage, and the collaborative ensemble framework has potential applicability to a broader class of online optimization problems.
Abstract
We investigate online convex optimization in non-stationary environments and choose dynamic regret as the performance measure, defined as the difference between cumulative loss incurred by the online algorithm and that of any feasible comparator sequence. Let $T$ be the time horizon and $P_T$ be the path length that essentially reflects the non-stationarity of environments, the state-of-the-art dynamic regret is $\mathcal{O}(\sqrt{T(1+P_T)})$. Although this bound is proved to be minimax optimal for convex functions, in this paper, we demonstrate that it is possible to further enhance the guarantee for some easy problem instances, particularly when online functions are smooth. Specifically, we introduce novel online algorithms that can exploit smoothness and replace the dependence on $T$ in dynamic regret with problem-dependent quantities: the variation in gradients of loss functions, the cumulative loss of the comparator sequence, and the minimum of these two terms. These quantities are at most $\mathcal{O}(T)$ while could be much smaller in benign environments. Therefore, our results are adaptive to the intrinsic difficulty of the problem, since the bounds are tighter than existing results for easy problems and meanwhile safeguard the same rate in the worst case. Notably, our proposed algorithms can achieve favorable dynamic regret with only one gradient per iteration, sharing the same gradient query complexity as the static regret minimization methods. To accomplish this, we introduce the collaborative online ensemble framework. The proposed framework employs a two-layer online ensemble to handle non-stationarity, and uses optimistic online learning and further introduces crucial correction terms to enable effective collaboration within the meta-base two layers, thereby attaining adaptivity. We believe the framework can be useful for broader problems.