Non-stationary Projection-free Online Learning with Dynamic and Adaptive Regret Guarantees

• URL: http://arxiv.org/abs/2305.11726v1
• Date: Fri, 19 May 2023 15:02:10 GMT
• Title: Non-stationary Projection-free Online Learning with Dynamic and Adaptive Regret Guarantees
• Authors: Yibo Wang, Wenhao Yang, Wei Jiang, Shiyin Lu, Bing Wang, Haihong Tang, Yuanyu Wan, Lijun Zhang
• Abstract summary: We investigate non-stationary projection-free online learning, and choose dynamic regret and adaptive regret to measure the performance. Our results are the first general-case dynamic regret bounds for projection-free online learning, and can recover the existing $mathcalO(T3/4)$ static regret. We propose a projection-free method to attain an $tildemathcalO(tau3/4)$ adaptive regret bound for any interval with length $tau.
• Score: 36.746745619968024
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Projection-free online learning has drawn increasing interest due to its efficiency in solving high-dimensional problems with complicated constraints. However, most existing projection-free online methods focus on minimizing the static regret, which unfortunately fails to capture the challenge of changing environments. In this paper, we investigate non-stationary projection-free online learning, and choose dynamic regret and adaptive regret to measure the performance. Specifically, we first provide a novel dynamic regret analysis for an existing projection-free method named $\text{BOGD}_\text{IP}$, and establish an $\mathcal{O}(T^{3/4}(1+P_T))$ dynamic regret bound, where $P_T$ denotes the path-length of the comparator sequence. Then, we improve the upper bound to $\mathcal{O}(T^{3/4}(1+P_T)^{1/4})$ by running multiple $\text{BOGD}_\text{IP}$ algorithms with different step sizes in parallel, and tracking the best one on the fly. Our results are the first general-case dynamic regret bounds for projection-free online learning, and can recover the existing $\mathcal{O}(T^{3/4})$ static regret by setting $P_T = 0$. Furthermore, we propose a projection-free method to attain an $\tilde{\mathcal{O}}(\tau^{3/4})$ adaptive regret bound for any interval with length $\tau$, which nearly matches the static regret over that interval. The essential idea is to maintain a set of $\text{BOGD}_\text{IP}$ algorithms dynamically, and combine them by a meta algorithm. Moreover, we demonstrate that it is also equipped with an $\mathcal{O}(T^{3/4}(1+P_T)^{1/4})$ dynamic regret bound. Finally, empirical studies verify our theoretical findings.

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