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.

翻译：无投影在线学习因在处理高维复杂约束问题时的高效性而日益受到关注。然而，现有大多数无投影在线方法主要关注最小化静态遗憾，这未能捕捉环境变化的挑战。本文研究非平稳无投影在线学习，并采用动态遗憾和自适应遗憾作为性能度量。具体而言，我们首先为现有的无投影方法$\text{BOGD}_\text{IP}$提供了一种新的动态遗憾分析，建立了$\mathcal{O}(T^{3/4}(1+P_T))$的动态遗憾界，其中$P_T$表示比较序列的路径长度。随后，通过并行运行多个具有不同步长的$\text{BOGD}_\text{IP}$算法，并在线追踪最优算法，我们将上界改进为$\mathcal{O}(T^{3/4}(1+P_T)^{1/4})$。我们的结果是无投影在线学习领域首个通用动态遗憾界，且通过设定$P_T = 0$可恢复现有的$\mathcal{O}(T^{3/4})$静态遗憾。此外，我们提出了一种无投影方法，对于任意长度为$\tau$的时间区间，实现了$\tilde{\mathcal{O}}(\tau^{3/4})$的自适应遗憾界，这几乎匹配该区间上的静态遗憾。其核心思想是动态维护一组$\text{BOGD}_\text{IP}$算法，并通过元算法进行组合。同时，我们证明该方法同样具有$\mathcal{O}(T^{3/4}(1+P_T)^{1/4})$的动态遗憾界。最后，实证研究验证了我们的理论发现。