We consider the setting of online convex optimization (OCO) with \textit{exp-concave} losses. The best regret bound known for this setting is $O(n\log{}T)$, where $n$ is the dimension and $T$ is the number of prediction rounds (treating all other quantities as constants and assuming $T$ is sufficiently large), and is attainable via the well-known Online Newton Step algorithm (ONS). However, ONS requires on each iteration to compute a projection (according to some matrix-induced norm) onto the feasible convex set, which is often computationally prohibitive in high-dimensional settings and when the feasible set admits a non-trivial structure. In this work we consider projection-free online algorithms for exp-concave and smooth losses, where by projection-free we refer to algorithms that rely only on the availability of a linear optimization oracle (LOO) for the feasible set, which in many applications of interest admits much more efficient implementations than a projection oracle. We present an LOO-based ONS-style algorithm, which using overall $O(T)$ calls to a LOO, guarantees in worst case regret bounded by $\widetilde{O}(n^{2/3}T^{2/3})$ (ignoring all quantities except for $n,T$). However, our algorithm is most interesting in an important and plausible low-dimensional data scenario: if the gradients (approximately) span a subspace of dimension at most $\rho$, $\rho << n$, the regret bound improves to $\widetilde{O}(\rho^{2/3}T^{2/3})$, and by applying standard deterministic sketching techniques, both the space and average additional per-iteration runtime requirements are only $O(\rho{}n)$ (instead of $O(n^2)$). This improves upon recently proposed LOO-based algorithms for OCO which, while having the same state-of-the-art dependence on the horizon $T$, suffer from regret/oracle complexity that scales with $\sqrt{n}$ or worse.

翻译：摘要：我们考虑具有\textit{指数凹}损失的在线凸优化（OCO）场景。该场景下已知的最佳遗憾界为$O(n\log T)$，其中$n$为维度，$T$为预测轮数（将所有其他量视为常数并假设$T$足够大），且可通过著名的在线牛顿步算法（ONS）实现。然而，ONS每次迭代需在可行凸集上执行投影（依据某种矩阵诱导范数），这在可行集具有非平凡结构的高维场景中通常计算成本高昂。本文针对指数凹和光滑损失提出无投影在线算法——所谓“无投影”指算法仅依赖可行集的线性优化预言机（LOO）可用性，而许多实际应用中LOO的实现效率远高于投影预言机。我们提出基于LOO的ONS风格算法，该算法总共调用$O(T)$次LOO时，最坏情况下遗憾界为$\widetilde{O}(n^{2/3}T^{2/3})$（忽略除$n$和$T$外的所有量）。但该算法在一重要且实际可行的低维数据场景中更具价值：若梯度（近似）张成维数至多为$\rho$（$\rho << n$）的子空间，则遗憾界可改进至$\widetilde{O}(\rho^{2/3}T^{2/3})$；通过应用标准确定性素描技术，空间和平均每轮额外运行时需求仅为$O(\rho n)$（而非$O(n^2)$）。这改进了近期提出的基于LOO的OCO算法——后者尽管在时间域$T$上具有相同的最优依赖度，但其遗憾/预言机复杂度需随$\sqrt{n}$或更差规模扩展。