The aim of this paper is to design computationally-efficient and optimal algorithms for the online and stochastic exp-concave optimization settings. Typical algorithms for these settings, such as the Online Newton Step (ONS), can guarantee a $O(d\ln T)$ bound on their regret after $T$ rounds, where $d$ is the dimension of the feasible set. However, such algorithms perform so-called generalized projections whenever their iterates step outside the feasible set. Such generalized projections require $\Omega(d^3)$ arithmetic operations even for simple sets such a Euclidean ball, making the total runtime of ONS of order $d^3 T$ after $T$ rounds, in the worst-case. In this paper, we side-step generalized projections by using a self-concordant barrier as a regularizer to compute the Newton steps. This ensures that the iterates are always within the feasible set without requiring projections. This approach still requires the computation of the inverse of the Hessian of the barrier at every step. However, using the stability properties of the Newton steps, we show that the inverse of the Hessians can be efficiently approximated via Taylor expansions for most rounds, resulting in a $O(d^2 T +d^\omega \sqrt{T})$ total computational complexity, where $\omega$ is the exponent of matrix multiplication. In the stochastic setting, we show that this translates into a $O(d^3/\epsilon)$ computational complexity for finding an $\epsilon$-suboptimal point, answering an open question by Koren 2013. We first show these new results for the simple case where the feasible set is a Euclidean ball. Then, to move to general convex set, we use a reduction to Online Convex Optimization over the Euclidean ball. Our final algorithm can be viewed as a more efficient version of ONS.

翻译：摘要：本文旨在针对在线与随机exp-concave优化场景设计计算高效且最优的算法。针对此类场景的典型算法（如在线牛顿步算法ONS）可保证经过T轮迭代后遗憾界为$O(d\ln T)$，其中$d$为可行集维度。然而，此类算法在迭代步超出可行集时需执行所谓广义投影操作。即使对于欧几里得球等简单集合，这类广义投影也需要$\Omega(d^3)$次算术运算，导致ONS算法在最坏情况下经过T轮迭代的总运行时间达到$d^3 T$阶。本文通过使用自和谐障碍函数作为正则化器计算牛顿步，从而绕开广义投影。该方法确保迭代点始终落在可行集内而无需投影。该方案仍需要在每步计算障碍函数海森矩阵的逆。然而，利用牛顿步的稳定性性质，我们证明大多数轮次中可通过泰勒展开有效近似海森矩阵的逆，从而将总计算复杂度降至$O(d^2 T + d^\omega \sqrt{T})$，其中$\omega$为矩阵乘法指数。在随机场景下，该复杂度转化为寻找$\epsilon$-次优点所需的$O(d^3/\epsilon)$计算量，解答了Koren（2013）提出的开放性问题。我们首先针对可行集为欧几里得球的简单情形证明这些新结果。随后，为推广至一般凸集，我们采用了一种归约至欧几里得球上的在线凸优化方法。最终算法可视为ONS的一种更高效率版本。