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：本文的目的是设计计算高效且最优化的计算算法, 用于在线和随机化的Excal- Excredition 优化设置。 这些设置的典型算法, 如在线 牛顿 Step (ONS), 可以在最坏的情况下, 保证在 $T 回合后, 他们的遗憾中, 美元是可行的 。 但是, 这些算法在它们超越可行设置时, 就会进行所谓的通用预测。 这种通用的预测甚至需要 $\ omega (d3) $ 的计算操作, 甚至简单设置 Eucliidean 球, 使 ONS 的运行时间在 $T 回合后 $d_ 3 T$。 在本文中, 我们使用一个自调的屏障屏障屏障障碍值, 向普通的设置一个不需预测值。 这个方法仍然需要计算最终屏障的逆向下方的 Exional 。 然而, 使用最稳定的 Edx 将 美元 的 Ex 的 Ex 版本, 我们用一个自动的 状态来显示 美元 。