论文标题

具有开环尺寸的Frank-Wolfe算法的加速度

Acceleration of Frank-Wolfe Algorithms with Open-Loop Step-Sizes

论文作者

Wirth, Elias, Kerdreux, Thomas, Pokutta, Sebastian

论文摘要

Frank-Wolfe算法(FW)是一种流行的一阶方法,用于解决依赖于线性最小化甲骨文而不是潜在昂贵的类似投影的甲骨文的约束凸优化问题。许多作品在使用线路搜索或短步骤时,在各种结构假设和特定FW变体的各种结构假设下都确定了加速的收敛速率,需要从目标函数中反馈。对于使用开环尺寸规则,又称具有预定的台阶尺寸的开环尺寸规则,算法在算法上非常简单且稳定时,对加速的收敛制度知之甚少。 Not only is FW with open-loop step-size rules not always subject to the same convergence rate lower bounds as FW with line-search or short-step, but in some specific cases, such as kernel herding in infinite dimensions, it has been empirically observed that FW with open-loop step-size rules enjoys to faster convergence rates than FW with line-search or short-step.我们对这种无法解释的现象提出了部分答案,它表征了一个通用设置,其fl fw具有开环尺寸规则的速度比线条搜索或短途搜索更快,并得出了一些通过开放式式台阶规则的FW加速融合结果。最后,我们证明具有开环尺寸的FW可以与基于动量的开环FW变体竞争。

Frank-Wolfe algorithms (FW) are popular first-order methods for solving constrained convex optimization problems that rely on a linear minimization oracle instead of potentially expensive projection-like oracles. Many works have identified accelerated convergence rates under various structural assumptions on the optimization problem and for specific FW variants when using line-search or short-step, requiring feedback from the objective function. Little is known about accelerated convergence regimes when utilizing open-loop step-size rules, a.k.a. FW with pre-determined step-sizes, which are algorithmically extremely simple and stable. Not only is FW with open-loop step-size rules not always subject to the same convergence rate lower bounds as FW with line-search or short-step, but in some specific cases, such as kernel herding in infinite dimensions, it has been empirically observed that FW with open-loop step-size rules enjoys to faster convergence rates than FW with line-search or short-step. We propose a partial answer to this unexplained phenomenon in kernel herding, characterize a general setting for which FW with open-loop step-size rules converges non-asymptotically faster than with line-search or short-step, and derive several accelerated convergence results for FW with open-loop step-size rules. Finally, we demonstrate that FW with open-loop step-sizes can compete with momentum-based open-loop FW variants.

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