Generalized self-concordance is a key property present in the objective function of many important learning problems. We establish the convergence rate of a simple Frank-Wolfe variant that uses the open-loop step size strategy $\gamma_t = 2/(t+2)$, obtaining a $\mathcal{O}(1/t)$ convergence rate for this class of functions in terms of primal gap and Frank-Wolfe gap, where $t$ is the iteration count. This avoids the use of second-order information or the need to estimate local smoothness parameters of previous work. We also show improved convergence rates for various common cases, e.g., when the feasible region under consideration is uniformly convex or polyhedral.

翻译：广义自和谐性是许多重要学习问题目标函数中存在的关键属性。我们建立了一种简单Frank-Wolfe变体的收敛速率，该变体采用开环步长策略$\gamma_t = 2/(t+2)$，在此类函数上关于原始间隙和Frank-Wolfe间隙获得了$\mathcal{O}(1/t)$的收敛速率，其中$t$为迭代次数。这避免了使用二阶信息或估计先前工作中局部平滑参数的需求。我们还展示了各种常见情况下的改进收敛速率，例如当所考虑的可行区域是均匀凸或多面体时。