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性质。我们建立了一种简单弗兰克-沃尔夫变体的收敛速率，该变体采用开环步长策略 $\gamma_t = 2/(t+2)$，针对此类函数在原始间隙和弗兰克-沃尔夫间隙方面获得 $\mathcal{O}(1/t)$ 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光滑参数的需求。我们还展示了在各种常见情形（例如，当所考虑可行域为一致凸或多面体时）下的改进收敛速率。