Stochastic Gradient Descent (SGD) is among the simplest and most popular methods in optimization. The convergence rate for SGD has been extensively studied and tight analyses have been established for the running average scheme, but the sub-optimality of the final iterate is still not well-understood. shamir2013stochastic gave the best known upper bound for the final iterate of SGD minimizing non-smooth convex functions, which is $O(\log T/\sqrt{T})$ for Lipschitz convex functions and $O(\log T/ T)$ with additional assumption on strongly convexity. The best known lower bounds, however, are worse than the upper bounds by a factor of $\log T$. harvey2019tight gave matching lower bounds but their construction requires dimension $d= T$. It was then asked by koren2020open how to characterize the final-iterate convergence of SGD in the constant dimension setting. In this paper, we answer this question in the more general setting for any $d\leq T$, proving $\Omega(\log d/\sqrt{T})$ and $\Omega(\log d/T)$ lower bounds for the sub-optimality of the final iterate of SGD in minimizing non-smooth Lipschitz convex and strongly convex functions respectively with standard step size schedules. Our results provide the first general dimension dependent lower bound on the convergence of SGD's final iterate, partially resolving a COLT open question raised by koren2020open. We also present further evidence to show the correct rate in one dimension should be $\Theta(1/\sqrt{T})$, such as a proof of a tight $O(1/\sqrt{T})$ upper bound for one-dimensional special cases in settings more general than koren2020open.
翻译:SGD 和 $O( log T/ t/ T) 的趋同率 已经进行了广泛研究, 并且已经为运行平均机程建立了严格的分析, 但最后迭代的亚最佳度仍然不完全理解。 shamir2013stochacast 给出了 SGD 最终迭代最小化非moot convex (SGD) 功能最著名的上限 。 20美元 (log T/\ sqrt{ T} ) 用于 Lipschitz convex 函数和 $O( log T/ T) 的趋同率 。 然而, 已知的最小值的下限比上限差, $T$。 hurve2019tight 给其构造需要维度 $d= T$。 然后通过 orren202020 开放来描述 SGD 最终的趋同值的趋同值 。 在本文中, 我们首先回答这个问题, 在任何 $GEO\\ drate 常规值 的直值中, ex slentrental ex ex ex dqration a deal deal deal deal ex ex ex ex deal deal dqrate a ex a ex ex a $ dqt ex a ex ex a ex a $ dal dex dqt ex a ex a ex a ex a ex a ex a ex a ex ex. slations a ex. slations a ex a ex a ex a ex.