On the Last Iterate Convergence of Momentum Methods
On the Last Iterate Convergence of Momentum Methods
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发表时间：
2021-02
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通讯作者：
Xiaoyun Li;Mingrui Liu;Francesco Orabona
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作者：
Xiaoyun Li;Mingrui Liu;Francesco Orabona
具有动量（SGDM）的SGD是一种广泛使用的算法系列，用于大规模优化机器学习问题。但是，当优化通用凸功能时，任何SGDM算法都不知道与普通SGD相比。此外，即使是最新的结果，也需要更改SGDM算法，例如迭代的平均值和对有限域的投影，这些域很少在实践中使用。在本文中，我们关注SGDM最后一次迭代的收敛速率。我们首次证明，对于任何恒定的动量因素，都存在Lipschitz和凸功能，而SGDM的最后一个迭代均具有$ \ omega的次优收敛速率（\ frac {\ ln t} {\ sqrt {\ sqrt {\ sqrt { $ t $迭代后的t}}）$。基于这一事实，我们研究了一类（自适应和非自适应）遵循基于调查的领导者的SGDM算法，并随着势头越来越多和缩小更新。对于这些算法，我们表明最后一个迭代具有最佳收敛$ O（\ frac {1} {\ sqrt {t}}）$，用于无约束的凸随机优化问题，而没有投影到有限域的域也不是$ t $的知识。此外，当与自适应步骤一起使用时，我们显示了基于FTRL的SGDM的各种结果。也显示了经验结果。
SGD with Momentum (SGDM) is a widely used family of algorithms for large-scale optimization of machine learning problems. Yet, when optimizing generic convex functions, no advantage is known for any SGDM algorithm over plain SGD. Moreover, even the most recent results require changes to the SGDM algorithms, like averaging of the iterates and a projection onto a bounded domain, which are rarely used in practice. In this paper, we focus on the convergence rate of the last iterate of SGDM. For the first time, we prove that for any constant momentum factor, there exists a Lipschitz and convex function for which the last iterate of SGDM suffers from a suboptimal convergence rate of $\Omega(\frac{\ln T}{\sqrt{T}})$ after $T$ iterations. Based on this fact, we study a class of (both adaptive and non-adaptive) Follow-The-Regularized-Leader-based SGDM algorithms with increasing momentum and shrinking updates. For these algorithms, we show that the last iterate has optimal convergence $O(\frac{1}{\sqrt{T}})$ for unconstrained convex stochastic optimization problems without projections onto bounded domains nor knowledge of $T$. Further, we show a variety of results for FTRL-based SGDM when used with adaptive stepsizes. Empirical results are shown as well.