SGD with Momentum (SGDM) is a widely used family of algorithms for largescale 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 nonadaptive) FollowTheRegularizedLeaderbased SGDM algorithms with \emph{increasing momentum} and \emph{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 FTRLbased SGDM when used with adaptive stepsizes. Empirical results are shown as well.
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Surrogate Losses for Online Learning of Stepsizes in Stochastic NonConvex Optimization
Stochastic Gradient Descent (SGD) has played a central role in machine learning. However, it requires a carefully handpicked stepsize for fast convergence, which is notoriously tedious and timeconsuming to tune. Over the last several years, a plethora of adaptive gradientbased algorithms have emerged to ameliorate this problem. In this paper, we propose new surrogate losses to cast the problem of learning the optimal stepsizes for the stochastic optimization of a nonconvex smooth objective function onto an online convex optimization problem. This allows the use of noregret online algorithms to compute optimal stepsizes on the fly. In turn, this results in a SGD algorithm with selftuned stepsizes that guarantees convergence rates that are automatically adaptive to the level of noise.
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 Award ID(s):
 1925930
 NSFPAR ID:
 10111513
 Date Published:
 Journal Name:
 Proceedings of the 36th International Conference on Machine Learning
 Volume:
 97
 Page Range / eLocation ID:
 7664  7672
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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