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Stability and Convergence of Stochastic Gradient Clipping: Beyond Lipschitz Continuity and Smoothness

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Stability and Convergence of Stochastic Gradient Clipping: Beyond Lipschitz Continuity and Smoothness Stability and Convergence of Stochastic Gradient Clipping: Beyond Lipschitz Continuity and Smoothness Vien V. Mai 1 Mikael Johansson 1 Abstract problems are at the core of many machine-learning appli- cations, and are often solved using stochastic (sub)gradient Stochastic gradient algorithms are often unsta- methods. In spite of their successes, stochastic gradient ble when applied to functions that do not have methods can be sensitive to their parameters (Nemirovski Lipschitz-continuous and/or bounded gradients. et al., 2009; Asi & Duchi, 2019a) and have severe instabil- Gradient clipping is a simple and effective tech- ity (unboundedness) problems when applied to functions nique to stabilize the training process for prob- that grow faster than quadratically in the decision vector x lems that are prone to the exploding gradient (Andradottir¨ , 1996; Asi & Duchi, 2019a). Consequently, a problem. Despite its widespread popularity, the careful (and sometimes time-consuming) parameter tuning convergence properties of the gradient clipping is often required for these methods to perform well in prac- heuristic are poorly understood, especially for tice. Even so, a good parameter selection is not sufficient to stochastic problems. This paper establishes both circumvent the instability issue on steep functions. qualitative and quantitative convergence results of the clipped stochastic (sub)gradient method Gradient clipping and the closely related gradient normal- (SGD) for non-smooth convex functions with ization technique are simple modifications to the underlying rapidly growing subgradients. Our analyses show algorithm to control the step length that an update can make that clipping enhances the stability of SGD and relative to the current iterate. These techniques enhance that the clipped SGD algorithm enjoys finite con- the stability of the optimization process, while adding es- vergence rates in many cases. We also study the sentially no extra cost to the original update. As a result, convergence of a clipped method with momen- gradient clipping has been a common choice in many ap- tum, which includes clipped SGD as a special plied domains of machine learning (Pascanu et al., 2013). case, for weakly convex problems under standard In this work, we consider gradient clipping applied to the assumptions. With a novel Lyapunov analysis, classical SGD method. Throughout the paper, we frequently we show that the proposed method achieves the use the following clipping operator best-known rate for the considered class of prob- lems, demonstrating the effectiveness of clipped n n γ clipγ : R ! R : x 7! min 1; x; methods also in this regime. Numerical results kxk2 confirm our theoretical developments. which is nothing else but the orthogonal projection onto the γ-ball. It is important to note that for noiseless gradients, clipping just changes the magnitude and does not effect the 1. Introduction search direction. However, in the stochastic setting, the We study stochastic optimization problems on the form expected value of the clipped stochastic gradient may point in a completely different direction than the true gradient. Z minimize f(x) := P [f(x; S)] = f(x; s)dP (s); (1) Clipped SGD To solve problem (1), we use an iterative n E x R n 2 S procedure that starts from x0 2 R and g0 2 @f(x0;S0) and generates a sequence of points x 2 n by repeating where S ∼ P is a random variable; f(x; s) is the instanta- k R the following steps for k = 0; 1; 2;:::: neous loss parameterized by x on a sample s 2 S. Such xk = xk − αkdk; dk = clip (gk) : (2) 1Division of Decision and Control Systems, EECS, KTH Royal +1 γk Institute of Technology, Stockholm, Sweden. Correspondence to: We refer to α as the kth stepsize and γ as the kth clipping V. V. Mai <[email protected]>, M. Johansson <[email protected]>. k k threshold, while gk = f 0(xk;Sk) is the kth stochastic sub- 1 Pmk i th gradient or its mini-batch version, gk = f (xk;S ) Proceedings of the 38 International Conference on Machine mk i=1 0 k Learning, PMLR 139, 2021. Copyright 2021 by the author(s). if multiple samples are used in each iteration. Stability and Convergence of Stochastic Gradient Clipping 1.1. Related work et al., 1998; Andradottir¨ , 1996) and discussed above. In (Asi & Duchi, 2019b), almost sure convergence of the so-called Our work is closely related to a number of topics which we truncated method is proven for convex functions that can briefly review below. grow polynomially from the the set of solutions. Gradient clipping Gradient clipping and normalization Weakly convex minimization The class of weakly con- were recognized early in the development of subgradient vex functions is broad, allowing for both non-smooth and methods as a useful tool to obtain convergence for rapidly non-convex objectives, and has favorable structures for growing convex functions (Shor, 1985; Ermoliev, 1988; algorithmic foundations and complexity theory. Earlier Alber et al., 1998). For the normalized method, the sem- works on weakly convex minimization (Nurminskii, 1973; inal work (Shor, 1985) establishes a convergence rate for Ruszczynski´ , 1987; Ermol’ev & Norkin, 1998) establish ? the quantity hgk= kgkk2 ; xk − x i without any assumptions qualitative convergence results for subgradient-based meth- on gk. By only requiring that subgradients are bounded ods. With the recent advances in statistical learning and on bounded sets, which always holds for continuous func- signal processing, there has been an emerging line of work tions, Alber et al.(1998) prove convergence in the objec- on this topic (see, e.g., (Duchi & Ruan, 2018; Davis & Grim- tive value for the clipped subgradient method. The work mer, 2019; Davis & Drusvyatskiy, 2019)). Convergence (Hazan et al., 2015) considers normalized gradient methods properties have been analyzed for many popular stochastic for quasi-convex and locally smooth functions. Recently, algorithms such as: model-based methods (Davis & Drusvy- the authors in (Zhang et al., 2019; 2020) analyze clipped atskiy, 2019; Duchi & Ruan, 2018; Asi & Duchi, 2019b); methods for twice-differentiable functions satisfying a more momentum extensions (Mai & Johansson, 2020); adaptive relaxed condition than the traditional L-smoothness. How- methods (Alacaoglu et al., 2020); and more. ever, much less is known in the stochastic setting. The work (Andradottir¨ , 1996) proposes a method that uses two inde- 1.2. Contributions pendent samples in each iteration and proves its almost sure convergence under the same growth condition used in (Alber The performance of stochastic (sub)gradient methods de- et al., 1998). Hazan et al.(2015) establish certain complex- pends heavily on how rapidly the underlying function is ity results for a mini-batch stochastic normalized gradient allowed to grow. Much convergence theory for these meth- method under some strong assumptions on the closeness ods hinges on the L-smoothness assumption for differen- of all the generated iterates to the optimal solution and the tiable functions or uniformly bounded subgradients for non- boundedness of all the individual mini-batch functions. In smooth ones. These conditions restrict the corresponding (Zhang et al., 2019; 2020), stochastic clipped SGD methods convergence rates to functions with at most quadratic and are analyzed under the assumption that the noise in the gra- linear growth, respectively. Beyond these well-behaved dient estimate is bounded for every x almost surely. Such classes, there is abundant evidence that SGD and its rela- assumption does not hold even for Gaussian noise in the tives may fail to converge. It is our goal in this work to show, gradients. Finally, we refer to (Cutkosky & Mehta, 2020; both theoretically and empirically, that the addition of the Curtis et al., 2019; Gorbunov et al., 2020) for recent theoret- clipping step greatly improves the convergence properties ical developments for clipped and normalized methods on of SGD. To that end, we make the following contributions: standard L-smooth problems. • We establish stability and convergence guarantees for clipped SGD on convex problems with arbitrary growth Robustness and stability The problems of robustness (exponential, super-exponential, etc.) of the subgradi- and stability in stochastic optimization have been empha- ents. We show that clipping coupled with standard mini- sized in many studies (see, e.g., (Nemirovski et al., 2009; batching suffices to guarantee almost sure convergence. Andradottir¨ , 1996; Asi & Duchi, 2019a;b) and references Even more, a finite convergence rate can also be obtained therein). Much recent work on this topic concentrates in this setting. around model-based algorithms that attempt to construct more accurate models of the objective than the linear one • We then turn to convex functions with polynomial growth provided by the stochastic subgradient. When such mod- and show that without the need for mini-batching, clipped els can be obtained and the resulting update steps can be SGD can essentially achieve the same optimal conver- performed efficiently, these methods often possess good gence rate as for stochastic strongly convex and Lipschitz stability properties and can be less sensitive to parameter se- continuous functions. lection than traditional stochastic subgradient methods. For example, the work (Asi & Duchi, 2019a) establishes almost • We consider a momentum extension of clipped SGD for sure convergence of stochastic (approximate) proximal point weakly convex minimization under standard growth con- methods under the arbitrary growth condition used in (Alber ditions. With a carefully constructed Lyapunov function, Stability and Convergence of Stochastic Gradient Clipping we are able to overcome the bias introduced by the clip- Assumption A3 (Finite variance). There exists a scalar ping step and preserve the best-known sample complexity σ > 0 such that: for this function class. h 2i 2 E kf 0(x; S) − f 0(x)k2 ≤ σ ; 8x 2 dom(f); Our experiments on phase retrieval, absolute linear regres- where f (x) = [f (x; S)] 2 @f(x).
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