Smoothed Variable Sample-size Accelerated Proximal Methods for Nonsmooth Stochastic Convex Programs

Jalilzadeh, Afrooz; Shanbhag, Uday V.; Blanchet, José

doi:10.48550/arxiv.1803.00718

Cited by 15 publications

(43 citation statements)

References 26 publications

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“…While stochastic counterpart of the proximal gradient method (and its accelerated counterpart) have received much attention [31,32,33,34], stochastic generalizations of the proximal-point method have been less studied. Koshal, Nedić and Shanbhag [4] presented one of the first instances of a stochastic iterative proximal point method for strictly monotone stochastic variational inequality problems and provided almost sure convergence.…”

Section: Preliminaries On Proximal-point Schemesmentioning

confidence: 99%

“…Rate statements (available for stochastic optimization) are summarized in Table 1. When the operator T may be cast as the sum of two operators A and B, there has been significant study of splitting methods [38,39,40,41] when the expectation-valued operator is single-valued in the optimization regime [31,32,33,34,42] and more generally [43,44] when A is Lipschitz and expectation-valued while B is maximal monotone. Sample-average approximation techniques have also been developed [45,46] as a form of approximation framework.…”

Section: Preliminaries On Proximal-point Schemesmentioning

confidence: 99%

“…We evaluate the solution quality of player i by the residual function res(x i ) = x i − B i,η (x −i ) and use res(x) = N i=1 res(x i )/N to denote the residual across multiple players in a game. [33,37] 1.3e-2 6.7 5.0e-4 0.26 [30,40] 3.5e-2 6.7 7.6e-4 0.27 [25,45] 5.0e-2 6.7 1.7e-3 0.26 [20,50] 6.6e-2 6.7 2.5e-3 0.26…”

Section: A Monotone Stochastic Bilevel Gamementioning

confidence: 99%

“…Suppose fi (x i , x −i ) = 1 2 d i (x i ) 2 + 3x i • N j=1 x j , where d i is randomly generated from U(0, 100), ∀i. The random parameter is a i (ω) ∼ U (33,37), ∀i. Algorithm parameters.…”

Section: A Monotone Stochastic Bilevel Gamementioning

confidence: 99%

“…We assume that there are N = 13 leaders and M = 10 followers. Furthermore, let C i be randomly generated from U(0, 100), ∀i = 1, • • • , N , c j = 50, ∀j = 1, • • • , M , b = 7 and a(ω) ∼ U(33,37) where U(l, u) denotes the uniform distribution on the interval [l, u].…”

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On the computation of equilibria in monotone and potential stochastic hierarchical games

Cui¹,

Shanbhag

2021

Preprint

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We consider a class of hierarchical noncooperative N−player games where the ith player solves a parametrized mathematical program with equilibrium constraints (MPEC) with the caveat that the implicit form of the ith player's in MPEC is convex in player strategy, given rival decisions. This represents a challenging class of games that subsumes multi-leader multifollower games in which player-specific problems are convex in an implicit sense, given rival decisions. We consider settings where player playoffs are expectation-valued with lower-level equilibrium constraints imposed in an almost-sure sense; i.e. player problems are parametrized stochastic MPECs. Few, if any, general purpose schemes exist for computing equilibria even for deterministic specializations of such games. We develop computational schemes in two distinct regimes: (a) Monotone regimes. When player-specific implicit problems are convex, then the necessary and sufficient equilibrium conditions are given by a stochastic inclusion. Under a monotonicity assumption on the operator, we develop a variance-reduced stochastic proximalpoint scheme that achieves deterministic rates of convergence in terms of solving proximalpoint problems in both monotone and strongly monotone regimes. In addition, the schemes are characterized by either optimal or near-optimal oracle (or sample) complexity guarantees. Finally, the generated sequences are shown to be convergent to a solution in an almost-sure sense in both monotone and strongly monotone regimes. Notably, the schemes do not impose any smoothness assumptions on player problems and allow for state-dependence in noise; (b) Potentiality. When the implicit form of the game admits a potential function, we develop an asynchronous relaxed inexact smoothed proximal best-response framework. However, any such avenue is impeded by the need to efficiently compute an approximate solution of an MPEC with a strongly convex implicit objective. To this end, we consider the smoothed counterpart of this game where each player's problem is smoothed via randomized smoothing. Notably, under suitable assumptions, we show that an η-smoothed game admits an η-approximate Nash equilibrium of the original game. Our proposed scheme produces a sequence that converges almost surely to an η-approximate Nash equilibrium in both relaxed and unrelaxed settings. This scheme is reliant on computing the proximal problem, a stochastic MPEC whose implicit form has a strongly convex objective, with increasing accuracy in finite-time. The smoothing framework allows for developing a variance-reduced zeroth-order scheme for such problems that admits a fast rate of convergence. Numerical studies on a class of multi-leader multi-follower games suggest that variance-reduced proximal schemes provide significantly better accuracy with far lower run-times. The relaxed best-response scheme scales well will problem size and generally displays more stability than its unrelaxed counterpart.

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Section: Preliminaries On Proximal-point Schemesmentioning

confidence: 99%

Section: Preliminaries On Proximal-point Schemesmentioning

confidence: 99%

Section: A Monotone Stochastic Bilevel Gamementioning

confidence: 99%

Section: A Monotone Stochastic Bilevel Gamementioning

confidence: 99%

mentioning

confidence: 99%

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On the computation of equilibria in monotone and potential stochastic hierarchical games

Cui¹,

Shanbhag

2021

Preprint

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Stochastic relaxed inertial forward-backward-forward splitting for monotone inclusions in Hilbert spaces

et al. 2022

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We consider monotone inclusions defined on a Hilbert space where the operator is given by the sum of a maximal monotone operator T and a single-valued monotone, Lipschitz continuous, and expectation-valued operator V. We draw motivation from the seminal work by Attouch and Cabot (Attouch in AMO 80:547–598, 2019, Attouch in MP 184: 243–287) on relaxed inertial methods for monotone inclusions and present a stochastic extension of the relaxed inertial forward–backward-forward method. Facilitated by an online variance reduction strategy via a mini-batch approach, we show that our method produces a sequence that weakly converges to the solution set. Moreover, it is possible to estimate the rate at which the discrete velocity of the stochastic process vanishes. Under strong monotonicity, we demonstrate strong convergence, and give a detailed assessment of the iteration and oracle complexity of the scheme. When the mini-batch is raised at a geometric (polynomial) rate, the rate statement can be strengthened to a linear (suitable polynomial) rate while the oracle complexity of computing an $$\epsilon $$ ϵ -solution improves to $${\mathcal {O}}(1/\epsilon )$$ O ( 1 / ϵ ) . Importantly, the latter claim allows for possibly biased oracles, a key theoretical advancement allowing for far broader applicability. By defining a restricted gap function based on the Fitzpatrick function, we prove that the expected gap of an averaged sequence diminishes at a sublinear rate of $${\mathcal {O}}(1/k)$$ O ( 1 / k ) while the oracle complexity of computing a suitably defined $$\epsilon $$ ϵ -solution is $${\mathcal {O}}(1/\epsilon ^{1+a})$$ O ( 1 / ϵ 1 + a ) where $$a > 1$$ a > 1 . Numerical results on two-stage games and an overlapping group Lasso problem illustrate the advantages of our method compared to competitors.

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Asynchronous variance-reduced block schemes for composite non-convex stochastic optimization: block-specific steplengths and adapted batch-sizes

Lei

Shanbhag

2020

Optimization Methods and Software

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We consider the minimization of a sum of an expectation-valued coordinate-wise L i -smooth nonconvex function and a nonsmooth block-separable convex regularizer. Prior schemes are characterized by the following shortcomings: (a) Steplengths require global knowledge of Lipschitz constants; (b) Batchsizes of gradients are centrally updated and require knowledge of the global clock; (c) a.s. convergence guarantees are unavailable; (d) Rates are inferior compared to deterministic counterparts. Specifically, (a) and (b) require coordination across blocks and necessitate global information, leading to potentially larger constants in the rate and the oracle complexity bounds, impeding decentralized implementations, and resulting in relatively poor empirical behavior. We address these shortcomings by proposing an asynchronous variance-reduced algorithm, where in each iteration, a single block is randomly chosen to update its estimates by a proximal variable sample-size stochastic gradient scheme, while the remaining blocks are kept invariant. Notably, each block employs a steplength that is in accordance with its blockspecific Lipschitz constant while block-specific batch-sizes are random variables updated at a rate that grows either at a geometric or polynomial rate with the (random) number of times that block is selected. We show that every limit point for almost every sample path is a stationary point and establish the ergodic non-asymptotic rate O(1/K). Iteration and oracle complexity to obtain an -stationary point are shown to be O(1/ ) and O(1/ 2 ), respectively. Furthermore, under a µ-proximal Polyak-Łojasiewicz (PL) condition with the batch size increasing at a geometric rate, we prove that the suboptimality diminishes at a geometric rate, the optimal deterministic rate while iteration and oracle complexity to obtain an -optimal solution are proven to be O((L max /µ) ln(1/ )) and O (L ave /µ)(1/ ) 1+c with c ≥ 0, respectively. In the single block setting, we obtain the optimal oracle complexity bound O(1/ ). In pursuit of less aggressive sampling rates, when the batch sizes increase at a polynomial rate of degree v ≥ 1, suboptimality decays at a corresponding polynomial rate while the iteration and oracle complexity to obtain an −optimal solution are provably O(v(1/ ) 1/v ) and O e v v 2v+1 (1/ ) 1+1/v , respectively. Finally, preliminary numerics support our theoretical findings, displaying significant improvements over schemes where steplengths are based on global Lipschitz constants. * The authors are with the

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Smoothed Variable Sample-size Accelerated Proximal Methods for Nonsmooth Stochastic Convex Programs

Cited by 15 publications

References 26 publications

On the computation of equilibria in monotone and potential stochastic hierarchical games

On the computation of equilibria in monotone and potential stochastic hierarchical games

Stochastic relaxed inertial forward-backward-forward splitting for monotone inclusions in Hilbert spaces

Asynchronous variance-reduced block schemes for composite non-convex stochastic optimization: block-specific steplengths and adapted batch-sizes

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