Convergence of adaptive algorithms for weakly convex constrained optimization

• URL: http://arxiv.org/abs/2006.06650v1
• Date: Thu, 11 Jun 2020 17:43:19 GMT
• Title: Convergence of adaptive algorithms for weakly convex constrained optimization
• Authors: Ahmet Alacaoglu, Yura Malitsky, Volkan Cevher
• Abstract summary: We prove the $mathcaltilde O(t-1/4)$ rate of convergence for the norm of the gradient of Moreau envelope. Our analysis works with mini-batch size of $1$, constant first and second order moment parameters, and possibly smooth optimization domains.
• Score: 59.36386973876765
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We analyze the adaptive first order algorithm AMSGrad, for solving a constrained stochastic optimization problem with a weakly convex objective. We prove the $\mathcal{\tilde O}(t^{-1/4})$ rate of convergence for the norm of the gradient of Moreau envelope, which is the standard stationarity measure for this class of problems. It matches the known rates that adaptive algorithms enjoy for the specific case of unconstrained smooth stochastic optimization. Our analysis works with mini-batch size of $1$, constant first and second order moment parameters, and possibly unbounded optimization domains. Finally, we illustrate the applications and extensions of our results to specific problems and algorithms.

Related papers

• A simple uniformly optimal method without line search for convex optimization [9.280355951055865]
  We show that line search is superfluous in attaining the optimal rate of convergence for solving a convex optimization problem whose parameters are not given a priori. We present a novel accelerated gradient descent type algorithm called AC-FGM that can achieve an optimal $mathcalO (1/k2)$ rate of convergence for smooth convex optimization.
  arXiv  Detail & Related papers  (2023-10-16T05:26:03Z)
• Accelerated First-Order Optimization under Nonlinear Constraints [73.2273449996098]
  We exploit between first-order algorithms for constrained optimization and non-smooth systems to design a new class of accelerated first-order algorithms. An important property of these algorithms is that constraints are expressed in terms of velocities instead of sparse variables.
  arXiv  Detail & Related papers  (2023-02-01T08:50:48Z)
• Adaptive Stochastic Optimisation of Nonconvex Composite Objectives [2.1700203922407493]
  We propose and analyse a family of generalised composite mirror descent algorithms. With adaptive step sizes, the proposed algorithms converge without requiring prior knowledge of the problem. We exploit the low-dimensional structure of the decision sets for high-dimensional problems.
  arXiv  Detail & Related papers  (2022-11-21T18:31:43Z)
• Near-Optimal Algorithms for Making the Gradient Small in Stochastic Minimax Optimization [14.719077076351377]
  We study the problem of finding a near-stationary point for smooth minimax optimization. We show that a novel algorithm called Recursive IteratioNRAIN achieves both convex-concave and strongly-concave cases.
  arXiv  Detail & Related papers  (2022-08-11T16:55:26Z)
• Recent Theoretical Advances in Non-Convex Optimization [56.88981258425256]
  Motivated by recent increased interest in analysis of optimization algorithms for non- optimization in deep networks and other problems in data, we give an overview of recent results of theoretical optimization algorithms for non- optimization.
  arXiv  Detail & Related papers  (2020-12-11T08:28:51Z)
• Adaptive extra-gradient methods for min-max optimization and games [35.02879452114223]
  We present a new family of minmax optimization algorithms that automatically exploit the geometry of the gradient data observed at earlier iterations. Thanks to this adaptation mechanism, the proposed method automatically detects whether the problem is smooth or not. It converges to an $varepsilon$-optimal solution within $mathcalO (1/varepsilon)$ iterations in smooth problems, and within $mathcalO (1/varepsilon)$ iterations in non-smooth ones.
  arXiv  Detail & Related papers  (2020-10-22T22:54:54Z)
• A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic [142.1492359556374]
  Bilevel optimization is a class of problems which exhibit a two-level structure. We propose a two-timescale approximation (TTSA) algorithm for tackling such a bilevel problem. We show that a two-timescale natural actor-critic policy optimization algorithm can be viewed as a special case of our TTSA framework.
  arXiv  Detail & Related papers  (2020-07-10T05:20:02Z)
• Exploiting Higher Order Smoothness in Derivative-free Optimization and Continuous Bandits [99.70167985955352]
  We study the problem of zero-order optimization of a strongly convex function. We consider a randomized approximation of the projected gradient descent algorithm. Our results imply that the zero-order algorithm is nearly optimal in terms of sample complexity and the problem parameters.
  arXiv  Detail & Related papers  (2020-06-14T10:42:23Z)
• Adaptive First-and Zeroth-order Methods for Weakly Convex Stochastic Optimization Problems [12.010310883787911]
  We analyze a new family of adaptive subgradient methods for solving an important class of weakly convex (possibly nonsmooth) optimization problems. Experimental results indicate how the proposed algorithms empirically outperform its zerothorder gradient descent and its design variant.
  arXiv  Detail & Related papers  (2020-05-19T07:44:52Z)
• Private Stochastic Convex Optimization: Optimal Rates in Linear Time [74.47681868973598]
  We study the problem of minimizing the population loss given i.i.d. samples from a distribution over convex loss functions. A recent work of Bassily et al. has established the optimal bound on the excess population loss achievable given $n$ samples. We describe two new techniques for deriving convex optimization algorithms both achieving the optimal bound on excess loss and using $O(minn, n2/d)$ gradient computations.
  arXiv  Detail & Related papers  (2020-05-10T19:52:03Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.