Stochastic Optimization with Constraints: A Non-asymptotic Instance-Dependent Analysis

• URL: http://arxiv.org/abs/2404.00042v1
• Date: Sun, 24 Mar 2024 14:45:11 GMT
• Title: Stochastic Optimization with Constraints: A Non-asymptotic Instance-Dependent Analysis
• Authors: Koulik Khamaru,
• Abstract summary: We analyze the behavior of a natural variance reduced proximal gradient (VRPG) algorithm for convex optimization under convex constraints. Our main result is a non-asymptotic guarantee for VRPG algorithm. We show that our guarantee captures the complexity of the loss function, the variability of the noise, and the geometry of the constraint set.
• Score: 2.1756081703276
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the problem of stochastic convex optimization under convex constraints. We analyze the behavior of a natural variance reduced proximal gradient (VRPG) algorithm for this problem. Our main result is a non-asymptotic guarantee for VRPG algorithm. Contrary to minimax worst case guarantees, our result is instance-dependent in nature. This means that our guarantee captures the complexity of the loss function, the variability of the noise, and the geometry of the constraint set. We show that the non-asymptotic performance of the VRPG algorithm is governed by the scaled distance (scaled by $\sqrt{N}$) between the solutions of the given problem and that of a certain small perturbation of the given problem -- both solved under the given convex constraints; here, $N$ denotes the number of samples. Leveraging a well-established connection between local minimax lower bounds and solutions to perturbed problems, we show that as $N \rightarrow \infty$, the VRPG algorithm achieves the renowned local minimax lower bound by H\`{a}jek and Le Cam up to universal constants and a logarithmic factor of the sample size.

Related papers

• A Sample Efficient Alternating Minimization-based Algorithm For Robust Phase Retrieval [56.67706781191521]
  In this work, we present a robust phase retrieval problem where the task is to recover an unknown signal. Our proposed oracle avoids the need for computationally spectral descent, using a simple gradient step and outliers.
  arXiv  Detail & Related papers  (2024-09-07T06:37:23Z)
• Universal Online Learning with Gradient Variations: A Multi-layer Online Ensemble Approach [57.92727189589498]
  We propose an online convex optimization approach with two different levels of adaptivity. We obtain $mathcalO(log V_T)$, $mathcalO(d log V_T)$ and $hatmathcalO(sqrtV_T)$ regret bounds for strongly convex, exp-concave and convex loss functions.
  arXiv  Detail & Related papers  (2023-07-17T09:55:35Z)
• Accelerated SGD for Non-Strongly-Convex Least Squares [14.010916616909743]
  We consider approximation for the least squares regression problem in the non-strongly convex setting. We present the first practical algorithm that achieves the optimal prediction error rates in terms of dependence on the noise of the problem.
  arXiv  Detail & Related papers  (2022-03-03T14:39:33Z)
• Accelerated and instance-optimal policy evaluation with linear function approximation [17.995515643150657]
  Existing algorithms fail to match at least one of these lower bounds. We develop an accelerated, variance-reduced fast temporal difference algorithm that simultaneously matches both lower bounds and attains a strong notion of instance-optimality.
  arXiv  Detail & Related papers  (2021-12-24T17:21:04Z)
• Instance-optimality in optimal value estimation: Adaptivity via variance-reduced Q-learning [99.34907092347733]
  We analyze the problem of estimating optimal $Q$-value functions for a discounted Markov decision process with discrete states and actions. Using a local minimax framework, we show that this functional arises in lower bounds on the accuracy on any estimation procedure. In the other direction, we establish the sharpness of our lower bounds, up to factors logarithmic in the state and action spaces, by analyzing a variance-reduced version of $Q$-learning.
  arXiv  Detail & Related papers  (2021-06-28T00:38:54Z)
• A Feasible Level Proximal Point Method for Nonconvex Sparse Constrained Optimization [25.73397307080647]
  We present a new model of a general convex or non objective machine machine objectives. We propose an algorithm that solves a constraint with gradually relaxed point levels of each subproblem. We demonstrate the effectiveness of our new numerical scale problems.
  arXiv  Detail & Related papers  (2020-10-23T05:24:05Z)
• Conditional gradient methods for stochastically constrained convex minimization [54.53786593679331]
  We propose two novel conditional gradient-based methods for solving structured convex optimization problems. The most important feature of our framework is that only a subset of the constraints is processed at each iteration. Our algorithms rely on variance reduction and smoothing used in conjunction with conditional gradient steps, and are accompanied by rigorous convergence guarantees.
  arXiv  Detail & Related papers  (2020-07-07T21:26:35Z)
• Dynamic Regret of Convex and Smooth Functions [93.71361250701075]
  We investigate online convex optimization in non-stationary environments. We choose the dynamic regret as the performance measure. We show that it is possible to further enhance the dynamic regret by exploiting the smoothness condition.
  arXiv  Detail & Related papers  (2020-07-07T14:10:57Z)
• 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)

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.