A Gradient Smoothed Functional Algorithm with Truncated Cauchy Random Perturbations for Stochastic Optimization

• URL: http://arxiv.org/abs/2208.00290v4
• Date: Fri, 30 Jun 2023 15:26:45 GMT
• Title: A Gradient Smoothed Functional Algorithm with Truncated Cauchy Random Perturbations for Stochastic Optimization
• Authors: Akash Mondal, Prashanth L. A., Shalabh Bhatnagar
• Abstract summary: We present a convex gradient algorithm for minimizing a smooth objective function that is an expectation over noisy cost samples. We also show that our algorithm avoids the ratelibria, implying convergence to local minima.
• Score: 10.820943271350442
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, we present a stochastic gradient algorithm for minimizing a smooth objective function that is an expectation over noisy cost samples, and only the latter are observed for any given parameter. Our algorithm employs a gradient estimation scheme with random perturbations, which are formed using the truncated Cauchy distribution from the delta sphere. We analyze the bias and variance of the proposed gradient estimator. Our algorithm is found to be particularly useful in the case when the objective function is non-convex, and the parameter dimension is high. From an asymptotic convergence analysis, we establish that our algorithm converges almost surely to the set of stationary points of the objective function and obtains the asymptotic convergence rate. We also show that our algorithm avoids unstable equilibria, implying convergence to local minima. Further, we perform a non-asymptotic convergence analysis of our algorithm. In particular, we establish here a non-asymptotic bound for finding an epsilon-stationary point of the non-convex objective function. Finally, we demonstrate numerically through simulations that the performance of our algorithm outperforms GSF, SPSA, and RDSA by a significant margin over a few non-convex settings and further validate its performance over convex (noisy) objectives.

Related papers

• A Mean-Field Analysis of Neural Stochastic Gradient Descent-Ascent for Functional Minimax Optimization [90.87444114491116]
  This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparametricized two-layer neural networks. We address (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural networks. Results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $O(alpha-1)$, measured in terms of the Wasserstein distance.
  arXiv  Detail & Related papers  (2024-04-18T16:46:08Z)
• A Stochastic-Gradient-based Interior-Point Algorithm for Solving Smooth Bound-Constrained Optimization Problems [12.29270365918848]
  The proposed algorithm is based on the subject-point unique constraints from other interior-point methods. It is shown that with a careful balance between the projection, step-size and sequence sequences, the proposed algorithm convergence guarantees in both numerical and deterministic settings.
  arXiv  Detail & Related papers  (2023-04-28T15:30:43Z)
• An Algebraically Converging Stochastic Gradient Descent Algorithm for Global Optimization [14.336473214524663]
  A key component in the algorithm is the randomness based on the value of the objective function. We prove the convergence of the algorithm with an algebra and tuning in the parameter space. We present several numerical examples to demonstrate the efficiency and robustness of the algorithm.
  arXiv  Detail & Related papers  (2022-04-12T16:27:49Z)
• Differentiable Annealed Importance Sampling and the Perils of Gradient Noise [68.44523807580438]
  Annealed importance sampling (AIS) and related algorithms are highly effective tools for marginal likelihood estimation. Differentiability is a desirable property as it would admit the possibility of optimizing marginal likelihood as an objective. We propose a differentiable algorithm by abandoning Metropolis-Hastings steps, which further unlocks mini-batch computation.
  arXiv  Detail & Related papers  (2021-07-21T17:10:14Z)
• Optimal Rates for Random Order Online Optimization [60.011653053877126]
  We study the citetgarber 2020online, where the loss functions may be chosen by an adversary, but are then presented online in a uniformly random order. We show that citetgarber 2020online algorithms achieve the optimal bounds and significantly improve their stability.
  arXiv  Detail & Related papers  (2021-06-29T09:48:46Z)
• Zeroth-Order Hybrid Gradient Descent: Towards A Principled Black-Box Optimization Framework [100.36569795440889]
  This work is on the iteration of zero-th-order (ZO) optimization which does not require first-order information. We show that with a graceful design in coordinate importance sampling, the proposed ZO optimization method is efficient both in terms of complexity as well as as function query cost.
  arXiv  Detail & Related papers  (2020-12-21T17:29:58Z)
• Asymptotic study of stochastic adaptive algorithm in non-convex landscape [2.1320960069210484]
  This paper studies some assumption properties of adaptive algorithms widely used in optimization and machine learning. Among them Adagrad and Rmsprop, which are involved in most of the blackbox deep learning algorithms.
  arXiv  Detail & Related papers  (2020-12-10T12:54:45Z)
• Stochastic optimization with momentum: convergence, fluctuations, and traps avoidance [0.0]
  In this paper, a general optimization procedure is studied, unifying several variants of the gradient descent such as, among others, the heavy ball method, the Nesterov Accelerated Gradient (S-NAG), and the widely used Adam method. The avoidance is studied as a noisy discretization of a non-autonomous ordinary differential equation.
  arXiv  Detail & Related papers  (2020-12-07T19:14:49Z)
• 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)
• Convergence of adaptive algorithms for weakly convex constrained optimization [59.36386973876765]
  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.
  arXiv  Detail & Related papers  (2020-06-11T17:43:19Z)
• Non-asymptotic bounds for stochastic optimization with biased noisy gradient oracles [8.655294504286635]
  We introduce biased gradient oracles to capture a setting where the function measurements have an estimation error. Our proposed oracles are in practical contexts, for instance, risk measure estimation from a batch of independent and identically distributed simulation.
  arXiv  Detail & Related papers  (2020-02-26T12:53:04Z)

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.