Moreau-Yoshida Variational Transport: A General Framework For Solving Regularized Distributional Optimization Problems

• URL: http://arxiv.org/abs/2307.16358v2
• Date: Sat, 10 Aug 2024 22:23:54 GMT
• Title: Moreau-Yoshida Variational Transport: A General Framework For Solving Regularized Distributional Optimization Problems
• Authors: Dai Hai Nguyen, Tetsuya Sakurai,
• Abstract summary: We consider a general optimization problem of minimizing a composite objective functional defined over a class probability distributions. We propose a novel method, dubbed as Moreau-Yoshida Variational Transport (MYVT), for solving the regularized distributional optimization problem.
• Score: 3.038642416291856
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider a general optimization problem of minimizing a composite objective functional defined over a class of probability distributions. The objective is composed of two functionals: one is assumed to possess the variational representation and the other is expressed in terms of the expectation operator of a possibly nonsmooth convex regularizer function. Such a regularized distributional optimization problem widely appears in machine learning and statistics, such as proximal Monte-Carlo sampling, Bayesian inference and generative modeling, for regularized estimation and generation. We propose a novel method, dubbed as Moreau-Yoshida Variational Transport (MYVT), for solving the regularized distributional optimization problem. First, as the name suggests, our method employs the Moreau-Yoshida envelope for a smooth approximation of the nonsmooth function in the objective. Second, we reformulate the approximate problem as a concave-convex saddle point problem by leveraging the variational representation, and then develope an efficient primal-dual algorithm to approximate the saddle point. Furthermore, we provide theoretical analyses and report experimental results to demonstrate the effectiveness of the proposed method.

Related papers

• Distributed Fractional Bayesian Learning for Adaptive Optimization [7.16937736207894]
  This paper considers a distributed adaptive optimization problem, where all agents only have access to their local cost functions with a common unknown parameter. We aim to provide valuable insights for addressing parameter uncertainty in distributed optimization problems and simultaneously find the optimal solution.
  arXiv  Detail & Related papers  (2024-04-17T13:09:33Z)
• An Inexact Halpern Iteration with Application to Distributionally Robust Optimization [9.529117276663431]
  We investigate the inexact variants of the scheme in both deterministic and deterministic convergence settings. We show that by choosing the inexactness appropriately, the inexact schemes admit an $O(k-1) convergence rate in terms of the (expected) residue norm.
  arXiv  Detail & Related papers  (2024-02-08T20:12:47Z)
• Moreau Envelope ADMM for Decentralized Weakly Convex Optimization [55.2289666758254]
  This paper proposes a proximal variant of the alternating direction method of multipliers (ADMM) for distributed optimization. The results of our numerical experiments indicate that our method is faster and more robust than widely-used approaches.
  arXiv  Detail & Related papers  (2023-08-31T14:16:30Z)
• Variational Refinement for Importance Sampling Using the Forward Kullback-Leibler Divergence [77.06203118175335]
  Variational Inference (VI) is a popular alternative to exact sampling in Bayesian inference. Importance sampling (IS) is often used to fine-tune and de-bias the estimates of approximate Bayesian inference procedures. We propose a novel combination of optimization and sampling techniques for approximate Bayesian inference.
  arXiv  Detail & Related papers  (2021-06-30T11:00:24Z)
• Loss function based second-order Jensen inequality and its application to particle variational inference [112.58907653042317]
  Particle variational inference (PVI) uses an ensemble of models as an empirical approximation for the posterior distribution. PVI iteratively updates each model with a repulsion force to ensure the diversity of the optimized models. We derive a novel generalization error bound and show that it can be reduced by enhancing the diversity of models.
  arXiv  Detail & Related papers  (2021-06-09T12:13:51Z)
• Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solution [8.465228064780748]
  We prove that computing the Wasserstein distance between a discrete probability measure supported on two points is already #P-hard. We introduce a distributionally robust dual optimal transport problem whose objective function is smoothed with the most adverse disturbance distributions. We show that smoothing the dual objective function is equivalent to regularizing the primal objective function.
  arXiv  Detail & Related papers  (2021-03-10T18:53:59Z)
• Stochastic Learning Approach to Binary Optimization for Optimal Design of Experiments [0.0]
  We present a novel approach to binary optimization for optimal experimental design (OED) for Bayesian inverse problems governed by mathematical models such as partial differential equations. The OED utility function, namely, the regularized optimality gradient, is cast into an objective function in the form of an expectation over a Bernoulli distribution. The objective is then solved by using a probabilistic optimization routine to find an optimal observational policy.
  arXiv  Detail & Related papers  (2021-01-15T03:54:12Z)
• Variational Transport: A Convergent Particle-BasedAlgorithm for Distributional Optimization [106.70006655990176]
  A distributional optimization problem arises widely in machine learning and statistics. We propose a novel particle-based algorithm, dubbed as variational transport, which approximately performs Wasserstein gradient descent. We prove that when the objective function satisfies a functional version of the Polyak-Lojasiewicz (PL) (Polyak, 1963) and smoothness conditions, variational transport converges linearly.
  arXiv  Detail & Related papers  (2020-12-21T18:33:13Z)
• Robust, Accurate Stochastic Optimization for Variational Inference [68.83746081733464]
  We show that common optimization methods lead to poor variational approximations if the problem is moderately large. Motivated by these findings, we develop a more robust and accurate optimization framework by viewing the underlying algorithm as producing a Markov chain.
  arXiv  Detail & Related papers  (2020-09-01T19:12:11Z)
• Distributed Averaging Methods for Randomized Second Order Optimization [54.51566432934556]
  We consider distributed optimization problems where forming the Hessian is computationally challenging and communication is a bottleneck. We develop unbiased parameter averaging methods for randomized second order optimization that employ sampling and sketching of the Hessian. We also extend the framework of second order averaging methods to introduce an unbiased distributed optimization framework for heterogeneous computing systems.
  arXiv  Detail & Related papers  (2020-02-16T09:01:18Z)

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.