Efficient Proximal Algorithms for Non-Smooth Optimization and Sampling

Core Concepts

The authors develop efficient proximal algorithms to solve non-smooth convex optimization problems and draw samples from non-smooth log-concave distributions. The core of their algorithms is an efficient implementation of the proximal map using the regularized cutting-plane method, which is then applied in both optimization and sampling settings.

Abstract

The paper considers two specific settings for the non-smooth objective/potential function f: Semi-smooth setting: f satisfies a H??lder-continuity condition on the (sub)gradient f'. Composite setting: f is a finite sum of semi-smooth components. For optimization, the authors develop an Adaptive Proximal Bundle Method (APBM) that uses the regularized cutting-plane method to solve the proximal subproblems. APBM is a universal method that does not require any problem-dependent parameters as input. The authors establish the iteration-complexity of APBM for both semi-smooth and composite settings. For sampling, the authors propose an efficient realization of the Restricted Gaussian Oracle (RGO), which is a key component in the Alternating Sampling Framework (ASF) for sampling from log-concave distributions. The RGO implementation is based on rejection sampling using the regularized cutting-plane method. The authors establish the complexity of this proximal sampling oracle and then combine it with ASF to obtain a proximal sampling algorithm with non-asymptotic complexity bounds for semi-smooth and composite potentials.

Stats

The authors use the following key metrics and figures in their analysis: Lipschitz constant Lα and H??lder exponent α for semi-smooth functions Number of semi-smooth components n and their corresponding Lipschitz constants Lαi and H??lder exponents αi for composite functions Iteration complexity bounds for the proximal map implementation, optimization algorithm, and sampling algorithm

Quotes

"The core of both proximal optimization and sampling algorithms is a proximal map of f." "Sampling can be viewed as an optimization over the manifold of probability distributions." "The alternating sampling framework (ASF) introduced in [23] is a generic framework for sampling from a distribution πX(x) ∝exp(−f(x))."

Key Insights Distilled From

Proximal Oracles for Optimization and Sampling

by Jiaming Lian... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2404.02239.pdf

Deeper Inquiries

How can the proposed proximal algorithms be extended to handle non-convex or weakly convex objective/potential functions

The proposed proximal algorithms can be extended to handle non-convex or weakly convex objective/potential functions by incorporating techniques that are suitable for such functions. For non-convex functions, one approach could be to utilize stochastic gradient descent (SGD) or variants like mini-batch SGD to optimize the objective function. These methods can handle non-convexity by iteratively updating the parameters in the direction of the negative gradient. Additionally, techniques like momentum, adaptive learning rates, and regularization can be incorporated to improve convergence and prevent overfitting. For weakly convex functions, methods like proximal gradient descent or proximal bundle methods can be adapted to handle the optimization process. These methods involve solving a sequence of subproblems that approximate the objective function using cutting-plane methods or other optimization techniques. By carefully selecting step sizes and updating rules, these methods can efficiently optimize weakly convex functions while ensuring convergence to a local minimum.

What are the potential applications of the developed proximal optimization and sampling methods in real-world problems

The developed proximal optimization and sampling methods have various potential applications in real-world problems across different domains. In machine learning, these methods can be used for training models with non-smooth loss functions, such as in deep learning for image recognition or natural language processing tasks. The ability to handle non-smooth functions efficiently can lead to faster convergence and better performance of machine learning models. In optimization problems in logistics and supply chain management, the proximal algorithms can be applied to optimize complex systems with non-smooth cost functions or constraints. This can help in improving resource allocation, scheduling, and decision-making processes in logistics operations. In finance, the developed methods can be utilized for portfolio optimization, risk management, and trading strategies where the objective functions or utility functions may exhibit non-smooth behavior. By efficiently optimizing and sampling from these functions, financial institutions can make better investment decisions and manage risks effectively.

Can the techniques used in the analysis of the proximal map implementation be applied to other types of non-smooth functions beyond the semi-smooth and composite settings considered in this work

The techniques used in the analysis of the proximal map implementation for semi-smooth and composite settings can be applied to other types of non-smooth functions beyond the ones considered in this work. For example, the regularized cutting-plane method and adaptive stepsize strategies can be generalized to handle functions with different levels of smoothness or non-smoothness. By adapting the concepts of proximal optimization and sampling to different types of non-smooth functions, researchers and practitioners can address a wider range of optimization and sampling problems in various fields. The key lies in understanding the properties of the non-smooth functions and designing algorithms that can efficiently handle their optimization and sampling requirements. This flexibility and adaptability make the techniques developed in this work valuable for a diverse set of applications in data science, optimization, and machine learning.