Adaptive Proximal Algorithms for Convex Optimization with Local Lipschitz Gradient

Core Concepts

Efficiently optimize convex functions using adaptive proximal algorithms without backtracking.

Abstract

The article introduces adaPGM, an adaptive proximal gradient method that eliminates the need for linesearch in convex optimization. It adapts step sizes based on local smoothness estimates, improving efficiency and handling nonsmooth terms. The method is extended to a primal-dual setting, introducing adaPDM for more general problems. The algorithm avoids evaluating the linear operator norm by incorporating a backtracking procedure efficiently. Numerical simulations demonstrate the effectiveness of these adaptive algorithms compared to traditional methods.

Stats

Backtracking linesearch is avoided in convex optimization.
AdaPGM adapts step sizes based on local smoothness estimates.
AdaPDM extends the method to handle more general problems.
The algorithm efficiently avoids evaluating the linear operator norm.

Quotes

"AdaPGM adapts step sizes based on local smoothness estimates."
"Numerical simulations demonstrate the effectiveness of the proposed algorithms."

Key Insights Distilled From

Adaptive proximal algorithms for convex optimization under local Lipschitz continuity of the gradient

by Puya Latafat... at arxiv.org 03-14-2024

https://arxiv.org/pdf/2301.04431.pdf

Deeper Inquiries

How does adaPGM compare to traditional gradient descent methods

AdaPGM differs from traditional gradient descent methods in its adaptive stepsize selection based on local smoothness estimates. Unlike traditional methods that use a fixed stepsize or require backtracking to adjust the stepsize, adaPGM dynamically adjusts the stepsize during optimization without the need for backtracking. This adaptability allows adaPGM to potentially converge faster by taking larger steps when appropriate and smaller steps when necessary, leading to improved convergence rates compared to traditional gradient descent methods.

What are the implications of eliminating backtracking in optimization algorithms

The elimination of backtracking in optimization algorithms, as seen in adaptive algorithms like adaPGM, has several implications:

Efficiency: Backtracking requires additional function evaluations and can be computationally expensive. By eliminating this process, adaptive algorithms can reduce computational costs and improve efficiency.
Simplicity: Removing backtracking simplifies the algorithm implementation and reduces the complexity of parameter tuning.
Faster Convergence: Adaptive algorithms can adaptively adjust their stepsizes based on local smoothness estimates, potentially leading to faster convergence rates compared to traditional methods with fixed stepsizes or backtracking.

How can adaptive algorithms like adaPDM be applied in real-world scenarios beyond numerical simulations

Adaptive algorithms like AdaPDM have various real-world applications beyond numerical simulations:

Machine Learning: In machine learning tasks such as training deep neural networks or optimizing model parameters, adaptive algorithms can efficiently handle complex optimization problems with nonsmooth components.
Signal Processing: Adaptive algorithms are useful in signal processing applications where optimization problems involve constraints or nonsmooth functions.
Control Systems: In control systems design and optimization, adaptive algorithms can help optimize system performance while considering constraints and non-smooth objectives.
Finance: Adaptive algorithms are valuable in financial modeling and portfolio optimization where objective functions may be non-convex or contain nonsmooth terms.

By applying these adaptive techniques in real-world scenarios, practitioners can benefit from improved convergence rates, reduced computational costs, and more efficient optimization processes across various domains.