Gradient Descent Methods for Regularized Optimization
Filip Nikolovski, Irena Stojkovska, Katerina Hadzi-Velkova Saneva, Zoran Hadzi-Velkov
TL;DR
The paper addresses regularized optimization, where non-differentiable terms like $\ell^1$ hinder plain gradient descent. It analyzes GD and proximal GD, derives convergence guarantees for Lipschitz-smooth and μ-strongly convex settings, and shows how proximal GD with a closed-form LASSO prox enables efficient handling of $\ell^1$ penalties through soft-thresholding. A novel variable step-size proximal GD method is proposed, estimating a local Lipschitz constant $L_k$ to set $\lambda_k$ and adjust step sizes adaptively, with theoretical discussion and empirical validation. Numerical results on synthetic and real data demonstrate that the variable-step approach reduces iteration counts and runtime relative to constant step sizes, while preserving comparable overall complexity and outperforming Adam for the tested sparse regression tasks. The work advances practical first-order methods for sparse and low-rank recovery by integrating local smoothness awareness into proximal optimization.
Abstract
Regularization is a widely recognized technique in mathematical optimization. It can be used to smooth out objective functions, refine the feasible solution set, or prevent overfitting in machine learning models. Due to its simplicity and robustness, the gradient descent (GD) method is one of the primary methods used for numerical optimization of differentiable objective functions. However, GD is not well-suited for solving $\ell^1$ regularized optimization problems since these problems are non-differentiable at zero, causing iteration updates to oscillate or fail to converge. Instead, a more effective version of GD, called the proximal gradient descent employs a technique known as soft-thresholding to shrink the iteration updates toward zero, thus enabling sparsity in the solution. Motivated by the widespread applications of proximal GD in sparse and low-rank recovery across various engineering disciplines, we provide an overview of the GD and proximal GD methods for solving regularized optimization problems. Furthermore, this paper proposes a novel algorithm for the proximal GD method that incorporates a variable step size. Unlike conventional proximal GD, which uses a fixed step size based on the global Lipschitz constant, our method estimates the Lipschitz constant locally at each iteration and uses its reciprocal as the step size. This eliminates the need for a global Lipschitz constant, which can be impractical to compute. Numerical experiments we performed on synthetic and real-data sets show notable performance improvement of the proposed method compared to the conventional proximal GD with constant step size, both in terms of number of iterations and in time requirements.