Slowly Varying Regression under Sparsity

TL;DR

This paper introduces a novel framework for slowly varying sparse regression, reformulating the estimation problem as a binary convex optimization, and providing efficient algorithms with proven optimality and scalability for high-dimensional data.

Contribution

It develops a new relaxation technique transforming the non-convex problem into a binary convex form, enabling efficient and provably optimal solutions for slowly varying sparse regression models.

Findings

Outperforms existing methods in accuracy and speed.

Scalable to thousands of parameters.

Provides open-source implementation.

Abstract

We present the framework of slowly varying regression under sparsity, allowing sparse regression models to exhibit slow and sparse variations. The problem of parameter estimation is formulated as a mixed-integer optimization problem. We demonstrate that it can be precisely reformulated as a binary convex optimization problem through a novel relaxation technique. This relaxation involves a new equality on Moore-Penrose inverses, convexifying the non-convex objective function while matching the original objective on all feasible binary points. This enables us to efficiently solve the problem to provable optimality using a cutting plane-type algorithm. We develop a highly optimized implementation of this algorithm, substantially improving upon the asymptotic computational complexity of a straightforward implementation. Additionally, we propose a fast heuristic method that guarantees a feasible solution and, as empirically illustrated, produces high-quality warm-start solutions for the binary optimization problem. To tune the framework's hyperparameters, we suggest a practical procedure relying on binary search that, under certain assumptions, is guaranteed to recover the true model parameters. On both synthetic and real-world datasets, we demonstrate that the resulting algorithm outperforms competing formulations in comparable times across various metrics, including estimation accuracy, predictive power, and computational time. The algorithm is highly scalable, allowing us to train models with thousands of parameters. Our implementation is available open-source at https://github.com/vvdigalakis/SSVRegression.git.