New ABGD algorithm achieves near-optimal piecewise linear regression in high dimensions

Haitham Kanj and Kiryung Lee from Ohio State University have published a paper on arXiv (2605.06959) introducing the Adaptive Block Gradient Descent (ABGD) algorithm, a parametric approach to piecewise linear regression in high-dimensional settings. The key innovation is representing piecewise linear functions as the difference of two max-affine (DoMA) functions, which allows ABGD to efficiently learn complex, nonlinear patterns while maintaining theoretical guarantees. The algorithm is initialized using a prior method designed for max-affine functions, and the authors provide a non-asymptotic local convergence analysis under sub-Gaussian covariate and noise distributions.

ABGD achieves linear convergence to an ε-accurate estimate with a sample complexity of Õ(d max(σ_z/ε,1)^2), where σ_z² is noise variance. This rate is proven minimax optimal up to logarithmic factors. In the noiseless case, exact recovery requires only Õ(d) samples, a dramatic improvement over previous approaches. Synthetic experiments confirm the theoretical guarantees, and tests on real-world datasets show ABGD matches or outperforms existing state-of-the-art methods like deep learning and kernel methods. This work has significant implications for machine learning applications requiring interpretable yet powerful nonlinear models, such as robotics, finance, and scientific computing.