Explicit Bounds on Hoffman Constants for Sparse Polyhedral Regularizers

Abstract

Hoffman constants quantify the distance to the solution set of a polyhedral system in terms of its residual; they control error bounds, geometric conditioning, the convergence rate of first-order iterative algorithms, and the stability of regression estimators to target perturbations. Despite their importance, explicit values of Hoffman constants are rarely available due to their computational complexity. In this paper, we derive explicit bounds on the Hoffman constant for a class of structured polyhedral systems arising in sparse regularization. We show that the constants admit a decomposition into spectral, geometric, and interaction components, which together determine their scaling behavior. This framework explains the gap between worst-case dimension-dependent bounds and the favorable conditioning observed empirically in structured problems. As a primary application, we analyze $\ell_1$-regularized least squares, demonstrating that the effective constants depend on low-dimensional active subspaces rather than the ambient dimension. We then extend to the elastic net and adaptive $\ell_1$-regularized formulations, demonstrating how the underlying optimization geometry is altered by the $\ell_2$ penalty and reweighting scheme. Furthermore, we relate our results to classical statistical assumptions and show that the restricted eigenvalue condition bounds only the primal curvature of the active set, leaving the dual interaction geometry unconstrained and is thus insufficient alone for fast theoretical convergence rates. We further show standard $\ell_\infty$ coherence conditions do not yield dimension-independent Euclidean bounds to close this gap. Our theoretical results show that Hoffman constants admit explicit, interpretable forms in structured regression settings and, in practice, provide actionable tools for post-hoc feature pruning and a priori optimal hyperparameter selection.