Implicit Regularization for Optimal Sparse Recovery

TL;DR

This paper demonstrates that specific implicit regularization strategies in gradient descent can achieve optimal sparse recovery rates in underdetermined linear systems, adapting to problem difficulty and outperforming explicit methods.

Contribution

It introduces an implicit regularization scheme with particular initialization, step size, and stopping rules that attains minimax optimality and adapts to instance difficulty in sparse recovery.

Findings

Achieves minimax optimal sparse recovery rates.

Adapts to instance difficulty with high SNR.

Exhibits a phase transition from hard to easy instances.

Abstract

We investigate implicit regularization schemes for gradient descent methods applied to unpenalized least squares regression to solve the problem of reconstructing a sparse signal from an underdetermined system of linear measurements under the restricted isometry assumption. For a given parametrization yielding a non-convex optimization problem, we show that prescribed choices of initialization, step size and stopping time yield a statistically and computationally optimal algorithm that achieves the minimax rate with the same cost required to read the data up to poly-logarithmic factors. Beyond minimax optimality, we show that our algorithm adapts to instance difficulty and yields a dimension-independent rate when the signal-to-noise ratio is high enough. Key to the computational efficiency of our method is an increasing step size scheme that adapts to refined estimates of the true solution. We validate our findings with numerical experiments and compare our algorithm against explicit $\ell_{1}$ penalization. Going from hard instances to easy ones, our algorithm is seen to undergo a phase transition, eventually matching least squares with an oracle knowledge of the true support.