Implicit regularization for optimal sparse recovery

Conference item

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 ℓ1 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.

Authors: Vaškevičius, T; Kanade, V; Rebeschini, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Neural Information Processing Systems Foundation
• Host title: Advances in Neural Information Processing Systems 32 (NIPS 2019)
• Pages: 2968-2979
• Publication date: 2019-12-14
• Acceptance date: 2019-09-03
• Event title: 33rd Conference on Neural Information Processing Systems (NeurIPS 2019)
• Event location: Vancouver, Canada
• Event website: https://nips.cc/Conferences/2019
• Event start date: 2019-12-08
• Event end date: 2020-12-14
• Language: English