UCL ELLIS

Implicit Regularization for Optimal Sparse Recovery

Speaker	Varun Kanade
Affiliation	University of Oxford
Date	Friday, 15 November 2019
Time	13:00-14:00
Location	Zoom
Link	Roberts Building G06 Sir Ambrose Fleming LT
Event series	DeepMind/ELLIS CSML Seminar Series
Abstract	We present an implicit regularization scheme 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 parameterization 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. 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. (based on joint work with Patrick Rebeschini and Tomas Vaskevicius) Varun Kanade is an associate professor at University of Oxford in the Department of Computer Science. He has been a Simons Postdoctoral Fellow at the University of California, Berkeley and a FSMP postdoctoral fellow at ENS, Paris. He obtained his Ph.D. from Harvard University in 2012. His research interests are in machine learning and theoretical computer science.
Biography