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)
- Speaker: Varun Kanade, University of Oxford
- Friday 29 May 2020, 14:00–15:00
- Venue: https://zoom.us/j/95022384263?pwd=N3Z6elB2Vy9Jajd6azlCNjFHQVlKdz09.
- Series: Statistics; organiser: Dr Sergio Bacallado.