Implicit Regularization for Optimal Sparse Recovery
This provides a statistically and computationally optimal method for sparse signal recovery, which is incremental as it builds on implicit regularization techniques but offers specific gains in efficiency and adaptability.
The paper tackles the problem of reconstructing a sparse signal from underdetermined linear measurements by using implicit regularization in gradient descent for unpenalized least squares regression, achieving minimax optimal rates with computational efficiency comparable to reading the data up to poly-logarithmic factors and adapting to instance difficulty.
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.