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SUMMARY:Implicit Regularization for Optimal Sparse Recovery LOCATION:CoRE A 301 DESCRIPTION;ENCODING=QUOTED-PRINTABLE:
Abstract:
We present an implicit regularization s
cheme for gradient descent methods applied to unpenalized least squares reg
ression to solve the problem of reconstructing a sparse signal from an unde
rdetermined system of linear measurements under the restricted isometry ass
umption. For a given parameterization yielding a non-convex optimization pr
oblem, we show that prescribed choices of initialization, step size and sto
pping 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 al
gorithm adapts to
instance difficulty and yields a dimension-independe
nt rate when the signal-to-noise ratio is high enough. We validate our find
ings with numerical experiments and compare our algorithm against explicit
$ell_{1}$ penalization. Going from hard instances to easy ones, our algorit
hm is seen to undergo a phase transition, eventually matching least squares
with an oracle knowledge of the true support.
Based on joint work wi th Patrick Rebeschini and Tomas Vaskevicius.
DTSTAMP:20240329T004811Z DTSTART;TZID=America/New_York:20191002T094500 SEQUENCE:0 TRANSP:OPAQUE END:VEVENT END:VCALENDAR