  We contribute to the growing body of knowledge on more powerful and adaptive
stepsizes for convex optimization, empowered by local curvature information. We
do not go the route of fully-fledged second-order methods which require the
expensive computation of the Hessian. Instead, our key observation is that, for
some problems (e.g., when minimizing the sum of squares of absolutely convex
functions), certain local curvature information is readily available, and can
be used to obtain surprisingly powerful matrix-valued stepsizes, and meaningful
theory. In particular, we develop three new methods$\unicode{x2013}$LCD1, LCD2
and LCD3$\unicode{x2013}$where the abbreviation stands for local curvature
descent. While LCD1 generalizes gradient descent with fixed stepsize, LCD2
generalizes gradient descent with Polyak stepsize. Our methods enhance these
classical gradient descent baselines with local curvature information, and our
theory recovers the known rates in the special case when no curvature
information is used. Our last method, LCD3, is a variable metric version of
LCD2; this feature leads to a closed-form expression for the iterates. Our
empirical results are encouraging, and show that the local curvature descent
improves upon gradient descent.
