  Applications abound in which optimization problems must be repeatedly solved,
each time with new (but similar) data. Analytic optimization algorithms can be
hand-designed to provably solve these problems in an iterative fashion. On one
hand, data-driven algorithms can "learn to optimize" (L2O) with much fewer
iterations and similar cost per iteration as general-purpose optimization
algorithms. On the other hand, unfortunately, many L2O algorithms lack converge
guarantees. To fuse the advantages of these approaches, we present a Safe-L2O
framework. Safe-L2O updates incorporate a safeguard to guarantee convergence
for convex problems with proximal and/or gradient oracles. The safeguard is
simple and computationally cheap to implement, and it is activated only when
the data-driven L2O updates would perform poorly or appear to diverge. This
yields the numerical benefits of employing machine learning to create rapid L2O
algorithms while still guaranteeing convergence. Our numerical examples show
convergence of Safe-L2O algorithms, even when the provided data is not from the
distribution of training data.
