1,855 research outputs found
Revisiting Projection-Free Optimization for Strongly Convex Constraint Sets
We revisit the Frank-Wolfe (FW) optimization under strongly convex constraint
sets. We provide a faster convergence rate for FW without line search, showing
that a previously overlooked variant of FW is indeed faster than the standard
variant. With line search, we show that FW can converge to the global optimum,
even for smooth functions that are not convex, but are quasi-convex and
locally-Lipschitz. We also show that, for the general case of (smooth)
non-convex functions, FW with line search converges with high probability to a
stationary point at a rate of , as long as the
constraint set is strongly convex -- one of the fastest convergence rates in
non-convex optimization.Comment: Extended version of paper accepted at AAAI-19, 19 pages, 10 figure
Convex optimization over intersection of simple sets: improved convergence rate guarantees via an exact penalty approach
We consider the problem of minimizing a convex function over the intersection
of finitely many simple sets which are easy to project onto. This is an
important problem arising in various domains such as machine learning. The main
difficulty lies in finding the projection of a point in the intersection of
many sets. Existing approaches yield an infeasible point with an
iteration-complexity of for nonsmooth problems with no
guarantees on the in-feasibility. By reformulating the problem through exact
penalty functions, we derive first-order algorithms which not only guarantees
that the distance to the intersection is small but also improve the complexity
to and for smooth functions. For
composite and smooth problems, this is achieved through a saddle-point
reformulation where the proximal operators required by the primal-dual
algorithms can be computed in closed form. We illustrate the benefits of our
approach on a graph transduction problem and on graph matching
Riemannian Optimization via Frank-Wolfe Methods
We study projection-free methods for constrained Riemannian optimization. In
particular, we propose the Riemannian Frank-Wolfe (RFW) method. We analyze
non-asymptotic convergence rates of RFW to an optimum for (geodesically) convex
problems, and to a critical point for nonconvex objectives. We also present a
practical setting under which RFW can attain a linear convergence rate. As a
concrete example, we specialize Rfw to the manifold of positive definite
matrices and apply it to two tasks: (i) computing the matrix geometric mean
(Riemannian centroid); and (ii) computing the Bures-Wasserstein barycenter.
Both tasks involve geodesically convex interval constraints, for which we show
that the Riemannian "linear oracle" required by RFW admits a closed-form
solution; this result may be of independent interest. We further specialize RFW
to the special orthogonal group and show that here too, the Riemannian "linear
oracle" can be solved in closed form. Here, we describe an application to the
synchronization of data matrices (Procrustes problem). We complement our
theoretical results with an empirical comparison of Rfw against
state-of-the-art Riemannian optimization methods and observe that RFW performs
competitively on the task of computing Riemannian centroids.Comment: Under Review. Largely revised version, including an extended
experimental section and an application to the special orthogonal group and
the Procrustes proble
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
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