2 research outputs found
Universal Algorithms: Beyond the Simplex
The bulk of universal algorithms in the online convex optimisation literature
are variants of the Hedge (exponential weights) algorithm on the simplex. While
these algorithms extend to polytope domains by assigning weights to the
vertices, this process is computationally unfeasible for many important classes
of polytopes where the number of vertices depends exponentially on the
dimension . In this paper we show the Subgradient algorithm is universal,
meaning it has regret in the antagonistic setting and
pseudo-regret in the i.i.d setting, with two main advantages over Hedge: (1)
The update step is more efficient as the action vectors have length only
rather than ; and (2) Subgradient gives better performance if the cost
vectors satisfy Euclidean rather than sup-norm bounds. This paper extends the
authors' recent results for Subgradient on the simplex. We also prove the same
and bounds when the domain is the unit ball. To the
authors' knowledge this is the first instance of these bounds on a domain other
than a polytope.Comment: 1 figure, 40 page
Reusing Combinatorial Structure: Faster Iterative Projections over Submodular Base Polytopes
Optimization algorithms such as projected Newton's method, FISTA, mirror
descent and its variants enjoy near-optimal regret bounds and convergence
rates, but suffer from a computational bottleneck of computing "projections''
in potentially each iteration (e.g., regret of online mirror
descent). On the other hand, conditional gradient variants solve a linear
optimization in each iteration, but result in suboptimal rates (e.g.,
regret of online Frank-Wolfe). Motivated by this trade-off in
runtime v/s convergence rates, we consider iterative projections of close-by
points over widely-prevalent submodular base polytopes . We develop a
toolkit to speed up the computation of projections using both discrete and
continuous perspectives. We subsequently adapt the away-step Frank-Wolfe
algorithm to use this information and enable early termination. For the special
case of cardinality based submodular polytopes, we improve the runtime of
computing certain Bregman projections by a factor of . Our
theoretical results show orders of magnitude reduction in runtime in
preliminary computational experiments