20,065 research outputs found
Subsampling Algorithms for Semidefinite Programming
We derive a stochastic gradient algorithm for semidefinite optimization using
randomization techniques. The algorithm uses subsampling to reduce the
computational cost of each iteration and the subsampling ratio explicitly
controls granularity, i.e. the tradeoff between cost per iteration and total
number of iterations. Furthermore, the total computational cost is directly
proportional to the complexity (i.e. rank) of the solution. We study numerical
performance on some large-scale problems arising in statistical learning.Comment: Final version, to appear in Stochastic System
Low-Rank Inducing Norms with Optimality Interpretations
Optimization problems with rank constraints appear in many diverse fields
such as control, machine learning and image analysis. Since the rank constraint
is non-convex, these problems are often approximately solved via convex
relaxations. Nuclear norm regularization is the prevailing convexifying
technique for dealing with these types of problem. This paper introduces a
family of low-rank inducing norms and regularizers which includes the nuclear
norm as a special case. A posteriori guarantees on solving an underlying rank
constrained optimization problem with these convex relaxations are provided. We
evaluate the performance of the low-rank inducing norms on three matrix
completion problems. In all examples, the nuclear norm heuristic is
outperformed by convex relaxations based on other low-rank inducing norms. For
two of the problems there exist low-rank inducing norms that succeed in
recovering the partially unknown matrix, while the nuclear norm fails. These
low-rank inducing norms are shown to be representable as semi-definite
programs. Moreover, these norms have cheaply computable proximal mappings,
which makes it possible to also solve problems of large size using first-order
methods
- …