42 research outputs found
Sublinear classical and quantum algorithms for general matrix games
We investigate sublinear classical and quantum algorithms for matrix games, a
fundamental problem in optimization and machine learning, with provable
guarantees. Given a matrix , sublinear algorithms
for the matrix game
were previously known only for two special cases: (1) being the
-norm unit ball, and (2) being either the -
or the -norm unit ball. We give a sublinear classical algorithm that
can interpolate smoothly between these two cases: for any fixed ,
we solve the matrix game where is a -norm unit ball
within additive error in time . We
also provide a corresponding sublinear quantum algorithm that solves the same
task in time with a
quadratic improvement in both and . Both our classical and quantum
algorithms are optimal in the dimension parameters and up to
poly-logarithmic factors. Finally, we propose sublinear classical and quantum
algorithms for the approximate Carath\'eodory problem and the -margin
support vector machines as applications.Comment: 16 pages, 2 figures. To appear in the Thirty-Fifth AAAI Conference on
Artificial Intelligence (AAAI 2021
Input/output-to-state stability of switched systems under restricted switching
This paper deals with input/output-to-state stability (IOSS) of
continuous-time switched nonlinear systems. Given a family of systems, possibly
containing unstable dynamics, and a set of restrictions on admissible switches
between the subsystems and admissible dwell times on the subsystems, we
identify a class of switching signals that obeys these restrictions and
preserves stability of the resulting switched system. The primary apparatus for
our analysis is multiple Lyapunov-like functions. Input-to-state stability
(ISS) and global asymptotic stability (GAS) of switched systems under
pre-specified restrictions on switching signals fall as special cases of our
results when no outputs (resp., also inputs) are considered.Comment: 14 pages, no figur
A Consistent Regularization Approach for Structured Prediction
We propose and analyze a regularization approach for structured prediction
problems. We characterize a large class of loss functions that allows to
naturally embed structured outputs in a linear space. We exploit this fact to
design learning algorithms using a surrogate loss approach and regularization
techniques. We prove universal consistency and finite sample bounds
characterizing the generalization properties of the proposed methods.
Experimental results are provided to demonstrate the practical usefulness of
the proposed approach.Comment: 39 pages, 2 Tables, 1 Figur