12,380 research outputs found
Escaping the Local Minima via Simulated Annealing: Optimization of Approximately Convex Functions
We consider the problem of optimizing an approximately convex function over a
bounded convex set in using only function evaluations. The
problem is reduced to sampling from an \emph{approximately} log-concave
distribution using the Hit-and-Run method, which is shown to have the same
complexity as sampling from log-concave distributions. In
addition to extend the analysis for log-concave distributions to approximate
log-concave distributions, the implementation of the 1-dimensional sampler of
the Hit-and-Run walk requires new methods and analysis. The algorithm then is
based on simulated annealing which does not relies on first order conditions
which makes it essentially immune to local minima.
We then apply the method to different motivating problems. In the context of
zeroth order stochastic convex optimization, the proposed method produces an
-minimizer after noisy function
evaluations by inducing a -approximately log concave
distribution. We also consider in detail the case when the "amount of
non-convexity" decays towards the optimum of the function. Other applications
of the method discussed in this work include private computation of empirical
risk minimizers, two-stage stochastic programming, and approximate dynamic
programming for online learning.Comment: 27 page
Second-Order Kernel Online Convex Optimization with Adaptive Sketching
Kernel online convex optimization (KOCO) is a framework combining the
expressiveness of non-parametric kernel models with the regret guarantees of
online learning. First-order KOCO methods such as functional gradient descent
require only time and space per iteration, and, when the only
information on the losses is their convexity, achieve a minimax optimal
regret. Nonetheless, many common losses in kernel
problems, such as squared loss, logistic loss, and squared hinge loss posses
stronger curvature that can be exploited. In this case, second-order KOCO
methods achieve regret, which
we show scales as , where
is the effective dimension of the problem and is usually much smaller than
. The main drawback of second-order methods is their
much higher space and time complexity. In this paper, we
introduce kernel online Newton step (KONS), a new second-order KOCO method that
also achieves regret. To address the
computational complexity of second-order methods, we introduce a new matrix
sketching algorithm for the kernel matrix , and show that for
a chosen parameter our Sketched-KONS reduces the space and time
complexity by a factor of to space and
time per iteration, while incurring only times more regret
Parameter estimation in softmax decision-making models with linear objective functions
With an eye towards human-centered automation, we contribute to the
development of a systematic means to infer features of human decision-making
from behavioral data. Motivated by the common use of softmax selection in
models of human decision-making, we study the maximum likelihood parameter
estimation problem for softmax decision-making models with linear objective
functions. We present conditions under which the likelihood function is convex.
These allow us to provide sufficient conditions for convergence of the
resulting maximum likelihood estimator and to construct its asymptotic
distribution. In the case of models with nonlinear objective functions, we show
how the estimator can be applied by linearizing about a nominal parameter
value. We apply the estimator to fit the stochastic UCL (Upper Credible Limit)
model of human decision-making to human subject data. We show statistically
significant differences in behavior across related, but distinct, tasks.Comment: In pres
Implicit Langevin Algorithms for Sampling From Log-concave Densities
For sampling from a log-concave density, we study implicit integrators
resulting from -method discretization of the overdamped Langevin
diffusion stochastic differential equation. Theoretical and algorithmic
properties of the resulting sampling methods for and a
range of step sizes are established. Our results generalize and extend prior
works in several directions. In particular, for , we prove
geometric ergodicity and stability of the resulting methods for all step sizes.
We show that obtaining subsequent samples amounts to solving a strongly-convex
optimization problem, which is readily achievable using one of numerous
existing methods. Numerical examples supporting our theoretical analysis are
also presented
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