163 research outputs found
Fast MCMC sampling algorithms on polytopes
We propose and analyze two new MCMC sampling algorithms, the Vaidya walk and
the John walk, for generating samples from the uniform distribution over a
polytope. Both random walks are sampling algorithms derived from interior point
methods. The former is based on volumetric-logarithmic barrier introduced by
Vaidya whereas the latter uses John's ellipsoids. We show that the Vaidya walk
mixes in significantly fewer steps than the logarithmic-barrier based Dikin
walk studied in past work. For a polytope in defined by
linear constraints, we show that the mixing time from a warm start is bounded
as , compared to the mixing time
bound for the Dikin walk. The cost of each step of the Vaidya walk is of the
same order as the Dikin walk, and at most twice as large in terms of constant
pre-factors. For the John walk, we prove an
bound on its mixing time and conjecture
that an improved variant of it could achieve a mixing time of
. Additionally, we propose variants
of the Vaidya and John walks that mix in polynomial time from a deterministic
starting point. The speed-up of the Vaidya walk over the Dikin walk are
illustrated in numerical examples.Comment: 86 pages, 9 figures, First two authors contributed equall
Efficiently Sampling the PSD Cone with the Metric Dikin Walk
Semi-definite programs represent a frontier of efficient computation. While
there has been much progress on semi-definite optimization, with moderate-sized
instances currently solvable in practice by the interior-point method, the
basic problem of sampling semi-definite solutions remains a formidable
challenge. The direct application of known polynomial-time algorithms for
sampling general convex bodies to semi-definite sampling leads to a
prohibitively high running time. In addition, known general methods require an
expensive rounding phase as pre-processing. Here we analyze the Dikin walk, by
first adapting it to general metrics, then devising suitable metrics for the
PSD cone with affine constraints. The resulting mixing time and per-step
complexity are considerably smaller, and by an appropriate choice of the
metric, the dependence on the number of constraints can be made
polylogarithmic. We introduce a refined notion of self-concordant matrix
functions and give rules for combining different metrics. Along the way, we
further develop the theory of interior-point methods for sampling.Comment: 54 page
Randomized Control in Performance Analysis and Empirical Asset Pricing
The present article explores the application of randomized control techniques
in empirical asset pricing and performance evaluation. It introduces geometric
random walks, a class of Markov chain Monte Carlo methods, to construct
flexible control groups in the form of random portfolios adhering to investor
constraints. The sampling-based methods enable an exploration of the
relationship between academically studied factor premia and performance in a
practical setting. In an empirical application, the study assesses the
potential to capture premias associated with size, value, quality, and momentum
within a strongly constrained setup, exemplified by the investor guidelines of
the MSCI Diversified Multifactor index. Additionally, the article highlights
issues with the more traditional use case of random portfolios for drawing
inferences in performance evaluation, showcasing challenges related to the
intricacies of high-dimensional geometry.Comment: 57 pages, 7 figures, 2 table
Faster Convex Optimization: Simulated Annealing with an Efficient Universal Barrier
This paper explores a surprising equivalence between two seemingly-distinct
convex optimization methods. We show that simulated annealing, a well-studied
random walk algorithms, is directly equivalent, in a certain sense, to the
central path interior point algorithm for the the entropic universal barrier
function. This connection exhibits several benefits. First, we are able improve
the state of the art time complexity for convex optimization under the
membership oracle model. We improve the analysis of the randomized algorithm of
Kalai and Vempala by utilizing tools developed by Nesterov and Nemirovskii that
underly the central path following interior point algorithm. We are able to
tighten the temperature schedule for simulated annealing which gives an
improved running time, reducing by square root of the dimension in certain
instances. Second, we get an efficient randomized interior point method with an
efficiently computable universal barrier for any convex set described by a
membership oracle. Previously, efficiently computable barriers were known only
for particular convex sets
Dynamic Neuromechanical Sets for Locomotion
Most biological systems employ multiple redundant actuators, which is a complicated problem of controls and analysis. Unless assumptions about how the brain and body work together, and assumptions about how the body prioritizes tasks are applied, it is not possible to find the actuator controls. The purpose of this research is to develop computational tools for the analysis of arbitrary musculoskeletal models that employ redundant actuators. Instead of relying primarily on optimization frameworks and numerical methods or task prioritization schemes used typically in biomechanics to find a singular solution for actuator controls, tools for feasible sets analysis are instead developed to find the bounds of possible actuator controls. Previously in the literature, feasible sets analysis has been used in order analyze models assuming static poses. Here, tools that explore the feasible sets of actuator controls over the course of a dynamic task are developed. The cost-function agnostic methods of analysis developed in this work run parallel and in concert with other methods of analysis such as principle components analysis, muscle synergies theory and task prioritization. Researchers and healthcare professionals can gain greater insights into decision making during behavioral tasks by layering these other tools on top of feasible sets analysis
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