69,896 research outputs found
Stochastic Optimization with Importance Sampling
Uniform sampling of training data has been commonly used in traditional
stochastic optimization algorithms such as Proximal Stochastic Gradient Descent
(prox-SGD) and Proximal Stochastic Dual Coordinate Ascent (prox-SDCA). Although
uniform sampling can guarantee that the sampled stochastic quantity is an
unbiased estimate of the corresponding true quantity, the resulting estimator
may have a rather high variance, which negatively affects the convergence of
the underlying optimization procedure. In this paper we study stochastic
optimization with importance sampling, which improves the convergence rate by
reducing the stochastic variance. Specifically, we study prox-SGD (actually,
stochastic mirror descent) with importance sampling and prox-SDCA with
importance sampling. For prox-SGD, instead of adopting uniform sampling
throughout the training process, the proposed algorithm employs importance
sampling to minimize the variance of the stochastic gradient. For prox-SDCA,
the proposed importance sampling scheme aims to achieve higher expected dual
value at each dual coordinate ascent step. We provide extensive theoretical
analysis to show that the convergence rates with the proposed importance
sampling methods can be significantly improved under suitable conditions both
for prox-SGD and for prox-SDCA. Experiments are provided to verify the
theoretical analysis.Comment: 29 page
Sweeping process by prox-regular sets in Riemannian Hilbert manifolds
In this paper, we deal with sweeping processes on (possibly
infinite-dimensional) Riemannian Hilbert manifolds. We extend the useful
notions (proximal normal cone, prox-regularity) already defined in the setting
of a Hilbert space to the framework of such manifolds. Especially we introduce
the concept of local prox-regularity of a closed subset in accordance with the
geometrical features of the ambient manifold and we check that this regularity
implies a property of hypomonotonicity for the proximal normal cone. Moreover
we show that the metric projection onto a locally prox-regular set is
single-valued in its neighborhood. Then under some assumptions, we prove the
well-posedness of perturbed sweeping processes by locally prox-regular sets.Comment: 27 page
Thermal food processing computation software
The objective of this research consisted of developing the two following thermal food processing software: “F-CALC” is software developed to carry out thermal process calculations based on the well-known Ball's formula method, and “OPT-PROx” is software for thermal food processing optimization based on variable retort temperature processing and global optimization technique. Time-temperature data loaded from Excel-file is used by “F-CALC” software to evaluate the heat penetration parameters jh and fh, as well as to compute process lethality for given process time or vice versa. The possibility of computing the process time and lethality for broken heating curves is included. The diversity of thermal food processing optimization problems with different objectives and required constraints are solvable by “OPT-PROx” software. The adaptive random search algorithm coupled with penalty functions approach, and the finite difference method with cubic spline approximation are utilized by “OPT-PROx” for simulation and optimization thermal food processes. The possibility of estimating the thermal diffusivity coefficient based on the mean squared error function minimization is included. The “OPT-PROx” software was successfully tested on the real thermal food processing problems, namely in the case of total process time minimization with a constraint for average and surface retentions the “OPT-PROx” demonstrates significant advantage over the traditional constant temperature processes in terms of process time and final product quality. The developed user friendly dialogue and used numerical procedures make the “F-CALC” and “OPT-PROx” software extremely useful for food scientists (research and education) and engineers (real thermal food process evaluation and optimization)
Proximal Stochastic Newton-type Gradient Descent Methods for Minimizing Regularized Finite Sums
In this work, we generalized and unified recent two completely different
works of Jascha \cite{sohl2014fast} and Lee \cite{lee2012proximal} respectively
into one by proposing the \textbf{prox}imal s\textbf{to}chastic
\textbf{N}ewton-type gradient (PROXTONE) method for optimizing the sums of two
convex functions: one is the average of a huge number of smooth convex
functions, and the other is a non-smooth convex function. While a set of
recently proposed proximal stochastic gradient methods, include MISO,
Prox-SDCA, Prox-SVRG, and SAG, converge at linear rates, the PROXTONE
incorporates second order information to obtain stronger convergence results,
that it achieves a linear convergence rate not only in the value of the
objective function, but also in the \emph{solution}. The proof is simple and
intuitive, and the results and technique can be served as a initiate for the
research on the proximal stochastic methods that employ second order
information.Comment: arXiv admin note: text overlap with arXiv:1309.2388, arXiv:1403.4699
by other author
Solving Variational Inequalities with Monotone Operators on Domains Given by Linear Minimization Oracles
The standard algorithms for solving large-scale convex-concave saddle point
problems, or, more generally, variational inequalities with monotone operators,
are proximal type algorithms which at every iteration need to compute a
prox-mapping, that is, to minimize over problem's domain the sum of a
linear form and the specific convex distance-generating function underlying the
algorithms in question. Relative computational simplicity of prox-mappings,
which is the standard requirement when implementing proximal algorithms,
clearly implies the possibility to equip with a relatively computationally
cheap Linear Minimization Oracle (LMO) able to minimize over linear forms.
There are, however, important situations where a cheap LMO indeed is available,
but where no proximal setup with easy-to-compute prox-mappings is known. This
fact motivates our goal in this paper, which is to develop techniques for
solving variational inequalities with monotone operators on domains given by
Linear Minimization Oracles. The techniques we develope can be viewed as a
substantial extension of the proposed in [5] method of nonsmooth convex
minimization over an LMO-represented domain
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