1,100 research outputs found
Lipschitz gradients for global optimization in a one-point-based partitioning scheme
A global optimization problem is studied where the objective function
is a multidimensional black-box function and its gradient satisfies the
Lipschitz condition over a hyperinterval with an unknown Lipschitz constant
. Different methods for solving this problem by using an a priori given
estimate of , its adaptive estimates, and adaptive estimates of local
Lipschitz constants are known in the literature. Recently, the authors have
proposed a one-dimensional algorithm working with multiple estimates of the
Lipschitz constant for (the existence of such an algorithm was a
challenge for 15 years). In this paper, a new multidimensional geometric method
evolving the ideas of this one-dimensional scheme and using an efficient
one-point-based partitioning strategy is proposed. Numerical experiments
executed on 800 multidimensional test functions demonstrate quite a promising
performance in comparison with popular DIRECT-based methods.Comment: 25 pages, 4 figures, 5 tables. arXiv admin note: text overlap with
arXiv:1103.205
Deterministic global optimization using space-filling curves and multiple estimates of Lipschitz and Holder constants
In this paper, the global optimization problem with
being a hyperinterval in and satisfying the Lipschitz condition
with an unknown Lipschitz constant is considered. It is supposed that the
function can be multiextremal, non-differentiable, and given as a
`black-box'. To attack the problem, a new global optimization algorithm based
on the following two ideas is proposed and studied both theoretically and
numerically. First, the new algorithm uses numerical approximations to
space-filling curves to reduce the original Lipschitz multi-dimensional problem
to a univariate one satisfying the H\"{o}lder condition. Second, the algorithm
at each iteration applies a new geometric technique working with a number of
possible H\"{o}lder constants chosen from a set of values varying from zero to
infinity showing so that ideas introduced in a popular DIRECT method can be
used in the H\"{o}lder global optimization. Convergence conditions of the
resulting deterministic global optimization method are established. Numerical
experiments carried out on several hundreds of test functions show quite a
promising performance of the new algorithm in comparison with its direct
competitors.Comment: 26 pages, 10 figures, 4 table
A simple parameter-free and adaptive approach to optimization under a minimal local smoothness assumption
We study the problem of optimizing a function under a \emph{budgeted number
of evaluations}. We only assume that the function is \emph{locally} smooth
around one of its global optima. The difficulty of optimization is measured in
terms of 1) the amount of \emph{noise} of the function evaluation and 2)
the local smoothness, , of the function. A smaller results in smaller
optimization error. We come with a new, simple, and parameter-free approach.
First, for all values of and , this approach recovers at least the
state-of-the-art regret guarantees. Second, our approach additionally obtains
these results while being \textit{agnostic} to the values of both and .
This leads to the first algorithm that naturally adapts to an \textit{unknown}
range of noise and leads to significant improvements in a moderate and
low-noise regime. Third, our approach also obtains a remarkable improvement
over the state-of-the-art SOO algorithm when the noise is very low which
includes the case of optimization under deterministic feedback (). There,
under our minimal local smoothness assumption, this improvement is of
exponential magnitude and holds for a class of functions that covers the vast
majority of functions that practitioners optimize (). We show that our
algorithmic improvement is borne out in experiments as we empirically show
faster convergence on common benchmarks
An Efficient Global Optimization Algorithm with Adaptive Estimates of the Local Lipschitz Constants
In this work, we present a new deterministic partition-based Global
Optimization (GO) algorithm that uses estimates of the local Lipschitz
constants associated with different sub-regions of the domain of the objective
function. The estimates of the local Lipschitz constants associated with each
partition are the result of adaptively balancing the global and local
information obtained so far from the algorithm, given in terms of absolute
slopes. We motivate a coupling strategy with local optimization algorithms to
accelerate the convergence speed of the proposed approach. In the end, we
compare our approach HALO (Hybrid Adaptive Lipschitzian Optimization) with
respect to popular GO algorithms using hundreds of test functions. From the
numerical results, the performance of HALO is very promising and can extend our
arsenal of efficient procedures for attacking challenging real-world GO
problems. The Python code of HALO is publicly available on GitHub.
https://github.com/dannyzx/HAL
Application of reduced-set pareto-lipschitzian optimization to truss optimization
In this paper, a recently proposed global Lipschitz optimization algorithm Pareto-Lipschitzian Optimization with Reduced-set (PLOR) is further developed, investigated and applied to truss optimization problems. Partition patterns of the PLOR algorithm are similar to those of DIviding RECTangles (DIRECT), which was widely applied to different real-life problems. However here a set of all Lipschitz constants is reduced to just two: the maximal and the minimal ones. In such a way the PLOR approach is independent of any user-defined parameters and balances equally local and global search during the optimization process. An expanded list of other well-known DIRECT-type algorithms is used in investigation and experimental comparison using the standard test problems and truss optimization problems. The experimental investigation shows that the PLOR algorithm gives very competitive results to other DIRECT-type algorithms using standard test problems and performs pretty well on real truss optimization problems
Learning to Approximate a Bregman Divergence
Bregman divergences generalize measures such as the squared Euclidean
distance and the KL divergence, and arise throughout many areas of machine
learning. In this paper, we focus on the problem of approximating an arbitrary
Bregman divergence from supervision, and we provide a well-principled approach
to analyzing such approximations. We develop a formulation and algorithm for
learning arbitrary Bregman divergences based on approximating their underlying
convex generating function via a piecewise linear function. We provide
theoretical approximation bounds using our parameterization and show that the
generalization error for metric learning using our framework
matches the known generalization error in the strictly less general Mahalanobis
metric learning setting. We further demonstrate empirically that our method
performs well in comparison to existing metric learning methods, particularly
for clustering and ranking problems.Comment: 19 pages, 4 figure
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