328,563 research outputs found
Global Optimization of Gaussian processes
Gaussian processes~(Kriging) are interpolating data-driven models that are
frequently applied in various disciplines. Often, Gaussian processes are
trained on datasets and are subsequently embedded as surrogate models in
optimization problems. These optimization problems are nonconvex and global
optimization is desired. However, previous literature observed computational
burdens limiting deterministic global optimization to Gaussian processes
trained on few data points. We propose a reduced-space formulation for
deterministic global optimization with trained Gaussian processes embedded. For
optimization, the branch-and-bound solver branches only on the degrees of
freedom and McCormick relaxations are propagated through explicit Gaussian
process models. The approach also leads to significantly smaller and
computationally cheaper subproblems for lower and upper bounding. To further
accelerate convergence, we derive envelopes of common covariance functions for
GPs and tight relaxations of acquisition functions used in Bayesian
optimization including expected improvement, probability of improvement, and
lower confidence bound. In total, we reduce computational time by orders of
magnitude compared to state-of-the-art methods, thus overcoming previous
computational burdens. We demonstrate the performance and scaling of the
proposed method and apply it to Bayesian optimization with global optimization
of the acquisition function and chance-constrained programming. The Gaussian
process models, acquisition functions, and training scripts are available
open-source within the "MeLOn - Machine Learning Models for Optimization"
toolbox~(https://git.rwth-aachen.de/avt.svt/public/MeLOn)
Optimization of Stochastic Discrete Event Simulation Models
Many systems in logistics can be adequately modeled using stochastic discrete
event simulation models. Often these models are used to find a good or optimal
configuration of the system. This implies that optimization algorithms have to
be coupled with the models. Optimization of stochastic
simulation models is a challenging research topic since the approaches should
be efficient, reliable and should provide some guarantee to find at least in
the limiting case with a runtime going to infinite the optimal solution with a
probability converging to 1.
The talk gives an overview on the state of the art in simulation
optimization. It shows that hybrid algorithms combining global and local
optimization methods are currently the best class of optimization approaches
in the area and it outlines the need for the development of software tools
including available algorithms
Fast global convergence of gradient methods for high-dimensional statistical recovery
Many statistical -estimators are based on convex optimization problems
formed by the combination of a data-dependent loss function with a norm-based
regularizer. We analyze the convergence rates of projected gradient and
composite gradient methods for solving such problems, working within a
high-dimensional framework that allows the data dimension \pdim to grow with
(and possibly exceed) the sample size \numobs. This high-dimensional
structure precludes the usual global assumptions---namely, strong convexity and
smoothness conditions---that underlie much of classical optimization analysis.
We define appropriately restricted versions of these conditions, and show that
they are satisfied with high probability for various statistical models. Under
these conditions, our theory guarantees that projected gradient descent has a
globally geometric rate of convergence up to the \emph{statistical precision}
of the model, meaning the typical distance between the true unknown parameter
and an optimal solution . This result is substantially
sharper than previous convergence results, which yielded sublinear convergence,
or linear convergence only up to the noise level. Our analysis applies to a
wide range of -estimators and statistical models, including sparse linear
regression using Lasso (-regularized regression); group Lasso for block
sparsity; log-linear models with regularization; low-rank matrix recovery using
nuclear norm regularization; and matrix decomposition. Overall, our analysis
reveals interesting connections between statistical precision and computational
efficiency in high-dimensional estimation
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