64 research outputs found
Adaptive and Safe Bayesian Optimization in High Dimensions via One-Dimensional Subspaces
Bayesian optimization is known to be difficult to scale to high dimensions,
because the acquisition step requires solving a non-convex optimization problem
in the same search space. In order to scale the method and keep its benefits,
we propose an algorithm (LineBO) that restricts the problem to a sequence of
iteratively chosen one-dimensional sub-problems that can be solved efficiently.
We show that our algorithm converges globally and obtains a fast local rate
when the function is strongly convex. Further, if the objective has an
invariant subspace, our method automatically adapts to the effective dimension
without changing the algorithm. When combined with the SafeOpt algorithm to
solve the sub-problems, we obtain the first safe Bayesian optimization
algorithm with theoretical guarantees applicable in high-dimensional settings.
We evaluate our method on multiple synthetic benchmarks, where we obtain
competitive performance. Further, we deploy our algorithm to optimize the beam
intensity of the Swiss Free Electron Laser with up to 40 parameters while
satisfying safe operation constraints
Uniform Lipschitz functions on the triangular lattice have logarithmic variations
Uniform integer-valued Lipschitz functions on a domain of size of the
triangular lattice are shown to have variations of order . The
level lines of such functions form a loop model on the edges of the
hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for
the loop O(2) model is constructed as a thermodynamic limit and is shown to be
unique. It contains only finite loops and has properties indicative of
scale-invariance: macroscopic loops appearing at every scale. The existence of
the infinite-volume measure carries over to height functions pinned at the
origin; the uniqueness of the Gibbs measure does not. The proof is based on a
representation of the loop model via a pair of spin configurations that
are shown to satisfy the FKG inequality. We prove RSW-type estimates for a
certain connectivity notion in the aforementioned spin model.Comment: Compared to v1: Theorem 1.3 (uniqueness of the Gibbs measure) is
added; proof of Theorem 3.2 (delocalization) is significantly shortened; more
details added in Section 4 (proof of the dichotomy theorem
Generating random quantum channels
Several techniques of generating random quantum channels, which act on the
set of -dimensional quantum states, are investigated. We present three
approaches to the problem of sampling of quantum channels and show under which
conditions they become mathematically equivalent, and lead to the uniform,
Lebesgue measure on the convex set of quantum operations. We compare their
advantages and computational complexity and demonstrate which of them is
particularly suitable for numerical investigations. Additional results focus on
the spectral gap and other spectral properties of random quantum channels and
their invariant states. We compute mean values of several quantities
characterizing a given quantum channel, including its unitarity, the average
output purity and the -norm coherence of a channel, averaged over the entire
set of the quantum channels with respect to the uniform measure. An ensemble of
classical stochastic matrices obtained due to super-decoherence of random
quantum stochastic maps is analyzed and their spectral properties are studied
using the Bloch representation of a classical probability vector.Comment: 29 pages, 7 figure
Strongly Correlated Random Interacting Processes
The focus of the workshop was to discuss the recent developments and future research directions in the area of large scale random interacting processes, with main emphasis in models where local microscopic interactions either produce strong correlations at macroscopic levels, or generate non-equilibrium dynamics. This report contains extended abstracts of the presentations, which featured research in several directions including selfinteracting random walks, spatially growing processes, strongly dependent percolation, spin systems with long-range order, and random permutations
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