Query Complexity of Derivative-Free Optimization

This paper provides lower bounds on the convergence rate of Derivative Free Optimization (DFO) with noisy function evaluations, exposing a fundamental and unavoidable gap between the performance of algorithms with access to gradients and those with access to only function evaluations. However, there are situations in which DFO is unavoidable, and for such situations we propose a new DFO algorithm that is proved to be near optimal for the class of strongly convex objective functions. A distinctive feature of the algorithm is that it uses only Boolean-valued function comparisons, rather than function evaluations. This makes the algorithm useful in an even wider range of applications, such as optimization based on paired comparisons from human subjects, for example. We also show that regardless of whether DFO is based on noisy function evaluations or Boolean-valued function comparisons, the convergence rate is the same

Highly-Smooth Zero-th Order Online Optimization Vianney Perchet

The minimization of convex functions which are only available through partial and noisy information is a key methodological problem in many disciplines. In this paper we consider convex optimization with noisy zero-th order information, that is noisy function evaluations at any desired point. We focus on problems with high degrees of smoothness, such as logistic regression. We show that as opposed to gradient-based algorithms, high-order smoothness may be used to improve estimation rates, with a precise dependence of our upper-bounds on the degree of smoothness. In particular, we show that for infinitely differentiable functions, we recover the same dependence on sample size as gradient-based algorithms, with an extra dimension-dependent factor. This is done for both convex and strongly-convex functions, with finite horizon and anytime algorithms. Finally, we also recover similar results in the online optimization setting.Comment: Conference on Learning Theory (COLT), Jun 2016, New York, United States. 201

Benchmarking the NEWUOA on the BBOB-2009 Noisy Testbed

International audienceThe NEWUOA which belongs to the class of Derivative-Free optimization algorithms is benchmarked on the BBOB-2009 noisy testbed. A multistart strategy is applied with a maximum number of function evaluations of 10^4 times the search space dimension

BOSH:Bayesian Optimization by Sampling Hierarchically

Deployments of Bayesian Optimization (BO) for functions with stochastic evaluations, such as parameter tuning via cross validation and simulation optimization, typically optimize an average of a fixed set of noisy realizations of the objective function. However, disregarding the true objective function in this manner finds a high-precision optimum of the wrong function. To solve this problem, we propose Bayesian Optimization by Sampling Hierarchically (BOSH), a novel BO routine pairing a hierarchical Gaussian process with an information-theoretic framework to generate a growing pool of realizations as the optimization progresses. We demonstrate that BOSH provides more efficient and higher-precision optimization than standard BO across synthetic benchmarks, simulation optimization, reinforcement learning and hyper-parameter tuning tasks

Stochastic Derivative-Free Optimization of Noisy Functions

Optimization problems with numerical noise arise from the growing use of computer simulation of complex systems. This thesis concerns the development, analysis and applications of randomized derivative-free optimization (DFO) algorithms for noisy functions. The first contribution is the introduction of DFO-VASP, an algorithm for solving the problem of finding the optimal volumetric alignment of protein structures. Our method compensates for noisy, variable-time volume evaluations and warm-starts the search for globally optimal superposition. These techniques enable DFO-VASP to generate practical and accurate superpositions in a timely manner. The second algorithm, STARS, is aimed at solving general noisy optimization problems and employs a random search framework while dynamically adjusting the smoothing step-size using noise information. rate analysis of this algorithm is provided in both additive and multiplicative noise settings. STARS outperforms randomized zero-order methods in both additive and multiplicative settings and has an advantage of being insensitive to the level noise in terms of number of function evaluations and final objective value. The third contribution is a trust-region model-based algorithm STORM, that relies on constructing random models and estimates that are sufficiently accurate with high probability. This algorithm is shown to converge with probability one. Numerical experiments show that STORM outperforms other stochastic DFO methods in solving noisy functions