Space-filling designs such as Low Discrepancy Sequence (LDS), Latin Hypercube Sampling (LHS) and Jittered Sampling (JS) were proposed for fully parallel hyperparameter search, and were shown to be more effective than random and grid search. We prove that LHS and JS outperform random search only by a constant factor. Consequently, we introduce a new sampling approach based on the reshaping of the search distribution, and we show both theoretically and numerically that it leads to significant gains over random search. Two methods are proposed for the reshaping: Recentering (when the distribution of the optimum is known), and Cauchy transformation (when the distribution of the optimum is unknown). The proposed methods are first validated on artificial experiments and simple real-world tests on clustering and Salmon mappings. Then we demonstrate that they drive performance improvement in a wide range of expensive artificial intelligence tasks, namely attend/infer/repeat, video next frame segmentation forecasting and progressive generative adversarial networks.

This paper studies the optimization of strategies in the context of possibly randomized two players zero-sum games with incomplete information. We compare 5 algorithms for tuning the parameters of strategies over a benchmark of 12 games. A first evolutionary approach consists in designing a highly randomized opponent (called naive opponent) and optimizing the parametric strategy against it; a second one is optimizing iteratively the strategy, i.e. constructing a sequence of strategies starting from the naive one. 2 versions of coevolutions, real and approximate, are also tested as well as a seed method. The coevolution methods were performing well, but results were not stable from one game to another. In spite of its simplicity, the seed method, which can be seen as an extremal version of coevolution, works even when nothing else works. Incidentally, these methods brought out some unexpected strategies for some games, such as Batawaf or the game of War, which seem, at first view, purely random games without any structured actions possible for the players or Guess Who, where a dichotomy between the characters seems to be the most reasonable strategy. All source codes of games are written in Matlab/Octave and are freely available for download.

Derivative Free Optimization is known to be an efficient and robust method to tackle the black-box optimization problem. When it comes to noisy functions, classical comparison-based algorithms are slower than gradient-based algorithms. For quadratic functions, Evolutionary Algorithms without large mutations have a simple regret at best O(1/√N) when N is the number of function evaluations, whereas stochastic gradient descent can reach (tightly) a simple regret in O(1/N). It has been conjectured that gradient approximation by finite differences (hence, not a comparison-based method) is necessary for reaching such a O(1/N). We answer this conjecture in the negative, providing a comparison-based algorithm as good as gradient methods, i.e. reaching O(1/N) - under the condition, however, that the noise is Gaussian. Experimental results confirm the O(1/N) simple regret, i.e., squared rate compared to many published results at O(1/√N).

The performance measure of an algorithm is a crucial part of its analysis. The performance can be determined by the study on the convergence rate of the algorithm in question. It is necessary to study some (hopefully convergent) sequence that will measure how "good" is the approximated optimum compared to the real optimum. The concept of Regret is widely used in the bandit literature for assessing the performance of an algorithm. The same concept is also used in the framework of optimization algorithms, sometimes under other names or without a specific name. And the numerical evaluation of convergence rate of noisy algorithms often involves approximations of regrets. We discuss here two types of approximations of Simple Regret used in practice for the evaluation of algorithms for noisy optimization. We use specific algorithms of different nature and the noisy sphere function to show the following results. The approximation of Simple Regret, termed here Approximate Simple Regret, used in some optimization testbeds, fails to estimate the Simple Regret convergence rate. We also discuss a recent new approximation of Simple Regret, that we term Robust Simple Regret, and show its advantages and disadvantages.

We study mathematically and experimentally the convergence rate of differential evolution and particle swarm optimization for simple unimodal functions. Due to parallelization concerns, the focus is on lower bounds on the runtime, i.e. upper bounds on the speed-up, as a function of the population size. Two cases are particularly relevant: A population size of the same order of magnitude as the dimension and larger population sizes. We use the branching factor as a tool for proving bounds and get, as upper bounds, a linear speed-up for a population size similar to the dimension, and a logarithmic speed-up for larger population sizes. We then propose parametrizations for differential evolution and particle swarm optimization that reach these bounds.

We consider the problem of optimizing functions corrupted with additive noise. It is known that Evolutionary Algorithms can reach a Simple Regret O(1/√n) within logarithmic factors, when n is the number of function evaluations. Here, Simple Regret at evaluation $n$ is the difference between the evaluation of the function at the current recommendation point of the algorithm and at the real optimum. We show mathematically that this bound is tight, for any family of functions that includes sphere functions, at least for a wide set of Evolution Strategies without large mutations.