Noisy Optimization: Fast Convergence Rates with Comparison-Based Algorithms

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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).