We consider the problem of noisy gradient-free minimization of the k-th order partial derivative of a $\beta$-H{\"o}lder function supported on a d-dimensional cube. We show that T ^{($\beta$+d+k)/(2$\beta$+d)} log(T )^{(\beta-k)/(2\beta+d)} is a non-asymptotic minimax rate of the T step cumulative regret for all $\beta$ \ge 0. In the special case k = 0, our results cover the problem of noisy gradient-free minimization of $\beta$-H{\"o}lder functions, closing the existing gap between the known upper and lower bounds. We show that a minimizer of a suitably chosen local polynomial estimator is rate-optimal. The minimax optimal upper bound is achieved under the passive design, that is, when the query points are i.i.d. Thus, there is no advantage in considering sequential designs when it is only known that f is a $\beta$-H{\"o}lder function with no additional property. We propose an algorithm feasible in polynomial time that constructs a proxy of the minimizer of the local polynomial estimator. The procedure requires computing the estimator on auxiliary random points. The resulting polynomial time algorithm matches the lower bound.
Théo Paquier, A. Tsybakov, F. Portier et al.· 0 citations
We study pointwise convergence rates of purely random partition estimators in nonparametric regression, where the partition -- into hyper-rectangles by purely random trees, or into Voronoi cells by prototype rules -- is built independently of the responses. Our analysis rests on a single geometric criterion, shape regularity, relating the diameter of a cell to its volume, which is shown by Bettinger, Portier and Saumard (2026) to be necessary and sufficient, up to logarithmic factors, for achieving the minimax rate $n^{-1/(d+2)}$. We show that centered and uniform trees are not shape-regular -- their cells'aspect ratio grows exponentially with the number of splits with probability bounded away from zero -- explaining the super-logarithmic corrections in their error bounds, whereas Mondrian trees, whose splits adapt to the current cell geometry, are shape-regular in probability and attain the minimax rate. The same analysis applied to Voronoi partitions yields the first pointwise concentration bounds for Proto-NN, resolving an open problem of Gy\"orfi and Weiss (2021), and shows that OptiNet achieves the minimax rate with markedly better success probability -- even almost surely, for a suitable choice of parameters -- thanks to its $\eta$-net construction.
J'er'emy Bettinger, F. Portier, Adrien Saumard· 0 citations
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