Julien Chhor, Suzanne Sigalla, Alexandre B. Tsybakov
In the nonparametric regression setting, we construct an estimator which is a continuous function interpolating the data points with high probability, while attaining minimax optimal rates under mean squared risk on the scale of Hölder classes adaptively to the unknown smoothness.
Arya Akhavan, Alexandre B. Tsybakov
We address the problem of zero-order optimization from noisy observations for an objective function satisfying the Polyak-Łojasiewicz or the strong convexity condition. Additionally, we assume that the objective function has an additive structure and satisfies a higher-order smoothness property, characterized by the Hölder family of functions. The additive model for Hölder classes of functions is well-studied in the literature on nonparametric function estimation, where it is shown that such a model benefits from a substantial improvement of the estimation accuracy compared to the Hölder model without additive structure. We study this established framework in the context of gradient-free optimization. We propose a randomized gradient estimator that, when plugged into a gradient descent algorithm, allows one to achieve minimax optimal optimization error of the order $dT^{-(β-1)/β}$, where $d$ is the dimension of the problem, $T$ is the number of queries and $β\ge 2$ is the Hölder degree of smoothness. We conclude that, in contrast to nonparametric estimation problems, no substantial gain of accuracy can be achieved when using additive models in gradient-free optimization.
Arya Akhavan, Karim Lounici, Massimiliano Pontil et al.
We study the contextual continuum bandits problem, where the learner sequentially receives a side information vector and has to choose an action in a convex set, minimizing a function associated with the context. The goal is to minimize all the underlying functions for the received contexts, leading to the contextual notion of regret, which is stronger than the standard static regret. Assuming that the objective functions are $γ$-Hölder with respect to the contexts, $0<γ\le 1,$ we demonstrate that any algorithm achieving a sub-linear static regret can be extended to achieve a sub-linear contextual regret. We prove a static-to-contextual regret conversion theorem that provides an upper bound for the contextual regret of the output algorithm as a function of the static regret of the input algorithm. We further study the implications of this general result for three fundamental cases of dependency of the objective function on the action variable: (a) Lipschitz bandits, (b) convex bandits, (c) strongly convex and smooth bandits. For Lipschitz bandits and $γ=1,$ combining our results with the lower bound of Slivkins (2014), we prove that the minimax optimal contextual regret for the noise-free adversarial setting is achieved. Then, we prove that in the presence of noise, the contextual regret rate as a function of the number of queries is the same for convex bandits as it is for strongly convex and smooth bandits. Lastly, we present a minimax lower bound, implying two key facts. First, obtaining a sub-linear contextual regret may be impossible over functions that are not continuous with respect to the context. Second, for convex bandits and strongly convex and smooth bandits, the algorithms that we propose achieve, up to a logarithmic factor, the minimax optimal rate of contextual regret as a function of the number of queries.
Arya Akhavan, Massimiliano Pontil, Alexandre B. Tsybakov
We study the problem of zero-order optimization of a strongly convex function. The goal is to find the minimizer of the function by a sequential exploration of its values, under measurement noise. We study the impact of higher order smoothness properties of the function on the optimization error and on the cumulative regret. To solve this problem we consider a randomized approximation of the projected gradient descent algorithm. The gradient is estimated by a randomized procedure involving two function evaluations and a smoothing kernel. We derive upper bounds for this algorithm both in the constrained and unconstrained settings and prove minimax lower bounds for any sequential search method. Our results imply that the zero-order algorithm is nearly optimal in terms of sample complexity and the problem parameters. Based on this algorithm, we also propose an estimator of the minimum value of the function achieving almost sharp oracle behavior. We compare our results with the state-of-the-art, highlighting a number of key improvements.
Mikhail Belkin, Alexander Rakhlin, Alexandre B. Tsybakov
We show that learning methods interpolating the training data can achieve optimal rates for the problems of nonparametric regression and prediction with square loss.
Alexandre Belloni, Mathieu Rosenbaum, Alexandre B. Tsybakov
Several new estimation methods have been recently proposed for the linear regression model with observation error in the design. Different assumptions on the data generating process have motivated different estimators and analysis. In particular, the literature considered (1) observation errors in the design uniformly bounded by some $\bar δ$, and (2) zero mean independent observation errors. Under the first assumption, the rates of convergence of the proposed estimators depend explicitly on $\bar δ$, while the second assumption has been applied when an estimator for the second moment of the observational error is available. This work proposes and studies two new estimators which, compared to other procedures for regression models with errors in the design, exploit an additional $l_{\infty}$-norm regularization. The first estimator is applicable when both (1) and (2) hold but does not require an estimator for the second moment of the observational error. The second estimator is applicable under (2) and requires an estimator for the second moment of the observation error. Importantly, we impose no assumption on the accuracy of this pilot estimator, in contrast to the previously known procedures. As the recent proposals, we allow the number of covariates to be much larger than the sample size. We establish the rates of convergence of the estimators and compare them with the bounds obtained for related estimators in the literature. These comparisons show interesting insights on the interplay of the assumptions and the achievable rates of convergence.
Alexander Rakhlin, Karthik Sridharan, Alexandre B. Tsybakov
We consider the random design regression model with square loss. We propose a method that aggregates empirical minimizers (ERM) over appropriately chosen random subsets and reduces to ERM in the extreme case, and we establish sharp oracle inequalities for its risk. We show that, under the $\varepsilon^{-p}$ growth of the empirical $\varepsilon$-entropy, the excess risk of the proposed method attains the rate $n^{-2/(2+p)}$ for $p\in(0,2)$ and $n^{-1/p}$ for $p>2$ where $n$ is the sample size. Furthermore, for $p\in(0,2)$, the excess risk rate matches the behavior of the minimax risk of function estimation in regression problems under the well-specified model. This yields a conclusion that the rates of statistical estimation in well-specified models (minimax risk) and in misspecified models (minimax regret) are equivalent in the regime $p\in(0,2)$. In other words, for $p\in(0,2)$ the problem of statistical learning enjoys the same minimax rate as the problem of statistical estimation. On the contrary, for $p>2$ we show that the rates of the minimax regret are, in general, slower than for the minimax risk. Our oracle inequalities also imply the $v\log(n/v)/n$ rates for Vapnik-Chervonenkis type classes of dimension $v$ without the usual convexity assumption on the class; we show that these rates are optimal. Finally, for a slightly modified method, we derive a bound on the excess risk of $s$-sparse convex aggregation improving that of Lounici [Math. Methods Statist. 16 (2007) 246-259] and providing the optimal rate.