Exploiting Higher Order Smoothness in Derivative-free Optimization and Continuous Bandits

Abstract

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.

Small Errors in Random Zeroth Order Optimization are Imaginary

“…All these results are similar to the rates as in [NS17]. In fact, these rates are sharper than the noise-free rates as provided in [APT20], where higher-order smoothness -although for the perturbed setting -is explicitly taken into account. The key difference with all of the aforementioned works is that we can select our smoothing parameter as being for example of the form δ k = δ/(1 + k) without the fear of numerical problems.…”

Improved Exploiting Higher Order Smoothness in Derivative-free Optimization and Continuous Bandit

“…2. Note that the results of this work have better dependency ε(N ) or N (ε) than Gasnikov's one-point method only if β > 2 else another technique in Theorem 1 is better (see [8] or Theorem 5.1 in [1]). The result in this work is achieved using both kernel smoothing technique and measure concentration inequalities.…”

“…3. The lower bound for strongly convex case is got under conditions γ ≥ N − 1 /2+ 1 /β (otherwise it is better to use convex methods) and (see [1]) 2γ ≤ max x∈Q ∇f (x) .…”

“…, but Akhavan, Pontil and Tsybakov [1] found error in Lemma 2 in [2] where factor d of dimension (n in our notation) is missing.…”

One-Point Gradient-Free Methods for Composite Optimization with Applications to Distributed Optimization

“…Gradient-free methods. Let us highlight the main works devoted to the zeroth-order methods: for two-point feedback [19,17,7,11], for one-point feedback [2,9,1,18].…”