arXiv subjects

Clement Lezane

Publications and source records attributed to Clement Lezane.

2 recordsLinked to original sources

arXiv2024

Differentially Private Algorithms for Linear Queries via Stochastic Convex Optimization

This article establishes a method to answer a finite set of linear queries on a given dataset while ensuring differential privacy. To achieve this, we formulate the corresponding task as a saddle-point problem, i.e. an optimization problem whose solution corresponds to a distribution minimizing the difference between answers to the linear queries based on the true distribution and answers from a differentially private distribution. Against this background, we establish two new algorithms for corresponding differentially private data release: the first is based on the differentially private Frank-Wolfe method, the second combines randomized smoothing with stochastic convex optimization techniques for a solution to the saddle-point problem. While previous works assess the accuracy of differentially private algorithms with reference to the empirical data distribution, a key contribution of our work is a more natural evaluation of the proposed algorithms' accuracy with reference to the true data-generating distribution.

stat.ME↗

arXiv2024

Accelerated Mirror Descent for Non-Euclidean Star-convex Functions

Acceleration for non-convex functions is a fundamental challenge in optimisation. We revisit star-convex functions, which are strictly unimodal on all lines through a minimizer. [1] accelerate unconstrained star-convex minimization of functions that are smooth with respect to the Euclidean norm. To do so, they add a certain binary search step to gradient descent. In this paper, we accelerate unconstrained star-convex minimization of functions that are weakly smooth with respect to an arbitrary norm. We add a binary search step to mirror descent, generalize the approach and refine its complexity analysis. We prove that our algorithms have sharp convergence rates for star-convex functions with $\alpha$-Holder continuous gradients and demonstrate that our rates are nearly optimal for $p$-norms. [1] Near-Optimal Methods for Minimizing Star-Convex Functions and Beyond, Hinder Oliver and Sidford Aaron and Sohoni Nimit

math.OC↗