Kwassi Joseph Dzahini

I arrived at Polytechnique Montréal for a Ph.D. in Mathematics in January 2017 after my graduate studies at Université Lille 1. My Ph.D research focuses on stochastique blackbox optimization. I'm expected to defend my Ph.D. by the end of the summer 2020.

Supervisor: Sébastien Le Digabel, from Polytechnique Montréal

Co-supervisor: Michael Kokkolaras, from McGill university

Contact: liljoseph92@yahoo.com

This work introduces StoMADS, a stochastic variant of the mesh adaptive direct-search (MADS) algorithm originally developed for deterministic blackbox optimization. StoMADS considers the unconstrained optimization of an objective function f whose values can be computed only through a blackbox corrupted by some random noise following an unknown distribution. The proposed method is based on an algorithmic framework similar to that of MADS and uses random estimates of function values obtained from stochastic observations since the exact deterministic computable version of f is not available. Such estimates are required to be accurate with a sufficiently large but fixed probability and satisfy a variance condition. The ability of the proposed algorithm to generate an asymptotically dense set of search directions is then exploited to show convergence to a Clarke stationary point of f with probability one, using martingale theory.

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This work presents the convergence rate analysis of stochastic variants of the broad class of direct-search methods of directional type. It introduces an algorithm designed to optimize differentiable objective functions f whose values can only be computed through a stochastically noisy blackbox. The proposed stochastic directional direct-search (SDDS) algorithm accepts new iterates by imposing a sufficient decrease condition on so called probabilistic estimates of the corresponding unavailable objective function values. The accuracy of such estimates is required to hold with a sufficiently large but fixed probability β. The analysis of this method utilizes an existing supermartingale-based framework proposed for the convergence rates analysis of stochastic optimization methods that use adaptive step sizes. It aims to show that the expected number of iterations required to drive the norm of the gradient of f below a given threshold ϵ is bounded in O(ϵ^(−p/min(p−1,1))/(2β−1)) with p>1. Unlike prior analysis using the same aforementioned framework such as those of stochastic trust-region methods and stochastic line search methods, SDDS does not use any gradient information to find descent directions. However, its convergence rate is similar to those of both latter methods with a dependence on ϵ that also matches that of the broad class of deterministic directional direct-search methods which accept new iterates by imposing a sufficient decrease condition.

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Current project: the paper will be available soon.