Stochastic Tangential Pareto Dynamics Provably Samples the Whole Pareto Set
Résumé
The framework of stochastic multi-objective programming allows for the inclusion of uncertainties in multi-objective optimization problems at the cost of transforming the set of objectives into a set of expectations of random quantities. The stochastic multigradient descent algorithm (SMGDA) gives a solution to these types of problems using only noisy gradient information. However, a bias in the algorithm causes it to converge to only a subset of the whole Pareto front, limiting its use. We analyze the source of this bias and prove the convergence of SMGDA to a stationary point in the nonconvex L-lipschitz smooth case. First, based on this analysis, we propose to reduce the bias of the stochastic multi-gradient calculation using an exponential smoothing technique. We then propose a novel approach to exploring the whole Pareto set by combining the debiased stochastic multigradient with an additive non-vanishing noise that guides the dynamics of the iterates tangential to the Pareto set. We finish by proving that our algorithm, Stochastic Tangential Pareto Dynamics (STPD), generates samples concentrated on the whole Pareto set.
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