Convex analysis in normed spaces and metric projections onto convex bodies

Balestro, Vitor; Martini, Horst; Teixeira, Ralph

doi:10.48550/arxiv.1908.08742

Cited by 1 publication

(1 citation statement)

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“…This partial derivative involves differentiation of a metric projection onto a convex set, which has been studied extensively in the literature of convex analysis [PR70; Zaj84; AT14]. Recently, Balestro et al [BMT19] established that for distance functions given by smooth norms, the derivative of metric projection for any z / ∈ S is given by: ∇ρ(z, S) = ∇ min…”

Section: D3 First-order Optimizationmentioning

confidence: 99%

Preference learning along multiple criteria: A game-theoretic perspective

Bhatia¹,

Pananjady²,

Dragan³

et al. 2021

Preprint

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The literature on ranking from ordinal data is vast, and there are several ways to aggregate overall preferences from pairwise comparisons between objects. In particular, it is well known that any Nash equilibrium of the zero-sum game induced by the preference matrix defines a natural solution concept (winning distribution over objects) known as a von Neumann winner. Many real-world problems, however, are inevitably multi-criteria, with different pairwise preferences governing the different criteria. In this work, we generalize the notion of a von Neumann winner to the multi-criteria setting by taking inspiration from Blackwell's approachability. Our framework allows for non-linear aggregation of preferences across criteria, and generalizes the linearization-based approach from multi-objective optimization.From a theoretical standpoint, we show that the Blackwell winner of a multi-criteria problem instance can be computed as the solution to a convex optimization problem. Furthermore, given random samples of pairwise comparisons, we show that a simple "plug-in" estimator achieves near-optimal minimax sample complexity. Finally, we showcase the practical utility of our framework in a user study on autonomous driving, where we find that the Blackwell winner outperforms the von Neumann winner for the overall preferences.

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Section: D3 First-order Optimizationmentioning

confidence: 99%