Neural Information Processing Systems http://nips.cc/

However, an emerging issue in allocation problems is data privacy.

In this paper, we first propose a flexible framework for safe screening based on the Fenchel-Rockafellar duality and then deriveastrong safe screening rule for norm-regularized least squares using the proposed framework.

We then follow the proof of Theorem 3 in Farnia and Tse [2016]. Our formulation differs from Nowak-Vila et al. [2020] in the fact that we allow probabilistic prediction to be ground truth. Proposition 4. Let G be a multi-graph. We follow the proof of Friesen [2019] for simple graphs. Proposition 5. Let G be a multi-graph.

Multimarginal optimal transport (MOT) has gained increasing attention in recent years, notably due to its relevance in machine learning and statistics, where one seeks to jointly compare and align multiple probability distributions. This paper presents a unified and complete Kantorovich duality theory for MOT problem on general Polish product spaces with bounded continuous cost function. For marginal compact spaces, the duality identity is derived through a convex-analytic reformulation, that identifies the dual problem as a Fenchel-Rockafellar conjugate. We obtain dual attainment and show that optimal potentials may always be chosen in the class of $c$-conjugate families, thereby extending classical two-marginal conjugacy principle into a genuinely multimarginal setting. In non-compact setting, where direct compactness arguments are unavailable, we recover duality via a truncation-tightness procedure based on weak compactness of multimarginal transference plans and boundedness of the cost. We prove that the dual value is preserved under restriction to compact subsets and that admissible dual families can be regularized into uniformly bounded $c$-conjugate potentials. The argument relies on a refined use of $c$-splitting sets and their equivalence with multimarginal $c$-cyclical monotonicity. We then obtain dual attainment and exact primal-dual equality for MOT on arbitrary Polish spaces, together with a canonical representation of optimal dual potentials by $c$-conjugacy. These results provide a structural foundation for further developments in probabilistic and statistical analysis of MOT, including stability, differentiability, and asymptotic theory under marginal perturbations.