We study the price of anarchy of first-price auctions in the autobidding world, where bidders can be either utility maximizers (i.e., traditional bidders) or value maximizers (i.e., autobidders).

Expectation Maximization (EM) is among the most popular algorithms for maximum likelihood estimation, but it is generally only guaranteed to find its stationary points of the log-likelihood objective. The goal of this article is to present theoretical and empirical evidence that over-parameterization can help EM avoid spurious local optima in the log-likelihood. We consider the problem of estimating the mean vectors of a Gaussian mixture model in a scenario where the mixing weights are known. Our study shows that the global behavior of EM, when one uses an over-parameterized model in which the mixing weights are treated as unknown, is better than that when one uses the (correct) model with the mixing weights fixed to the known values. For symmetric Gaussians mixtures with two components, we prove that introducing the (statistically redundant) weight parameters enables EM to find the global maximizer of the log-likelihood starting from almost any initial mean parameters, whereas EM without this over-parameterization may very often fail. For other Gaussian mixtures, we provide empirical evidence that shows similar behavior. Our results corroborate the value of over-parameterization in solving non-convex optimization problems, previously observed in other domains.

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

In classic auction theory, reserve prices are known to be effective for improving revenue for the auctioneer against quasi-linear utility maximizing bidders.

Theonlyexception wasthenumber oflatentcomponents used for the Walker-2Dbody-partsdomain, as we found empirically thatk = 5led to saturation of the learning process early on.

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.