Using the calculus of variations, we prove the following structure theorem for noise stable partitions: a partition of $n$-dimensional Euclidean space into $m$ disjoint sets of fixed Gaussian volumes that maximize their noise stability must be $(m-1)$-dimensional, if $m-1\leq n$. In particular, the maximum noise stability of a partition of $m$ sets in $\mathbb{R}^{n}$ of fixed Gaussian volumes is constant for all $n$ satisfying $n\geq m-1$. From this result, we obtain: (i) A proof of the Plurality is Stablest Conjecture for $3$ candidate elections, for all correlation parameters $\rho$ satisfying $0<\rho<\rho_{0}$, where $\rho_{0}>0$ is a fixed constant (that does not depend on the dimension $n$), when each candidate has an equal chance of winning. (ii) A variational proof of Borell's Inequality (corresponding to the case $m=2$). The structure theorem answers a question of De-Mossel-Neeman and of Ghazi-Kamath-Raghavendra. Item (i) is the first proof of any case of the Plurality is Stablest Conjecture of Khot-Kindler-Mossel-O'Donnell (2005) for fixed $\rho$, with the case $\rho\to1^{-}$ being solved recently. Item (i) is also the first evidence for the optimality of the Frieze-Jerrum semidefinite program for solving MAX-3-CUT, assuming the Unique Games Conjecture. Without the assumption that each candidate has an equal chance of winning in (i), the Plurality is Stablest Conjecture is known to be false.
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We revisit the problem of Stone duality for lattices with various quasioperators, first studied in [14], presenting a fresh duality result. The new result is an improvement over that of [14] in two important respects. First, the axiomatization of frames in [14] was rather cumbersome and it is now simplified, partly by incorporating Gehrke's proposal [8] of section stability for relations. Second, morphisms are redefined so as to preserve Galois stable (and co-stable) sets and we rely for this, partly again, on Goldblatt's [11] recently proposed definition of bounded morphisms for polarities, though we need to strengthen the definition in order to get a Stone duality result. In studying the dual algebraic structures associated to polarities with relations we demonstrate that stable/co-stable set operators result as the Galois closure of the restriction of classical (though sorted) image operators generated by the frame relations to Galois stable/co-stable sets. This provides a proof, at the representation level, that non-distributive logics can be viewed as fragments of sorted, residuated (poly)modal logics, a research direction initiated in [16,17].
We study housing markets as introduced by Shapley and Scarf (1974). We investigate the computational complexity of various questions regarding the situation of an agent $a$ in a housing market $H$: we show that it is $\mathsf{NP}$-hard to find an allocation in the core of $H$ where (i) $a$ receives a certain house, (ii) $a$ does not receive a certain house, or (iii) $a$ receives a house other than her own. We prove that the core of housing markets respects improvement in the following sense: given an allocation in the core of $H$ where agent $a$ receives a house $h$, if the value of the house owned by $a$ increases, then the resulting housing market admits an allocation where $a$ receives either $h$, or a house that she prefers to $h$; moreover, such an allocation can be found efficiently. We further show an analogous result in the Stable Roommates setting by proving that stable matchings in a one-sided market also respect improvement.
Phase-type (PH) distributions are a popular tool for the analysis of univariate risks in numerous actuarial applications. Their multivariate counterparts (MPH$^\ast$), however, have not seen such a proliferation, due to lack of explicit formulas and complicated estimation procedures. A simple construction of multivariate phase-type distributions -- mPH -- is proposed for the parametric description of multivariate risks, leading to models of considerable probabilistic flexibility and statistical tractability. The main idea is to start different Markov processes at the same state, and allow them to evolve independently thereafter, leading to dependent absorption times. By dimension augmentation arguments, this construction can be cast into the umbrella of MPH$^\ast$ class, but enjoys explicit formulas which the general specification lacks, including common measures of dependence. Moreover, it is shown that the class is still rich enough to be dense on the set of multivariate risks supported on the positive orthant, and it is the smallest known sub-class to have this property. In particular, the latter result provides a new short proof of the denseness of the MPH$^\ast$ class. In practice this means that the mPH class allows for modeling of bivariate risks with any given correlation or copula. We derive an EM algorithm for its statistical estimation, and illustrate it on bivariate insurance data. Extensions to more general settings are outlined.
We study the limiting behavior of the familywise error rate (FWER) of the Bonferroni procedure in a multiple testing problem. We establish that, in the equicorrelated normal setup, the FWER of Bonferroni's method tends to zero asymptotically (i.e for a sufficiently large number of hypotheses) for any positive equicorrelation. We extend this result for generalized familywise error rates.
This paper deals with the problem of graph matching or network alignment for Erd\H{o}s--R\'enyi graphs, which can be viewed as a noisy average-case version of the graph isomorphism problem. Let $G$ and $G'$ be $G(n, p)$ Erd\H{o}s--R\'enyi graphs marginally, identified with their adjacency matrices. Assume that $G$ and $G'$ are correlated such that $\mathbb{E}[G_{ij} G'_{ij}] = p(1-\alpha)$. For a permutation $\pi$ representing a latent matching between the vertices of $G$ and $G'$, denote by $G^\pi$ the graph obtained from permuting the vertices of $G$ by $\pi$. Observing $G^\pi$ and $G'$, we aim to recover the matching $\pi$. In this work, we show that for every $\varepsilon \in (0,1]$, there is $n_0>0$ depending on $\varepsilon$ and absolute constants $\alpha_0, R > 0$ with the following property. Let $n \ge n_0$, $(1+\varepsilon) \log n \le np \le n^{\frac{1}{R \log \log n}}$, and $0 < \alpha < \min(\alpha_0,\varepsilon/4)$. There is a polynomial-time algorithm $F$ such that $\mathbb{P}\{F(G^\pi,G')=\pi\}=1-o(1)$. This is the first polynomial-time algorithm that recovers the exact matching between vertices of correlated Erd\H{o}s--R\'enyi graphs with constant correlation with high probability. The algorithm is based on comparison of partition trees associated with the graph vertices.
Consider the computations at a node in a message passing algorithm. Assume that the node has incoming and outgoing messages $\mathbf{x} = (x_1, x_2, \ldots, x_n)$ and $\mathbf{y} = (y_1, y_2, \ldots, y_n)$, respectively. In this paper, we investigate a class of structures that can be adopted by the node for computing $\mathbf{y}$ from $\mathbf{x}$, where each $y_j, j = 1, 2, \ldots, n$ is computed via a binary tree with leaves $\mathbf{x}$ excluding $x_j$. We make three main contributions regarding this class of structures. First, we prove that the minimum complexity of such a structure is $3n - 6$, and if a structure has such complexity, its minimum latency is $\delta + \lceil \log(n-2^{\delta}) \rceil$ with $\delta = \lfloor \log(n/2) \rfloor$, where the logarithm always takes base two. Second, we prove that the minimum latency of such a structure is $\lceil \log(n-1) \rceil$, and if a structure has such latency, its minimum complexity is $n \log(n-1)$ when $n-1$ is a power of two. Third, given $(n, \tau)$ with $\tau \geq \lceil \log(n-1) \rceil$, we propose a construction for a structure which we conjecture to have the minimum complexity among structures with latencies at most $\tau$. Our construction method runs in $O(n^3 \log^2(n))$ time, and the obtained structure has complexity at most (generally much smaller than) $n \lceil \log(n) \rceil - 2$.
We study the algorithm of Gurvich, Khachyian and Karzanov (GKK algorithm) when it is ran over mean-payoff games with no simple cycle of weight zero. We propose a new symmetric analysis, lowering the $O(n^2 N)$ upper-bound of Pisaruk on the number of iterations down to $N + Ep + Em$, which is smaller than $nN$, where $n$ is the number of vertices, $N$ is the largest absolute value of a weight, and $Ep$ and $Em$ are respectively the largest finite energy and dual-energy values of the game. Since each iteration is computed in $O(m)$, this improves on the state of the art pseudopolynomial $O(mnN)$ runtime bound of Brim, Chaloupka, Doyen, Gentilini and Raskin, by taking into account the structure of the game graph. We complement our result by showing that the analysis of Dorfman, Kaplan and Zwick also applies to the GKK algorithm, which is thus also subject to the state of the art combinatorial runtime bound of $O(m 2^{n/2})$.
Reinforcement learning, mathematically described by Markov Decision Problems, may be approached either through dynamic programming or policy search. Actor-critic algorithms combine the merits of both approaches by alternating between steps to estimate the value function and policy gradient updates. Due to the fact that the updates exhibit correlated noise and biased gradient updates, only the asymptotic behavior of actor-critic is known by connecting its behavior to dynamical systems. This work puts forth a new variant of actor-critic that employs Monte Carlo rollouts during the policy search updates, which results in controllable bias that depends on the number of critic evaluations. As a result, we are able to provide for the first time the convergence rate of actor-critic algorithms when the policy search step employs policy gradient, agnostic to the choice of policy evaluation technique. In particular, we establish conditions under which the sample complexity is comparable to stochastic gradient method for non-convex problems or slower as a result of the critic estimation error, which is the main complexity bottleneck. These results hold in continuous state and action spaces with linear function approximation for the value function. We then specialize these conceptual results to the case where the critic is estimated by Temporal Difference, Gradient Temporal Difference, and Accelerated Gradient Temporal Difference. These learning rates are then corroborated on a navigation problem involving an obstacle, providing insight into the interplay between optimization and generalization in reinforcement learning.
The existence of simple, uncoupled no-regret dynamics that converge to correlated equilibria in normal-form games is a celebrated result in the theory of multi-agent systems. Specifically, it has been known for more than 20 years that when all players seek to minimize their internal regret in a repeated normal-form game, the empirical frequency of play converges to a normal-form correlated equilibrium. Extensive-form (that is, tree-form) games generalize normal-form games by modeling both sequential and simultaneous moves, as well as private information. Because of the sequential nature and presence of partial information in the game, extensive-form correlation has significantly different properties than the normal-form counterpart, many of which are still open research directions. Extensive-form correlated equilibrium (EFCE) has been proposed as the natural extensive-form counterpart to normal-form correlated equilibrium. However, it was currently unknown whether EFCE emerges as the result of uncoupled agent dynamics. In this paper, we give the first uncoupled no-regret dynamics that converge to the set of EFCEs in $n$-player general-sum extensive-form games with perfect recall. First, we introduce a notion of trigger regret in extensive-form games, which extends that of internal regret in normal-form games. When each player has low trigger regret, the empirical frequency of play is close to an EFCE. Then, we give an efficient no-trigger-regret algorithm. Our algorithm decomposes trigger regret into local subproblems at each decision point for the player, and constructs a global strategy of the player from the local solutions at each decision point.
Human parsing is for pixel-wise human semantic understanding. As human bodies are underlying hierarchically structured, how to model human structures is the central theme in this task. Focusing on this, we seek to simultaneously exploit the representational capacity of deep graph networks and the hierarchical human structures. In particular, we provide following two contributions. First, three kinds of part relations, i.e., decomposition, composition, and dependency, are, for the first time, completely and precisely described by three distinct relation networks. This is in stark contrast to previous parsers, which only focus on a portion of the relations and adopt a type-agnostic relation modeling strategy. More expressive relation information can be captured by explicitly imposing the parameters in the relation networks to satisfy the specific characteristics of different relations. Second, previous parsers largely ignore the need for an approximation algorithm over the loopy human hierarchy, while we instead address an iterative reasoning process, by assimilating generic message-passing networks with their edge-typed, convolutional counterparts. With these efforts, our parser lays the foundation for more sophisticated and flexible human relation patterns of reasoning. Comprehensive experiments on five datasets demonstrate that our parser sets a new state-of-the-art on each.
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