A fundamental challenge in multiagent systems is to design local control algorithms to ensure a desirable collective behavior. The information available to the agents, gathered either through communication or sensing, naturally restricts the achievable performance. Hence, it is fundamental to identify what piece of information is valuable and can be exploited to design control laws with enhanced performance guarantees. This paper studies the case when such information is uncertain or inaccessible for a class of submodular resource allocation problems termed covering problems. In the first part of this paper, we pinpoint a fundamental risk-reward tradeoff faced by the system operator when conditioning the control design on a valuable but uncertain piece of information, which we refer to as the cardinality, that represents the maximum number of agents that can simultaneously select any given resource. Building on this analysis, we propose a distributed algorithm that allows agents to learn the cardinality while adjusting their behavior over time. This algorithm is proved to perform on par or better to the optimal design obtained when the exact cardinality is known a priori .
The problem of designing a distribution rule so as to maximise @math has been studied in @cite_15 and @cite_7 . Both works impose a natural constraint on the admissible @math , requiring @math and @math to be non-increasing. The optimal distribution rule is explicitly derived in the former work, while the latter shows how @math is fully characterised by a single scalar quantity @math defined in , measuring how fast the distribution rule @math decreases. We intend to build upon these results, which are summarised in the following proposition. Given @math and a distribution rule @math , we define j k-1 j f(j)-f(j+1),(k-1) f(k) ,.
@cite_15 @cite_7
2277973662 2082401067 2611972874 2107037617
This paper introduces a parallel and distributed algorithm for solving the following minimization problem with linear constraints: @math minimizef1(x1)+ź+fN(xN)subject toA1x1+ź+ANxN=c,x1źX1,ź,xNźXN,where @math Nź2, @math fi are convex functions, @math Ai are matrices, and @math Xi are feasible sets for variable @math xi. Our algorithm extends the alternating direction method of multipliers (ADMM) and decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This paper shows that the classic ADMM can be extended to the N-block Jacobi fashion and preserve convergence in the following two cases: (i) matrices @math Ai are mutually near-orthogonal and have full column-rank, or (ii) proximal terms are added to the N subproblems (but without any assumption on matrices @math Ai). In the latter case, certain proximal terms can let the subproblem be solved in more flexible and efficient ways. We show that @math źxk+1-xkźM2 converges at a rate of o(1 k) where M is a symmetric positive semi-definte matrix. Since the parameters used in the convergence analysis are conservative, we introduce a strategy for automatically tuning the parameters to substantially accelerate our algorithm in practice. We implemented our algorithm (for the case ii above) on Amazon EC2 and tested it on basis pursuit problems with >300 GB of distributed data. This is the first time that successfully solving a compressive sensing problem of such a large scale is reported. Let B i be deterministic real symmetric m × m matrices, and ξ i be independent random scalars with zero mean and “of order of one” (e.g., @math ). We are interested to know under what conditions “typical norm” of the random matrix @math is of order of 1. An evident necessary condition is @math , which, essentially, translates to @math ; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of S N is @math : @math for all Ω > 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints @math , where F is quadratic in X = (X 1,... ,X k ). We show that when F is convex in every one of X j , a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices @math . We study the problem of efficiently optimizing submodular functions under cardinality constraints in distributed setting. Recently, several distributed algorithms for this problem have been introduced which either achieve a sub-optimal solution or they run in super-constant number of rounds of computation. Unlike previous work, we aim to design distributed algorithms in multiple rounds with almost optimal approximation guarantees at the cost of outputting a larger number of elements. Toward this goal, we present a distributed algorithm that, for any e > 0 and any constant r, outputs a set S of O(rk e1 r) items in r rounds, and achieves a (1-e)-approximation of the value of the optimum set with k items. This is the first distributed algorithm that achieves an approximation factor of (1-e) running in less than log 1 e number of rounds. We also prove a hardness result showing that the output of any 1-e approximation distributed algorithm limited to one distributed round should have at least Ω(k e) items. In light of this hardness result, our distributed algorithm in one round, r = 1, is asymptotically tight in terms of the output size. We support the theoretical guarantees with an extensive empirical study of our algorithm showing that achieving almost optimum solutions is indeed possible in a few rounds for large-scale real datasets. Abstract Stanley posed the problem of minimizing the functional H(f)= ∫ 0 ∞ dx ∫ 0 f(x) dy log (f (x) −y+f −1 (y) − x) over nonincreasing nonnegative f on (0, ∞) of integral unity. We show that the minimum is unique and has the value − 1 2 , as was conjectured by Stanley. The minimizing function f 0 , H(f 0 ) = − 1 2 , is given parametrically by f 0 (x)=( 2 π )(sin θ −θ cos θ), x =f o (x) + 2 cos θ, 0 ⩽ θ ⩽ π (2) for 0 ⩽ x ⩽ 2; and f 0 ( x ) = 0 for x ⩾ 2. Closely related unpublished results have been obtained by Hammersley. We also find the minimum of H ( f ) subject to the constraints f (0) ⩽ a and f −1 (0) = inf ( x : f ( x ) = 0) ⩽ b where a and b are given. Proofs of the results for the case of constraints are complicated and will be given elsewhere. Let λ n be the shape of the random Young tableau with n unit squares obtained from sampling from the Schensted distribution where P(λ n = n! π 2 (λ n ) (3) where π ( λ n ) is the product of the n hook lengths of λ n . Consider the stochastic processes λ n (t)=( 1 n 1 2 )λ(t n 1 2 , n ⩾ 1, t ⩾ 0 (4) where λ n ( t ) is the height of the tableau λ n at a horizontal distance t from the corner. We show λ n → f 0 in the sense of weak convergence in a certain metric, where f 0 is the deterministic function in (2). Let l ( σ n ) denote the length of the longest increasing subsequence of a random permutation σ n of 1,2,…, n . Hammersley showed that l(ω n ) n 1 2 →c in probability, n→∞ (6) Schensted showed that l ( σ n ) has the same distribution as λ n (0) under the distribution (3) on λ n . It has long been conjectured (apparently first by Baer and Brock) that c = 2. We show here that c ⩾ 2 as a by-product of (5).