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We study distribution-free property testing and learning problems where the unknown probability distribution is a product distribution over ℝ^d. For many important classes of functions, such as intersections of halfspaces, polynomial threshold functions, convex sets, and k-alternating functions, the known algorithms either have complexity that depends on the support size of the distribution, or are proven to work only for specific examples of product distributions. We introduce a general method, which we call downsampling, that resolves these issues. Downsampling uses a notion of "rectilinear isoperimetry" for product distributions, which further strengthens the connection between isoperimetry, testing and learning. Using this technique, we attain new efficient distribution-free algorithms under product distributions on ℝ^d: 1) A simpler proof for non-adaptive, one-sided monotonicity testing of functions [n]^d → {0,1}, and improved sample complexity for testing monotonicity over unknown product distributions, from O(d⁷) [Black, Chakrabarty, & Seshadhri, SODA 2020] to O(d³). 2) Polynomial-time agnostic learning algorithms for functions of a constant number of halfspaces, and constant-degree polynomial threshold functions; 3) An exp{O(dlog(dk))}-time agnostic learning algorithm, and an exp{O(dlog(dk))}-sample tolerant tester, for functions of k convex sets; and a 2^O(d) sample-based one-sided tester for convex sets; 4) An exp{O(k√d)}-time agnostic learning algorithm for k-alternating functions, and a sample-based tolerant tester with the same complexity.

Weak submodularity is a natural relaxation of the diminishing return property, which is equivalent to submodularity. Weak submodularity has been used to show that many (monotone) functions that arise in practice can be efficiently maximized with provable guarantees. In this work we introduce two natural generalizations of weak submodularity for non-monotone functions. We show that an efficient randomized greedy algorithm has provable approximation guarantees for maximizing these functions subject to a cardinality constraint. We then provide a more refined analysis that takes into account that the weak submodularity parameter may change (sometimes improving) throughout the execution of the algorithm. This leads to improved approximation guarantees in some settings. We provide applications of our results for monotone and non-monotone maximization problems.

In monotone submodular function maximization, approximation guarantees based on the curvature of the objective function have been extensively studied in the literature. However, the notion of curvature is often pessimistic, and we rarely obtain improved approximation guarantees, even for very simple objective functions. In this paper, we provide a novel approximation guarantee by extracting an M^{natural}-concave function h:2^E -> R_+, a notion in discrete convex analysis, from the objective function f:2^E -> R_+. We introduce a novel notion called the M^{natural}-concave curvature of a given set function f, which measures how much f deviates from an M^{natural}-concave function, and show that we can obtain a (1-gamma/e-epsilon)-approximation to the problem of maximizing f under a cardinality constraint in polynomial time, where gamma is the value of the M^{natural}-concave curvature and epsilon > 0 is an arbitrary constant. Then, we show that we can obtain nontrivial approximation guarantees for various problems by applying the proposed algorithm.

We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle the case where the minimization is done over a sphere. The analysis of our algorithms is obtained by combining results from graph limit theory, along with a novel spectral decomposition of matrices. Specifically, we prove that a matrix A can be decomposed into a structured part and a pseudorandom part, where the structured part is a block matrix with a polylogarithmic number of blocks, such that in each block all the entries are the same, and the pseudorandom part has a small spectral norm, achieving better error bound than the existing decomposition theorem of Frieze and Kannan (FOCS'96). As an additional application of the decomposition theorem, we give a sublinear-time approximation algorithm for computing the top singular values of a matrix.

In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in the streaming setting. In particular, the elements arrive sequentially and at any point of time, the algorithm has access only to a small fraction of the data stored in primary memory. For this problem, we propose a (0.363-epsilon)-approximation algorithm, requiring only a single pass through the data; moreover, we propose a (0.4-epsilon)-approximation algorithm requiring a constant number of passes through the data. The required memory space of both algorithms depends only on the size of the knapsack capacity and epsilon.

International Computer Music Conference Proceedings: vol. 2002, (dlps) bbp2372.2002.048, http://hdl.handle.net/2027/spo.bbp2372.2002.048, This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. Please contact mpub-help@umich.edu to use this work in a way not covered by the license.

A temporal constraint language G is a set of relations with first-order definitions in (Q; <). Let CSP(G) denote the set of constraint satisfaction problem instances with relations from G. CSP(G) admits robust approximation if, for any e >= 0, given a (1-e)-satisfiable instance of CSP(G), we can compute an assignment that satisfies at least a (1-f(e))-fraction of constraints in polynomial time. Here, f(e) is some function satisfying f(0)=0 and f(e) goes 0 as e goes 0. Firstly, we give a qualitative characterization of robust approximability: Assuming the Unique Games Conjecture, we give a necessary and sufficient condition on G under which CSP(G) admits robust approximation. Secondly, we give a quantitative characterization of robust approximability: Assuming the Unique Games Conjecture, we precisely characterize how f(e) depends on e for each G. We show that our robust approximation algorithms can be run in almost linear time.

We study the constraint satisfaction problem over the point algebra. In this problem, an instance consists of a set of variables and a set of binary constraints of forms (x < y), (x <= y), (x \neq y) or (x = y). Then, the objective is to assign integers to variables so as to satisfy as many constraints as possible.This problem contains many important problems such as Correlation Clustering, Maximum Acyclic Subgraph, and Feedback Arc Set. We first give an exact algorithm that runs in O^*(3^{\frac{log 5}{log 6}n}) time, which improves the previous best O^*(3^n) obtained by a standard dynamic programming. Our algorithm combines the dynamic programming with the split-and-list technique. The split-and-list technique involves matrix products and we make use of sparsity of matrices to speed up the computation. As for approximation, we give a 0.4586-approximation algorithm when the objective is maximizing the number of satisfied constraints, and give an O(log n log log n)-approximation algorithm when the objective is minimizing the number of unsatisfied constraints.