Provably Good Solutions to the Knapsack Problem via Neural Networks of Bounded Size

The development of a satisfying and rigorous mathematical understanding of the performance of neural networks is a major challenge in artificial intelligence. Against this background, we study the expressive power of neural networks through the example of the classical NP-hard Knapsack Problem. Our main contribution is a class of recurrent neural networks (RNNs) with rectified linear units that are iteratively applied to each item of a Knapsack instance and thereby compute optimal or provably good solution values. We show that an RNN of depth four and width depending quadratically on the profit of an optimum Knapsack solution is sufficient to find optimum Knapsack solutions. We also prove the following tradeoff between the size of an RNN and the quality of the computed Knapsack solution: for Knapsack instances consisting of $n$ items, an RNN of depth five and width $w$ computes a solution of value at least $1-\mathcal{O}(n^2/\sqrt{w})$ times the optimum solution value. Our results build upon a classical dynamic programming formulation of the Knapsack Problem as well as a careful rounding of profit values that are also at the core of the well-known fully polynomial-time approximation scheme for the Knapsack Problem. A carefully conducted computational study qualitatively supports our theoretical size bounds. Finally, we point out that our results can be generalized to many other combinatorial optimization problems that admit dynamic programming solution methods, such as various Shortest Path Problems, the Longest Common Subsequence Problem, and the Traveling Salesperson Problem.Comment: A short version of this paper appears in the proceedings of AAAI 202

Distributed PCP Theorems for Hardness of Approximation in P

We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment $x \in \{0,1\}^n$ to a CNF formula $\varphi$ is shared between two parties, where Alice knows $x_1, \dots, x_{n/2}$, Bob knows $x_{n/2+1},\dots,x_n$, and both parties know $\varphi$. The goal is to have Alice and Bob jointly write a PCP that $x$ satisfies $\varphi$, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic variant, where the players are helped by Merlin, a third party who knows all of $x$. Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that there are no truly-subquadratic approximation algorithms for Bichromatic Maximum Inner Product over {0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first two problems we obtain nearly-polynomial factors of $2^{(\log n)^{1-o(1)}}$; only $(1+o(1))$-factor lower bounds (under SETH) were known before

Inexact Block Coordinate Descent Algorithms for Nonsmooth Nonconvex Optimization

In this paper, we propose an inexact block coordinate descent algorithm for large-scale nonsmooth nonconvex optimization problems. At each iteration, a particular block variable is selected and updated by inexactly solving the original optimization problem with respect to that block variable. More precisely, a local approximation of the original optimization problem is solved. The proposed algorithm has several attractive features, namely, i) high flexibility, as the approximation function only needs to be strictly convex and it does not have to be a global upper bound of the original function; ii) fast convergence, as the approximation function can be designed to exploit the problem structure at hand and the stepsize is calculated by the line search; iii) low complexity, as the approximation subproblems are much easier to solve and the line search scheme is carried out over a properly constructed differentiable function; iv) guaranteed convergence of a subsequence to a stationary point, even when the objective function does not have a Lipschitz continuous gradient. Interestingly, when the approximation subproblem is solved by a descent algorithm, convergence of a subsequence to a stationary point is still guaranteed even if the approximation subproblem is solved inexactly by terminating the descent algorithm after a finite number of iterations. These features make the proposed algorithm suitable for large-scale problems where the dimension exceeds the memory and/or the processing capability of the existing hardware. These features are also illustrated by several applications in signal processing and machine learning, for instance, network anomaly detection and phase retrieval

Degree of separability of bipartite quantum states

We investigate the problem of finding the optimal convex decomposition of a bipartite quantum state into a separable part and a positive remainder, in which the weight of the separable part is maximal. This weight is naturally identified with the degree of separability of the state. In a recent work, the problem was solved for two-qubit states using semidefinite programming. In this paper, we describe a procedure to obtain the optimal decomposition of a bipartite state of any finite dimension via a sequence of semidefinite relaxations. The sequence of decompositions thus obtained is shown to converge to the optimal one. This provides, for the first time, a systematic method to determine the so-called optimal Lewenstein-Sanpera decomposition of any bipartite state. Numerical results are provided to illustrate this procedure, and the special case of rank-2 states is also discussed.Comment: 11 pages, 7 figures, submitted to PR

A parametric integer programming algorithm for bilevel mixed integer programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader's variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case it yields a ``better than fully polynomial time'' approximation scheme with running time polynomial in the logarithm of the relative precision. For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of variables is fixed.Comment: 11 page