An official website of the United States government
Quadratic Unconstrained Binary Optimization (QUBO) problem becomes an attractive and valuable optimization problem formulation in that it can easily transform into a variety of other combinatorial optimization problems such as Graph/number Partition, Max-Cut, SAT, Vertex Coloring, TSP, etc. Some of these problems are NP-hard and widely applied in industry and scientific research. Meanwhile, QUBO has been discovered to be compatible with two emerging computing paradigms, neuromorphic computing, and quantum computing, with tremendous potential to speed up future optimization solvers. In this paper, we propose a novel neuromorphic computing paradigm that employs multiple collaborative spiking neural networks to solve QUBO problems. Each SNN conducts a local stochastic gradient descent search and shares the global best solutions periodically to perform a meta-heuristic search for optima. We simulate our model and compare it to a single SNN solver and a mult-SNN solver without collaboration. Through tests on benchmark problems, the proposed method is demonstrated to be more efficient and effective in searching for QUBO optima. Specifically, it exhibits x10 and x15-20 speedup respectively on the multi-SNN solver without collaboration and the single-SNN solver.
more »
« less
Partition Crossover can Linearize Optima Lattices of k-bounded Pseudo-Boolean Functions
When Partition Crossover is used to recombine two parents which are local optima, the ospring are all local optima in the smallest hyperplane subspace that contains the two parents. The ospring can also be organized into a non-planar hypercube "lattice." Fur- thermore, all of the ospring can be evaluated using a simple linear equation. When a child of Partition Crossover is a local optimum in the full search space, the linear equation exactly determines its evaluation. When a child of Partition Crossover can be improved by local search, the linear equation is an upper bound on the evaluation of the associated local optimum when minimizing. This theoret- ical result holds for all k-bounded Pseudo-Boolean optimization problems, including MAX-kSAT, QUBO problems, as well as ran- dom and adjacent NK landscapes. These linear equations provide a stronger explanation as to why the "Big Valley" distribution of local optima exists.
more »
« less
- Award ID(s):
- 1908866
- PAR ID:
- 10495865
- Publisher / Repository:
- ACM
- Date Published:
- Journal Name:
- ACM Foundations of Genetic Algorithms Conference
- ISSN:
- 979-8-4007-0202-0
- ISBN:
- 979-8-4007-0202-0
- Format(s):
- Medium: X
- Location:
- Potsdam Germany
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Crossover designs play an increasingly important role in precision medicine. We show the search of an optimal crossover design can be formulated as a convex optimization problem and convex optimization tools, such as CVX, can be directly used to search for an optimal crossover design. We first demonstrate how to transform crossover design problems into convex optimization problems and show CVX can effortlessly find optimal crossover designs that coincide with a few theoretical crossover optimal designs in the literature. The proposed approach is especially useful when it becomes problematic to construct optimal designs analytically for complicated models. We then apply CVX to find crossover designs for models with auto-correlated error structures or when the information matrices may be singular and analytical answers are unavailable. We also construct N-of-1 trials frequently used in precision medicine to estimate treatment effects on the individuals or to estimate average treatment effects, including finding dual-objective optimal crossover designs.more » « less
-
Low-rank matrix recovery is a fundamental problem in machine learning with numerous applications. In practice, the problem can be solved by convex optimization namely nuclear norm minimization, or by non-convex optimization as it is well-known that for low-rank matrix problems like matrix sensing and matrix completion, all local optima of the natural non-convex objectives are also globally optimal under certain ideal assumptions. In this paper, we study new approaches for matrix sensing in a semi-random model where an adversary can add any number of arbitrary sensing matrices. More precisely, the problem is to recover a low-rank matrix $$X^\star$$ from linear measurements $$b_i = \langle A_i, X^\star \rangle$$, where an unknown subset of the sensing matrices satisfies the Restricted Isometry Property (RIP) and the rest of the $$A_i$$'s are chosen adversarially. It is known that in the semi-random model, existing non-convex objectives can have bad local optima. To fix this, we present a descent-style algorithm that provably recovers the ground-truth matrix $$X^\star$$. For the closely-related problem of semi-random matrix completion, prior work [CG18] showed that all bad local optima can be eliminated by reweighting the input data. However, the analogous approach for matrix sensing requires reweighting a set of matrices to satisfy RIP, which is a condition that is NP-hard to check. Instead, we build on the framework proposed in [KLL$^+$23] for semi-random sparse linear regression, where the algorithm in each iteration reweights the input based on the current solution, and then takes a weighted gradient step that is guaranteed to work well locally. Our analysis crucially exploits the connection between sparsity in vector problems and low-rankness in matrix problems, which may have other applications in obtaining robust algorithms for sparse and low-rank problems.more » « less
-
Singh, M (Ed.)Discrete optimization problems arise in many biological contexts and, in many cases, we seek to make inferences from the optimal solutions. However, the number of optimal solutions is frequently very large and making inferences from any single solution may result in conclusions that are not supported by other optimal solutions. We describe a general approach for efficiently (polynomial time) and exactly (without sampling) computing statistics on the space of optimal solutions. These statistics provide insights into the space of optimal solutions that can be used to support the use of a single optimum (e.g., when the optimal solutions are similar) or justify the need for selecting multiple optima (e.g., when the solution space is large and diverse) from which to make inferences. We demonstrate this approach on two well-known problems and identify the properties of these problems that make them amenable to this method.more » « less
-
We consider the problem of enumerating optimal solutions for two hypergraph k-partitioning problems, namely, Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. The input in hypergraph k-partitioning problems is a hypergraph [Formula: see text] with positive hyperedge costs along with a fixed positive integer k. The goal is to find a partition of V into k nonempty parts [Formula: see text]—known as a k-partition—so as to minimize an objective of interest. (1) If the objective of interest is the maximum cut value of the parts, then the problem is known as Minmax-Hypergraph-k-Partition. A subset of hyperedges is a minmax-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Minmax-Hypergraph-k-Partition. (2) If the objective of interest is the total cost of hyperedges crossing the k-partition, then the problem is known as Hypergraph-k-Cut. A subset of hyperedges is a min-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Hypergraph-k-Cut. We give the first polynomial bound on the number of minmax-k-cut-sets and a polynomial-time algorithm to enumerate all of them in hypergraphs for every fixed k. Our technique is strong enough to also enable an [Formula: see text]-time deterministic algorithm to enumerate all min-k-cut-sets in hypergraphs, thus improving on the previously known [Formula: see text]-time deterministic algorithm, in which n is the number of vertices and p is the size of the hypergraph. The correctness analysis of our enumeration approach relies on a structural result that is a strong and unifying generalization of known structural results for Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. We believe that our structural result is likely to be of independent interest in the theory of hypergraphs (and graphs). Funding: All authors were supported by NSF AF 1814613 and 1907937.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3594805.3607129
