In an optimal design problem, we are given a set of linear experiments v1,…,vn∈Rd and k≥d, and our goal is to select a set or a multiset S⊆[n] of size k such that Φ((∑i∈Sviv⊤i)−1) is minimized. When Φ(M)=Determinant(M)1/d, the problem is known as the D-optimal design problem, and when Φ(M)=Trace(M), it is known as the A-optimal design problem. One of the most common heuristics used in practice to solve these problems is the local search heuristic, also known as the Fedorov’s exchange method (Fedorov, 1972). This is due to its simplicity and its empirical performance (Cook and Nachtrheim, 1980; Millermore »
Combinatorial Algorithms for Optimal Design
In an optimal design problem, we are given a set of linear experiments v1,…,vn∈Rd and k≥d, and our goal is to select a set or a multiset S⊆[n] of size k such that Φ((∑i∈Sviv⊤i)−1) is minimized. When Φ(M)=Determinant(M)1/d, the problem is known as the D-optimal design problem, and when Φ(M)=Trace(M), it is known as the A-optimal design problem. One of the most common heuristics used in practice to solve these problems is the local search heuristic, also known as the Fedorov’s exchange method (Fedorov, 1972). This is due to its simplicity and its empirical performance (Cook and Nachtrheim, 1980; Miller and Nguyen, 1994; Atkinson et al., 2007). However, despite its wide usage no theoretical bound has been proven for this algorithm. In this paper, we bridge this gap and prove approximation guarantees for the local search algorithms for D-optimal design and A-optimal design problems. We show that the local search algorithms are asymptotically optimal when kd is large. In addition to this, we also prove similar approximation guarantees for the greedy algorithms for D-optimal design and A-optimal design problems when kd is large.
- Award ID(s):
- 1910423
- Publication Date:
- NSF-PAR ID:
- 10196079
- Journal Name:
- Conference on Learning Theory, COLT 2019
- Volume:
- 99
- Page Range or eLocation-ID:
- 2210-2258
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We study the A-optimal design problem where we are given vectors υ1, …, υn ∊ ℝd, an integer k ≥ d, and the goal is to select a set S of k vectors that minimizes the trace of (∑i∊Svivi⊺)−1. Traditionally, the problem is an instance of optimal design of experiments in statistics [35] where each vector corresponds to a linear measurement of an unknown vector and the goal is to pick k of them that minimize the average variance of the error in the maximum likelihood estimate of the vector being measured. The problem also finds applications in sensor placementmore »
-
We study the classic Maximum Independent Set problem under the notion of stability introduced by Bilu and Linial (2010): a weighted instance of Independent Set is γ-stable if it has a unique optimal solution that remains the unique optimal solution under multiplicative perturbations of the weights by a factor of at most γ ≥ 1. The goal then is to efficiently recover this “pronounced” optimal solution exactly. In this work, we solve stable instances of Independent Set on several classes of graphs: we improve upon previous results by solving \tilde{O}(∆/sqrt(log ∆))-stable instances on graphs of maximum degree ∆, (k −more »
-
We study the classic set cover problem from the perspective of sub-linear algorithms. Given access to a collection of m sets over n elements in the query model, we show that sub-linear algorithms derived from existing techniques have almost tight query complexities. On one hand, first we show an adaptation of the streaming algorithm presented in [17] to the sub-linear query model, that returns an α-approximate cover using Õ(m(n/k)^1/(α–1) + nk) queries to the input, where k denotes the value of a minimum set cover. We then complement this upper bound by proving that for lower values of k, themore »
-
Experimental design is a classical area in statistics and has also found new applications in machine learning. In the combinatorial experimental design problem, the aim is to estimate an unknown m-dimensional vector x from linear measurements where a Gaussian noise is introduced in each measurement. The goal is to pick k out of the given n experiments so as to make the most accurate estimate of the unknown parameter x. Given a set S of chosen experiments, the most likelihood estimate x0 can be obtained by a least squares computation. One of the robust measures of error estimation is themore »