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  1. Experimental design is a classical statistics problem, and its aim is to estimate an unknown vector from linear measurements where a Gaussian noise is introduced in each measurement. For the combinatorial experimental design problem, the goal is to pick a subset of experiments so as to make the most accurate estimate of the unknown parameters. In this paper, we will study one of the most robust measures of error estimation—the D-optimality criterion, which corresponds to minimizing the volume of the confidence ellipsoid for the estimation error. The problem gives rise to two natural variants depending on whether repetitions of experimentsmore »are allowed or not. We first propose an approximation algorithm with a 1/e-approximation for the D-optimal design problem with and without repetitions, giving the first constant-factor approximation for the problem. We then analyze another sampling approximation algorithm and prove that it is asymptotically optimal. Finally, for D-optimal design with repetitions, we study a different algorithm proposed by the literature and show that it can improve this asymptotic approximation ratio. All the sampling algorithms studied in this paper are shown to admit polynomial-time deterministic implementations.« less
  2. Dimensionality reduction is a classical technique widely used for data analysis. One foundational instantiation is Principal Component Analysis (PCA), which minimizes the average reconstruction error. In this paper, we introduce the multi-criteria dimensionality reduction problem where we are given multiple objectives that need to be optimized simultaneously. As an application, our model captures several fairness criteria for dimensionality reduction such as the Fair-PCA problem introduced by Samadi et al. [NeurIPS18] and the Nash Social Welfare (NSW) problem. In the Fair-PCA problem, the input data is divided into k groups, and the goal is to find a single d-dimensional representation formore »all groups for which the maximum reconstruction error of any one group is minimized. In NSW the goal is to maximize the product of the individual variances of the groups achieved by the common low-dimensinal space. Our main result is an exact polynomial-time algorithm for the two-criteria dimensionality reduction problem when the two criteria are increasing concave functions. As an application of this result, we obtain a polynomial time algorithm for Fair-PCA for k=2 groups, resolving an open problem of Samadi et al.[NeurIPS18], and a polynomial time algorithm for NSW objective for k=2 groups. We also give approximation algorithms for k>2. Our technical contribution in the above results is to prove new low-rank properties of extreme point solutions to semi-definite programs. We conclude with the results of several experiments indicating improved performance and generalized application of our algorithm on real-world datasets.« less
  3. 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 »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.« less