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1. Learning and Sampling of Atomic Interventions from Observations

   Bhattacharyya, Arnab; Gayen, Sutanu; Kandasamy, Saravanan; Maran, Ashwin; Vinodchandran, N. V. (July 2020, Proceedings of the 37th International Conference on Machine Learning)

   We study the problem of efficiently estimating the effect of an intervention on a single variable using observational samples. Our goal is to give algorithms with polynomial time and sample complexity in a non-parametric setting. Tian and Pearl (AAAI ’02) have exactly characterized the class of causal graphs for which causal effects of atomic interventions can be identified from observational data. We make their result quantitative. Suppose 𝒫 is a causal model on a set V of n observable variables with respect to a given causal graph G, and let do(x) be an identifiable intervention on a variable X. We show that assuming that G has bounded in-degree and bounded c-components (k) and that the observational distribution satisfies a strong positivity condition: (i) [Evaluation] There is an algorithm that outputs with probability 2/3 an evaluator for a distribution P^ that satisfies TV(P(V | do(x)), P^(V)) < eps using m=O (n/eps^2) samples from P and O(mn) time. The evaluator can return in O(n) time the probability P^(v) for any assignment v to V. (ii) [Sampling] There is an algorithm that outputs with probability 2/3 a sampler for a distribution P^ that satisfies TV(P(V | do(x)), P^(V)) < eps using m=O (n/eps^2) samples from P and O(mn) time. The sampler returns an iid sample from P^ with probability 1 in O(n) time. We extend our techniques to estimate P(Y | do(x)) for a subset Y of variables of interest. We also show lower bounds for the sample complexity, demonstrating that our sample complexity has optimal dependence on the parameters n and eps, as well as if k=1 on the strong positivity parameter. 

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2. Learning mixtures of gaussians with censored data

   Aragam, Bryon; Tai, Wai_Ming (July 2023, Proceedings of the 40th International Conference on Machine Learning)

   We study the problem of learning mixtures of Gaussians with censored data. Statistical learning with censored data is a classical problem, with numerous practical applications, however, finite-sample guarantees for even simple latent variable models such as Gaussian mixtures are missing. Formally, we are given censored data from a mixture of univariate Gaussians $$\sum_{i=1}^k w_i \mathcal{N}(\mu_i,\sigma^2),$$ i.e. the sample is observed only if it lies inside a set $$S$$. The goal is to learn the weights $$w_i$$ and the means $$\mu_i$$. We propose an algorithm that takes only $$\frac{1}{\varepsilon^{O(k)}}$$ samples to estimate the weights $$w_i$$ and the means $$\mu_i$$ within $$\varepsilon$$ error. 

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3. Causal Inference Despite Limited Global Confounding via Mixture Models

   Gordon, S.; Mazaheri, B.; Rabani, Y.; Schulman, L. (April 2023, Proceedings of Machine Learning Research)

   van der Schaar, M.; Zhang, C.; Janzing, D. (Ed.)

   A Bayesian Network is a directed acyclic graph (DAG) on a set of n random variables (the vertices); a Bayesian Network Distribution (BND) is a probability distribution on the random variables that is Markovian on the graph. A finite k-mixture of such models is graphically represented by a larger graph which has an additional “hidden” (or “latent”) random variable U, ranging in {1,...,k}, and a directed edge from U to every other vertex. Models of this type are fundamental to causal inference, where U models an unobserved confounding effect of multiple populations, obscuring the causal relationships in the observable DAG. By solving the mixture problem and recovering the joint probability distribution with U, traditionally unidentifiable causal relationships become identifiable. Using a reduction to the more well-studied “product” case on empty graphs, we give the first algorithm to learn mixtures of non-empty DAGs. 

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4. Unit-disk range searching and applications

   https://doi.org/10.20382/jocg.v14i1a13  

   Wang, Haitao (December 2023, Journal of computational geometry)

   Given a set $$P$$ of $$n$$ points in the plane, we consider the problem of computing the number of points of $$P$$ in a query unit disk (i.e., all query disks have the same radius). We show that the main techniques for simplex range searching in the plane can be adapted to this problem. For example, by adapting Matoušek's results, we can build a data structure of $O(n)$ space in $$O(n^{1+\delta})$$ time (for any $$\delta>0$$) so that each query can be answered in $$O(\sqrt{n})$$ time; alternatively, we can build a data structure of $$O(n^2/\log^2 n)$$ space with $$O(n^{1+\delta})$$ preprocessing time (for any $$\delta>0$$) and $$O(\log n)$$ query time. Our techniques lead to improvements for several other classical problems in computational geometry. 1. Given a set of $$n$$ unit disks and a set of $$n$$ points in the plane, the batched unit-disk range counting problem is to compute for each disk the number of points in it. Previous work [Katz and Sharir, 1997] solved the problem in $$O(n^{4/3}\log n)$$ time. We give a new algorithm of $$O(n^{4/3})$$ time, which is optimal as it matches an $$\Omega(n^{4/3})$$-time lower bound. For small $$\chi$$, where $$\chi$$ is the number of pairs of unit disks that intersect, we further improve the algorithm to $$O(n^{2/3}\chi^{1/3}+n^{1+\delta})$$ time, for any $$\delta>0$$. 2. The above result immediately leads to an $$O(n^{4/3})$$ time optimal algorithm for counting the intersecting pairs of circles for a set of $$n$$ unit circles in the plane. The previous best algorithms solve the problem in $$O(n^{4/3}\log n)$$ deterministic time [Katz and Sharir, 1997] or in $$O(n^{4/3}\log^{2/3} n)$$ expected time by a randomized algorithm [Agarwal, Pellegrini, and Sharir, 1993]. 3. Given a set $$P$$ of $$n$$ points in the plane and an integer $$k$$, the distance selection problem is to find the $$k$$-th smallest distance among all pairwise distances of $$P$$. The problem can be solved in $$O(n^{4/3}\log^2 n)$$ deterministic time [Katz and Sharir, 1997] or in $$O(n\log n+n^{2/3}k^{1/3}\log^{5/3}n)$$ expected time by a randomized algorithm [Chan, 2001]. Our new randomized algorithm runs in $$O(n\log n +n^{2/3}k^{1/3}\log n)$$ expected time. 4. Given a set $$P$$ of $$n$$ points in the plane, the discrete $$2$$-center problem is to compute two smallest congruent disks whose centers are in $$P$$ and whose union covers $$P$$. An $$O(n^{4/3}\log^5 n)$$-time algorithm was known [Agarwal, Sharir, and Welzl, 1998]. Our techniques yield a deterministic algorithm of $$O(n^{4/3}\log^{10/3} n\cdot (\log\log n)^{O(1)})$$ time and a randomized algorithm of $$O(n^{4/3}\log^3 n\cdot (\log\log n)^{1/3})$$ expected time. 

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5. Simple and efficient pseudorandom genera- tors from Gaussian processes.

   Chattopadhyay, Eshan; De, Anindya; Servedio, Rocco (July 2019, Leibniz international proceedings in informatics)

   Abstract We show that a very simple pseudorandom generator fools intersections of k linear threshold functions (LTFs) and arbitrary functions of k LTFs over n-dimensional Gaussian space. The two analyses of our PRG (for intersections versus arbitrary functions of LTFs) are quite different from each other and from previous analyses of PRGs for functions of halfspaces. Our analysis for arbitrary functions of LTFs establishes bounds on the Wasserstein distance between Gaussian random vectors with similar covariance matrices, and combines these bounds with a conversion from Wasserstein distance to "union-of-orthants" distance from [Xi Chen et al., 2014]. Our analysis for intersections of LTFs uses extensions of the classical Sudakov-Fernique type inequalities, which give bounds on the difference between the expectations of the maxima of two Gaussian random vectors with similar covariance matrices. For all values of k, our generator has seed length O(log n) + poly(k) for arbitrary functions of k LTFs and O(log n) + poly(log k) for intersections of k LTFs. The best previous result, due to [Gopalan et al., 2010], only gave such PRGs for arbitrary functions of k LTFs when k=O(log log n) and for intersections of k LTFs when k=O((log n)/(log log n)). Thus our PRG achieves an O(log n) seed length for values of k that are exponentially larger than previous work could achieve. By combining our PRG over Gaussian space with an invariance principle for arbitrary functions of LTFs and with a regularity lemma, we obtain a deterministic algorithm that approximately counts satisfying assignments of arbitrary functions of k general LTFs over {0,1}^n in time poly(n) * 2^{poly(k,1/epsilon)} for all values of k. This algorithm has a poly(n) runtime for k =(log n)^c for some absolute constant c>0, while the previous best poly(n)-time algorithms could only handle k = O(log log n). For intersections of LTFs, by combining these tools with a recent PRG due to [R. O'Donnell et al., 2018], we obtain a deterministic algorithm that can approximately count satisfying assignments of intersections of k general LTFs over {0,1}^n in time poly(n) * 2^{poly(log k, 1/epsilon)}. This algorithm has a poly(n) runtime for k =2^{(log n)^c} for some absolute constant c>0, while the previous best poly(n)-time algorithms for intersections of k LTFs, due to [Gopalan et al., 2010], could only handle k=O((log n)/(log log n)). 

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