Motivated by applications in Optimization, Game Theory, and the training of Generative Adversarial Networks, the convergence properties of first order methods in min-max problems have received extensive study. It has been recognized that they may cycle, and there is no good understanding of their limit points when they do not. When they converge, do they converge to local min-max solutions? We characterize the limit points of two basic first order methods, namely Gradient Descent/Ascent (GDA) and Optimistic Gradient Descent Ascent (OGDA). We show that both dynamics avoid unstable critical points for almost all initializations. Moreover, for small step sizes and under mild assumptions, the set of OGDA-stable critical points is a superset of GDA-stable critical points, which is a superset of local min-max solutions (strict in some cases). The connecting thread is that the behavior of these dynamics can be studied from a dynamical systems perspective.

We consider testing and learning problems on causal Bayesian networks as defined by Pearl (Pearl, 2009). Given a causal Bayesian network M on a graph with n discrete variables and bounded in-degree and bounded ``confounded components'', we show that O(log n) interventions on an unknown causal Bayesian network X on the same graph, and O(n/epsilon^2) samples per intervention, suffice to efficiently distinguish whether X=M or whether there exists some intervention under which X and M are farther than epsilon in total variation distance. We also obtain sample/time/intervention efficient algorithms for: (i) testing the identity of two unknown causal Bayesian networks on the same graph; and (ii) learning a causal Bayesian network on a given graph. Although our algorithms are non-adaptive, we show that adaptivity does not help in general: Omega(log n) interventions are necessary for testing the identity of two unknown causal Bayesian networks on the same graph, even adaptively. Our algorithms are enabled by a new subadditivity inequality for the squared Hellinger distance between two causal Bayesian networks.

We analyze linear independence of rank one tensors produced by tensor powers of randomly perturbed vectors. This enables efficient decomposition of sums of high-order tensors. Our analysis builds upon [BCMV14] but allows for a wider range of perturbation models, including discrete ones. We give an application to recovering assemblies of neurons. Assemblies are large sets of neurons representing specific memories or concepts. The size of the intersection of two assemblies has been shown in experiments to represent the extent to which these memories co-occur or these concepts are related; the phenomenon is called association of assemblies. This suggests that an animal's memory is a complex web of associations, and poses the problem of recovering this representation from cognitive data. Motivated by this problem, we study the following more general question: Can we reconstruct the Venn diagram of a family of sets, given the sizes of their l-wise intersections? We show that as long as the family of sets is randomly perturbed, it is enough for the number of measurements to be polynomially larger than the number of nonempty regions of the Venn diagram to fully reconstruct the diagram.

Asynchronous Gibbs sampling has been recently shown to be fast-mixing and an accurate method for estimating probabilities of events on a small number of variables of a graphical model satisfying Dobrushin's condition~\cite{DeSaOR16}. We investigate whether it can be used to accurately estimate expectations of functions of {\em all the variables} of the model. Under the same condition, we show that the synchronous (sequential) and asynchronous Gibbs samplers can be coupled so that the expected Hamming distance between their (multivariate) samples remains bounded by $O(\tau \log n),$ where $n$ is the number of variables in the graphical model, and $\tau$ is a measure of the asynchronicity. A similar bound holds for any constant power of the Hamming distance. Hence, the expectation of any function that is Lipschitz with respect to a power of the Hamming distance, can be estimated with a bias that grows logarithmically in $n$. Going beyond Lipschitz functions, we consider the bias arising from asynchronicity in estimating the expectation of polynomial functions of all variables in the model. Using recent concentration of measure results~\cite{DaskalakisDK17,GheissariLP17,GotzeSS18}, we show that the bias introduced by the asynchronicity is of smaller order than the standard deviation of the function value already present in the true model. We perform experiments on a multi-processor machine to empirically illustrate our theoretical findings.

We consider testing and learning problems on causal Bayesian networks as defined by Pearl (Pearl, 2009). Given a causal Bayesian network M on a graph with n discrete variables and bounded in-degree and bounded ``confounded components'', we show that O(log n) interventions on an unknown causal Bayesian network X on the same graph, and O(n/epsilon^2) samples per intervention, suffice to efficiently distinguish whether X=M or whether there exists some intervention under which X and M are farther than epsilon in total variation distance. We also obtain sample/time/intervention efficient algorithms for: (i) testing the identity of two unknown causal Bayesian networks on the same graph; and (ii) learning a causal Bayesian network on a given graph. Although our algorithms are non-adaptive, we show that adaptivity does not help in general: Omega(log n) interventions are necessary for testing the identity of two unknown causal Bayesian networks on the same graph, even adaptively. Our algorithms are enabled by a new subadditivity inequality for the squared Hellinger distance between two causal Bayesian networks.

Asynchronous Gibbs sampling has been recently shown to be fast-mixing and an accurate method for estimating probabilities of events on a small number of variables of a graphical model satisfying Dobrushin's condition~\cite{DeSaOR16}. We investigate whether it can be used to accurately estimate expectations of functions of {\em all the variables} of the model. Under the same condition, we show that the synchronous (sequential) and asynchronous Gibbs samplers can be coupled so that the expected Hamming distance between their (multivariate) samples remains bounded by $O(\tau \log n),$ where $n$ is the number of variables in the graphical model, and $\tau$ is a measure of the asynchronicity. A similar bound holds for any constant power of the Hamming distance. Hence, the expectation of any function that is Lipschitz with respect to a power of the Hamming distance, can be estimated with a bias that grows logarithmically in $n$. Going beyond Lipschitz functions, we consider the bias arising from asynchronicity in estimating the expectation of polynomial functions of all variables in the model. Using recent concentration of measure results, we show that the bias introduced by the asynchronicity is of smaller order than the standard deviation of the function value already present in the true model. We perform experiments on a multi-processor machine to empirically illustrate our theoretical findings.

We provide an efficient algorithm for the classical problem, going back to Galton, Pearson, and Fisher, of estimating, with arbitrary accuracy the parameters of a multivariate normal distribution from truncated samples. Truncated samples from a $d$-variate normal ${\cal N}(\mathbf{\mu},\mathbf{\Sigma})$ means a samples is only revealed if it falls in some subset $S \subseteq \mathbb{R}^d$; otherwise the samples are hidden and their count in proportion to the revealed samples is also hidden. We show that the mean $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$ can be estimated with arbitrary accuracy in polynomial-time, as long as we have oracle access to $S$, and $S$ has non-trivial measure under the unknown $d$-variate normal distribution. Additionally we show that without oracle access to $S$, any non-trivial estimation is impossible.

Motivated by applications in Game Theory, Optimization, and Generative Adversarial Networks, recent work of Daskalakis et al. and Liang and Stokes has established that a variant of the widely used Gradient Descent/Ascent procedure, called "Optimistic Gradient Descent/Ascent (OGDA)", exhibits last-iterate convergence to saddle points in {\em unconstrained} convex-concave min-max optimization problems. We show that the same holds true in the more general problem of {\em constrained} min-max optimization under a variant of the Multiplicative-Weights-Update method called "Optimistic Multiplicative-Weights Update (OMWU)". The generality of the constrained problem, which in particular captures all Linear Programming, requires fundamentally different techniques for analyzing the progress of OMWU towards min-max solutions. We show that OMWU monotonically improves the Kullback-Leibler divergence of the current iterate to the (appropriately normalized) min-max solution until it enters a neighborhood of the solution. Inside that neighborhood we show that OMWU becomes a contracting map converging to the exact solution. We experiment with zero-sum games to measure how the convergence rate scales with the dimension.

Motivated by applications in Optimization, Game Theory, and the training of Generative Adversarial Networks, the convergence properties of first order methods in min-max problems have received extensive study. It has been recognized that they may cycle, and there is no good understanding of their limit points when they do not. When they converge, do they converge to local min-max solutions? We characterize the limit points of two basic first order methods, namely Gradient Descent/Ascent (GDA) and Optimistic Gradient Descent Ascent (OGDA). We show that both dynamics avoid unstable critical points for almost all initializations. Moreover, for small step sizes and under mild assumptions, the set of \{OGDA\}-stable critical points is a superset of \{GDA\}-stable critical points, which is a superset of local min-max solutions (strict in some cases). The connecting thread is that the behavior of these dynamics can be studied from a dynamical systems perspective.

We consider testing and learning problems on causal Bayesian networks as defined by Pearl (Pearl, 2009). Given a causal Bayesian network $\mathcal{M}$ on a graph with $n$ discrete variables and bounded in-degree and bounded `confounded components', we show that $O(\log n)$ interventions on an unknown causal Bayesian network $\mathcal{X}$ on the same graph, and $\tilde{O}(n/\epsilon^2)$ samples per intervention, suffice to efficiently distinguish whether $\mathcal{X}=\mathcal{M}$ or whether there exists some intervention under which $\mathcal{X}$ and $\mathcal{M}$ are farther than $\epsilon$ in total variation distance. We also obtain sample/time/intervention efficient algorithms for: (i) testing the identity of two unknown causal Bayesian networks on the same graph; and (ii) learning a causal Bayesian network on a given graph. Although our algorithms are non-adaptive, we show that adaptivity does not help in general: $\Omega(\log n)$ interventions are necessary for testing the identity of two unknown causal Bayesian networks on the same graph, even adaptively. Our algorithms are enabled by a new subadditivity inequality for the squared Hellinger distance between two causal Bayesian networks.