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In this work we consider the approximability of $\textsf{Max-CSP}(f)$ in the context of sketching algorithms and completely characterize the approximability of all Boolean CSPs. Specifically, given $f$, $\gamma$ and $\beta$ we show that either (1) the $(\gamma,\beta)$-approximation version of $\textsf{Max-CSP}(f)$ has a linear sketching algorithm using $O(\log n)$ space, or (2) for every $\epsilon > 0$ the $(\gamma-\epsilon,\beta+\epsilon)$-approximation version of $\textsf{Max-CSP}(f)$ requires $\Omega(\sqrt{n})$ space for any sketching algorithm. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of $k=2$ where we get a dichotomy and the case when the satisfying assignments of $f$ support a distribution on $\{-1,1\}^k$ with uniform marginals. Our positive results show wider applicability of bias-based algorithms used previously by [GVV17] and [CGV20] by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [KKS15], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results.

Imitation Learning from observation describes policy learning in a similar way to human learning. An agent's policy is trained by observing an expert performing a task. While many state-only imitation learning approaches are based on adversarial imitation learning, one main drawback is that adversarial training is often unstable and lacks a reliable convergence estimator. If the true environment reward is unknown and cannot be used to select the best-performing model, this can result in bad real-world policy performance. We propose a non-adversarial learning-from-observations approach, together with an interpretable convergence and performance metric. Our training objective minimizes the Kulback-Leibler divergence (KLD) between the policy and expert state transition trajectories which can be optimized in a non-adversarial fashion. Such methods demonstrate improved robustness when learned density models guide the optimization. We further improve the sample efficiency by rewriting the KLD minimization as the Soft Actor Critic objective based on a modified reward using additional density models that estimate the environment's forward and backward dynamics. Finally, we evaluate the effectiveness of our approach on well-known continuous control environments and show state-of-the-art performance while having a reliable performance estimator compared to several recent learning-from-observation methods.

Deep learning frameworks such as TensorFlow and PyTorch provide a productive interface for expressing and training a deep neural network (DNN) model on a single device or using data parallelism. Still, they may not be flexible or efficient enough in training emerging large models on distributed devices, which require more sophisticated parallelism beyond data parallelism. Plugins or wrappers have been developed to strengthen these frameworks for model or pipeline parallelism, but they complicate the usage and implementation of distributed deep learning. Aiming at a simple, neat redesign of distributed deep learning frameworks for various parallelism paradigms, we present OneFlow, a novel distributed training framework based on an SBP (split, broadcast and partial-value) abstraction and the actor model. SBP enables much easier programming of data parallelism and model parallelism than existing frameworks, and the actor model provides a succinct runtime mechanism to manage the complex dependencies imposed by resource constraints, data movement and computation in distributed deep learning. We demonstrate the general applicability and efficiency of OneFlow for training various large DNN models with case studies and extensive experiments. The results show that OneFlow outperforms many well-known customized libraries built on top of the state-of-the-art frameworks. The code of OneFlow is available at: //github.com/Oneflow-Inc/oneflow.

In this paper, we introduce reduced-bias estimators for the estimation of the tail index of a Pareto-type distribution. This is achieved through the use of a regularised weighted least squares with an exponential regression model for log-spacings of top order statistics. The asymptotic properties of the proposed estimators are investigated analytically and found to be asymptotically unbiased, consistent and normally distributed. Also, the finite sample behaviour of the estimators are studied through a simulations theory. The proposed estimators were found to yield low bias and MSE. In addition, the proposed estimators are illustrated through the estimation of the tail index of the underlying distribution of claims from the insurance industry.

We consider the problem of estimating high-dimensional covariance matrices of $K$-populations or classes in the setting where the sample sizes are comparable to the data dimension. We propose estimating each class covariance matrix as a distinct linear combination of all class sample covariance matrices. This approach is shown to reduce the estimation error when the sample sizes are limited, and the true class covariance matrices share a somewhat similar structure. We develop an effective method for estimating the coefficients in the linear combination that minimize the mean squared error under the general assumption that the samples are drawn from (unspecified) elliptically symmetric distributions possessing finite fourth-order moments. To this end, we utilize the spatial sign covariance matrix, which we show (under rather general conditions) to be an asymptotically unbiased estimator of the normalized covariance matrix as the dimension grows to infinity. We also show how the proposed method can be used in choosing the regularization parameters for multiple target matrices in a single class covariance matrix estimation problem. We assess the proposed method via numerical simulation studies including an application in global minimum variance portfolio optimization using real stock data.

We consider generative adversarial networks (GAN) for estimating parameters in a deep generative model. The data-generating distribution is assumed to concentrate around some low-dimensional structure, making the target distribution singular to the Lebesgue measure. Under this assumption, we obtain convergence rates of a GAN type estimator with respect to the Wasserstein metric. The convergence rate depends only on the noise level, intrinsic dimension and smoothness of the underlying structure. Furthermore, the rate is faster than that obtained by likelihood approaches, which provides insights into why GAN approaches perform better in many real problems. A lower bound of the minimax optimal rate is also investigated.

For real symmetric matrices that are accessible only through matrix vector products, we present Monte Carlo estimators for computing the diagonal elements. Our probabilistic bounds for normwise absolute and relative errors apply to Monte Carlo estimators based on random Rademacher, sparse Rademacher, normalized and unnormalized Gaussian vectors, and to vectors with bounded fourth moments. The novel use of matrix concentration inequalities in our proofs represents a systematic model for future analyses. Our bounds mostly do not depend on the matrix dimension, target different error measures than existing work, and imply that the accuracy of the estimators increases with the diagonal dominance of the matrix. An application to derivative-based global sensitivity metrics corroborates this, as do numerical experiments on synthetic test matrices. We recommend against the use in practice of sparse Rademacher vectors, which are the basis for many randomized sketching and sampling algorithms, because they tend to deliver barely a digit of accuracy even under large sampling amounts.

This paper uses techniques from Random Matrix Theory to find the ideal training-testing data split for a simple linear regression with m data points, each an independent n-dimensional multivariate Gaussian. It defines "ideal" as satisfying the integrity metric, i.e. the empirical model error is the actual measurement noise, and thus fairly reflects the value or lack of same of the model. This paper is the first to solve for the training and test size for any model in a way that is truly optimal. The number of data points in the training set is the root of a quartic polynomial Theorem 1 derives which depends only on m and n; the covariance matrix of the multivariate Gaussian, the true model parameters, and the true measurement noise drop out of the calculations. The critical mathematical difficulties were realizing that the problems herein were discussed in the context of the Jacobi Ensemble, a probability distribution describing the eigenvalues of a known random matrix model, and evaluating a new integral in the style of Selberg and Aomoto. Mathematical results are supported with thorough computational evidence. This paper is a step towards automatic choices of training/test set sizes in machine learning.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.