# Stochastic gradients methods for statistical inference

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Statistical inference, such as hypothesis testing and calculating a confidence interval, is an important tool for accessing uncertainty in machine learning and statistical problems. Stochastic gradient methods, such as stochastic gradient descent (SGD), have recently been successfully applied to point estimation in large scale machine learning problems. In this work, we present novel stochastic gradient methods for statistical inference in large scale machine learning problems. Unregularized M -estimation using SGD. Using SGD with a fixed step size, we demonstrate that the average of such SGD sequences can be used for statistical inference, after proper scaling. An intuitive analysis using the Ornstein-Uhlenbeck process suggests that such averages are asymptotically normal. From a practical perspective, our SGD-based inference procedure is a first order method, and is well-suited for large scale problems. To show its merits, we apply it to both synthetic and real datasets, and demonstrate that its accuracy is comparable to classical statistical methods, while requiring potentially far less computation. Approximate Newton-based statistical inference using only stochastic gradients for unregularized M -estimation. We present a novel inference framework for convex empirical risk minimization, using approximate stochastic Newton steps. The proposed algorithm is based on the notion of finite differences and allows the approximation of a Hessian-vector product from first-order information. In theory, our method efficiently computes the statistical error covariance in M -estimation for unregularized convex learning problems, without using exact second order information, or resampling the entire data set. In practice, we demonstrate the effectiveness of our framework on large-scale machine learning problems, that go even beyond convexity: as a highlight, our work can be used to detect certain adversarial attacks on neural networks. High dimensional linear regression statistical inference using only stochastic gra- dients. As an extension of the approximate Newton-based statistical inference algorithm for unregularized problems, we present a similar algorithm, using only stochastic gradients, for statistical inference in high dimensional linear regression, where the number of features is much larger than the number of samples. Stochastic gradient methods for time series analysis. We present a novel stochastic gradient descent algorithm for time series analysis, which correctly captures correlation structures in a time series dataset during optimization. Instead of uniformly sampling indices in vanilla SGD, we uniformly sample contiguous blocks of indices, where the block length depends on the dataset.