More Like this

1. Estimation of the covariance structure of heavy-tailed distributions

   Minsker, Stanislav; Wei, Xiaohan (December 2017, NIPS)

   We propose and analyze a new estimator of the covariance matrix that admits strong theoretical guarantees under weak assumptions on the underlying distribution, such as existence of moments of only low order. While estimation of covariance matrices corresponding to sub-Gaussian distributions is well-understood, much less in known in the case of heavy-tailed data. As K. Balasubramanian and M. Yuan write, "data from real-world experiments oftentimes tend to be corrupted with outliers and/or exhibit heavy tails. In such cases, it is not clear that those covariance matrix estimators .. remain optimal" and "what are the other possible strategies to deal with heavy tailed distributions warrant further studies." We make a step towards answering this question and prove tight deviation inequalities for the proposed estimator that depend only on the parameters controlling the intrinsic dimension'' associated to the covariance matrix (as opposed to the dimension of the ambient space); in particular, our results are applicable in the case of high-dimensional observations. 

   more » « less
2. Learning Linear Models Using Distributed Iterative Hessian Sketching

   Wang, Han; Anderson, James (July 2022, Proceedings of Machine Learning Research)

   Firoozi, R.; Mehr, N.; Yel, E.; Antonova, R; Bohg, J.; Schwager, M.; Kochenderfer, M. (Ed.)

   This work considers the problem of learning the Markov parameters of a linear system from ob- served data. Recent non-asymptotic system identification results have characterized the sample complexity of this problem in the single and multi-rollout setting. In both instances, the number of samples required in order to obtain acceptable estimates can produce optimization problems with an intractably large number of decision variables for a second-order algorithm. We show that a randomized and distributed Newton algorithm based on Hessian-sketching can produce ε-optimal solutions and converges geometrically. Moreover, the algorithm is trivially parallelizable. Our re- sults hold for a variety of sketching matrices and we illustrate the theory with numerical examples. 

   more » « less
3. Estimating the number and effect sizes of non-null hypotheses

   Brennan, Jennifer; Vinayak, Ramya K.; Jamieson, Kevin (July 2020, Proceedings of Machine Learning Research)

   We study the problem of estimating the distribution of effect sizes (the mean of the test statistic under the alternate hypothesis) in a multiple testing setting. Knowing this distribution allows us to calculate the power (type II error) of any experimental design. We show that it is possible to estimate this distribution using an inexpensive pilot experiment, which takes significantly fewer samples than would be required by an experiment that identified the discoveries. Our estimator can be used to guarantee the number of discoveries that will be made using a given experimental design in a future experiment. We prove that this simple and computationally efficient estimator enjoys a number of favorable theoretical properties, and demonstrate its effectiveness on data from a gene knockout experiment on influenza inhibition in Drosophila. 

   more » « less
4. Estimating the number and effect sizes of non-null hypotheses

   Brennan, Jennifer; Vinayak, Ramya K.; Jamieson, Kevin (January 2020, Proceedings of Machine Learning Research)

   We study the problem of estimating the distribution of effect sizes (the mean of the test statistic under the alternate hypothesis) in a multiple testing setting. Knowing this distribution allows us to calculate the power (type II error) of any experimental design. We show that it is possible to estimate this distribution using an inexpensive pilot experiment, which takes significantly fewer samples than would be required by an experiment that identified the discoveries. Our estimator can be used to guarantee the number of discoveries that will be made using a given experimental design in a future experiment. We prove that this simple and computationally efficient estimator enjoys a number of favorable theoretical properties, and demonstrate its effectiveness on data from a gene knockout experiment on influenza inhibition in Drosophila. 

   more » « less
5. Optimal Algorithms for Augmented Testing of Discrete Distributions

   Aliakbarpour, Maryam; Indyk, Piotr; Rubinfeld, Ronitt; Silwal, Sandeep (December 2024, Advances in Neural Information Processing Systems 38: Annual Conference on Neural Information Processing Systems (NeurIPS 2024))

   We consider the problem of hypothesis testing for discrete distributions. In the standard model, where we have sample access to an underlying distribution p, extensive research has established optimal bounds for uniformity testing, identity testing (goodness of fit), and closeness testing (equivalence or two-sample testing). We explore these problems in a setting where a predicted data distribution, possibly derived from historical data or predictive machine learning models, is available. We demonstrate that such a predictor can indeed reduce the number of samples required for all three property testing tasks. The reduction in sample complexity depends directly on the predictor’s quality, measured by its total variation distance from p. A key advantage of our algorithms is their adaptability to the precision of the prediction. Specifically, our algorithms can self-adjust their sample complexity based on the accuracy of the available prediction, operating without any prior knowledge of the estimation’s accuracy (i.e. they are consistent). Additionally, we never use more samples than the standard approaches require, even if the predictions provide no meaningful information (i.e. they are also robust). We provide lower bounds to indicate that the improvements in sample complexity achieved by our algorithms are information-theoretically optimal. Furthermore, experimental results show that the performance of our algorithms on real data significantly exceeds our worst-case guarantees for sample complexity, demonstrating the practicality of our approach. 

   more » « less