Browsing by Subject "Statistical inference"
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Item Dirty statistical models(2012-05) Jalali, Ali, 1982-; Sanghavi, Sujay Rajendra, 1979-; Caramanis, Constantine; Ghosh, Joydeep; Dhillon, Inderjit; Ravikumar, PradeepIn fields across science and engineering, we are increasingly faced with problems where the number of variables or features we need to estimate is much larger than the number of observations. Under such high-dimensional scaling, for any hope of statistically consistent estimation, it becomes vital to leverage any potential structure in the problem such as sparsity, low-rank structure or block sparsity. However, data may deviate significantly from any one such statistical model. The motivation of this thesis is: can we simultaneously leverage more than one such statistical structural model, to obtain consistency in a larger number of problems, and with fewer samples, than can be obtained by single models? Our approach involves combining via simple linear superposition, a technique we term dirty models. The idea is very simple: while any one structure might not capture the data, a superposition of structural classes might. Dirty models thus searches for a parameter that can be decomposed into a number of simpler structures such as (a) sparse plus block-sparse, (b) sparse plus low-rank and (c) low-rank plus block-sparse. In this thesis, we propose dirty model based algorithms for different problems such as multi-task learning, graph clustering and time-series analysis with latent factors. We analyze these algorithms in terms of the number of observations we need to estimate the variables. These algorithms are based on convex optimization and sometimes they are relatively slow. We provide a class of low-complexity greedy algorithms that not only can solve these optimizations faster, but also guarantee the solution. Other than theoretical results, in each case, we provide experimental results to illustrate the power of dirty models.Item Interpretable random structure for non-standard data with applications to biomedical and social science studies(2022-05-02) Liu, Zhengqing; Müller, Peter, 1963 August 9-; Sarkar, Abhra; Somer-Topcu, Zeynep; Walker, Stephen G; Zigler, Corwin MIn this dissertation, we introduce examples of non-standard data that occur in statistical inference for biomedical and social science research. We develop novel statistical methodology that aims to make interpretable and practically meaningful inference from the datasets. The first project proposes a hypothesis testing procedure applied to phylogenetic trees that represent evolutionary paths of seasonal influenza strains. The method quantifies the change of genetic composition of seasonal influenza over years and serves as a crucial step to inform vaccine selection. The second project uses information from atmospherical studies that describe the movement and dispersion of pollutants emitted from coal powered factory. The goal is to make valid causal statement on how intervention applied at factories may affect the health outcome of surrounding community, which is an important step to make policy recommendation. The third project proposes a joint model to analyze tweet contents posted by UK general election candidates. The proposed method effectively borrows information from external source, such as concurrent newspaper. The joint model describes the types of issues candidates tend to focus on, given their demographic information and the election constituency they represent.Item Learning from aggregated data(2019-02-11) Bhowmik, Avradeep; Ghosh, Joydeep; Vikalo, Haris; Sanghavi, Sujay; Dimakis, Georgios-Alexandros; Koyejo, OluwasanmiData aggregation is ubiquitous in modern life. Due to various reasons like privacy, scalability, robustness, etc., ground truth data is often subjected to aggregation before being released to the public, or utilised by researchers and analysts. Learning from aggregated data is a challenging problem that requires significant algorithmic innovation, since naive application of standard techniques to aggregated data is vulnerable to the ecological fallacy. In this work, we explore three different versions of this setting. First, we tackle the problem of using generalised linear models when features/covariates are fully observed but the targets are only available as histograms- a common scenario in the healthcare domain where many datasets contain both non-sensitive attributes like age, sex, zip-code, etc., as well as privacy sensitive attributes like healthcare records. We introduce an efficient algorithm that uses alternating data imputation and GLM estimation steps to learn predictive models in this setting. Next, we look at the problem of learning sparse linear models when both features and targets are in aggregated form, specified as empirical estimates of group-wise means computed over different sub-groups of the population. We show that if the true sub-populations are heterogeneous enough, the optimal sparse parameter can be recovered within an arbitrarily small tolerance even in the presence of noise, provided the empirical estimates are obtained from a sufficiently large number of observations. Third, we tackle the scenario of predictive modelling with data that is subjected to spatio-temporal aggregation. We show that by formulating the problem in the frequency domain, we can bypass the mathematical and representational challenges that arise due to non-uniform aggregation, misaligned sampling periods and aliasing. We introduce a novel algorithm that uses restricted Fourier transforms to estimate a linear model which, when applied to spatio-temporally aggregated data, has a generalisation error that is provably close to the optimal performance by the best possible linear model that can be learned from the non-aggregated data set. We then focus our attention on the complementary problem that involves designing aggregation strategies that can allow learning, as well as developing algorithmic techniques that can use only the aggregates to train a model that works on individual samples. We motivate our methods by using the example of Gaussian regression, and subsequently extend our techniques to subsume binary classifiers and generalised linear models. We deonstrate the effectiveness of our techniques with empirical evaluation on data from healthcare and telecommunication. Finally, we present a concrete example of our methods applied to a real life practical problem. Specifically, we consider an application in the domain of online advertising where the complexity of bidding strategies require accurate estimates of most probable cost-per-click or CPC incurred by advertisers, but the data used for training these CPC prediction models are only available as aggregated invoices supplied by an ad publisher on a daily or hourly basis. We introduce a novel learning framework that can use aggregates computed at varying levels of granularity for building individual-level predictive models. We generalise our modelling and algorithmic framework to handle data from diverse domains, and extend our techniques to cover arbitrary aggregation paradigms like sliding windows and overlapping/non-uniform aggregation. We show empirical evidence for the efficacy of our techniques with experiments on both synthetic data and real data from the online advertising domain as well as healthcare to demonstrate the wider applicability of our framework.Item Stochastic gradients methods for statistical inference(2019-05) Li, Tianyang, Ph. D.; Caramanis, Constantine; Dimakis, Alexandros; Huang, Qixing; Kyrillidis, AnastasiosStatistical 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.