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Register a new userEstimation of Accurate and Calibrated Uncertainties in Deterministic models
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In this paper we focus on the problem of assigning uncertainties to single-point predictions generated by a deterministic model that outputs a continuous variable. This problem applies to any state-of-the-art physics or engineering models that have a computational cost that does not readily allow to run ensembles and to estimate the uncertainty associated to single-point predictions. Essentially, we devise a method to easily transform a deterministic prediction into a probabilistic one. We show that for doing so, one has to compromise between the accuracy and the reliability (calibration) of such a probabilistic model. Hence, we introduce a cost function that encodes their trade-off. We use the Continuous Rank Probability Score to measure accuracy and we derive an analytic formula for the reliability, in the case of forecasts of continuous scalar variables expressed in terms of Gaussian distributions. The new Accuracy-Reliability cost function is then used to estimate the input-dependent variance, given a black-box mean function, by solving a two-objective optimization problem. The simple philosophy behind this strategy is that predictions based on the estimated variances should not only be accurate, but also reliable (i.e. statistical consistent with observations). Conversely, early works based on the minimization of classical cost functions, such as the negative log probability density, cannot simultaneously enforce both accuracy and reliability. We show several examples both with synthetic data, where the underlying hidden noise can accurately be recovered, and with large real-world datasets.
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When the data are stored in a distributed manner, direct application of traditional statistical inference procedures is often prohibitive due to communication cost and privacy concerns. This paper develops and investigates two Communication-Efficient Accurate Statistical Estimators (CEASE), implemented through iterative algorithms for distributed optimization. In each iteration, node machines carry out computation in parallel and communicate with the central processor, which then broadcasts aggregated information to node machines for new updates. The algorithms adapt to the similarity among loss functions on node machines, and converge rapidly when each node machine has large enough sample size. Moreover, they do not require good initialization and enjoy linear converge guarantees under general conditions. The contraction rate of optimization errors is presented explicitly, with dependence on the local sample size unveiled. In addition, the improved statistical accuracy per iteration is derived. By regarding the proposed method as a multi-step statistical estimator, we show that statistical efficiency can be achieved in finite steps in typical statistical applications. In addition, we give the conditions under which the one-step CEASE estimator is statistically efficient. Extensive numerical experiments on both synthetic and real data validate the theoretical results and demonstrate the superior performance of our algorithms.
Estimators computed from adaptively collected data do not behave like their non-adaptive brethren. Rather, the sequential dependence of the collection policy can lead to severe distributional biases that persist even in the infinite data limit. We develop a general method -- $mathbf{W}$-decorrelation -- for transforming the bias of adaptive linear regression estimators into variance. The method uses only coarse-grained information about the data collection policy and does not need access to propensity scores or exact knowledge of the policy. We bound the finite-sample bias and variance of the $mathbf{W}$-estimator and develop asymptotically correct confidence intervals based on a novel martingale central limit theorem. We then demonstrate the empirical benefits of the generic $mathbf{W}$-decorrelation procedure in two different adaptive data settings: the multi-armed bandit and the autoregressive time series.
Mixtures-of-Experts models and their maximum likelihood estimation (MLE) via the EM algorithm have been thoroughly studied in the statistics and machine learning literature. They are subject of a growing investigation in the context of modeling with high-dimensional predictors with regularized MLE. We examine MoE with Gaussian gating network, for clustering and regression, and propose an $ell_1$-regularized MLE to encourage sparse models and deal with the high-dimensional setting. We develop an EM-Lasso algorithm to perform parameter estimation and utilize a BIC-like criterion to select the model parameters, including the sparsity tuning hyperparameters. Experiments conducted on simulated data show the good performance of the proposed regularized MLE compared to the standard MLE with the EM algorithm.
Mixture of Experts (MoE) are successful models for modeling heterogeneous data in many statistical learning problems including regression, clustering and classification. Generally fitted by maximum likelihood estimation via the well-known EM algorithm, their application to high-dimensional problems is still therefore challenging. We consider the problem of fitting and feature selection in MoE models, and propose a regularized maximum likelihood estimation approach that encourages sparse solutions for heterogeneous regression data models with potentially high-dimensional predictors. Unlike state-of-the art regularized MLE for MoE, the proposed modelings do not require an approximate of the penalty function. We develop two hybrid EM algorithms: an Expectation-Majorization-Maximization (EM/MM) algorithm, and an EM algorithm with coordinate ascent algorithm. The proposed algorithms allow to automatically obtaining sparse solutions without thresholding, and avoid matrix inversion by allowing univariate parameter updates. An experimental study shows the good performance of the algorithms in terms of recovering the actual sparse solutions, parameter estimation, and clustering of heterogeneous regression data.
We propose a novel supervised multi-class/single-label classifier that maps training data onto a linearly separable latent space with a simplex-like geometry. This approach allows us to transform the classification problem into a well-defined regression problem. For its solution we can choose suitable distance metrics in feature space and regression models predicting latent space coordinates. A benchmark on various artificial and real-world data sets is used to demonstrate the calibration qualities and prediction performance of our classifier.
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