Traditional nonparametric estimation methods often lead to a slow convergence rate in large dimensions and require unrealistically enormous sizes of datasets for reliable conclusions. We develop an approach based on mixed gradients, either observed or estimated, to effectively estimate the function at near-parametric convergence rates. The novel approach and computational algorithm could lead to methods useful to practitioners in many areas of science and engineering. Our theoretical results reveal a behavior universal to this class of nonparametric estimation problems. We explore a general setting involving tensor product spaces and build upon the smoothing spline analysis of variance (SS-ANOVA) framework. For $d$-dimensional models under full interaction, the optimal rates with gradient information on $p$ covariates are identical to those for the $(d-p)$-interaction models without gradients and, therefore, the models are immune to the "curse of interaction". For additive models, the optimal rates using gradient information are root-$n$, thus achieving the "parametric rate". We demonstrate aspects of the theoretical results through synthetic and real data applications.
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Estimating the expectations of functionals applied to sums of random variables (RVs) is a well-known problem encountered in many challenging applications. Generally, closed-form expressions of these quantities are out of reach. A naive Monte Carlo simulation is an alternative approach. However, this method requires numerous samples for rare event problems. Therefore, it is paramount to use variance reduction techniques to develop fast and efficient estimation methods. In this work, we use importance sampling (IS), known for its efficiency in requiring fewer computations to achieve the same accuracy requirements. We propose a state-dependent IS scheme based on a stochastic optimal control formulation, where the control is dependent on state and time. We aim to calculate rare event quantities that could be written as an expectation of a functional of the sums of independent RVs. The proposed algorithm is generic and can be applied without restrictions on the univariate distributions of RVs or the functional applied to the sum. We apply this approach to the log-normal distribution to compute the left tail and cumulative distribution of the ratio of independent RVs. For each case, we numerically demonstrate that the proposed state-dependent IS algorithm compares favorably to most well-known estimators dealing with similar problems.
Preferential sampling is a common feature in geostatistics and occurs when the locations to be sampled are chosen based on information about the phenomena under study. In this case, point pattern models are commonly used as the probability law for the distribution of the locations. However, analytic intractability of the point process likelihood prevents its direct calculation. Many Bayesian (and non-Bayesian) approaches in non-parametric model specifications handle this difficulty with approximation-based methods. These approximations involve errors that are difficult to quantify and can lead to biased inference. This paper presents an approach for performing exact Bayesian inference for this setting without the need for model approximation. A qualitatively minor change on the traditional model is proposed to circumvent the likelihood intractability. This change enables the use of an augmented model strategy. Recent work on Bayesian inference for point pattern models can be adapted to the geostatistics setting and renders computational tractability for exact inference for the proposed methodology. Estimation of model parameters and prediction of the response at unsampled locations can then be obtained from the joint posterior distribution of the augmented model. Simulated studies showed good quality of the proposed model for estimation and prediction in a variety of preferentiality scenarios. The performance of our approach is illustrated in the analysis of real datasets and compares favourably against approximation-based approaches. The paper is concluded with comments regarding extensions of and improvements to the proposed methodology.
We present an alternating least squares type numerical optimization scheme to estimate conditionally-independent mixture models in $\mathbb{R}^n$, with minimal additional distributional assumptions. Following the method of moments, we tackle a coupled system of low-rank tensor decomposition problems. The steep costs associated with high-dimensional tensors are avoided, through the development of specialized tensor-free operations. Numerical experiments illustrate the performance of the algorithm and its applicability to various models and applications. In many cases the results exhibit improved reliability over the expectation-maximization algorithm, with similar time and storage costs. We also provide some supporting theory, establishing identifiability and local linear convergence.
The objective of this paper is to investigate a new numerical method for the approximation of the self-diffusion matrix of a tagged particle process defined on a grid. While standard numerical methods make use of long-time averages of empirical means of deviations of some stochastic processes, and are thus subject to statistical noise, we propose here a tensor method in order to compute an approximation of the solution of a high-dimensional quadratic optimization problem, which enables to obtain a numerical approximation of the self-diffusion matrix. The tensor method we use here relies on an iterative scheme which builds low-rank approximations of the quantity of interest and on a carefully tuned variance reduction method so as to evaluate the various terms arising in the functional to minimize. In particular, we numerically observe here that it is much less subject to statistical noise than classical approaches.
In randomized experiments and observational studies, weighting methods are often used to generalize and transport treatment effect estimates to a target population. Traditional methods construct the weights by separately modeling the treatment assignment and study selection probabilities and then multiplying functions (e.g., inverses) of their estimates. However, these estimated multiplicative weights may not produce adequate covariate balance and can be highly variable, resulting in biased and unstable estimators, especially when there is limited covariate overlap across populations or treatment groups. To address these limitations, we propose a general weighting approach that weights each treatment group towards the target population in a single step. We present a framework and provide a justification for this one-step approach in terms of generic probability distributions. We show a formal connection between our method and inverse probability and inverse odds weighting. By construction, the proposed approach balances covariates and produces stable estimators. We show that our estimator for the target average treatment effect is consistent, asymptotically Normal, multiply robust, and semiparametrically efficient. We demonstrate the performance of this approach using a simulation study and a randomized case study on the effects of physician racial diversity on preventive healthcare utilization among Black men in California.
Online imitation learning is the problem of how best to mimic expert demonstrations, given access to the environment or an accurate simulator. Prior work has shown that in the infinite sample regime, exact moment matching achieves value equivalence to the expert policy. However, in the finite sample regime, even if one has no optimization error, empirical variance can lead to a performance gap that scales with $H^2 / N$ for behavioral cloning and $H / \sqrt{N}$ for online moment matching, where $H$ is the horizon and $N$ is the size of the expert dataset. We introduce the technique of replay estimation to reduce this empirical variance: by repeatedly executing cached expert actions in a stochastic simulator, we compute a smoother expert visitation distribution estimate to match. In the presence of general function approximation, we prove a meta theorem reducing the performance gap of our approach to the parameter estimation error for offline classification (i.e. learning the expert policy). In the tabular setting or with linear function approximation, our meta theorem shows that the performance gap incurred by our approach achieves the optimal $\widetilde{O} \left( \min({H^{3/2}} / {N}, {H} / {\sqrt{N}} \right)$ dependency, under significantly weaker assumptions compared to prior work. We implement multiple instantiations of our approach on several continuous control tasks and find that we are able to significantly improve policy performance across a variety of dataset sizes.
Bayesian optimisation (BO) algorithms have shown remarkable success in applications involving expensive black-box functions. Traditionally BO has been set as a sequential decision-making process which estimates the utility of query points via an acquisition function and a prior over functions, such as a Gaussian process. Recently, however, a reformulation of BO via density-ratio estimation (BORE) allowed reinterpreting the acquisition function as a probabilistic binary classifier, removing the need for an explicit prior over functions and increasing scalability. In this paper, we present a theoretical analysis of BORE's regret and an extension of the algorithm with improved uncertainty estimates. We also show that BORE can be naturally extended to a batch optimisation setting by recasting the problem as approximate Bayesian inference. The resulting algorithms come equipped with theoretical performance guarantees and are assessed against other batch and sequential BO baselines in a series of experiments.
Digital sensors can lead to noisy results under many circumstances. To be able to remove the undesired noise from images, proper noise modeling and an accurate noise parameter estimation is crucial. In this project, we use a Poisson-Gaussian noise model for the raw-images captured by the sensor, as it fits the physical characteristics of the sensor closely. Moreover, we limit ourselves to the case where observed (noisy), and ground-truth (noise-free) image pairs are available. Using such pairs is beneficial for the noise estimation and is not widely studied in literature. Based on this model, we derive the theoretical maximum likelihood solution, discuss its practical implementation and optimization. Further, we propose two algorithms based on variance and cumulant statistics. Finally, we compare the results of our methods with two different approaches, a CNN we trained ourselves, and another one taken from literature. The comparison between all these methods shows that our algorithms outperform the others in terms of MSE and have good additional properties.
This work provides a theoretical analysis for optimally solving the pose estimation problem using total least squares for vector observations from landmark features, which is central to applications involving simultaneous localization and mapping. First, the optimization process is formulated with observation vectors extracted from point-cloud features. Then, error-covariance expressions are derived. The attitude and position estimates obtained via the derived optimization process are proven to reach the bounds defined by the Cram\'er-Rao lower bound under the small-angle approximation of attitude errors. A fully populated observation noise-covariance matrix is assumed as the weight in the cost function to cover the most general case of the sensor uncertainty. This includes more generic correlations in the errors than previous cases involving an isotropic noise assumption. The proposed solution is verified using Monte Carlo simulations and an experiment with an actual LIDAR to validate the error-covariance analysis.
Derivatives are a key nonparametric functional in wide-ranging applications where the rate of change of an unknown function is of interest. In the Bayesian paradigm, Gaussian processes (GPs) are routinely used as a flexible prior for unknown functions, and are arguably one of the most popular tools in many areas. However, little is known about the optimal modelling strategy and theoretical properties when using GPs for derivatives. In this article, we study a plug-in strategy by differentiating the posterior distribution with GP priors for derivatives of any order. This practically appealing plug-in GP method has been previously perceived as suboptimal and degraded, but this is not necessarily the case. We provide posterior contraction rates for plug-in GPs and establish that they remarkably adapt to derivative orders. We show that the posterior measure of the regression function and its derivatives, with the same choice of hyperparameter that does not depend on the order of derivatives, converges at the minimax optimal rate up to a logarithmic factor for functions in certain classes. This to the best of our knowledge provides the first positive result for plug-in GPs in the context of inferring derivative functionals, and leads to a practically simple nonparametric Bayesian method with guided hyperparameter tuning for simultaneously estimating the regression function and its derivatives. Simulations show competitive finite sample performance of the plug-in GP method. A climate change application on analyzing the global sea-level rise is discussed.
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