Integral Operator Approaches for Scattered Data Fitting on Spheres

This paper focuses on scattered data fitting problems on spheres. We study the approximation performance of a class of weighted spectral filter algorithms, including Tikhonov regularization, Landaweber iteration, spectral cut-off, and iterated Tikhonov, in fitting noisy data with possibly unbounded random noise. For the analysis, we develop an integral operator approach that can be regarded as an extension of the widely used sampling inequality approach and norming set method in the community of scattered data fitting. After providing an equivalence between the operator differences and quadrature rules, we succeed in deriving optimal Sobolev-type error estimates of weighted spectral filter algorithms. Our derived error estimates do not suffer from the saturation phenomenon for Tikhonov regularization in the literature, native-space-barrier for existing error analysis and adapts to different embedding spaces. We also propose a divide-and-conquer scheme to equip weighted spectral filter algorithms to reduce their computational burden and present the optimal approximation error bounds.

Weighted Spectral Filters for Kernel Interpolation on Spheres: Estimates of Prediction Accuracy for Noisy Data

Spherical radial-basis-based kernel interpolation abounds in image sciences including geophysical image reconstruction, climate trends description and image rendering due to its excellent spatial localization property and perfect approximation performance. However, in dealing with noisy data, kernel interpolation frequently behaves not so well due to the large condition number of the kernel matrix and instability of the interpolation process. In this paper, we introduce a weighted spectral filter approach to reduce the condition number of the kernel matrix and then stabilize kernel interpolation. The main building blocks of the proposed method are the well developed spherical positive quadrature rules and high-pass spectral filters. Using a recently developed integral operator approach for spherical data analysis, we theoretically demonstrate that the proposed weighted spectral filter approach succeeds in breaking through the bottleneck of kernel interpolation, especially in fitting noisy data. We provide optimal approximation rates of the new method to show that our approach does not compromise the predicting accuracy. Furthermore, we conduct both toy simulations and two real-world data experiments with synthetically added noise in geophysical image reconstruction and climate image processing to verify our theoretical assertions and show the feasibility of the weighted spectral filter approach.

Adaptive Parameter Selection for Kernel Ridge Regression

This paper focuses on parameter selection issues of kernel ridge regression (KRR). Due to special spectral properties of KRR, we find that delicate subdivision of the parameter interval shrinks the difference between two successive KRR estimates. Based on this observation, we develop an early-stopping type parameter selection strategy for KRR according to the so-called Lepskii-type principle. Theoretical verifications are presented in the framework of learning theory to show that KRR equipped with the proposed parameter selection strategy succeeds in achieving optimal learning rates and adapts to different norms, providing a new record of parameter selection for kernel methods.

Lifting the Veil: Unlocking the Power of Depth in Q-learning

With the help of massive data and rich computational resources, deep Q-learning has been widely used in operations research and management science and has contributed to great success in numerous applications, including recommender systems, supply chains, games, and robotic manipulation. However, the success of deep Q-learning lacks solid theoretical verification and interpretability. The aim of this paper is to theoretically verify the power of depth in deep Q-learning. Within the framework of statistical learning theory, we rigorously prove that deep Q-learning outperforms its traditional version by demonstrating its good generalization error bound. Our results reveal that the main reason for the success of deep Q-learning is the excellent performance of deep neural networks (deep nets) in capturing the special properties of rewards namely, spatial sparseness and piecewise constancy, rather than their large capacities. In this paper, we make fundamental contributions to the field of reinforcement learning by answering to the following three questions: Why does deep Q-learning perform so well? When does deep Q-learning perform better than traditional Q-learning? How many samples are required to achieve a specific prediction accuracy for deep Q-learning? Our theoretical assertions are verified by applying deep Q-learning in the well-known beer game in supply chain management and a simulated recommender system.

Distributed Uncertainty Quantification of Kernel Interpolation on Spheres

For radial basis function (RBF) kernel interpolation of scattered data, Schaback in 1995 proved that the attainable approximation error and the condition number of the underlying interpolation matrix cannot be made small simultaneously. He referred to this finding as an "uncertainty relation", an undesirable consequence of which is that RBF kernel interpolation is susceptible to noisy data. In this paper, we propose and study a distributed interpolation method to manage and quantify the uncertainty brought on by interpolating noisy spherical data of non-negligible magnitude. We also present numerical simulation results showing that our method is practical and robust in terms of handling noisy data from challenging computing environments.

Adaptive Distributed Kernel Ridge Regression: A Feasible Distributed Learning Scheme for Data Silos

Data silos, mainly caused by privacy and interoperability, significantly constrain collaborations among different organizations with similar data for the same purpose. Distributed learning based on divide-and-conquer provides a promising way to settle the data silos, but it suffers from several challenges, including autonomy, privacy guarantees, and the necessity of collaborations. This paper focuses on developing an adaptive distributed kernel ridge regression (AdaDKRR) by taking autonomy in parameter selection, privacy in communicating non-sensitive information, and the necessity of collaborations in performance improvement into account. We provide both solid theoretical verification and comprehensive experiments for AdaDKRR to demonstrate its feasibility and effectiveness. Theoretically, we prove that under some mild conditions, AdaDKRR performs similarly to running the optimal learning algorithms on the whole data, verifying the necessity of collaborations and showing that no other distributed learning scheme can essentially beat AdaDKRR under the same conditions. Numerically, we test AdaDKRR on both toy simulations and two real-world applications to show that AdaDKRR is superior to other existing distributed learning schemes. All these results show that AdaDKRR is a feasible scheme to defend against data silos, which are highly desired in numerous application regions such as intelligent decision-making, pricing forecasting, and performance prediction for products.

Optimal Approximation and Learning Rates for Deep Convolutional Neural Networks

This paper focuses on approximation and learning performance analysis for deep convolutional neural networks with zero-padding and max-pooling. We prove that, to approximate $r$-smooth function, the approximation rates of deep convolutional neural networks with depth $L$ are of order $ (L^2/\log L)^{-2r/d} $, which is optimal up to a logarithmic factor. Furthermore, we deduce almost optimal learning rates for implementing empirical risk minimization over deep convolutional neural networks.

Deep Convolutional Neural Networks with Zero-Padding: Feature Extraction and Learning

This paper studies the performance of deep convolutional neural networks (DCNNs) with zero-padding in feature extraction and learning. After verifying the roles of zero-padding in enabling translation-equivalence, and pooling in its translation-invariance driven nature, we show that with similar number of free parameters, any deep fully connected networks (DFCNs) can be represented by DCNNs with zero-padding. This demonstrates that DCNNs with zero-padding is essentially better than DFCNs in feature extraction. Consequently, we derive universal consistency of DCNNs with zero-padding and show its translation-invariance in the learning process. All our theoretical results are verified by numerical experiments including both toy simulations and real-data running.

Sketching with Spherical Designs for Noisy Data Fitting on Spheres

This paper proposes a sketching strategy based on spherical designs, which is applied to the classical spherical basis function approach for massive spherical data fitting. We conduct theoretical analysis and numerical verifications to demonstrate the feasibility of the proposed { sketching} strategy. From the theoretical side, we prove that sketching based on spherical designs can reduce the computational burden of the spherical basis function approach without sacrificing its approximation capability. In particular, we provide upper and lower bounds for the proposed { sketching} strategy to fit noisy data on spheres. From the experimental side, we numerically illustrate the feasibility of the sketching strategy by showing its comparable fitting performance with the spherical basis function approach. These interesting findings show that the proposed sketching strategy is capable of fitting massive and noisy data on spheres.

Kernel-Based Distributed Q-Learning: A Scalable Reinforcement Learning Approach for Dynamic Treatment Regimes

In recent years, large amounts of electronic health records (EHRs) concerning chronic diseases, such as cancer, diabetes, and mental disease, have been collected to facilitate medical diagnosis. Modeling the dynamic properties of EHRs related to chronic diseases can be efficiently done using dynamic treatment regimes (DTRs), which are a set of sequential decision rules. While Reinforcement learning (RL) is a widely used method for creating DTRs, there is ongoing research in developing RL algorithms that can effectively handle large amounts of data. In this paper, we present a novel approach, a distributed Q-learning algorithm, for generating DTRs. The novelties of our research are as follows: 1) From a methodological perspective, we present a novel and scalable approach for generating DTRs by combining distributed learning with Q-learning. The proposed approach is specifically designed to handle large amounts of data and effectively generate DTRs. 2) From a theoretical standpoint, we provide generalization error bounds for the proposed distributed Q-learning algorithm, which are derived within the framework of statistical learning theory. These bounds quantify the relationships between sample size, prediction accuracy, and computational burden, providing insights into the performance of the algorithm. 3) From an applied perspective, we demonstrate the effectiveness of our proposed distributed Q-learning algorithm for DTRs by applying it to clinical cancer treatments. The results show that our algorithm outperforms both traditional linear Q-learning and commonly used deep Q-learning in terms of both prediction accuracy and computation cost.