On the rates of convergence for learning with convolutional neural networks

We study approximation and learning capacities of convolutional neural networks (CNNs) with one-side zero-padding and multiple channels. Our first result proves a new approximation bound for CNNs with certain constraint on the weights. Our second result gives new analysis on the covering number of feed-forward neural networks with CNNs as special cases. The analysis carefully takes into account the size of the weights and hence gives better bounds than the existing literature in some situations. Using these two results, we are able to derive rates of convergence for estimators based on CNNs in many learning problems. In particular, we establish minimax optimal convergence rates of the least squares based on CNNs for learning smooth functions in the nonparametric regression setting. For binary classification, we derive convergence rates for CNN classifiers with hinge loss and logistic loss. It is also shown that the obtained rates for classification are minimax optimal in some common settings.

Nonlinear functional regression by functional deep neural network with kernel embedding

With the rapid development of deep learning in various fields of science and technology, such as speech recognition, image classification, and natural language processing, recently it is also widely applied in the functional data analysis (FDA) with some empirical success. However, due to the infinite dimensional input, we need a powerful dimension reduction method for functional learning tasks, especially for the nonlinear functional regression. In this paper, based on the idea of smooth kernel integral transformation, we propose a functional deep neural network with an efficient and fully data-dependent dimension reduction method. The architecture of our functional net consists of a kernel embedding step: an integral transformation with a data-dependent smooth kernel; a projection step: a dimension reduction by projection with eigenfunction basis based on the embedding kernel; and finally an expressive deep ReLU neural network for the prediction. The utilization of smooth kernel embedding enables our functional net to be discretization invariant, efficient, and robust to noisy observations, capable of utilizing information in both input functions and responses data, and have a low requirement on the number of discrete points for an unimpaired generalization performance. We conduct theoretical analysis including approximation error and generalization error analysis, and numerical simulations to verify these advantages of our functional net.

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.

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.

Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks

Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on surfaces, including spheres. However, theoretical understanding of the numerical performance of PINNs, especially PINNs on surfaces or manifolds, is still lacking. In this paper, we establish rigorous analysis of the physics-informed convolutional neural network (PICNN) for solving PDEs on the sphere. By using and improving the latest approximation results of deep convolutional neural networks and spherical harmonic analysis, we prove an upper bound for the approximation error with respect to the Sobolev norm. Subsequently, we integrate this with innovative localization complexity analysis to establish fast convergence rates for PICNN. Our theoretical results are also confirmed and supplemented by our experiments. In light of these findings, we explore potential strategies for circumventing the curse of dimensionality that arises when solving high-dimensional PDEs.

Classification with Deep Neural Networks and Logistic Loss

Deep neural networks (DNNs) trained with the logistic loss (i.e., the cross entropy loss) have made impressive advancements in various binary classification tasks. However, generalization analysis for binary classification with DNNs and logistic loss remains scarce. The unboundedness of the target function for the logistic loss is the main obstacle to deriving satisfying generalization bounds. In this paper, we aim to fill this gap by establishing a novel and elegant oracle-type inequality, which enables us to deal with the boundedness restriction of the target function, and using it to derive sharp convergence rates for fully connected ReLU DNN classifiers trained with logistic loss. In particular, we obtain optimal convergence rates (up to log factors) only requiring the H\"older smoothness of the conditional class probability $\eta$ of data. Moreover, we consider a compositional assumption that requires $\eta$ to be the composition of several vector-valued functions of which each component function is either a maximum value function or a H\"older smooth function only depending on a small number of its input variables. Under this assumption, we derive optimal convergence rates (up to log factors) which are independent of the input dimension of data. This result explains why DNN classifiers can perform well in practical high-dimensional classification problems. Besides the novel oracle-type inequality, the sharp convergence rates given in our paper also owe to a tight error bound for approximating the natural logarithm function near zero (where it is unbounded) by ReLU DNNs. In addition, we justify our claims for the optimality of rates by proving corresponding minimax lower bounds. All these results are new in the literature and will deepen our theoretical understanding of classification with DNNs.

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.

Rates of Approximation by ReLU Shallow Neural Networks

Neural networks activated by the rectified linear unit (ReLU) play a central role in the recent development of deep learning. The topic of approximating functions from H\"older spaces by these networks is crucial for understanding the efficiency of the induced learning algorithms. Although the topic has been well investigated in the setting of deep neural networks with many layers of hidden neurons, it is still open for shallow networks having only one hidden layer. In this paper, we provide rates of uniform approximation by these networks. We show that ReLU shallow neural networks with $m$ hidden neurons can uniformly approximate functions from the H\"older space $W_\infty^r([-1, 1]^d)$ with rates $O((\log m)^{\frac{1}{2} +d}m^{-\frac{r}{d}\frac{d+2}{d+4}})$ when $r<d/2 +2$. Such rates are very close to the optimal one $O(m^{-\frac{r}{d}})$ in the sense that $\frac{d+2}{d+4}$ is close to $1$, when the dimension $d$ is large.

Learning Theory of Distribution Regression with Neural Networks

In this paper, we aim at establishing an approximation theory and a learning theory of distribution regression via a fully connected neural network (FNN). In contrast to the classical regression methods, the input variables of distribution regression are probability measures. Then we often need to perform a second-stage sampling process to approximate the actual information of the distribution. On the other hand, the classical neural network structure requires the input variable to be a vector. When the input samples are probability distributions, the traditional deep neural network method cannot be directly used and the difficulty arises for distribution regression. A well-defined neural network structure for distribution inputs is intensively desirable. There is no mathematical model and theoretical analysis on neural network realization of distribution regression. To overcome technical difficulties and address this issue, we establish a novel fully connected neural network framework to realize an approximation theory of functionals defined on the space of Borel probability measures. Furthermore, based on the established functional approximation results, in the hypothesis space induced by the novel FNN structure with distribution inputs, almost optimal learning rates for the proposed distribution regression model up to logarithmic terms are derived via a novel two-stage error decomposition technique.