An interesting observation in artificial neural networks is their favorable generalization error despite typically being extremely overparameterized. It is well known that classical statistical learning methods often result in vacuous generalization errors in the case of overparameterized neural networks. Adopting the recently developed Neural Tangent (NT) kernel theory, we prove uniform generalization bounds for overparameterized neural networks in kernel regimes, when the true data generating model belongs to the reproducing kernel Hilbert space (RKHS) corresponding to the NT kernel. Importantly, our bounds capture the exact error rates depending on the differentiability of the activation functions. In order to establish these bounds, we propose the information gain of the NT kernel as a measure of complexity of the learning problem. Our analysis uses a Mercer decomposition of the NT kernel in the basis of spherical harmonics and the decay rate of the corresponding eigenvalues. As a byproduct of our results, we show the equivalence between the RKHS corresponding to the NT kernel and its counterpart corresponding to the Mat\'ern family of kernels, that induces a very general class of models. We further discuss the implications of our analysis for some recent results on the regret bounds for reinforcement learning algorithms, which use overparameterized neural networks.
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Deep Ritz methods (DRM) have been proven numerically to be efficient in solving partial differential equations. In this paper, we present a convergence rate in $H^{1}$ norm for deep Ritz methods for Laplace equations with Dirichlet boundary condition, where the error depends on the depth and width in the deep neural networks and the number of samples explicitly. Further we can properly choose the depth and width in the deep neural networks in terms of the number of training samples. The main idea of the proof is to decompose the total error of DRM into three parts, that is approximation error, statistical error and the error caused by the boundary penalty. We bound the approximation error in $H^{1}$ norm with $\mathrm{ReLU}^{2}$ networks and control the statistical error via Rademacher complexity. In particular, we derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with $\mathrm{ReLU}^{2}$ network, which is of immense independent interest. We also analysis the error inducing by the boundary penalty method and give a prior rule for tuning the penalty parameter.
For the differential privacy under the sub-Gamma noise, we derive the asymptotic properties of a class of network models with binary values with a general link function. In this paper, we release the degree sequences of the binary networks under a general noisy mechanism with the discrete Laplace mechanism as a special case. We establish the asymptotic result including both consistency and asymptotically normality of the parameter estimator when the number of parameters goes to infinity in a class of network models. Simulations and a real data example are provided to illustrate asymptotic results.
This paper introduces a new neural network based prior for real valued functions on $\mathbb R^d$ which, by construction, is more easily and cheaply scaled up in the domain dimension $d$ compared to the usual Karhunen-Lo\`eve function space prior. The new prior is a Gaussian neural network prior, where each weight and bias has an independent Gaussian prior, but with the key difference that the variances decrease in the width of the network in such a way that the resulting function is almost surely well defined in the limit of an infinite width network. We show that in a Bayesian treatment of inferring unknown functions, the induced posterior over functions is amenable to Monte Carlo sampling using Hilbert space Markov chain Monte Carlo (MCMC) methods. This type of MCMC is popular, e.g. in the Bayesian Inverse Problems literature, because it is stable under mesh refinement, i.e. the acceptance probability does not shrink to $0$ as more parameters of the function's prior are introduced, even ad infinitum. In numerical examples we demonstrate these stated competitive advantages over other function space priors. We also implement examples in Bayesian Reinforcement Learning to automate tasks from data and demonstrate, for the first time, stability of MCMC to mesh refinement for these type of problems.
A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.
To theoretically understand the behavior of trained deep neural networks, it is necessary to study the dynamics induced by gradient methods from a random initialization. However, the nonlinear and compositional structure of these models make these dynamics difficult to analyze. To overcome these challenges, large-width asymptotics have recently emerged as a fruitful viewpoint and led to practical insights on real-world deep networks. For two-layer neural networks, it has been understood via these asymptotics that the nature of the trained model radically changes depending on the scale of the initial random weights, ranging from a kernel regime (for large initial variance) to a feature learning regime (for small initial variance). For deeper networks more regimes are possible, and in this paper we study in detail a specific choice of "small" initialization corresponding to ''mean-field'' limits of neural networks, which we call integrable parameterizations (IPs). First, we show that under standard i.i.d. zero-mean initialization, integrable parameterizations of neural networks with more than four layers start at a stationary point in the infinite-width limit and no learning occurs. We then propose various methods to avoid this trivial behavior and analyze in detail the resulting dynamics. In particular, one of these methods consists in using large initial learning rates, and we show that it is equivalent to a modification of the recently proposed maximal update parameterization $\mu$P. We confirm our results with numerical experiments on image classification tasks, which additionally show a strong difference in behavior between various choices of activation functions that is not yet captured by theory.
In this paper we present results on asymptotic characteristics of multivariate function classes in the uniform norm. Our main interest is the approximation of functions with mixed smoothness parameter not larger than $1/2$. Our focus will be on the behavior of the best $m$-term trigonometric approximation as well as the decay of Kolmogorov and entropy numbers in the uniform norm. It turns out that these quantities share a few fundamental abstract properties like their behavior under real interpolation, such that they can be treated simultaneously. We start with proving estimates on finite rank convolution operators with range in a step hyperbolic cross. These results imply bounds for the corresponding function space embeddings by a well-known decomposition technique. The decay of Kolmogorov numbers have direct implications for the problem of sampling recovery in $L_2$ in situations where recent results in the literature are not applicable since the corresponding approximation numbers are not square summable.
The rapid recent progress in machine learning (ML) has raised a number of scientific questions that challenge the longstanding dogma of the field. One of the most important riddles is the good empirical generalization of overparameterized models. Overparameterized models are excessively complex with respect to the size of the training dataset, which results in them perfectly fitting (i.e., interpolating) the training data, which is usually noisy. Such interpolation of noisy data is traditionally associated with detrimental overfitting, and yet a wide range of interpolating models -- from simple linear models to deep neural networks -- have recently been observed to generalize extremely well on fresh test data. Indeed, the recently discovered double descent phenomenon has revealed that highly overparameterized models often improve over the best underparameterized model in test performance. Understanding learning in this overparameterized regime requires new theory and foundational empirical studies, even for the simplest case of the linear model. The underpinnings of this understanding have been laid in very recent analyses of overparameterized linear regression and related statistical learning tasks, which resulted in precise analytic characterizations of double descent. This paper provides a succinct overview of this emerging theory of overparameterized ML (henceforth abbreviated as TOPML) that explains these recent findings through a statistical signal processing perspective. We emphasize the unique aspects that define the TOPML research area as a subfield of modern ML theory and outline interesting open questions that remain.
Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.
We study the problem of training deep neural networks with Rectified Linear Unit (ReLU) activiation function using gradient descent and stochastic gradient descent. In particular, we study the binary classification problem and show that for a broad family of loss functions, with proper random weight initialization, both gradient descent and stochastic gradient descent can find the global minima of the training loss for an over-parameterized deep ReLU network, under mild assumption on the training data. The key idea of our proof is that Gaussian random initialization followed by (stochastic) gradient descent produces a sequence of iterates that stay inside a small perturbation region centering around the initial weights, in which the empirical loss function of deep ReLU networks enjoys nice local curvature properties that ensure the global convergence of (stochastic) gradient descent. Our theoretical results shed light on understanding the optimization of deep learning, and pave the way to study the optimization dynamics of training modern deep neural networks.
The model parameters of convolutional neural networks (CNNs) are determined by backpropagation (BP). In this work, we propose an interpretable feedforward (FF) design without any BP as a reference. The FF design adopts a data-centric approach. It derives network parameters of the current layer based on data statistics from the output of the previous layer in a one-pass manner. To construct convolutional layers, we develop a new signal transform, called the Saab (Subspace Approximation with Adjusted Bias) transform. It is a variant of the principal component analysis (PCA) with an added bias vector to annihilate activation's nonlinearity. Multiple Saab transforms in cascade yield multiple convolutional layers. As to fully-connected (FC) layers, we construct them using a cascade of multi-stage linear least squared regressors (LSRs). The classification and robustness (against adversarial attacks) performances of BP- and FF-designed CNNs applied to the MNIST and the CIFAR-10 datasets are compared. Finally, we comment on the relationship between BP and FF designs.
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