We describe an algorithm that learns two-layer residual units using rectified linear unit (ReLU) activation: suppose the input $\mathbf{x}$ is from a distribution with support space $\mathbb{R}^d$ and the ground-truth generative model is a residual unit of this type, given by $\mathbf{y} = \boldsymbol{B}^\ast\left[\left(\boldsymbol{A}^\ast\mathbf{x}\right)^+ + \mathbf{x}\right]$, where ground-truth network parameters $\boldsymbol{A}^\ast \in \mathbb{R}^{d\times d}$ represent a full-rank matrix with nonnegative entries and $\boldsymbol{B}^\ast \in \mathbb{R}^{m\times d}$ is full-rank with $m \geq d$ and for $\boldsymbol{c} \in \mathbb{R}^d$, $[\boldsymbol{c}^{+}]_i = \max\{0, c_i\}$. We design layer-wise objectives as functionals whose analytic minimizers express the exact ground-truth network in terms of its parameters and nonlinearities. Following this objective landscape, learning residual units from finite samples can be formulated using convex optimization of a nonparametric function: for each layer, we first formulate the corresponding empirical risk minimization (ERM) as a positive semi-definite quadratic program (QP), then we show the solution space of the QP can be equivalently determined by a set of linear inequalities, which can then be efficiently solved by linear programming (LP). We further prove the strong statistical consistency of our algorithm, and demonstrate its robustness and sample efficiency through experimental results on synthetic data and a set of benchmark regression datasets.
Real world data often exhibit low-dimensional geometric structures, and can be viewed as samples near a low-dimensional manifold. This paper studies nonparametric regression of Hölder functions on low-dimensional manifolds using deep ReLU networks. Suppose $n$ training data are sampled from a Hölder function in $\mathcal{H}^{s,α}$ supported on a $d$-dimensional Riemannian manifold isometrically embedded in $\mathbb{R}^D$, with sub-gaussian noise. A deep ReLU network architecture is designed to estimate the underlying function from the training data. The mean squared error of the empirical esti…
Purdue University（普渡大学）；Hong Kong University of Science and Technology（香港科技大学）；East China Normal University（华东师范大学）
Overparametrized neural networks trained by gradient descent (GD) can provably overfit any training data. However, the generalization guarantee may not hold for noisy data. From a nonparametric perspective, this paper studies how well overparametrized neural networks can recover the true target function in the presence of random noises. We establish a lower bound on the $L_2$ estimation error with respect to the GD iterations, which is away from zero without a delicate scheme of early stopping. In turn, through a comprehensive analysis of $\ell_2$-regularized GD trajectories, we prove that for…
We study finite sample expressivity, i.e., memorization power of ReLU networks. Recent results require $N$ hidden nodes to memorize/interpolate arbitrary $N$ data points. In contrast, by exploiting depth, we show that 3-layer ReLU networks with $Ω(\sqrt{N})$ hidden nodes can perfectly memorize most datasets with $N$ points. We also prove that width $Θ(\sqrt{N})$ is necessary and sufficient for memorizing $N$ data points, proving tight bounds on memorization capacity. The sufficiency result can be extended to deeper networks; we show that an $L$-layer network with $W$ parameters in the hidden l…
We study in this paper the rate of convergence for learning densities under the Generative Adversarial Networks (GAN) framework, borrowing insights from nonparametric statistics. We introduce an improved GAN estimator that achieves a faster rate, through simultaneously leveraging the level of smoothness in the target density and the evaluation metric, which in theory remedies the mode collapse problem reported in the literature. A minimax lower bound is constructed to show that when the dimension is large, the exponent in the rate for the new GAN estimator is near optimal. One can view our res…
We introduce a neural-network architecture, termed the constrained recurrent sparse autoencoder (CRsAE), that solves convolutional dictionary learning problems, thus establishing a link between dictionary learning and neural networks. Specifically, we leverage the interpretation of the alternating-minimization algorithm for dictionary learning as an approximate Expectation-Maximization algorithm to develop autoencoders that enable the simultaneous training of the dictionary and regularization parameter (ReLU bias). The forward pass of the encoder approximates the sufficient statistics of the E…
Institute of Environmental and Agricultural Biology (X-BIO), University of Tyumen（秋明大学生态与农业生物学研究所（X-BIO））；University of Mississippi（密西西比大学）
We consider regression estimation with modified ReLU neural networks in which network weight matrices are first modified by a function $α$ before being multiplied by input vectors. We give an example of continuous, piecewise linear function $α$ for which the empirical risk minimizers over the classes of modified ReLU networks with $l_1$ and squared $l_2$ penalties attain, up to a logarithmic factor, the minimax rate of prediction of unknown $β$-smooth function.
Consider the multivariate nonparametric regression model. It is shown that estimators based on sparsely connected deep neural networks with ReLU activation function and properly chosen network architecture achieve the minimax rates of convergence (up to $\log n$-factors) under a general composition assumption on the regression function. The framework includes many well-studied structural constraints such as (generalized) additive models. While there is a lot of flexibility in the network architecture, the tuning parameter is the sparsity of the network. Specifically, we consider large networks…
High dimensional nonparametric regression is an inherently difficult problem with known lower bounds depending exponentially in dimension. A popular strategy to alleviate this curse of dimensionality has been to use additive models of \emph{first order}, which model the regression function as a sum of independent functions on each dimension. Though useful in controlling the variance of the estimate, such models are often too restrictive in practical settings. Between non-additive models which often have large variance and first order additive models which have large bias, there has been little…
Transfer Learning in Nonparametric Regression with Deep ReLU Networks
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Washington University in St. Louis（圣路易斯华盛顿大学）；Hong Kong University of Science and Technology（香港科技大学）
We investigate efficient learning from higher-order graph convolution and learning directly from adjacency matrices for node classification. We revisit the scaled graph residual network and remove ReLU activation from residual layers and apply a single weight matrix at each residual layer. We show that the resulting model lead to new graph convolution models as a polynomial of the normalized adjacency matrix, the residual weight matrix, and the residual scaling parameter. Additionally, we propose adaptive learning between directly graph polynomial convolution models and learning directly from …
Recent results in the literature indicate that a residual network (ResNet) composed of a single residual block outperforms linear predictors, in the sense that all local minima in its optimization landscape are at least as good as the best linear predictor. However, these results are limited to a single residual block (i.e., shallow ResNets), instead of the deep ResNets composed of multiple residual blocks. We take a step towards extending this result to deep ResNets. We start by two motivating examples. First, we show that there exist datasets for which all local minima of a fully-connected R…
Minimum Block Width for Universal Approximation by Residual Neural Networks with Inner Width One
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology（湖北省工程建模与科学计算重点实验室）
Royal Institute of Technology (KTH)（皇家理工学院（KTH））；Örebro University（厄勒布鲁大学）
Optimizing parameters with momentum, normalizing data values, and using rectified linear units (ReLUs) are popular choices in neural network (NN) regression. Although ReLUs are popular, they can collapse to a constant function and "die", effectively removing their contribution from the model. While some mitigations are known, the underlying reasons of ReLUs dying during optimization are currently poorly understood. In this paper, we consider the effects of target normalization and momentum on dying ReLUs. We find empirically that unit variance targets are well motivated and that ReLUs die more…
Distribution forecast can quantify forecast uncertainty and provide various forecast scenarios with their corresponding estimated probabilities. Accurate distribution forecast is crucial for planning - for example when making production capacity or inventory allocation decisions. We propose a practical and robust distribution forecast framework that relies on backtest-based bootstrap and adaptive residual selection. The proposed approach is robust to the choice of the underlying forecasting model, accounts for uncertainty around the input covariates, and relaxes the independence between residu…
Exact ReLU realization of affine one-dimensional refinement iterates via residual memory and offset frames
National University of Mongolia（蒙古国国立大学）；McGill University（麦吉尔大学）；Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences（蒙古科学院数学与数字技术研究所）
Nonparametric Classification on Low Dimensional Manifolds using Overparameterized Convolutional Residual Networks
Georgia Institute of Technology（佐治亚理工学院）；UC Santa Barbara（加州大学圣巴巴拉分校）；Northwestern University（西北大学）；University of Tokyo（东京大学）；Princeton University（普林斯顿大学）；UC San Diego（加州大学圣迭戈分校）
Beijing University of Aeronautics and Astronautics（北京航空航天大学）；University of Utah（犹他大学）；Chongqing University（重庆大学）；Beihang University（北京航空航天大学）
Challenges in multi-fidelity modeling relate to accuracy, uncertainty estimation and high-dimensionality. A novel additive structure is introduced in which the highest fidelity solution is written as a sum of the lowest fidelity solution and residuals between the solutions at successive fidelity levels, with Gaussian process priors placed over the low fidelity solution and each of the residuals. The resulting model is equipped with a closed-form solution for the predictive posterior, making it applicable to advanced, high-dimensional tasks that require uncertainty estimation. Its advantages ar…
Exact ReLU realization of binary affine refinement iterates via reflection folding and cone switching
McGill University（麦吉尔大学）；National University of Mongolia（蒙古国立大学）；Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences（蒙古科学院数学与数字技术研究所）
Exact Loop Controllers for ReLU Realization of Homogeneous Curve Refinements
McGill University（麦吉尔大学）；National University of Mongolia（蒙古国国立大学）；Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences（蒙古科学院数学与数字技术研究所）
We establish finite-sample guarantees for a polynomial-time algorithm for learning a nonlinear, nonparametric directed acyclic graphical (DAG) model from data. The analysis is model-free and does not assume linearity, additivity, independent noise, or faithfulness. Instead, we impose a condition on the residual variances that is closely related to previous work on linear models with equal variances. Compared to an optimal algorithm with oracle knowledge of the variable ordering, the additional cost of the algorithm is linear in the dimension $d$ and the number of samples $n$. Finally, we compa…
The residual network is now one of the most effective structures in deep learning, which utilizes the skip connections to ``guarantee" the performance will not get worse. However, the non-convexity of the neural network makes it unclear whether the skip connections do provably improve the learning ability since the nonlinearity may create many local minima. In some previous works \cite{freeman2016topology}, it is shown that despite the non-convexity, the loss landscape of the two-layer ReLU network has good properties when the number $m$ of hidden nodes is very large. In this paper, we follow …
The Rectified Linear Unit is currently a state-of-the-art activation function in deep convolutional neural networks. To combat ReLU's dying neuron problem, we propose the Parametric Variational Linear Unit (PVLU), which adds a sinusoidal function with trainable coefficients to ReLU. Along with introducing nonlinearity and non-zero gradients across the entire real domain, PVLU acts as a mechanism of fine-tuning when implemented in the context of transfer learning. On a simple, non-transfer sequential CNN, PVLU substitution allowed for relative error decreases of 16.3% and 11.3% (without and wit…
