The point estimates of ReLU classification networks---arguably the most widely used neural network architecture---have been shown to yield arbitrarily high confidence far away from the training data. This architecture, in conjunction with a maximum a posteriori estimation scheme, is thus not calibrated nor robust. Approximate Bayesian inference has been empirically demonstrated to improve predictive uncertainty in neural networks, although the theoretical analysis of such Bayesian approximations is limited. We theoretically analyze approximate Gaussian distributions on the weights of ReLU networks and show that they fix the overconfidence problem. Furthermore, we show that even a simplistic, thus cheap, Bayesian approximation, also fixes these issues. This indicates that a sufficient condition for a calibrated uncertainty on a ReLU network is "to be a bit Bayesian". These theoretical results validate the usage of last-layer Bayesian approximation and motivate a range of a fidelity-cost trade-off. We further validate these findings empirically via various standard experiments using common deep ReLU networks and Laplace approximations.
University of Tübingen（蒂宾根大学）；MPI for Intelligent Systems, Tübingen（蒂宾根马克斯·普朗克智能系统研究所）
A Bayesian treatment can mitigate overconfidence in ReLU nets around the training data. But far away from them, ReLU Bayesian neural networks (BNNs) can still underestimate uncertainty and thus be asymptotically overconfident. This issue arises since the output variance of a BNN with finitely many features is quadratic in the distance from the data region. Meanwhile, Bayesian linear models with ReLU features converge, in the infinite-width limit, to a particular Gaussian process (GP) with a variance that grows cubically so that no asymptotic overconfidence can occur. While this may seem of mos…
In practice, multi-task learning (through learning features shared among tasks) is an essential property of deep neural networks (NNs). While infinite-width limits of NNs can provide good intuition for their generalization behavior, the well-known infinite-width limits of NNs in the literature (e.g., neural tangent kernels) assume specific settings in which wide ReLU-NNs behave like shallow Gaussian Processes with a fixed kernel. Consequently, in such settings, these NNs lose their ability to benefit from multi-task learning in the infinite-width limit. In contrast, we prove that optimizing wi…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；University of Guelph（圭尔夫大学）
Theoretical results show that neural networks can be approximated by Gaussian processes in the infinite-width limit. However, for fully connected networks, it has been previously shown that for any fixed network width, $n$, the Gaussian approximation gets worse as the network depth, $d$, increases. Given that modern networks are deep, this raises the question of how well modern architectures, like ResNets, are captured by the infinite-width limit. To provide a better approximation, we study ReLU ResNets in the infinite-depth-and-width limit, where both depth and width tend to infinity as their…
University of Utah（犹他大学）；Mitsubishi Electric Research Labs（三菱电机研究实验室）
Two networks are equivalent if they produce the same output for any given input. In this paper, we study the possibility of transforming a deep neural network to another network with a different number of units or layers, which can be either equivalent, a local exact approximation, or a global linear approximation of the original network. On the practical side, we show that certain rectified linear units (ReLUs) can be safely removed from a network if they are always active or inactive for any valid input. If we only need an equivalent network for a smaller domain, then more units can be remov…
Amortized Simulation-Based Inference in Generalized Bayes via Neural Posterior Estimation
Department of Statistics, University of Oxford, Oxford, United Kingdom（英国牛津大学统计系）；Department of Statistics, University of Auckland, Auckland, New Zealand（新西兰奥克兰大学统计系）
the University of Sydney（悉尼大学）；JD Explore Academy, JD.com Inc.（京东探索研究院）；University of Social Sciences（社会科学大学）；Systems Research Institute of the Polish Academy of Sciences（波兰科学院系统研究所）；Tsinghua University（清华大学）
Bayesian neural networks (BNNs) have become a principal approach to alleviate overconfident predictions in deep learning, but they often suffer from scaling issues due to a large number of distribution parameters. In this paper, we discover that the first layer of a deep network possesses multiple disparate optima when solely retrained. This indicates a large posterior variance when the first layer is altered by a Bayesian layer, which motivates us to design a spatial-temporal-fusion BNN (STF-BNN) for efficiently scaling BNNs to large models: (1) first normally train a neural network from scra…
We consider the problem of learning an unknown ReLU network with respect to Gaussian inputs and obtain the first nontrivial results for networks of depth more than two. We give an algorithm whose running time is a fixed polynomial in the ambient dimension and some (exponentially large) function of only the network's parameters. Our bounds depend on the number of hidden units, depth, spectral norm of the weight matrices, and Lipschitz constant of the overall network (we show that some dependence on the Lipschitz constant is necessary). We also give a bound that is doubly exponential in the size…
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…
Before training a neural net, a classic rule of thumb is to randomly initialize the weights so the variance of activations is preserved across layers. This is traditionally interpreted using the total variance due to randomness in both weights \emph{and} samples. Alternatively, one can interpret the rule of thumb as preservation of the variance over samples for a fixed network. The two interpretations differ little for a shallow net, but the difference is shown to grow with depth for a deep ReLU net by decomposing the total variance into the network-averaged sum of the sample variance and squa…
University of California, Berkeley（加州大学伯克利分校）；Google（谷歌）
We establish conditions under which gradient descent applied to fixed-width deep networks drives the logistic loss to zero, and prove bounds on the rate of convergence. Our analysis applies for smoothed approximations to the ReLU, such as Swish and the Huberized ReLU, proposed in previous applied work. We provide two sufficient conditions for convergence. The first is simply a bound on the loss at initialization. The second is a data separation condition used in prior analyses.
Understanding the underlying mechanisms that enable the empirical successes of deep neural networks is essential for further improving their performance and explaining such networks. Towards this goal, a specific question is how to explain the "surprising" behavior of the same over-parametrized deep neural networks that can generalize well on real datasets and at the same time "memorize" training samples when the labels are randomized. In this paper, we demonstrate that deep ReLU networks generalize from training samples to new points via piece-wise linear interpolation. We provide a quantifie…
National University of Singapore（新加坡国立大学）；Purdue University（普渡大学）
This paper concentrates on the approximation power of deep feed-forward neural networks in terms of width and depth. It is proved by construction that ReLU networks with width $\mathcal{O}\big(\max\{d\lfloor N^{1/d}\rfloor,\, N+2\}\big)$ and depth $\mathcal{O}(L)$ can approximate a Hölder continuous function on $[0,1]^d$ with an approximation rate $\mathcal{O}\big(λ\sqrt{d} (N^2L^2\ln N)^{-α/d}\big)$, where $α\in (0,1]$ and $λ>0$ are Hölder order and constant, respectively. Such a rate is optimal up to a constant in terms of width and depth separately, while existing results are only nearly op…
Baylor College of Medicine（贝勒医学院）；Rice University（莱斯大学）
Understanding the learning dynamics and inductive bias of neural networks (NNs) is hindered by the opacity of the relationship between NN parameters and the function represented. We propose reparametrizing ReLU NNs as continuous piecewise linear splines. Using this spline lens, we study learning dynamics in shallow univariate ReLU NNs, finding unexpected insights and explanations for several perplexing phenomena. We develop a surprisingly simple and transparent view of the structure of the loss surface, including its critical and fixed points, Hessian, and Hessian spectrum. We also show that s…
We consider a deep ReLU / Leaky ReLU student network trained from the output of a fixed teacher network of the same depth, with Stochastic Gradient Descent (SGD). The student network is \emph{over-realized}: at each layer $l$, the number $n_l$ of student nodes is more than that ($m_l$) of teacher. Under mild conditions on dataset and teacher network, we prove that when the gradient is small at every data sample, each teacher node is \emph{specialized} by at least one student node \emph{at the lowest layer}. For two-layer network, such specialization can be achieved by training on any dataset o…
We present a framework to derive upper bounds on the number of regions that feed-forward neural networks with ReLU activation functions are affine linear on. It is based on an inductive analysis that keeps track of the number of such regions per dimensionality of their images within the layers. More precisely, the information about the number regions per dimensionality is pushed through the layers starting with one region of the input dimension of the neural network and using a recursion based on an analysis of how many regions per output dimensionality a subsequent layer with a certain width …
Singapore University of Technology and Design（新加坡科技设计大学）；Stanford University（斯坦福大学）；SUTD（新加坡科技设计大学）
Understanding the representational power of Deep Neural Networks (DNNs) and how their structural properties (e.g., depth, width, type of activation unit) affect the functions they can compute, has been an important yet challenging question in deep learning and approximation theory. In a seminal paper, Telgarsky highlighted the benefits of depth by presenting a family of functions (based on simple triangular waves) for which DNNs achieve zero classification error, whereas shallow networks with fewer than exponentially many nodes incur constant error. Even though Telgarsky's work reveals the lim…
Understanding the fundamental mechanism behind the success of deep neural networks is one of the key challenges in the modern machine learning literature. Despite numerous attempts, a solid theoretical analysis is yet to be developed. In this paper, we develop a novel unified framework to reveal a hidden regularization mechanism through the lens of convex optimization. We first show that the training of multiple three-layer ReLU sub-networks with weight decay regularization can be equivalently cast as a convex optimization problem in a higher dimensional space, where sparsity is enforced via a…
Sharp description of local minima in the loss landscape of high-dimensional two-layer ReLU neural networks
Physics, Chalmers University of Technology；University of Gothenburg；Ecole Normale Superieure, PSL \& CNRS；Engineering, Chalmers University of Technology；School of Computer Science；Applied Mathematics, University of the Witwatersrand
Implicit deep learning has recently become popular in the machine learning community since these implicit models can achieve competitive performance with state-of-the-art deep networks while using significantly less memory and computational resources. However, our theoretical understanding of when and how first-order methods such as gradient descent (GD) converge on \textit{nonlinear} implicit networks is limited. Although this type of problem has been studied in standard feed-forward networks, the case of implicit models is still intriguing because implicit networks have \textit{infinitely} m…
We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in $L^2$. As a model class, we consider the set $\mathcal{E}^β(\mathbb R^d)$ of possibly discontinuous piecewise $C^β$ functions $f : [-1/2, 1/2]^d \to \mathbb R$, where the different smooth regions of $f$ are separated by $C^β$ hypersurfaces. For dimension $d \geq 2$, regularity $β> 0$, and accuracy $\varepsilon > 0$, we construct artificial neural networks with ReLU activation function that approximate functions from $\m…
Nowadays, online learning is an appealing learning paradigm, which is of great interest in practice due to the recent emergence of large scale applications such as online advertising placement and online web ranking. Standard online learning assumes a finite number of samples while in practice data is streamed infinitely. In such a setting gradient descent with a diminishing learning rate does not work. We first introduce regret with rolling window, a new performance metric for online streaming learning, which measures the performance of an algorithm on every fixed number of contiguous samples…
For fixed training data and network parameters in the other layers the L1 loss of a ReLU neural network as a function of the first layer's parameters is a piece-wise affine function. We use the Deep ReLU Simplex algorithm to iteratively minimize the loss monotonically on adjacent vertices and analyze the trajectory of these vertex positions. We empirically observe that in a neighbourhood around a local minimum, the iterations behave differently such that conclusions on loss level and proximity of the local minimum can be made before it has been found: Firstly the loss seems to decay exponentia…
University of Queensland（昆士兰大学）；International Computer Science Institute（国际计算机科学研究所）
An interesting approach to analyzing neural networks that has received renewed attention is to examine the equivalent kernel of the neural network. This is based on the fact that a fully connected feedforward network with one hidden layer, a certain weight distribution, an activation function, and an infinite number of neurons can be viewed as a mapping into a Hilbert space. We derive the equivalent kernels of MLPs with ReLU or Leaky ReLU activations for all rotationally-invariant weight distributions, generalizing a previous result that required Gaussian weight distributions. Additionally, th…
We describe the convex semi-infinite dual of the two-layer vector-output ReLU neural network training problem. This semi-infinite dual admits a finite dimensional representation, but its support is over a convex set which is difficult to characterize. In particular, we demonstrate that the non-convex neural network training problem is equivalent to a finite-dimensional convex copositive program. Our work is the first to identify this strong connection between the global optima of neural networks and those of copositive programs. We thus demonstrate how neural networks implicitly attempt to sol…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
A new loss function is proposed for neural networks on classification tasks which extends the hinge loss by assigning gradients to its critical points. We will show that for a linear classifier on linearly separable data with fixed step size, the margin of this modified hinge loss converges to the $\ell_2$ max-margin at the rate of $\mathcal{O}( 1/t )$. This rate is fast when compared with the $\mathcal{O}(1/\log t)$ rate of exponential losses such as the logistic loss. Furthermore, empirical results suggest that this increased convergence speed carries over to ReLU networks.
Texas A&M University（得克萨斯农工大学）；University of Toronto（多伦多大学）
We prove the precise scaling, at finite depth and width, for the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network. The standard deviation is exponential in the ratio of network depth to width. Thus, even in the limit of infinite overparameterization, the NTK is not deterministic if depth and width simultaneously tend to infinity. Moreover, we prove that for such deep and wide networks, the NTK has a non-trivial evolution during training by showing that the mean of its first SGD update is also exponential in the ratio of network depth to width. This is…
National University of Singapore（新加坡国立大学）；University of Maryland, College Park（马里兰大学帕克分校）
This paper develops simple feed-forward neural networks that achieve the universal approximation property for all continuous functions with a fixed finite number of neurons. These neural networks are simple because they are designed with a simple, computable, and continuous activation function $σ$ leveraging a triangular-wave function and the softsign function. We first prove that $σ$-activated networks with width $36d(2d+1)$ and depth $11$ can approximate any continuous function on a $d$-dimensional hypercube within an arbitrarily small error. Hence, for supervised learning and its related re…
Department of Applied Mathematics, Hong Kong Polytechnic University（应用数学系，香港理工大学）；Department of Mathematics, National University of Singapore（数学系，新加坡国立大学）；Department of Mathematics and Statistics, Old Dominion University（数学与统计学系，老 Dominion 大学）
Deep neural networks' remarkable ability to correctly fit training data when optimized by gradient-based algorithms is yet to be fully understood. Recent theoretical results explain the convergence for ReLU networks that are wider than those used in practice by orders of magnitude. In this work, we take a step towards closing the gap between theory and practice by significantly improving the known theoretical bounds on both the network width and the convergence time. We show that convergence to a global minimum is guaranteed for networks with widths quadratic in the sample size and linear in t…
Columbia University（哥伦比亚大学）；Google（谷歌）；Google Research（谷歌研究院）
We propose a novel low-rank initialization framework for training low-rank deep neural networks -- networks where the weight parameters are re-parameterized by products of two low-rank matrices. The most successful prior existing approach, spectral initialization, draws a sample from the initialization distribution for the full-rank setting and then optimally approximates the full-rank initialization parameters in the Frobenius norm with a pair of low-rank initialization matrices via singular value decomposition. Our method is inspired by the insight that approximating the function correspondi…
We study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity. Our main theorem is that the logarithm of the $\ell_2$ norm of such a product applied to any fixed vector is asymptotically Gaussian. The fluctuations we find can be thought of as a finite temperature correction to the limit in which first the size and then the number of matrices tend to infinity. Depending on the scaling limit considered, the mean and variance of the limiting Gaussian depend only on either the first two or the first four moments of the mea…
Deep convolutional neural networks trained on large datsets have emerged as an intriguing alternative for compressing images and solving inverse problems such as denoising and compressive sensing. However, it has only recently been realized that even without training, convolutional networks can function as concise image models, and thus regularize inverse problems. In this paper, we provide further evidence for this finding by studying variations of convolutional neural networks that map few weight parameters to an image. The networks we consider only consist of convolutional operations, with …
Upper and lower bounds for the Lipschitz constant of random neural networks
Mathematical Institute for Machine Learning（机器学习数学研究所）；Data Science (MIDS), Catholic University of Eichstätt--Ingolstadt (KU)（数据科学（MIDS），埃希施泰特-英戈尔施塔特天主教大学）；Chair of Mathematics --- Statistics（数学-统计学教授职位）
