The expressiveness of deep neural network (DNN) is a perspective to understandthe surprising performance of DNN. The number of linear regions, i.e. pieces thata piece-wise-linear function represented by a DNN, is generally used to measurethe expressiveness. And the upper bound of regions number partitioned by a rec-tifier network, instead of the number itself, is a more practical measurement ofexpressiveness of a rectifier DNN. In this work, we propose a new and tighter up-per bound of regions number. Inspired by the proof of this upper bound and theframework of matrix computation in Hinz & Van de Geer (2019), we propose ageneral computational approach to compute a tight upper bound of regions numberfor theoretically any network structures (e.g. DNN with all kind of skip connec-tions and residual structures). Our experiments show our upper bound is tighterthan existing ones, and explain why skip connections and residual structures canimprove network performance.
The standard approach to supervised classification involves the minimization of a log-loss as an upper bound to the classification error. While this is a tight bound early on in the optimization, it overemphasizes the influence of incorrectly classified examples far from the decision boundary. Updating the upper bound during the optimization leads to improved classification rates while transforming the learning into a sequence of minimization problems. In addition, in the context where the classifier is part of a larger system, this modification makes it possible to link the performance of the…
We study reinforcement learning in non-episodic factored Markov decision processes (FMDPs). We propose two near-optimal and oracle-efficient algorithms for FMDPs. Assuming oracle access to an FMDP planner, they enjoy a Bayesian and a frequentist regret bound respectively, both of which reduce to the near-optimal bound $\widetilde{O}(DS\sqrt{AT})$ for standard non-factored MDPs. We propose a tighter connectivity measure, factored span, for FMDPs and prove a lower bound that depends on the factored span rather than the diameter $D$. In order to decrease the gap between lower and upper bounds, we…
It has been shown that a message passing neural networks (MPNNs), a popular family of neural networks for graph-structured data, are at most as expressive as the first-order Weisfeiler-Leman (1-WL) graph isomorphism test, which has motivated the development of more expressive architectures. In this work, we analyze if the limited expressiveness is actually a limiting factor for MPNNs and other WL-based models in standard graph datasets. Interestingly, we find that the expressiveness of WL is sufficient to identify almost all graphs in most datasets. Moreover, we find that the classification ac…
University College London（伦敦大学学院）；Massachusetts Institute of Technology（麻省理工学院）
Generalization error bounds are essential to understanding machine learning algorithms. This paper presents novel expected generalization error upper bounds based on the average joint distribution between the output hypothesis and each input training sample. Multiple generalization error upper bounds based on different information measures are provided, including Wasserstein distance, total variation distance, KL divergence, and Jensen-Shannon divergence. Due to the convexity of the information measures, the proposed bounds in terms of Wasserstein distance and total variation distance are show…
University of Oxford（牛津大学）；Tsinghua University（清华大学）
In large-scale classification problems, the data set always be faced with frequent updates when a part of the data is added to or removed from the original data set. In this case, conventional incremental learning, which updates an existing classifier by explicitly modeling the data modification, is more efficient than retraining a new classifier from scratch. However, sometimes, we are more interested in determining whether we should update the classifier or performing some sensitivity analysis tasks. To deal with these such tasks, we propose an algorithm to make rational inferences about the…
Neural networks are commonly used in safety-critical real-world applications. Unfortunately, the predicted output is often highly sensitive to small, and possibly imperceptible, changes to the input data. Proving that either no such adversarial examples exist, or providing a concrete instance, is therefore crucial to ensure safe applications. As enumerating and testing all potential adversarial examples is computationally infeasible, verification techniques have been developed to provide mathematically sound proofs of their absence using overestimations of the network activations. We propose a…
University of Toronto（多伦多大学）；Vector Institute for Artificial Intelligence（向量人工智能研究所）
Deep neural networks (DNNs) have emerged as a popular mathematical tool for function approximation due to their capability of modelling highly nonlinear functions. Their applications range from image classification and natural language processing to learning-based control. Despite their empirical successes, there is still a lack of theoretical understanding of the representative power of such deep architectures. In this work, we provide a theoretical analysis of the expressiveness of fully-connected, feedforward DNNs with 1-Lipschitz activation functions. In particular, we characterize the exp…
Kuang-Chi Institute of Advanced Technology（光启高等理工研究院）；ZenRhyme Consulting Services（ZenRhyme咨询服务公司）
Stochastic variational inference (SVI) plays a key role in Bayesian deep learning. Recently various divergences have been proposed to design the surrogate loss for variational inference. We present a simple upper bound of the evidence as the surrogate loss. This evidence upper bound (EUBO) equals to the log marginal likelihood plus the KL-divergence between the posterior and the proposal. We show that the proposed EUBO is tighter than previous upper bounds introduced by $χ$-divergence or $α$-divergence. To facilitate scalable inference, we present the numerical approximation of the gradient of…
Institute for Computational and Mathematical Engineering, Stanford University（斯坦福大学计算与数学工程研究所）；Department of Computer Science, Stanford University（斯坦福大学计算机科学系）
Strong worst-case performance bounds for episodic reinforcement learning exist but fortunately in practice RL algorithms perform much better than such bounds would predict. Algorithms and theory that provide strong problem-dependent bounds could help illuminate the key features of what makes a RL problem hard and reduce the barrier to using RL algorithms in practice. As a step towards this we derive an algorithm for finite horizon discrete MDPs and associated analysis that both yields state-of-the art worst-case regret bounds in the dominant terms and yields substantially tighter bounds if the…
Technische Universität Darmstadt（达姆施塔特工业大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
The Nadaraya-Watson kernel estimator is among the most popular nonparameteric regression technique thanks to its simplicity. Its asymptotic bias has been studied by Rosenblatt in 1969 and has been reported in a number of related literature. However, Rosenblatt's analysis is only valid for infinitesimal bandwidth. In contrast, we propose in this paper an upper bound of the bias which holds for finite bandwidths. Moreover, contrarily to the classic analysis we allow for discontinuous first order derivative of the regression function, we extend our bounds for multidimensional domains and we inclu…
California Institute of Technology（加州理工学院）；NVIDIA（英伟达）
Certified robustness is a desirable property for deep neural networks in safety-critical applications, and popular training algorithms can certify robustness of a neural network by computing a global bound on its Lipschitz constant. However, such a bound is often loose: it tends to over-regularize the neural network and degrade its natural accuracy. A tighter Lipschitz bound may provide a better tradeoff between natural and certified accuracy, but is generally hard to compute exactly due to non-convexity of the network. In this work, we propose an efficient and trainable \emph{local} Lipschitz…
Weakly Time-Coupled Approximation of Markov Decision Processes
Information and Decision Sciences Department, University of Illinois at Chicago（伊利诺伊大学芝加哥分校信息与决策科学系）；Industrial Engineering and Decision Analytics, Hong Kong University of Science and Technology（香港科学与技术大学工业工程与决策分析系）
Automating statistical modelling is a challenging problem in artificial intelligence. The Automatic Statistician takes a first step in this direction, by employing a kernel search algorithm with Gaussian Processes (GP) to provide interpretable statistical models for regression problems. However this does not scale due to its $O(N^3)$ running time for the model selection. We propose Scalable Kernel Composition (SKC), a scalable kernel search algorithm that extends the Automatic Statistician to bigger data sets. In doing so, we derive a cheap upper bound on the GP marginal likelihood that sandwi…
Wuhan University（武汉大学）；Shanghai Jiao Tong University（上海交通大学）；University of New South Wales（新南威尔士大学）；Harbin Institute of Technology（哈尔滨工业大学）；Mohamed bin Zayed University of AI（穆罕默德·本·扎耶德人工智能大学）
The research for characterizing GNN expressiveness attracts much attention as graph neural networks achieve a champion in the last five years. The number of linear regions has been considered a good measure for the expressivity of neural networks with piecewise linear activation. In this paper, we present some estimates for the number of linear regions of the classic graph convolutional networks (GCNs) with one layer and multiple-layer scenarios. In particular, we obtain an optimal upper bound for the maximum number of linear regions for one-layer GCNs, and the upper and lower bounds for multi…
Australian National University（澳大利亚国立大学）；ETH Zürich（苏黎世联邦理工学院）；NICTA（国家信息与通信技术研究院）
We study upper and lower bounds on the sample-complexity of learning near-optimal behaviour in finite-state discounted Markov Decision Processes (MDPs). For the upper bound we make the assumption that each action leads to at most two possible next-states and prove a new bound for a UCRL-style algorithm on the number of time-steps when it is not Probably Approximately Correct (PAC). The new lower bound strengthens previous work by being both more general (it applies to all policies) and tighter. The upper and lower bounds match up to logarithmic factors.
In this paper, we reformulate the forest representation learning approach as an additive model which boosts the augmented feature instead of the prediction. We substantially improve the upper bound of generalization gap from $\mathcal{O}(\sqrt\frac{\ln m}{m})$ to $\mathcal{O}(\frac{\ln m}{m})$, while $λ$ - the margin ratio between the margin standard deviation and the margin mean is small enough. This tighter upper bound inspires us to optimize the margin distribution ratio $λ$. Therefore, we design the margin distribution reweighting approach (mdDF) to achieve small ratio $λ$ by boosting the …
University of Southern California（南加州大学）；University of California, San Diego（加州大学圣地亚哥分校）
The problem of adaptive sampling for estimating probability mass functions (pmf) uniformly well is considered. Performance of the sampling strategy is measured in terms of the worst-case mean squared error. A Bayesian variant of the existing upper confidence bound (UCB) based approaches is proposed. It is shown analytically that the performance of this Bayesian variant is no worse than the existing approaches. The posterior distribution on the pmfs in the Bayesian setting allows for a tighter computation of upper confidence bounds which leads to significant performance gains in practice. Using…
In this work, we present a novel upper bound of target error to address the problem for unsupervised domain adaptation. Recent studies reveal that a deep neural network can learn transferable features which generalize well to novel tasks. Furthermore, a theory proposed by Ben-David et al. (2010) provides a upper bound for target error when transferring the knowledge, which can be summarized as minimizing the source error and distance between marginal distributions simultaneously. However, common methods based on the theory usually ignore the joint error such that samples from different classes…
Bucknell University（巴克内尔大学）；The University of Utah（犹他大学）
We can compare the expressiveness of neural networks that use rectified linear units (ReLUs) by the number of linear regions, which reflect the number of pieces of the piecewise linear functions modeled by such networks. However, enumerating these regions is prohibitive and the known analytical bounds are identical for networks with same dimensions. In this work, we approximate the number of linear regions through empirical bounds based on features of the trained network and probabilistic inference. Our first contribution is a method to sample the activation patterns defined by ReLUs using uni…
Department of Electrical Engineering, Tsinghua University（清华大学电子工程系）
The developments of deep neural networks (DNN) in recent years have ushered a brand new era of artificial intelligence. DNNs are proved to be excellent in solving very complex problems, e.g., visual recognition and text understanding, to the extent of competing with or even surpassing people. Despite inspiring and encouraging success of DNNs, thorough theoretical analyses still lack to unravel the mystery of their magics. The design of DNN structure is dominated by empirical results in terms of network depth, number of neurons and activations. A few of remarkable works published recently in an…
An important class of distance metrics proposed for training generative adversarial networks (GANs) is the integral probability metric (IPM), in which the neural net distance captures the practical GAN training via two neural networks. This paper investigates the minimax estimation problem of the neural net distance based on samples drawn from the distributions. We develop the first known minimax lower bound on the estimation error of the neural net distance, and an upper bound tighter than an existing bound on the estimator error for the empirical neural net distance. Our lower and upper boun…
We study the sequential general online regression, known also as the sequential probability assignments, under logarithmic loss when compared against a broad class of experts. We focus on obtaining tight, often matching, lower and upper bounds for the sequential minimax regret that are defined as the excess loss it incurs over a class of experts. After proving a general upper bound, we consider some specific classes of experts from Lipschitz class to bounded Hessian class and derive matching lower and upper bounds with provably optimal constants. Our bounds work for a wide range of values of t…
