The expressive power of neural networks is important for understanding deep learning. Most existing works consider this problem from the view of the depth of a network. In this paper, we study how width affects the expressiveness of neural networks. Classical results state that depth-bounded (e.g. depth-2) networks with suitable activation functions are universal approximators. We show a universal approximation theorem for width-bounded ReLU networks: width-(n + 4) ReLU networks, where n is the input dimension, are universal approximators. Moreover, except for a measure zero set, all functions cannot be approximated by width-n ReLU networks, which exhibits a phase transition. Several recent works demonstrate the benefits of depth by proving the depth-efficiency of neural networks. That is, there are classes of deep networks which cannot be realized by any shallow network whose size is no more than an exponential bound. Here we pose the dual question on the width-efficiency of ReLU networks: Are there wide networks that cannot be realized by narrow networks whose size is not substantially larger? We show that there exist classes of wide networks which cannot be realized by any narrow network whose depth is no more than a polynomial bound. On the other hand, we demonstrate by extensive experiments that narrow networks whose size exceed the polynomial bound by a constant factor can approximate wide and shallow network with high accuracy. Our results provide more comprehensive evidence that depth may be more effective than width for the expressiveness of ReLU networks.

This paper explores image caption generation using conditional variational autoencoders (CVAEs).Standard CVAEs with a fixed Gaussian prior yield descriptions with too little variability. Instead, we propose two models that explicitly structure the latent space around K components corresponding to different types of image content, and combine components to create priors for images that contain multiple types of content simultaneously (e.g., several kinds of objects). Our first model uses a Gaussian Mixture model (GMM) prior, while the second one defines a novel Additive Gaussian (AG) prior that linearly combines component means. We show that both models produce captions that are more diverse and more accurate than a strong LSTM baseline or a "vanilla" CVAE with a fixed Gaussian prior, with AG-CVAE showing particular promise.

Algorithm-dependent generalization error bounds are central to statistical learning theory. A learning algorithm may use a large hypothesis space, but the limited number of iterations controls its model capacity and generalization error. The impacts of stochastic gradient methods on generalization error for non-convex learning problems not only have important theoretical consequences, but are also critical to generalization errors of deep learning. In this paper, we study the generalization errors of Stochastic Gradient Langevin Dynamics (SGLD) with non-convex objectives. Two theories are proposed with non-asymptotic discrete-time analysis, using Stability and PAC-Bayesian results respectively. The stability-based theory obtains a bound of $O\left(\frac{1}{n}L\sqrt{\beta T_k}\right)$, where $L$ is uniform Lipschitz parameter, $\beta$ is inverse temperature, and $T_k$ is aggregated step sizes. For PAC-Bayesian theory, though the bound has a slower $O(1/\sqrt{n})$ rate, the contribution of each step is shown with an exponentially decaying factor by imposing $\ell^2$ regularization, and the uniform Lipschitz constant is also replaced by actual norms of gradients along trajectory. Our bounds have no implicit dependence on dimensions, norms or other capacity measures of parameter, which elegantly characterizes the phenomenon of "Fast Training Guarantees Generalization" in non-convex settings. This is the first algorithm-dependent result with reasonable dependence on aggregated step sizes for non-convex learning, and has important implications to statistical learning aspects of stochastic gradient methods in complicated models such as deep learning.

In this paper, we consider efficient differentially private empirical risk minimization from the viewpoint of optimization algorithms. For strongly convex and smooth objectives, we prove that gradient descent with output perturbation not only achieves nearly optimal utility, but also significantly improves the running time of previous state-of-the-art private optimization algorithms, for both $\epsilon$-DP and $(\epsilon, \delta)$-DP. For non-convex but smooth objectives, we propose an RRPSGD (Random Round Private Stochastic Gradient Descent) algorithm, which provably converges to a stationary point with privacy guarantee. Besides the expected utility bounds, we also provide guarantees in high probability form. Experiments demonstrate that our algorithm consistently outperforms existing method in both utility and running time.

In this work we study the quantitative relation between the recursive teaching dimension (RTD) and the VC dimension (VCD) of concept classes of finite sizes. The RTD of a concept class $\mathcal C \subseteq \{0, 1\}^n$, introduced by Zilles et al. (2011), is a combinatorial complexity measure characterized by the worst-case number of examples necessary to identify a concept in $\mathcal C$ according to the recursive teaching model. For any finite concept class $\mathcal C \subseteq \{0,1\}^n$ with $\mathrm{VCD}(\mathcal C)=d$, Simon & Zilles (2015) posed an open problem $\mathrm{RTD}(\mathcal C) = O(d)$, i.e., is RTD linearly upper bounded by VCD? Previously, the best known result is an exponential upper bound $\mathrm{RTD}(\mathcal C) = O(d \cdot 2^d)$, due to Chen et al. (2016). In this paper, we show a quadratic upper bound: $\mathrm{RTD}(\mathcal C) = O(d^2)$, much closer to an answer to the open problem. We also discuss the challenges in fully solving the problem.

While neural machine translation (NMT) is making good progress in the past two years, tens of millions of bilingual sentence pairs are needed for its training. However, human labeling is very costly. To tackle this training data bottleneck, we develop a dual-learning mechanism, which can enable an NMT system to automatically learn from unlabeled data through a dual-learning game. This mechanism is inspired by the following observation: any machine translation task has a dual task, e.g., English-to-French translation (primal) versus French-to-English translation (dual); the primal and dual tasks can form a closed loop, and generate informative feedback signals to train the translation models, even if without the involvement of a human labeler. In the dual-learning mechanism, we use one agent to represent the model for the primal task and the other agent to represent the model for the dual task, then ask them to teach each other through a reinforcement learning process. Based on the feedback signals generated during this process (e.g., the language-model likelihood of the output of a model, and the reconstruction error of the original sentence after the primal and dual translations), we can iteratively update the two models until convergence (e.g., using the policy gradient methods). We call the corresponding approach to neural machine translation \emph{dual-NMT}. Experiments show that dual-NMT works very well on English$\leftrightarrow$French translation; especially, by learning from monolingual data (with 10\% bilingual data for warm start), it achieves a comparable accuracy to NMT trained from the full bilingual data for the French-to-English translation task.

People believe that depth plays an important role in success of deep neural networks (DNN). However, this belief lacks solid theoretical justifications as far as we know. We investigate role of depth from perspective of margin bound. In margin bound, expected error is upper bounded by empirical margin error plus Rademacher Average (RA) based capacity term. First, we derive an upper bound for RA of DNN, and show that it increases with increasing depth. This indicates negative impact of depth on test performance. Second, we show that deeper networks tend to have larger representation power (measured by Betti numbers based complexity) than shallower networks in multi-class setting, and thus can lead to smaller empirical margin error. This implies positive impact of depth. The combination of these two results shows that for DNN with restricted number of hidden units, increasing depth is not always good since there is a tradeoff between positive and negative impacts. These results inspire us to seek alternative ways to achieve positive impact of depth, e.g., imposing margin-based penalty terms to cross entropy loss so as to reduce empirical margin error without increasing depth. Our experiments show that in this way, we achieve significantly better test performance.

This paper presents a large-scale sparse coding algorithm to deal with the challenging problem of noise-robust semi-supervised learning over very large data with only few noisy initial labels. By giving an L1-norm formulation of Laplacian regularization directly based upon the manifold structure of the data, we transform noise-robust semi-supervised learning into a generalized sparse coding problem so that noise reduction can be imposed upon the noisy initial labels. Furthermore, to keep the scalability of noise-robust semi-supervised learning over very large data, we make use of both nonlinear approximation and dimension reduction techniques to solve this generalized sparse coding problem in linear time and space complexity. Finally, we evaluate the proposed algorithm in the challenging task of large-scale semi-supervised image classification with only few noisy initial labels. The experimental results on several benchmark image datasets show the promising performance of the proposed algorithm.

This paper presents a direct semantic analysis method for learning the correlation matrix between visual and textual words from socially tagged images. In the literature, to improve the traditional visual bag-of-words (BOW) representation, latent semantic analysis has been studied extensively for learning a compact visual representation, where each visual word may be related to multiple latent topics. However, these latent topics do not convey any true semantic information which can be understood by human. In fact, it remains a challenging problem how to recover the relationships between visual and textual words. Motivated by the recent advances in dealing with socially tagged images, we develop a direct semantic analysis method which can explicitly learn the correlation matrix between visual and textual words for social image classification. To this end, we formulate our direct semantic analysis from a graph-based learning viewpoint. Once the correlation matrix is learnt, we can readily first obtain a semantically refined visual BOW representation and then apply it to social image classification. Experimental results on two benchmark image datasets show the promising performance of the proposed method.

We consider accurately answering smooth queries while preserving differential privacy. A query is said to be $K$-smooth if it is specified by a function defined on $[-1,1]^d$ whose partial derivatives up to order $K$ are all bounded. We develop an $\epsilon$-differentially private mechanism for the class of $K$-smooth queries. The major advantage of the algorithm is that it outputs a synthetic database. In real applications, a synthetic database output is appealing. Our mechanism achieves an accuracy of $O (n^{-\frac{K}{2d+K}}/\epsilon )$, and runs in polynomial time. We also generalize the mechanism to preserve $(\epsilon, \delta)$-differential privacy with slightly improved accuracy. Extensive experiments on benchmark datasets demonstrate that the mechanisms have good accuracy and are efficient.