NeurIPS 2018 论文列表

We identify and study two common failure modes for early training in deep ReLU nets. For each, we give a rigorous proof of when it occurs and how to avoid it, for fully connected, convolutional, and residual architectures. We show that the first failure mode, exploding or vanishing mean activation length, can be avoided by initializing weights from a symmetric distribution with variance 2/fan-in and, for ResNets, by correctly scaling the residual modules. We prove that the second failure mode, exponentially large variance of activation length, never occurs in residual nets once the first failure mode is avoided. In contrast, for fully connected nets, we prove that this failure mode can happen and is avoided by keeping constant the sum of the reciprocals of layer widths. We demonstrate empirically the effectiveness of our theoretical results in predicting when networks are able to start training. In particular, we note that many popular initializations fail our criteria, whereas correct initialization and architecture allows much deeper networks to be trained.

Stochastic gradient descent (SGD) gives an optimal convergence rate when minimizing convex stochastic objectives $f(x)$. However, in terms of making the gradients small, the original SGD does not give an optimal rate, even when $f(x)$ is convex. If $f(x)$ is convex, to find a point with gradient norm $\varepsilon$, we design an algorithm SGD3 with a near-optimal rate $\tilde{O}(\varepsilon^{-2})$, improving the best known rate $O(\varepsilon^{-8/3})$. If $f(x)$ is nonconvex, to find its $\varepsilon$-approximate local minimum, we design an algorithm SGD5 with rate $\tilde{O}(\varepsilon^{-3.5})$, where previously SGD variants only achieve $\tilde{O}(\varepsilon^{-4})$. This is no slower than the best known stochastic version of Newton's method in all parameter regimes.

The question of which global minima are accessible by a stochastic gradient decent (SGD) algorithm with specific learning rate and batch size is studied from the perspective of dynamical stability. The concept of non-uniformity is introduced, which, together with sharpness, characterizes the stability property of a global minimum and hence the accessibility of a particular SGD algorithm to that global minimum. In particular, this analysis shows that learning rate and batch size play different roles in minima selection. Extensive empirical results seem to correlate well with the theoretical findings and provide further support to these claims.

When the linear measurements of an instance of low-rank matrix recovery satisfy a restricted isometry property (RIP) --- i.e. they are approximately norm-preserving --- the problem is known to contain no spurious local minima, so exact recovery is guaranteed. In this paper, we show that moderate RIP is not enough to eliminate spurious local minima, so existing results can only hold for near-perfect RIP. In fact, counterexamples are ubiquitous: every $x$ is the spurious local minimum of a rank-1 instance of matrix recovery that satisfies RIP. One specific counterexample has RIP constant $\delta=1/2$, but causes randomly initialized stochastic gradient descent (SGD) to fail 12\% of the time. SGD is frequently able to avoid and escape spurious local minima, but this empirical result shows that it can occasionally be defeated by their existence. Hence, while exact recovery guarantees will likely require a proof of no spurious local minima, arguments based solely on norm preservation will only be applicable to a narrow set of nearly-isotropic instances.

A widespread folklore for explaining the success of Convolutional Neural Networks (CNNs) is that CNNs use a more compact representation than the Fully-connected Neural Network (FNN) and thus require fewer training samples to accurately estimate their parameters. We initiate the study of rigorously characterizing the sample complexity of estimating CNNs. We show that for an $m$-dimensional convolutional filter with linear activation acting on a $d$-dimensional input, the sample complexity of achieving population prediction error of $\epsilon$ is $\widetilde{O(m/\epsilon^2)$, whereas the sample-complexity for its FNN counterpart is lower bounded by $\Omega(d/\epsilon^2)$ samples. Since, in typical settings $m \ll d$, this result demonstrates the advantage of using a CNN. We further consider the sample complexity of estimating a one-hidden-layer CNN with linear activation where both the $m$-dimensional convolutional filter and the $r$-dimensional output weights are unknown. For this model, we show that the sample complexity is $\widetilde{O}\left((m+r)/\epsilon^2\right)$ when the ratio between the stride size and the filter size is a constant. For both models, we also present lower bounds showing our sample complexities are tight up to logarithmic factors. Our main tools for deriving these results are a localized empirical process analysis and a new lemma characterizing the convolutional structure. We believe that these tools may inspire further developments in understanding CNNs.

Batch Normalization (BatchNorm) is a widely adopted technique that enables faster and more stable training of deep neural networks (DNNs). Despite its pervasiveness, the exact reasons for BatchNorm's effectiveness are still poorly understood. The popular belief is that this effectiveness stems from controlling the change of the layers' input distributions during training to reduce the so-called "internal covariate shift". In this work, we demonstrate that such distributional stability of layer inputs has little to do with the success of BatchNorm. Instead, we uncover a more fundamental impact of BatchNorm on the training process: it makes the optimization landscape significantly smoother. This smoothness induces a more predictive and stable behavior of the gradients, allowing for faster training.

We consider online linear regression: at each round, an adversary reveals a covariate vector, the learner predicts a real value, the adversary reveals a label, and the learner suffers the squared prediction error. The aim is to minimize the difference between the cumulative loss and that of the linear predictor that is best in hindsight. Previous work demonstrated that the minimax optimal strategy is easy to compute recursively from the end of the game; this requires the entire sequence of covariate vectors in advance. We show that, once provided with a measure of the scale of the problem, we can invert the recursion and play the minimax strategy without knowing the future covariates. Further, we show that this forward recursion remains optimal even against adaptively chosen labels and covariates, provided that the adversary adheres to a set of constraints that prevent misrepresentation of the scale of the problem. This strategy is horizon-independent in that the regret and minimax strategies depend on the size of the constraint set and not on the time-horizon, and hence it incurs no more regret than the optimal strategy that knows in advance the number of rounds of the game. We also provide an interpretation of the minimax algorithm as a follow-the-regularized-leader strategy with a data-dependent regularizer and obtain an explicit expression for the minimax regret.

Quantization is a promising technique to reduce the model size, memory footprint, and massive computation operations of recurrent neural networks (RNNs) for embedded devices with limited resources. Although extreme low-bit quantization has achieved impressive success on convolutional neural networks, it still suffers from huge accuracy degradation on RNNs with the same low-bit precision. In this paper, we first investigate the accuracy degradation on RNN models under different quantization schemes, and the distribution of tensor values in the full precision model. Our observation reveals that due to the difference between the distributions of weights and activations, different quantization methods are suitable for different parts of models. Based on our observation, we propose HitNet, a hybrid ternary recurrent neural network, which bridges the accuracy gap between the full precision model and the quantized model. In HitNet, we develop a hybrid quantization method to quantize weights and activations. Moreover, we introduce a sloping factor motivated by prior work on Boltzmann machine to activation functions, further closing the accuracy gap between the full precision model and the quantized model. Overall, our HitNet can quantize RNN models into ternary values, {-1, 0, 1}, outperforming the state-of-the-art quantization methods on RNN models significantly. We test it on typical RNN models, such as Long-Short-Term Memory (LSTM) and Gated Recurrent Units (GRU), on which the results outperform previous work significantly. For example, we improve the perplexity per word (PPW) of a ternary LSTM on Penn Tree Bank (PTB) corpus from 126 (the state-of-the-art result to the best of our knowledge) to 110.3 with a full precision model in 97.2, and a ternary GRU from 142 to 113.5 with a full precision model in 102.7.

We consider a high dimensional linear regression problem where the goal is to efficiently recover an unknown vector \beta^* from n noisy linear observations Y=X \beta^

We introduce a new RL problem where the agent is required to generalize to a previously-unseen environment characterized by a subtask graph which describes a set of subtasks and their dependencies. Unlike existing hierarchical multitask RL approaches that explicitly describe what the agent should do at a high level, our problem only describes properties of subtasks and relationships among them, which requires the agent to perform complex reasoning to find the optimal subtask to execute. To solve this problem, we propose a neural subtask graph solver (NSGS) which encodes the subtask graph using a recursive neural network embedding. To overcome the difficulty of training, we propose a novel non-parametric gradient-based policy, graph reward propagation, to pre-train our NSGS agent and further finetune it through actor-critic method. The experimental results on two 2D visual domains show that our agent can perform complex reasoning to find a near-optimal way of executing the subtask graph and generalize well to the unseen subtask graphs. In addition, we compare our agent with a Monte-Carlo tree search (MCTS) method showing that our method is much more efficient than MCTS, and the performance of NSGS can be further improved by combining it with MCTS.

Recently, graph neural networks (GNNs) have revolutionized the field of graph representation learning through effectively learned node embeddings, and achieved state-of-the-art results in tasks such as node classification and link prediction. However, current GNN methods are inherently flat and do not learn hierarchical representations of graphs---a limitation that is especially problematic for the task of graph classification, where the goal is to predict the label associated with an entire graph. Here we propose DiffPool, a differentiable graph pooling module that can generate hierarchical representations of graphs and can be combined with various graph neural network architectures in an end-to-end fashion. DiffPool learns a differentiable soft cluster assignment for nodes at each layer of a deep GNN, mapping nodes to a set of clusters, which then form the coarsened input for the next GNN layer. Our experimental results show that combining existing GNN methods with DiffPool yields an average improvement of 5-10% accuracy on graph classification benchmarks, compared to all existing pooling approaches, achieving a new state-of-the-art on four out of five benchmark datasets.

We present a novel extension of multi-output Gaussian processes for handling heterogeneous outputs. We assume that each output has its own likelihood function and use a vector-valued Gaussian process prior to jointly model the parameters in all likelihoods as latent functions. Our multi-output Gaussian process uses a covariance function with a linear model of coregionalisation form. Assuming conditional independence across the underlying latent functions together with an inducing variable framework, we are able to obtain tractable variational bounds amenable to stochastic variational inference. We illustrate the performance of the model on synthetic data and two real datasets: a human behavioral study and a demographic high-dimensional dataset.

Recent work has shown that fast, compact low-bitwidth neural networks can be surprisingly accurate. These networks use homogeneous binarization: all parameters in each layer or (more commonly) the whole model have the same low bitwidth (e.g., 2 bits). However, modern hardware allows efficient designs where each arithmetic instruction can have a custom bitwidth, motivating heterogeneous binarization, where every parameter in the network may have a different bitwidth. In this paper, we show that it is feasible and useful to select bitwidths at the parameter granularity during training. For instance a heterogeneously quantized version of modern networks such as AlexNet and MobileNet, with the right mix of 1-, 2- and 3-bit parameters that average to just 1.4 bits can equal the accuracy of homogeneous 2-bit versions of these networks. Further, we provide analyses to show that the heterogeneously binarized systems yield FPGA- and ASIC-based implementations that are correspondingly more efficient in both circuit area and energy efficiency than their homogeneous counterparts.

Large batch size training of Neural Networks has been shown to incur accuracy loss when trained with the current methods. The exact underlying reasons for this are still not completely understood. Here, we study large batch size training through the lens of the Hessian operator and robust optimization. In particular, we perform a Hessian based study to analyze exactly how the landscape of the loss function changes when training with large batch size. We compute the true Hessian spectrum, without approximation, by back-propagating the second derivative. Extensive experiments on multiple networks show that saddle-points are not the cause for generalization gap of large batch size training, and the results consistently show that large batch converges to points with noticeably higher Hessian spectrum. Furthermore, we show that robust training allows one to favor flat areas, as points with large Hessian spectrum show poor robustness to adversarial perturbation. We further study this relationship, and provide empirical and theoretical proof that the inner loop for robust training is a saddle-free optimization problem \textit{almost everywhere}. We present detailed experiments with five different network architectures, including a residual network, tested on MNIST, CIFAR-10/100 datasets.

Deep reinforcement learning could be used to learn dexterous robotic policies but it is challenging to transfer them to new robots with vastly different hardware properties. It is also prohibitively expensive to learn a new policy from scratch for each robot hardware due to the high sample complexity of modern state-of-the-art algorithms. We propose a novel approach called Hardware Conditioned Policies where we train a universal policy conditioned on a vector representation of robot hardware. We considered robots in simulation with varied dynamics, kinematic structure, kinematic lengths and degrees-of-freedom. First, we use the kinematic structure directly as the hardware encoding and show great zero-shot transfer to completely novel robots not seen during training. For robots with lower zero-shot success rate, we also demonstrate that fine-tuning the policy network is significantly more sample-efficient than training a model from scratch. In tasks where knowing the agent dynamics is important for success, we learn an embedding for robot hardware and show that policies conditioned on the encoding of hardware tend to generalize and transfer well. Videos of experiments are available at: https://sites.google.com/view/robot-transfer-hcp.

Variational Auto-Encoders (VAE) have become very popular techniques to perform inference and learning in latent variable models as they allow us to leverage the rich representational power of neural networks to obtain flexible approximations of the posterior of latent variables as well as tight evidence lower bounds (ELBO). Com- bined with stochastic variational inference, this provides a methodology scaling to large datasets. However, for this methodology to be practically efficient, it is neces- sary to obtain low-variance unbiased estimators of the ELBO and its gradients with respect to the parameters of interest. While the use of Markov chain Monte Carlo (MCMC) techniques such as Hamiltonian Monte Carlo (HMC) has been previously suggested to achieve this [23, 26], the proposed methods require specifying reverse kernels which have a large impact on performance. Additionally, the resulting unbiased estimator of the ELBO for most MCMC kernels is typically not amenable to the reparameterization trick. We show here how to optimally select reverse kernels in this setting and, by building upon Hamiltonian Importance Sampling (HIS) [17], we obtain a scheme that provides low-variance unbiased estimators of the ELBO and its gradients using the reparameterization trick. This allows us to develop a Hamiltonian Variational Auto-Encoder (HVAE). This method can be re-interpreted as a target-informed normalizing flow [20] which, within our context, only requires a few evaluations of the gradient of the sampled likelihood and trivial Jacobian calculations at each iteration.

We present a neurosymbolic framework for the lifelong learning of algorithmic tasks that mix perception and procedural reasoning. Reusing high-level concepts across domains and learning complex procedures are key challenges in lifelong learning. We show that a program synthesis approach that combines gradient descent with combinatorial search over programs can be a more effective response to these challenges than purely neural methods. Our framework, called HOUDINI, represents neural networks as strongly typed, differentiable functional programs that use symbolic higher-order combinators to compose a library of neural functions. Our learning algorithm consists of: (1) a symbolic program synthesizer that performs a type-directed search over parameterized programs, and decides on the library functions to reuse, and the architectures to combine them, while learning a sequence of tasks; and (2) a neural module that trains these programs using stochastic gradient descent. We evaluate HOUDINI on three benchmarks that combine perception with the algorithmic tasks of counting, summing, and shortest-path computation. Our experiments show that HOUDINI transfers high-level concepts more effectively than traditional transfer learning and progressive neural networks, and that the typed representation of networks significantly accelerates the search.

Asynchronous Gibbs sampling has been recently shown to be fast-mixing and an accurate method for estimating probabilities of events on a small number of variables of a graphical model satisfying Dobrushin's condition~\cite{DeSaOR16}. We investigate whether it can be used to accurately estimate expectations of functions of {\em all the variables} of the model. Under the same condition, we show that the synchronous (sequential) and asynchronous Gibbs samplers can be coupled so that the expected Hamming distance between their (multivariate) samples remains bounded by $O(\tau \log n),$ where $n$ is the number of variables in the graphical model, and $\tau$ is a measure of the asynchronicity. A similar bound holds for any constant power of the Hamming distance. Hence, the expectation of any function that is Lipschitz with respect to a power of the Hamming distance, can be estimated with a bias that grows logarithmically in $n$. Going beyond Lipschitz functions, we consider the bias arising from asynchronicity in estimating the expectation of polynomial functions of all variables in the model. Using recent concentration of measure results~\cite{DaskalakisDK17,GheissariLP17,GotzeSS18}, we show that the bias introduced by the asynchronicity is of smaller order than the standard deviation of the function value already present in the true model. We perform experiments on a multi-processor machine to empirically illustrate our theoretical findings.

Boltzmann machines (BMs) are appealing candidates for powerful priors in variational autoencoders (VAEs), as they are capable of capturing nontrivial and multi-modal distributions over discrete variables. However, non-differentiability of the discrete units prohibits using the reparameterization trick, essential for low-noise back propagation. The Gumbel trick resolves this problem in a consistent way by relaxing the variables and distributions, but it is incompatible with BM priors. Here, we propose the GumBolt, a model that extends the Gumbel trick to BM priors in VAEs. GumBolt is significantly simpler than the recently proposed methods with BM prior and outperforms them by a considerable margin. It achieves state-of-the-art performance on permutation invariant MNIST and OMNIGLOT datasets in the scope of models with only discrete latent variables. Moreover, the performance can be further improved by allowing multi-sampled (importance-weighted) estimation of log-likelihood in training, which was not possible with previous models.