Most popular optimizers for deep learning can be broadly categorized as adaptive methods (e.g. Adam) and accelerated schemes (e.g. stochastic gradient descent (SGD) with momentum). For many models such as convolutional neural networks (CNNs), adaptive methods typically converge faster but generalize worse compared to SGD; for complex settings such as generative adversarial networks (GANs), adaptive methods are typically the default because of their stability.We propose AdaBelief to simultaneously achieve three goals: fast convergence as in adaptive methods, good generalization as in SGD, and training stability. The intuition for AdaBelief is to adapt the stepsize according to the "belief" in the current gradient direction. Viewing the exponential moving average (EMA) of the noisy gradient as the prediction of the gradient at the next time step, if the observed gradient greatly deviates from the prediction, we distrust the current observation and take a small step; if the observed gradient is close to the prediction, we trust it and take a large step. We validate AdaBelief in extensive experiments, showing that it outperforms other methods with fast convergence and high accuracy on image classification and language modeling. Specifically, on ImageNet, AdaBelief achieves comparable accuracy to SGD. Furthermore, in the training of a GAN on Cifar10, AdaBelief demonstrates high stability and improves the quality of generated samples compared to a well-tuned Adam optimizer. Code is available at https://github.com/juntang-zhuang/Adabelief-Optimizer

Neural ordinary differential equations (NODEs) have recently attracted increasing attention; however, their empirical performance on benchmark tasks (e.g. image classification) are significantly inferior to discrete-layer models. We demonstrate an explanation for their poorer performance is the inaccuracy of existing gradient estimation methods: the adjoint method has numerical errors in reverse-mode integration; the naive method directly back-propagates through ODE solvers, but suffers from a redundantly deep computation graph when searching for the optimal stepsize. We propose the Adaptive Checkpoint Adjoint (ACA) method: in automatic differentiation, ACA applies a trajectory checkpoint strategy which records the forward-mode trajectory as the reverse-mode trajectory to guarantee accuracy; ACA deletes redundant components for shallow computation graphs; and ACA supports adaptive solvers. On image classification tasks, compared with the adjoint and naive method, ACA achieves half the error rate in half the training time; NODE trained with ACA outperforms ResNet in both accuracy and test-retest reliability. On time-series modeling, ACA outperforms competing methods. Finally, in an example of the three-body problem, we show NODE with ACA can incorporate physical knowledge to achieve better accuracy. We provide the PyTorch implementation of ACA: \url{https://github.com/juntang-zhuang/torch-ACA}.

While neural networks have shown impressive performance on large datasets, applying these models to tasks where little data is available remains a challenging problem. In this paper we propose to use feature transfer in a zero-shot experimental setting on the task of semantic parsing. We first introduce a new method for learning the shared space between multiple domains based on the prediction of the domain label for each example. Our experiments support the superiority of this method in a zero-shot experimental setting in terms of accuracy metrics compared to state-of-the-art techniques. In the second part of this paper we study the impact of individual domains and examples on semantic parsing performance. We use influence functions to this aim and investigate the sensitivity of domain-label classification loss on each example. Our findings reveal that cross-domain adversarial attacks identify useful examples for training even from the domains the least similar to the target domain. Augmenting our training data with these influential examples further boosts our accuracy at both the token and the sequence level.

This paper investigates message-passing algorithms for solving systems of linear equations in the Laplacian matrices of graphs and to compute electric flows. These two problems are fundamental primitives that arise in several domains such as computer science, electrical engineering, operations research, and machine learning. Despite the extensive literature on approximately solving these problems in quasi-linear time, the algorithms that have been proposed are typically centralized and involve multiple graph theoretic constructions or sampling mechanisms that make them difficult to implement and analyze. On the other hand, message-passing routines are distributed, simple, and easy to implement. In this paper we establish a framework to analyze message-passing algorithms to solve voltage and flow problems. We characterize the error committed by the algorithms in d-regular graphs with equal weights. We show that the convergence of the algorithms is controlled by the total variation distance between the distributions of non-backtracking random walks that start from neighbor nodes. More broadly, our analysis of message-passing introduces new insights to address generic optimization problems with constraints.

We investigate the fundamental principles that drive the development of scalable algorithms for network optimization. Despite the significant amount of work on parallel and decentralized algorithms in the optimization community, the methods that have been proposed typically rely on strict separability assumptions for objective function and constraints. Beside sparsity, these methods typically do not exploit the strength of the interaction between variables in the system. We propose a notion of correlation in constrained optimization that is based on the sensitivity of the optimal solution upon perturbations of the constraints. We develop a general theory of sensitivity of optimizers the extends beyond the infinitesimal setting. We present instances in network optimization where the correlation decays exponentially fast with respect to the natural distance in the network, and we design algorithms that can exploit this decay to yield dimension-free optimization. Our results are the first of their kind, and open new possibilities in the theory of local algorithms.

We study a new class of codes for lossy compression with the squared-error distortion criterion, designed using the statistical framework of high-dimensional linear regression. Codewords are linear combinations of subsets of columns of a design matrix. Called a Sparse Superposition or Sparse Regression codebook, this structure is motivated by an analogous construction proposed recently by Barron and Joseph for communication over an AWGN channel. For i.i.d Gaussian sources and minimum-distance encoding, we show that such a code can attain the Shannon rate-distortion function with the optimal error exponent, for all distortions below a specified value. It is also shown that sparse regression codes are robust in the following sense: a codebook designed to compress an i.i.d Gaussian source of variance $\sigma^2$ with (squared-error) distortion $D$ can compress any ergodic source of variance less than $\sigma^2$ to within distortion $D$. Thus the sparse regression ensemble retains many of the good covering properties of the i.i.d random Gaussian ensemble, while having having a compact representation in terms of a matrix whose size is a low-order polynomial in the block-length.

Linear regression models depend directly on the design matrix and its properties. Techniques that efficiently estimate model coefficients by partitioning rows of the design matrix are increasingly popular for large-scale problems because they fit well with modern parallel computing architectures. We propose a simple measure of {\em concordance} between a design matrix and a subset of its rows that estimates how well a subset captures the variance-covariance structure of a larger data set. We illustrate the use of this measure in a heuristic method for selecting row partition sizes that balance statistical and computational efficiency goals in real-world problems.

We propose computationally efficient encoders and decoders for lossy compression using a Sparse Regression Code. The codebook is defined by a design matrix and codewords are structured linear combinations of columns of this matrix. The proposed encoding algorithm sequentially chooses columns of the design matrix to successively approximate the source sequence. It is shown to achieve the optimal distortion-rate function for i.i.d Gaussian sources under the squared-error distortion criterion. For a given rate, the parameters of the design matrix can be varied to trade off distortion performance with encoding complexity. An example of such a trade-off as a function of the block length n is the following. With computational resource (space or time) per source sample of O((n/\log n)^2), for a fixed distortion-level above the Gaussian distortion-rate function, the probability of excess distortion decays exponentially in n. The Sparse Regression Code is robust in the following sense: for any ergodic source, the proposed encoder achieves the optimal distortion-rate function of an i.i.d Gaussian source with the same variance. Simulations show that the encoder has good empirical performance, especially at low and moderate rates.

We address the question of convergence in the loopy belief propagation (LBP) algorithm. Specifically, we relate convergence of LBP to the existence of a weak limit for a sequence of Gibbs measures defined on the LBP s associated computation tree.Using tools FROM the theory OF Gibbs measures we develop easily testable sufficient conditions FOR convergence.The failure OF convergence OF LBP implies the existence OF multiple phases FOR the associated Gibbs specification.These results give new insight INTO the mechanics OF the algorithm.

Gaussian belief propagation (GaBP) is an iterative algorithm for computing the mean of a multivariate Gaussian distribution, or equivalently, the minimum of a multivariate positive definite quadratic function. Sufficient conditions, such as walk-summability, that guarantee the convergence and correctness of GaBP are known, but GaBP may fail to converge to the correct solution given an arbitrary positive definite quadratic function. As was observed in previous work, the GaBP algorithm fails to converge if the computation trees produced by the algorithm are not positive definite. In this work, we will show that the failure modes of the GaBP algorithm can be understood via graph covers, and we prove that a parameterized generalization of the min-sum algorithm can be used to ensure that the computation trees remain positive definite whenever the input matrix is positive definite. We demonstrate that the resulting algorithm is closely related to other iterative schemes for quadratic minimization such as the Gauss-Seidel and Jacobi algorithms. Finally, we observe, empirically, that there always exists a choice of parameters such that the above generalization of the GaBP algorithm converges.