Bayesian learning via Stochastic Gradient Langevin Dynamics (SGLD) has been suggested for differentially private learning. While previous research provides differential privacy bounds for SGLD at the initial steps of the algorithm or when close to convergence, the question of what differential privacy guarantees can be made in between remains unanswered. This interim region is of great importance, especially for Bayesian neural networks, as it is hard to guarantee convergence to the posterior. This paper shows that using SGLD might result in unbounded privacy loss for this interim region, even when sampling from the posterior is as differentially private as desired.

The stochastic heavy ball method (SHB), also known as stochastic gradient descent (SGD) with Polyak's momentum, is widely used in training neural networks. However, despite the remarkable success of such algorithm in practice, its theoretical characterization remains limited. In this paper, we focus on neural networks with two and three layers and provide a rigorous understanding of the properties of the solutions found by SHB: \emph{(i)} stability after dropping out part of the neurons, \emph{(ii)} connectivity along a low-loss path, and \emph{(iii)} convergence to the global optimum. To achieve this goal, we take a mean-field view and relate the SHB dynamics to a certain partial differential equation in the limit of large network widths. This mean-field perspective has inspired a recent line of work focusing on SGD while, in contrast, our paper considers an algorithm with momentum. More specifically, after proving existence and uniqueness of the limit differential equations, we show convergence to the global optimum and give a quantitative bound between the mean-field limit and the SHB dynamics of a finite-width network. Armed with this last bound, we are able to establish the dropout-stability and connectivity of SHB solutions.

This paper studies the algorithmic stability and generalizability of decentralized stochastic gradient descent (D-SGD). We prove that the consensus model learned by D-SGD is $\mathcal{O}{(N^{-1}+m^{-1} +\lambda^2)}$-stable in expectation in the non-convex non-smooth setting, where $N$ is the total sample size, $m$ is the worker number, and $1+\lambda$ is the spectral gap that measures the connectivity of the communication topology. These results then deliver an $\mathcal{O}{(N^{-(1+\alpha)/2}+ m^{-(1+\alpha)/2}+\lambda^{1+\alpha} + \phi_{\mathcal{S}})}$ in-average generalization bound, which is non-vacuous even when $\lambda$ is closed to $1$, in contrast to vacuous as suggested by existing literature on the projected version of D-SGD. Our theory indicates that the generalizability of D-SGD is positively correlated with the spectral gap, and can explain why consensus control in initial training phase can ensure better generalization. Experiments of VGG-11 and ResNet-18 on CIFAR-10, CIFAR-100 and Tiny-ImageNet justify our theory. To our best knowledge, this is the first work on the topology-aware generalization of vanilla D-SGD. Code is available at https://github.com/Raiden-Zhu/Generalization-of-DSGD.

Byzantine machine learning (ML) aims to ensure the resilience of distributed learning algorithms to misbehaving (or Byzantine) machines. Although this problem received significant attention, prior works often assume the data held by the machines to be homogeneous, which is seldom true in practical settings. Data heterogeneity makes Byzantine ML considerably more challenging, since a Byzantine machine can hardly be distinguished from a non-Byzantine outlier. A few solutions have been proposed to tackle this issue, but these provide suboptimal probabilistic guarantees and fare poorly in practice. This paper closes the theoretical gap, achieving optimality and inducing good empirical results. In fact, we show how to automatically adapt existing solutions for (homogeneous) Byzantine ML to the heterogeneous setting through a powerful mechanism, we call nearest neighbor mixing (NNM), which boosts any standard robust distributed gradient descent variant to yield optimal Byzantine resilience under heterogeneity. We obtain similar guarantees (in expectation) by plugging NNM in the distributed stochastic heavy ball method, a practical substitute to distributed gradient descent. We obtain empirical results that significantly outperform state-of-the-art Byzantine ML solutions.

We study on-line gradient-descent learning in multilayer networks analytically and numerically. The training is based on randomly drawn inputs and their corresponding outputs as defined by a target rule. In the thermodynamic limit we derive deterministic differential equations for the order parameters of the problem which allow an exact calculation of the evolution of the generalization error. First we consider a single-layer perceptron with sigmoidal activation function learning a target rule defined by a network of the same architecture. For this model the generalization error decays exponentially with the number of training examples if the learning rate is sufficiently small.

In neural networks, continual learning results in gradient interference among sequential tasks, leading to catastrophic forgetting of old tasks while learning new ones. This issue is addressed in recent methods by storing the important gradient spaces for old tasks and updating the model orthogonally during new tasks. However, such restrictive orthogonal gradient updates hamper the learning capability of the new tasks resulting in sub-optimal performance. To improve new learning while minimizing forgetting, in this paper we propose a Scaled Gradient Projection (SGP) method, where we combine the orthogonal gradient projections with scaled gradient steps along the important gradient spaces for the past tasks. The degree of gradient scaling along these spaces depends on the importance of the bases spanning them. We propose an efficient method for computing and accumulating importance of these bases using the singular value decomposition of the input representations for each task. We conduct extensive experiments ranging from continual image classification to reinforcement learning tasks and report better performance with less training overhead than the state-of-the-art approaches.

Computational chemistry has become an important tool to predict and understand molecular properties and reactions. Even though recent years have seen a significant growth in new algorithms and computational methods that speed up quantum chemical calculations, the bottleneck for trajectory-based methods to study photoinduced processes is still the huge number of electronic structure calculations. In this work, we present an innovative solution, in which the amount of electronic structure calculations is drastically reduced, by employing machine learning algorithms and methods borrowed from the realm of artificial intelligence. However, applying these algorithms effectively requires finding optimal hyperparameters, which remains a challenge itself. Here we present an automated user-friendly framework, HOAX, to perform the hyperparameter optimization for neural networks, which bypasses the need for a lengthy manual process. The neural network generated potential energy surfaces (PESs) reduces the computational costs compared to the ab initio-based PESs. We perform a comparative investigation on the performance of different hyperparameter optimiziation algorithms, namely grid search, simulated annealing, genetic algorithm, and bayesian optimizer in finding the optimal hyperparameters necessary for constructing the well-performing neural network in order to fit the PESs of small organic molecules. Our results show that this automated toolkit not only facilitate a straightforward way to perform the hyperparameter optimization but also the resulting neural networks-based generated PESs are in reasonable agreement with the ab initio-based PESs.

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives.

Using gradient descent (GD) with fixed or decaying step-size is a standard practice in unconstrained optimization problems. However, when the loss function is only locally convex, such a step-size schedule artificially slows GD down as it cannot explore the flat curvature of the loss function. To overcome that issue, we propose to exponentially increase the step-size of the GD algorithm. Under homogeneous assumptions on the loss function, we demonstrate that the iterates of the proposed \emph{exponential step size gradient descent} (EGD) algorithm converge linearly to the optimal solution. Leveraging that optimization insight, we then consider using the EGD algorithm for solving parameter estimation under both regular and non-regular statistical models whose loss function becomes locally convex when the sample size goes to infinity. We demonstrate that the EGD iterates reach the final statistical radius within the true parameter after a logarithmic number of iterations, which is in stark contrast to a \emph{polynomial} number of iterations of the GD algorithm in non-regular statistical models. Therefore, the total computational complexity of the EGD algorithm is \emph{optimal} and exponentially cheaper than that of the GD for solving parameter estimation in non-regular statistical models while being comparable to that of the GD in regular statistical settings. To the best of our knowledge, it resolves a long-standing gap between statistical and algorithmic computational complexities of parameter estimation in non-regular statistical models. Finally, we provide targeted applications of the general theory to several classes of statistical models, including generalized linear models with polynomial link functions and location Gaussian mixture models.

We propose a learning rate adaptation scheme, called QLAB, for descent optimizers. We derive QLAB by optimizing the quadratic approximation of the loss function and QLAB can be combined with any optimizer who can provide the descent update direction. The computation of an adaptive learning rate with QLAB requires only computing an extra loss function value. We theoretically prove the convergence of the descent optimizers with QLAB. We demonstrate the effectiveness of QLAB in a range of optimization problems by combining with conclusively stochastic gradient descent, stochastic gradient descent with momentum, and Adam. The performance is validated on multi-layer neural networks, CNN, VGG-Net, ResNet and ShuffleNet with two datasets, MNIST and CIFAR10.