We consider multi-agent learning via online gradient descent (OGD) in a class of games called $\lambda$-cocoercive games, a broad class of games that admits many Nash equilibria and that properly includes strongly monotone games. We characterize the finite-time last-iterate convergence rate for joint OGD learning on $\lambda$-cocoercive games; further, building on this result, we develop a fully adaptive OGD learning algorithm that does not require any knowledge of the problem parameter (e.g., the cocoercive constant $\lambda$) and show, via a novel double-stopping-time technique, that this adaptive algorithm achieves the same finite-time last-iterate convergence rate as its non-adaptive counterpart. Subsequently, we extend OGD learning to the noisy gradient feedback case and establish last-iterate convergence results---first qualitative almost sure convergence, then quantitative finite-time convergence rates---all under non-decreasing step-sizes. These results fill in several gaps in the existing multi-agent online learning literature, where three aspects---finite-time convergence rates, non-decreasing step-sizes, and fully adaptive algorithms---have not been previously explored.

We propose a unifying view of two different Bayesian inference algorithms, Stochastic Gradient Markov Chain Monte Carlo (SG-MCMC) and Stein Variational Gradient Descent (SVGD), leading to improved and efficient novel sampling schemes. We show that SVGD combined with a noise term can be framed as a multiple chain SG-MCMC method. Instead of treating each parallel chain independently from others, our proposed algorithm implements a repulsive force between particles, avoiding collapse and facilitating a better exploration of the parameter space. We also show how the addition of this noise term is necessary to obtain a valid SG-MCMC sampler, a significant difference with SVGD. Experiments with both synthetic distributions and real datasets illustrate the benefits of the proposed scheme.

Nonconvex minimax problems appear frequently in emerging machine learning applications, such as generative adversarial networks and adversarial learning. Simple algorithms such as the gradient descent ascent (GDA) are the common practice for solving these nonconvex games and receive lots of empirical success. Yet, it is known that these vanilla GDA algorithms with constant step size can potentially diverge even in the convex setting. In this work, we show that for a subclass of nonconvex-nonconcave objectives satisfying a so-called two-sided Polyak-{\L}ojasiewicz inequality, the alternating gradient descent ascent (AGDA) algorithm converges globally at a linear rate and the stochastic AGDA achieves a sublinear rate. We further develop a variance reduced algorithm that attains a provably faster rate than AGDA when the problem has the finite-sum structure.

Modern machine learning systems often leverage data that are generated ubiquitously and seamlessly through devices such as smartphones, cameras, microphones, or user's weblogs, transaction logs, social media, etc. Much of this data is private, and releasing models trained on such data without serious privacy considerations can reveal sensitive information (Narayanan and Shmatikov, 2008; Sweeney, 1997). Consequently, much emphasis has been placed in recent years on machine learning under the constraints of a robust privacy guarantee. One such notion that has emerged as a de facto standard is that of differential privacy. Informally, differential privacy provides a quantitative assessment of how different are the outputs of a randomized algorithm when fed two very similar inputs. If small changes in the input do not manifest as drastically different outputs, then it is hard to discern much information about the inputs solely based on the outputs of the algorithm. In the context of machine learning, this implies that if the learning algorithm is not overly sensitive to any single datum in the training set, then releasing the trained model should preserve the privacy of the training data. This requirement, apriori, seems compatible with the goal of learning, which is to find a model that generalizes well on the population and does not overfit to the given training sample. It seems reasonable then to argue that privacy is not necessarily at odds with generalization, especially when large training sets are available.

Distributed stochastic gradient descent (SGD) is essential for scaling the machine learning algorithms to a large number of computing nodes. However, the infrastructures variability such as high communication delay or random node slowdown greatly impedes the performance of distributed SGD algorithm, especially in a wireless system or sensor networks. In this paper, we propose an algorithmic approach named Overlap-Local-SGD (and its momentum variant) to overlap the communication and computation so as to speedup the distributed training procedure. The approach can help to mitigate the straggler effects as well. We achieve this by adding an anchor model on each node. After multiple local updates, locally trained models will be pulled back towards the synchronized anchor model rather than communicating with others. Experimental results of training a deep neural network on CIFAR-10 dataset demonstrate the effectiveness of Overlap-Local-SGD. We also provide a convergence guarantee for the proposed algorithm under non-convex objective functions.

This work examines the convergence of stochastic gradient-based optimization algorithms that use early stopping based on a validation function. The form of early stopping we consider is that optimization terminates when the norm of the gradient of a validation function falls below a threshold. We derive conditions that guarantee this stopping rule is well-defined, and provide bounds on the expected number of iterations and gradient evaluations needed to meet this criterion. The guarantee accounts for the distance between the training and validation sets, measured with the Wasserstein distance. We develop the approach in the general setting of a first-order optimization algorithm, with possibly biased update directions subject to a geometric drift condition. We then derive bounds on the expected running time for early stopping variants of several algorithms, including stochastic gradient descent (SGD), decentralized SGD (DSGD), and the stochastic variance reduced gradient (SVRG) algorithm. Finally, we consider the generalization properties of the iterate returned by early stopping.

The minimization of the loss function is of paramount importance in deep neural networks. On the other hand, many popular optimization algorithms have been shown to correspond to some evolution equation of gradient flow type. Inspired by the numerical schemes used for general evolution equations we introduce a second order stochastic Runge Kutta method and show that it yields a consistent procedure for the minimization of the loss function. In addition it can be coupled, in an adaptive framework, with a Stochastic Gradient Descent (SGD) to adjust automatically the learning rate of the SGD, without the need of any additional information on the Hessian of the loss functional. The adaptive SGD, called SGD-G2, is successfully tested on standard datasets.

The inference of chemical reaction networks is an important task in understanding the chemical processes in life sciences and environment. Yet, only a few reaction systems are well-understood due to a large number of important reaction pathways involved but still unknown. Revealing unknown reaction pathways is an important task for scientific discovery that takes decades and requires lots of expert knowledge. This work presents a neural network approach for discovering unknown reaction pathways from concentration time series data. The neural network denoted as Chemical Reaction Neural Network (CRNN), is designed to be equivalent to chemical reaction networks by following the fundamental physics laws of the Law of Mass Action and Arrhenius Law. The CRNN is physically interpretable, and its weights correspond to the reaction pathways and rate constants of the chemical reaction network. Then, inferencing the reaction pathways and the rate constants are accomplished by training the equivalent CRNN via stochastic gradient descent. The approach precludes the need for expert knowledge in proposing candidate reactions, such that the inference is autonomous and applicable to new systems for which there is no existing empirical knowledge to propose reaction pathways. The physical interpretability also makes the CRNN not only capable of fitting the data for a given system but also developing knowledge of unknown pathways that could be generalized to similar chemical systems. Finally, the approach is applied to several chemical systems in chemical engineering and biochemistry to demonstrate its robustness and generality.

Second-order gradient methods are among the most powerful algorithms in mathematical optimization. Algorithms in this family use a preconditioner matrix to transform the gradient before applying each step. Classically, this involves computing or approximating the matrix of second-order derivatives, i.e, the Hessian, in the context of exact deterministic optimization (e.g., Fletcher, 2013; Lewis & Overton, 2013; Nocedal, 1980). In contrast, AdaGrad (Duchi et al., 2011) and related algorithms that target stochastic optimization use the covariance matrix of second-order gradient statistics to form the preconditioner. While second-order methods often have significantly better convergence properties than first-order methods, the size of typical problems prohibits their use in practice, as they require quadratic storage and cubic computation time for each gradient update. Thus, these methods not commonly seen in the present practice of optimization in machine learning, which is largely dominated by the simpler to implement first-order methods. Arguably, one of the greatest challenges of modern optimization is to bridge this gap between the theoretical and practical optimization and make second-order optimization more feasible to implement and deploy. In this paper, we attempt to contribute towards narrowing this gap between theory and practice, focusing on second-order adaptive methods. These methods can be thought of as full-matrix analogues of common adaptive algorithms of the family of AdaGrad (Duchi et al., 2011) and Adam (Kingma & Ba, 2014).

Common Stochastic Gradient MCMC methods approximate gradients by stochastic ones via uniformly subsampled data points. We propose that a non-uniform subsampling can reduce the variance introduced by the stochastic approximation, hence making the sampling of a target distribution more accurate. An exponentially weighted stochastic gradient approach (EWSG) is developed for this objective by matching the transition kernels of SG-MCMC methods respectively based on stochastic and batch gradients. A demonstration of EWSG combined with second-order Langevin equation for sampling purposes is provided. In our method, non-uniform subsampling is done efficiently via a Metropolis-Hasting chain on the data index, which is coupled to the sampling algorithm. The fact that our method has reduced local variance with high probability is theoretically analyzed. A non-asymptotic global error analysis is also presented. Numerical experiments based on both synthetic and real world data sets are also provided to demonstrate the efficacy of the proposed approaches. While statistical accuracy has improved, the speed of convergence was empirically observed to be at least comparable to the uniform version.