As learning algorithms are increasingly deployed in markets and other competitive environments, understanding their dynamics is becoming increasingly important. We study the limiting behavior of competitive agents employing gradient-based learning algorithms. Specifically, we introduce a general framework for competitive gradient-based learning that encompasses a wide breadth of learning algorithms including policy gradient reinforcement learning, gradient based bandits, and certain online convex optimization algorithms. We show that unlike the single agent case, gradient learning schemes in competitive settings do not necessarily correspond to gradient flows and, hence, it is possible for limiting behaviors like periodic orbits to exist. We introduce a new class of games, Morse-Smale games, that correspond to gradient-like flows. We provide guarantees that competitive gradient-based learning algorithms (both in the full information and gradient-free settings) avoid linearly unstable critical points (i.e. strict saddle points and unstable limit cycles). Since generic local Nash equilibria are not unstable critical points---that is, in a formal mathematical sense, almost all Nash equilibria are not strict saddles---these results imply that gradient-based learning almost surely does not get stuck at critical points that do not correspond to Nash equilibria. For Morse-Smale games, we show that competitive gradient learning converges to linearly stable cycles (which includes stable Nash equilibria) almost surely. Finally, we specialize these results to commonly used multi-agent learning algorithms and provide illustrative examples that demonstrate the wide range of limiting behaviors competitive gradient learning exhibits.

Emerging technologies and applications including Internet of Things (IoT), social networking, and crowd-sourcing generate large amounts of data at the network edge. Machine learning models are often built from the collected data, to enable the detection, classification, and prediction of future events. Due to bandwidth, storage, and privacy concerns, it is often impractical to send all the data to a centralized location. In this paper, we consider the problem of learning model parameters from data distributed across multiple edge nodes, without sending raw data to a centralized place. Our focus is on a generic class of machine learning models that are trained using gradient-descent based approaches. We analyze the convergence rate of distributed gradient descent from a theoretical point of view, based on which we propose a control algorithm that determines the best trade-off between local update and global parameter aggregation to minimize the loss function under a given resource budget. The performance of the proposed algorithm is evaluated via extensive experiments with real datasets, both on a networked prototype system and in a larger-scale simulated environment. The experimentation results show that our proposed approach performs near to the optimum with various machine learning models and different data distributions.

We investigate the effect of explicitly enforcing the Lipschitz continuity of neural networks. Our main hypothesis is that constraining the Lipschitz constant of a networks will have a regularising effect. To this end, we provide a simple technique for computing the Lipschitz constant of a feed forward neural network composed of commonly used layer types. This technique is then utilised to formulate training a Lipschitz continuous neural network as a constrained optimisation problem, which can be easily solved using projected stochastic gradient methods. Our evaluation study shows that, in isolation, our method performs comparatively to state-of-the-art regularisation techniques. Moreover, when combined with existing approaches to regularising neural networks the performance gains are cumulative.

With the development of big data technology, Gradient Boosting Decision Tree, i.e. GBDT, becomes one of the most important machine learning algorithms for its accurate output. However, the training process of GBDT needs a lot of computational resources and time. In order to accelerate the training process of GBDT, the asynchronous parallel sampling gradient boosting decision tree, abbr. asynch-SGBDT is proposed in this paper. Via introducing sampling, we adapt the numerical optimization process of traditional GBDT training process into stochastic optimization process and use asynchronous parallel stochastic gradient descent to accelerate the GBDT training process. Meanwhile, the theoretical analysis of asynch-SGBDT is provided by us in this paper. Experimental results show that GBDT training process could be accelerated by asynch-SGBDT. Our asynchronous parallel strategy achieves an almost linear speedup, especially for high-dimensional sparse datasets.

Anomaly Detection has several important applications. In this paper, our focus is on detecting anomalies in seller-reviewer data using tensor decomposition. While tensor-decomposition is mostly unsupervised, we formulate Bayesian semi-supervised tensor decomposition to take advantage of sparse labeled data. In addition, we use Polya-Gamma data augmentation for the semi-supervised Bayesian tensor decomposition. Finally, we show that the Polya-Gamma formulation simplifies calculation of the Fisher information matrix for partial natural gradient learning. Our experimental results show that our semi-supervised approach outperforms state of the art unsupervised baselines. And that the partial natural gradient learning outperforms stochastic gradient learning and Online-EM with sufficient statistics.

Techniques such as ensembling and distillation promise model quality improvements when paired with almost any base model. However, due to increased test-time cost (for ensembles) and increased complexity of the training pipeline (for distillation), these techniques are challenging to use in industrial settings. In this paper we explore a variant of distillation which is relatively straightforward to use as it does not require a complicated multi-stage setup or many new hyperparameters. Our first claim is that online distillation enables us to use extra parallelism to fit very large datasets about twice as fast. Crucially, we can still speed up training even after we have already reached the point at which additional parallelism provides no benefit for synchronous or asynchronous stochastic gradient descent. Two neural networks trained on disjoint subsets of the data can share knowledge by encouraging each model to agree with the predictions the other model would have made. These predictions can come from a stale version of the other model so they can be safely computed using weights that only rarely get transmitted. Our second claim is that online distillation is a cost-effective way to make the exact predictions of a model dramatically more reproducible. We support our claims using experiments on the Criteo Display Ad Challenge dataset, ImageNet, and the largest to-date dataset used for neural language modeling, containing $6\times 10^{11}$ tokens and based on the Common Crawl repository of web data.

The convergence speed of stochastic gradient descent (SGD) can be improved by actively selecting mini-batches. We explore sampling schemes where similar data points are less likely to be selected in the same mini-batch. In particular, we prove that such repulsive sampling schemes lowers the variance of the gradient estimator. This generalizes recent work on using Determinantal Point Processes (DPPs) for mini-batch diversification (Zhang et al., 2017) to the broader class of repulsive point processes. We first show that the phenomenon of variance reduction by diversified sampling generalizes in particular to non-stationary point processes. We then show that other point processes may be computationally much more efficient than DPPs. In particular, we propose and investigate Poisson Disk sampling---frequently encountered in the computer graphics community---for this task. We show empirically that our approach improves over standard SGD both in terms of convergence speed as well as final model performance.

Despite the widespread use of machine learning in adversarial settings such as computer security, recent studies have demonstrated vulnerabilities to evasion attacks---carefully crafted adversarial samples that closely resemble legitimate instances, but cause misclassification. In this paper, we examine the adequacy of the leading approach to generating adversarial samples---the gradient-descent approach. In particular (1) we perform extensive experiments on three datasets, MNIST, USPS and Spambase, in order to analyse the effectiveness of the gradient-descent method against non-linear support vector machines, and conclude that carefully reduced kernel smoothness can significantly increase robustness to the attack; (2) we demonstrate that separated inter-class support vectors lead to more secure models, and propose a quantity similar to margin that can efficiently predict potential susceptibility to gradient-descent attacks, before the attack is launched; and (3) we design a new adversarial sample construction algorithm based on optimising the multiplicative ratio of class decision functions.

Deep Neural Network (DNN) acoustic models often use discriminative sequence training that optimises an objective function that better approximates the word error rate (WER) than frame-based training. Sequence training is normally implemented using Stochastic Gradient Descent (SGD) or Hessian Free (HF) training. This paper proposes an alternative batch style optimisation framework that employs a Natural Gradient (NG) approach to traverse through the parameter space. By correcting the gradient according to the local curvature of the KL-divergence, the NG optimisation process converges more quickly than HF. Furthermore, the proposed NG approach can be applied to any sequence discriminative training criterion. The efficacy of the NG method is shown using experiments on a Multi-Genre Broadcast (MGB) transcription task that demonstrates both the computational efficiency and the accuracy of the resulting DNN models.

With deep convolutional neural networks (CNN) introduced by [1] and [2], computer vision tasks and more specifically image classification have made huge improvements in the years following [3]. CNN performances benefit a lot from big collections of annotated images like [4] or [5]. They are trained by optimizing a loss function with gradient descents computed on random mini-batches according to [6]. The method called stochastic gradient descent (SGD) has proved to be very efficient to train neural networks in general. However current CNN structures are extremely deep like the 100 layers ResNet of [7] and contains a lot of parameters (around 60M for Alexnet [3] and 130M for vgg [8]). Those structures involve heavy gradient computation times making the training on big data-sets very slow. Computation on GPU accelerates the training but requires huge local memory caches. Nevertheless the mini-batch optimization seems suitable for distributing the training over several threads. Many methods have been proposed like 1 [9, 10], which propose to distribute the batches over different threads called workers that periodically exchange information via a central thread to synchronize their models.