The success of adversarial formulations in machine learning has brought renewed motivation for smooth games. In this work, we focus on the class of stochastic Hamiltonian methods and provide the first convergence guarantees for certain classes of stochastic smooth games. We propose a novel unbiased estimator for the stochastic Hamiltonian gradient descent (SHGD) and highlight its benefits. Using tools from the optimization literature we show that SHGD converges linearly to the neighbourhood of a stationary point. To guarantee convergence to the exact solution, we analyze SHGD with a decreasing step-size and we also present the first stochastic variance reduced Hamiltonian method. Our results provide the first global non-asymptotic last-iterate convergence guarantees for the class of stochastic unconstrained bilinear games and for the more general class of stochastic games that satisfy a "sufficiently bilinear" condition, notably including some non-convex non-concave problems. We supplement our analysis with experiments on stochastic bilinear and sufficiently bilinear games, where our theory is shown to be tight, and on simple adversarial machine learning formulations.

We generalize the concept of maximum-margin classifiers (MMCs) to arbitrary norms and non-linear functions. Support Vector Machines (SVMs) are a special case of MMC. We find that MMCs can be formulated as Integral Probability Metrics (IPMs) or classifiers with some form of gradient norm penalty. This implies a direct link to a class of Generative adversarial networks (GANs) which penalize a gradient norm. We show that the Discriminator in Wasserstein, Standard, Least-Squares, and Hinge GAN with Gradient Penalty is an MMC. We explain why maximizing a margin may be helpful in GANs. We hypothesize and confirm experimentally that $L^\infty$-norm penalties with Hinge loss produce better GANs than $L^2$-norm penalties (based on common evaluation metrics). We derive the margins of Relativistic paired (Rp) and average (Ra) GANs.

Most stochastic optimization methods use gradients once before discarding them. While variance reduction methods have shown that reusing past gradients can be beneficial when there is a finite number of datapoints, they do not easily extend to the online setting. One issue is the staleness due to using past gradients. We propose to correct this staleness using the idea of implicit gradient transport (IGT) which transforms gradients computed at previous iterates into gradients evaluated at the current iterate without using the Hessian explicitly. In addition to reducing the variance and bias of our updates over time, IGT can be used as a drop-in replacement for the gradient estimate in a number of well-understood methods such as heavy ball or Adam. We show experimentally that it achieves state-of-the-art results on a wide range of architectures and benchmarks. Additionally, the IGT gradient estimator yields the optimal asymptotic convergence rate for online stochastic optimization in the restricted setting where the Hessians of all component functions are equal.

Many recent machine learning tools rely on differentiable game formulations. While several numerical methods have been proposed for these types of games, most of the work has been on convergence proofs or on upper bounds for the rate of convergence of those methods. In this work, we approach the question of fundamental iteration complexity by providing lower bounds. We generalise Nesterov's argument -- used in single-objective optimisation to derive a lower bound for a class of first-order black box optimisation algorithms -- to games. Moreover, we extend to games the p-SCLI framework used to derive spectral lower bounds for a large class of derivative-based single-objective optimisers. Finally, we propose a definition of the condition number arising from our lower bound analysis that matches the conditioning observed in upper bounds. Our condition number is more expressive than previously used definitions, as it covers a wide range of games, including bilinear games that lack strong convex-concavity.

We consider differentiable games: multi-objective minimization problems, where the goal is to find a Nash equilibrium. The machine learning community has recently started using extrapolation-based variants of the gradient method. A prime example is the extragradient, which yields linear convergence in cases like bilinear games, where the standard gradient method fails. The full benefits of extrapolation-based methods are not known: i) there is no unified analysis for a large class of games that includes both strongly monotone and bilinear games; ii) it is not known whether the rate achieved by extragradient can be improved, e.g. by considering multiple extrapolation steps. We answer these questions through new analysis of the extragradient's local and global convergence properties. Our analysis covers the whole range of settings between purely bilinear and strongly monotone games. It reveals that extragradient converges via different mechanisms at these extremes; in between, it exploits the most favorable mechanism for the given problem. We then present lower bounds on the rate of convergence for a wide class of algorithms with any number of extrapolations. Our bounds prove that the extragradient achieves the optimal rate in this class, and that our upper bounds are tight. Our precise characterization of the extragradient's convergence behavior in games shows that, unlike in convex optimization, the extragradient method may be much faster than the gradient method.

Machine learning promises methods that generalize well from finite labeled data. However, the brittleness of existing neural net approaches is revealed by notable failures, such as the existence of adversarial examples that are misclassified despite being nearly identical to a training example, or the inability of recurrent sequence-processing nets to stay on track without teacher forcing. We introduce a method, which we refer to as \emph{state reification}, that involves modeling the distribution of hidden states over the training data and then projecting hidden states observed during testing toward this distribution. Our intuition is that if the network can remain in a familiar manifold of hidden space, subsequent layers of the net should be well trained to respond appropriately. We show that this state-reification method helps neural nets to generalize better, especially when labeled data are sparse, and also helps overcome the challenge of achieving robust generalization with adversarial training.

Machine learning (ML) techniques are enjoying rapidly increasing adoption. However, designing and implementing the systems that support ML models in real-world deployments remains a significant obstacle, in large part due to the radically different development and deployment profile of modern ML methods, and the range of practical concerns that come with broader adoption. We propose to foster a new systems machine learning research community at the intersection of the traditional systems and ML communities, focused on topics such as hardware systems for ML, software systems for ML, and ML optimized for metrics beyond predictive accuracy. To do this, we describe a new conference, SysML, that explicitly targets research at the intersection of systems and machine learning with a program committee split evenly between experts in systems and ML, and an explicit focus on topics at the intersection of the two.

Recent literature has demonstrated promising results for training Generative Adversarial Networks by employing a set of discriminators, in contrast to the traditional game involving one generator against a single adversary. Such methods perform single-objective optimization on some simple consolidation of the losses, e.g. an arithmetic average. In this work, we revisit the multiple-discriminator setting by framing the simultaneous minimization of losses provided by different models as a multi-objective optimization problem. Specifically, we evaluate the performance of multiple gradient descent and the hypervolume maximization algorithm on a number of different datasets. Moreover, we argue that the previously proposed methods and hypervolume maximization can all be seen as variations of multiple gradient descent in which the update direction can be computed efficiently. Our results indicate that hypervolume maximization presents a better compromise between sample quality and computational cost than previous methods.

We revisit the bias-variance tradeoff for neural networks in light of modern empirical findings. The traditional bias-variance tradeoff in machine learning suggests that as model complexity grows, variance increases. Classical bounds in statistical learning theory point to the number of parameters in a model as a measure of model complexity, which means the tradeoff would indicate that variance increases with the size of neural networks. However, we empirically find that variance due to training set sampling is roughly constant (with both width and depth) in practice. Variance caused by the non-convexity of the loss landscape is different. We find that it decreases with width and increases with depth, in our setting. We provide theoretical analysis, in a simplified setting inspired by linear models, that is consistent with our empirical findings for width. We view bias-variance as a useful lens to study generalization through and encourage further theoretical explanation from this perspective. The traditional view in machine learning is that increasingly complex models achieve lower bias at the expense of higher variance. This balance between underfitting (high bias) and overfitting (high variance) is commonly known as the bias-variance tradeoff (Figure 1). In their landmark work that initially highlighted this bias-variance dilemma in machine learning, Geman et al. (1992) suggest that larger neural networks suffer from higher variance.

Recurrent neural networks are known for their notorious exploding and vanishing gradient problem (EVGP). This problem becomes more evident in tasks where the information needed to correctly solve them exist over long time scales, because EVGP prevents important gradient components from being back-propagated adequately over a large number of steps. We introduce a simple stochastic algorithm (h-detach) that is specific to LSTM optimization and targeted towards addressing this problem. Specifically, we show that when the LSTM weights are large, the gradient components through the linear path (cell state) in the LSTM computational graph get suppressed. Based on the hypothesis that these components carry information about long term dependencies (which we show empirically), their suppression can prevent LSTMs from capturing them. Our algorithm prevents gradients flowing through this path from getting suppressed, thus allowing the LSTM to capture such dependencies better. We show significant convergence and generalization improvements using our algorithm on various benchmark datasets. Recurrent Neural Networks (RNNs) (Rumelhart et al. (1986); Elman (1990)) are a class of neural network architectures used for modeling sequential data.