Smooth minimax games often proceed by simultaneous or alternating gradient updates. Although algorithms with alternating updates are commonly used in practice, the majority of existing theoretical analyses focus on simultaneous algorithms for convenience of analysis. In this paper, we study alternating gradient descent-ascent (Alt-GDA) in minimax games and show that Alt-GDA is superior to its simultaneous counterpart (Sim-GDA) in many settings. We prove that Alt-GDA achieves a near-optimal local convergence rate for strongly convex-strongly concave (SCSC) problems while Sim-GDA converges at a much slower rate. To our knowledge, this is the first result of any setting showing that Alt-GDA converges faster than Sim-GDA by more than a constant. We further adapt the theory of integral quadratic constraints (IQC) and show that Alt-GDA attains the same rate globally for a subclass of SCSC minimax problems. Empirically, we demonstrate that alternating updates speed up GAN training significantly and the use of optimism only helps for simultaneous algorithms.

Smooth game optimization has recently attracted great interest in machine learning as it generalizes the single-objective optimization paradigm. However, game dynamics is more complex due to the interaction between different players and is therefore fundamentally different from minimization, posing new challenges for algorithm design. Notably, it has been shown that negative momentum is preferred due to its ability to reduce oscillation in game dynamics. Nevertheless, existing analysis of negative momentum was restricted to simple bilinear games. In this paper, we extend the analysis to smooth and strongly-convex strongly-concave minimax games by taking the variational inequality formulation. By connecting Polyak’s momentum with Chebyshev polynomials, we show that negative momentum accelerates convergence of game dynamics locally, though with a suboptimal rate. To the best of our knowledge, this is the \emph{first work} that provides an explicit convergence rate for negative momentum in this setting.

Annealed importance sampling (AIS) and related algorithms are highly effective tools for marginal likelihood estimation, but are not fully differentiable due to the use of Metropolis-Hastings correction steps. Differentiability is a desirable property as it would admit the possibility of optimizing marginal likelihood as an objective using gradient-based methods. To this end, we propose Differentiable AIS (DAIS), a variant of AIS which ensures differentiability by abandoning the Metropolis-Hastings corrections. As a further advantage, DAIS allows for mini-batch gradients. We provide a detailed convergence analysis for Bayesian linear regression which goes beyond previous analyses by explicitly accounting for the sampler not having reached equilibrium. Using this analysis, we prove that DAIS is consistent in the full-batch setting and provide a sublinear convergence rate. Furthermore, motivated by the problem of learning from large-scale datasets, we study a stochastic variant of DAIS that uses mini-batch gradients. Surprisingly, stochastic DAIS can be arbitrarily bad due to a fundamental incompatibility between the goals of last-iterate convergence to the posterior and elimination of the accumulated stochastic error. This is in stark contrast with other settings such as gradient-based optimization and Langevin dynamics, where the effect of gradient noise can be washed out by taking smaller steps. This indicates that annealing-based marginal likelihood estimation with stochastic gradients may require new ideas.

The theory of integral quadratic constraints (IQCs) allows the certification of exponential convergence of interconnected systems containing nonlinear or uncertain elements. In this work, we adapt the IQC theory to study first-order methods for smooth and strongly-monotone games and show how to design tailored quadratic constraints to get tight upper bounds of convergence rates. Using this framework, we recover the existing bound for the gradient method~(GD), derive sharper bounds for the proximal point method~(PPM) and optimistic gradient method~(OG), and provide for the first time a global convergence rate for the negative momentum method~(NM) with an iteration complexity $\mathcal{O}(\kappa^{1.5})$, which matches its known lower bound. In addition, for time-varying systems, we prove that the gradient method with optimal step size achieves the fastest provable worst-case convergence rate with quadratic Lyapunov functions. Finally, we further extend our analysis to stochastic games and study the impact of multiplicative noise on different algorithms. We show that it is impossible for an algorithm with one step of memory to achieve acceleration if it only queries the gradient once per batch (in contrast with the stochastic strongly-convex optimization setting, where such acceleration has been demonstrated). However, we exhibit an algorithm which achieves acceleration with two gradient queries per batch.

The choice of batch-size in a stochastic optimization algorithm plays a substantial role for both optimization and generalization. Increasing the batch-size used typically improves optimization but degrades generalization. To address the problem of improving generalization while maintaining optimal convergence in large-batch training, we propose to add covariance noise to the gradients. We demonstrate that the learning performance of our method is more accurately captured by the structure of the covariance matrix of the noise rather than by the variance of gradients. Moreover, over the convex-quadratic, we prove in theory that it can be characterized by the Frobenius norm of the noise matrix. Our empirical studies with standard deep learning model-architectures and datasets shows that our method not only improves generalization performance in large-batch training, but furthermore, does so in a way where the optimization performance remains desirable and the training duration is not elongated.

Reducing the test time resource requirements of a neural network while preserving test accuracy is crucial for running inference on resource-constrained devices. To achieve this goal, we introduce a novel network reparameterization based on the Kronecker-factored eigenbasis (KFE), and then apply Hessian-based structured pruning methods in this basis. As opposed to existing Hessian-based pruning algorithms which do pruning in parameter coordinates, our method works in the KFE where different weights are approximately independent, enabling accurate pruning and fast computation. We demonstrate empirically the effectiveness of the proposed method through extensive experiments. In particular, we highlight that the improvements are especially significant for more challenging datasets and networks. With negligible loss of accuracy, an iterative-pruning version gives a 10x reduction in model size and a 8x reduction in FLOPs on wide ResNet32.

Increasing the batch size is a popular way to speed up neural network training, but beyond some critical batch size, larger batch sizes yield diminishing returns. In this work, we study how the critical batch size changes based on properties of the optimization algorithm, including acceleration and preconditioning, through two different lenses: large scale experiments and analysis using a simple noisy quadratic model (NQM). We experimentally demonstrate that optimization algorithms that employ preconditioning, specifically Adam and K-FAC, result in much larger critical batch sizes than stochastic gradient descent with momentum. We also demonstrate that the NQM captures many of the essential features of real neural network training, despite being drastically simpler to work with. The NQM predicts our results with preconditioned optimizers, previous results with accelerated gradient descent, and other results around optimal learning rates and large batch training, making it a useful tool to generate testable predictions about neural network optimization. We demonstrate empirically that the simple noisy quadratic model (NQM) displays many similarities to neural networks in terms of large-batch training. We prove analytical convergence results for the NQM model that predict such behavior and hence provide possible explanations and a better understanding for many large-batch training phenomena.

Natural gradient descent has proven very effective at mitigating the catastrophic effects of pathological curvature in the objective function, but little is known theoretically about its convergence properties, especially for \emph{non-linear} networks. In this work, we analyze for the first time the speed of convergence to global optimum for natural gradient descent on non-linear neural networks with the squared error loss. We identify two conditions which guarantee the global convergence: (1) the Jacobian matrix (of network's output for all training cases w.r.t the parameters) is full row rank and (2) the Jacobian matrix is stable for small perturbations around the initialization. For two-layer ReLU neural networks (i.e. with one hidden layer), we prove that these two conditions do hold throughout the training under the assumptions that the inputs do not degenerate and the network is over-parameterized. We further extend our analysis to more general loss function with similar convergence property. Lastly, we show that K-FAC, an approximate natural gradient descent method, also converges to global minima under the same assumptions.

The generalization properties of Gaussian processes depend heavily on the choice of kernel, and this choice remains a dark art. We present the Neural Kernel Network (NKN), a flexible family of kernels represented by a neural network. The NKN’s architecture is based on the composition rules for kernels, so that each unit of the network corresponds to a valid kernel. It can compactly approximate compositional kernel structures such as those used by the Automatic Statistician (Lloyd et al., 2014), but because the architecture is differentiable, it is end-to-end trainable with gradient- based optimization. We show that the NKN is universal for the class of stationary kernels. Empirically we demonstrate NKN’s pattern discovery and extrapolation abilities on several tasks that depend crucially on identifying the underlying structure, including time series and texture extrapolation, as well as Bayesian optimization.

Variational Bayesian neural nets combine the flexibility of deep learning with Bayesian uncertainty estimation. Unfortunately, there is a tradeoff between cheap but simple variational families (e.g. fully factorized) or expensive and complicated inference procedures. We show that natural gradient ascent with adaptive weight noise implicitly fits a variational posterior to maximize the evidence lower bound (ELBO). This insight allows us to train full-covariance, fully factorized, or matrix-variate Gaussian variational posteriors using noisy versions of natural gradient, Adam, and K-FAC, respectively, making it possible to scale up to modern-size ConvNets. On standard regression benchmarks, our noisy K-FAC algorithm makes better predictions and matches Hamiltonian Monte Carlo’s predictive variances better than existing methods. Its improved uncertainty estimates lead to more efficient exploration in active learning, and intrinsic motivation for reinforcement learning.