Stochastic gradient descent (SGD) is a popular and efficient method with wide applications in training deep neural nets and other nonconvex models. While the behavior of SGD is well understood in the convex learning setting, the existing theoretical results for SGD applied to nonconvex objective functions are far from mature. For example, existing results require to impose a nontrivial assumption on the uniform boundedness of gradients for all iterates encountered in the learning process, which is hard to verify in practical implementations. In this paper, we establish a rigorous theoretical foundation for SGD in nonconvex learning by showing that this boundedness assumption can be removed without affecting convergence rates. In particular, we establish sufficient conditions for almost sure convergence as well as optimal convergence rates for SGD applied to both general nonconvex objective functions and gradient-dominated objective functions. A linear convergence is further derived in the case with zero variances.

Many randomized algorithms in machine learning can be analyzed as some kind of stochastic process. For example, MCMC algorithms intentionally inject carefully designed randomness in order to sample from a desired target distribution. There is a second category of randomized algorithms for which the for which the goal is optimization rather than sampling, and the randomness is viewed as a price to pay for computational tractability. For example, stochastic gradient methods for large scale optimization use noisy estimates of a gradient because they are cheap. While such algorithms are not designed with the goal of sampling from a target distribution, an algorithm of this kind has random outputs, and its behavior is determined by the distribution of its output. Results in this paper provide tools for analyzing the convergence of such algorithms as stochastic processes.

Despite the non-convex nature of their loss functions, deep neural networks are known to generalize well when optimized with stochastic gradient descent (SGD). Recent work conjectures that SGD with proper configuration is able to find wide and flat local minima, which have been proposed to be associated with good generalization performance. In this paper, we observe that local minima of modern deep networks are more than being flat or sharp. Specifically, at a local minimum there exist many asymmetric directions such that the loss increases abruptly along one side, and slowly along the opposite side--we formally define such minima as asymmetric valleys. Under mild assumptions, we prove that for asymmetric valleys, a solution biased towards the flat side generalizes better than the exact minimizer. Further, we show that simply averaging the weights along the SGD trajectory gives rise to such biased solutions implicitly. This provides a theoretical explanation for the intriguing phenomenon observed by Izmailov et al. (2018). In addition, we empirically find that batch normalization (BN) appears to be a major cause for asymmetric valleys.

Diffusion approximation provides weak approximation for stochastic gradient descent algorithms in a finite time horizon. In this paper, we introduce new tools motivated by the backward error analysis of numerical stochastic differential equations into the theoretical framework of diffusion approximation, extending the validity of the weak approximation from finite to infinite time horizon. The new techniques developed in this paper enable us to characterize the asymptotic behavior of constant-step-size SGD algorithms for strongly convex objective functions, a goal previously unreachable within the diffusion approximation framework. Our analysis builds upon a truncated formal power expansion of the solution of a stochastic modified equation arising from diffusion approximation, where the main technical ingredient is a uniform-in-time weak error bound controlling the long-term behavior of the expansion coefficient functions near the global minimum. We expect these new techniques to greatly expand the range of applicability of diffusion approximation to cover wider and deeper aspects of stochastic optimization algorithms in data science.

Generalization error (also known as the out-of-sample error) measures how well the hypothesis obtained from the training data can generalize to previously unseen data. Obtaining tight generalization error bounds is central to statistical learning theory. In this paper, we study the generalization error bound in learning general non-convex objectives, which has attracted significant attention in recent years. In particular, we study the (algorithm-dependent) generalization bounds of various iterative gradient based methods. (1) We present a very simple and elementary proof of a recent result for stochastic gradient Langevin dynamics (SGLD), due to Mou et al. (2018). Our proof can be easily extended to obtain similar generalization bounds for several other variants of SGLD (e.g., with postprocessing, momentum, mini-batch, acceleration, and more general noises), and improves upon the recent results in Pensia et al. (2018). (2) By incorporating ideas from the PAC-Bayesian theory into the stability framework, we obtain tighter distribution-dependent (or data-dependent) generalization bounds. Our bounds provide an intuitive explanation for the phenomenon reported in Zhang et al. (2017a). (3) We also study the setting where the total loss is the sum of a bounded loss and an additional `l2 regularization term. We obtain new generalization bounds for the continuous Langevin dynamic in this setting by leveraging the tool of Log-Sobolev inequality. Our new bounds are more desirable when the noisy level of the process is not small, and do not grow when T approaches to infinity.

Minmax optimization, especially in its general nonconvex-nonconcave formulation, has found extensive applications in modern machine learning frameworks such as generative adversarial networks (GAN), adversarial training and multi-agent reinforcement learning. Gradient-based algorithms, in particular gradient descent ascent (GDA), are widely used in practice to solve these problems. Despite the practical popularity of GDA, however, its theoretical behavior has been considered highly undesirable. Indeed, apart from possiblity of non-convergence, recent results (Daskalakis and Panageas, 2018; Mazumdar and Ratliff, 2018; Adolphs et al., 2018) show that even when GDA converges, its stable limit points can be points that are not local Nash equilibria, thus not game-theoretically meaningful. In this paper, we initiate a discussion on the proper optimality measures for minmax optimization, and introduce a new notion of local optimality---local minmax---as a more suitable alternative to the notion of local Nash equilibrium. We establish favorable properties of local minmax points, and show, most importantly, that as the ratio of the ascent step size to the descent step size goes to infinity, stable limit points of GDA are exactly local minmax points up to degenerate points, demonstrating that all stable limit points of GDA have a game-theoretic meaning for minmax problems.

In many statistics and machine learning frameworks, stochastic optimization with high variance gradients has become an important problem. For example, the performance of Monte Carlo variational inference (MCVI) seriously depends on the variance of its stochastic gradient estimator. In this paper, we focused on this problem and proposed a novel framework of variance reduction using multi-level Monte Carlo (MLMC) method. The framework is naturally compatible with reparameterization gradient estimators, which are one of the efficient variance reduction techniques that use the reparameterization trick. We also proposed a novel MCVI algorithm for stochastic gradient estimation on MLMC method in which sample size $N$ is adaptively estimated according to the ratio of the variance and computational cost for each iteration. We furthermore proved that, in our method, the norm of the gradient could converge to $0$ asymptotically. Finally, we evaluated our method by comparing it with benchmark methods in several experiments and showed that our method was able to reduce gradient variance and sampling cost efficiently and be closer to the optimum value than the other methods were.

We consider decentralized stochastic optimization with the objective function (e.g. data samples for machine learning task) being distributed over $n$ machines that can only communicate to their neighbors on a fixed communication graph. To reduce the communication bottleneck, the nodes compress (e.g. quantize or sparsify) their model updates. We cover both unbiased and biased compression operators with quality denoted by $\omega \leq 1$ ($\omega=1$ meaning no compression). We (i) propose a novel gossip-based stochastic gradient descent algorithm, CHOCO-SGD, that converges at rate $\mathcal{O}\left(1/(nT) + 1/(T \delta^2 \omega)^2\right)$ for strongly convex objectives, where $T$ denotes the number of iterations and $\delta$ the eigengap of the connectivity matrix. Despite compression quality and network connectivity affecting the higher order terms, the first term in the rate, $\mathcal{O}(1/(nT))$, is the same as for the centralized baseline with exact communication. We (ii) present a novel gossip algorithm, CHOCO-GOSSIP, for the average consensus problem that converges in time $\mathcal{O}(1/(\delta^2\omega) \log (1/\epsilon))$ for accuracy $\epsilon > 0$. This is (up to our knowledge) the first gossip algorithm that supports arbitrary compressed messages for $\omega > 0$ and still exhibits linear convergence. We (iii) show in experiments that both of our algorithms do outperform the respective state-of-the-art baselines and CHOCO-SGD can reduce communication by at least two orders of magnitudes.

Stochastic Variance-Reduced Cubic regularization (SVRC) algorithms have received increasing attention due to its improved gradient/Hessian complexities (i.e., number of queries to stochastic gradient/Hessian oracles) to find local minima for nonconvex finite-sum optimization. However, it is unclear whether existing SVRC algorithms can be further improved. Moreover, the semi-stochastic Hessian estimator adopted in existing SVRC algorithms prevents the use of Hessian-vector product-based fast cubic subproblem solvers, which makes SVRC algorithms computationally intractable for high-dimensional problems. In this paper, we first present a Stochastic Recursive Variance-Reduced Cubic regularization method (SRVRC) using a recursively updated semi-stochastic gradient and Hessian estimators. It enjoys improved gradient and Hessian complexities to find an $(\epsilon, \sqrt{\epsilon})$-approximate local minimum, and outperforms the state-of-the-art SVRC algorithms. Built upon SRVRC, we further propose a Hessian-free SRVRC algorithm, namely SRVRC$_{\text{free}}$, which only requires stochastic gradient and Hessian-vector product computations, and achieves $\tilde O(dn\epsilon^{-2} \land d\epsilon^{-3})$ runtime complexity, where $n$ is the number of component functions in the finite-sum structure, $d$ is the problem dimension, and $\epsilon$ is the optimization precision. This outperforms the best-known runtime complexity $\tilde O(d\epsilon^{-3.5})$ achieved by stochastic cubic regularization algorithm proposed in Tripuraneni et al. 2018.

Adaptive data analysis is frequently criticized for its pessimistic generalization guarantees. The source of these pessimistic bounds is a model that permits arbitrary, possibly adversarial analysts that optimally use information to bias results. While being a central issue in the field, still lacking are notions of natural analysts that allow for more optimistic bounds faithful to the reality that typical analysts aren't adversarial. In this work, we propose notions of natural analysts that smoothly interpolate between the optimal non-adaptive bounds and the best-known adaptive generalization bounds. To accomplish this, we model the analyst's knowledge as evolving according to the rules of an unknown dynamical system that takes in revealed information and outputs new statistical queries to the data. This allows us to restrict the analyst through different natural control-theoretic notions. One such notion corresponds to a recency bias, formalizing an inability to arbitrarily use distant information. Another complementary notion formalizes an anchoring bias, a tendency to weight initial information more strongly. Both notions come with quantitative parameters that smoothly interpolate between the non-adaptive case and the fully adaptive case, allowing for a rich spectrum of intermediate analysts that are neither non-adaptive nor adversarial. Natural not only from a cognitive perspective, we show that our notions also capture standard optimization methods, like gradient descent in various settings. This gives a new interpretation to the fact that gradient descent tends to overfit much less than its adaptive nature might suggest.