Eliminating Sharp Minima from SGD with Truncated Heavy-tailed Noise

The empirical success of deep learning is often attributed to SGD's mysterious ability to avoid sharp local minima in the loss landscape, which is well known to lead to poor generalization. Recently, empirical evidence of heavy-tailed gradient noise was reported in many deep learning tasks; under the presence of heavy-tailed gradient noise, it can be shown that SGD can escape sharp local minima, providing a partial solution to the mystery. In this work, we analyze a popular variant of SGD where gradients are truncated above a fixed threshold. We show that it achieves a stronger notion of avoiding sharp minima; it can effectively eliminate sharp local minima entirely from its training trajectory. We characterize the dynamics of truncated SGD driven by heavy-tailed noises. First, we show that truncation threshold and width of the attraction field dictate the order of the first exit time from the associated local minimum. Moreover, when the objective function satisfies appropriate structural conditions, we prove that as the learning rate decreases the dynamics of heavy-tailed truncated SGD closely resemble those of a continuous-time Markov chain which never visits any sharp minima. We verify our theoretical results with numerical experiments and discuss the implications on the generalizability of SGD in deep learning.Comment: 92 pages (13 pages for the main paper and 79 pages for the supplementary materials), 7 figure

On the Sample Complexity and Metastability of Heavy-tailed Policy Search in Continuous Control

Reinforcement learning is a framework for interactive decision-making with incentives sequentially revealed across time without a system dynamics model. Due to its scaling to continuous spaces, we focus on policy search where one iteratively improves a parameterized policy with stochastic policy gradient (PG) updates. In tabular Markov Decision Problems (MDPs), under persistent exploration and suitable parameterization, global optimality may be obtained. By contrast, in continuous space, the non-convexity poses a pathological challenge as evidenced by existing convergence results being mostly limited to stationarity or arbitrary local extrema. To close this gap, we step towards persistent exploration in continuous space through policy parameterizations defined by distributions of heavier tails defined by tail-index parameter alpha, which increases the likelihood of jumping in state space. Doing so invalidates smoothness conditions of the score function common to PG. Thus, we establish how the convergence rate to stationarity depends on the policy's tail index alpha, a Holder continuity parameter, integrability conditions, and an exploration tolerance parameter introduced here for the first time. Further, we characterize the dependence of the set of local maxima on the tail index through an exit and transition time analysis of a suitably defined Markov chain, identifying that policies associated with Levy Processes of a heavier tail converge to wider peaks. This phenomenon yields improved stability to perturbations in supervised learning, which we corroborate also manifests in improved performance of policy search, especially when myopic and farsighted incentives are misaligned

Asymmetric Heavy Tails and Implicit Bias in Gaussian Noise Injections

Gaussian noise injections (GNIs) are a family of simple and widely-used regularisation methods for training neural networks, where one injects additive or multiplicative Gaussian noise to the network activations at every iteration of the optimisation algorithm, which is typically chosen as stochastic gradient descent (SGD). In this paper we focus on the so-called `implicit effect' of GNIs, which is the effect of the injected noise on the dynamics of SGD. We show that this effect induces an asymmetric heavy-tailed noise on SGD gradient updates. In order to model this modified dynamics, we first develop a Langevin-like stochastic differential equation that is driven by a general family of asymmetric heavy-tailed noise. Using this model we then formally prove that GNIs induce an `implicit bias', which varies depending on the heaviness of the tails and the level of asymmetry. Our empirical results confirm that different types of neural networks trained with GNIs are well-modelled by the proposed dynamics and that the implicit effect of these injections induces a bias that degrades the performance of networks.Comment: Main paper of 12 pages, followed by appendi

The Heavy-Tail Phenomenon in SGD

In recent years, various notions of capacity and complexity have been proposed for characterizing the generalization properties of stochastic gradient descent (SGD) in deep learning. Some of the popular notions that correlate well with the performance on unseen data are (i) the `flatness' of the local minimum found by SGD, which is related to the eigenvalues of the Hessian, (ii) the ratio of the stepsize $\eta$ to the batch-size $b$, which essentially controls the magnitude of the stochastic gradient noise, and (iii) the `tail-index', which measures the heaviness of the tails of the network weights at convergence. In this paper, we argue that these three seemingly unrelated perspectives for generalization are deeply linked to each other. We claim that depending on the structure of the Hessian of the loss at the minimum, and the choices of the algorithm parameters $\eta$ and $b$, the SGD iterates will converge to a \emph{heavy-tailed} stationary distribution. We rigorously prove this claim in the setting of quadratic optimization: we show that even in a simple linear regression problem with independent and identically distributed data whose distribution has finite moments of all order, the iterates can be heavy-tailed with infinite variance. We further characterize the behavior of the tails with respect to algorithm parameters, the dimension, and the curvature. We then translate our results into insights about the behavior of SGD in deep learning. We support our theory with experiments conducted on synthetic data, fully connected, and convolutional neural networks

On the Theoretical Properties of Noise Correlation in Stochastic Optimization

Studying the properties of stochastic noise to optimize complex non-convex functions has been an active area of research in the field of machine learning. Prior work has shown that the noise of stochastic gradient descent improves optimization by overcoming undesirable obstacles in the landscape. Moreover, injecting artificial Gaussian noise has become a popular idea to quickly escape saddle points. Indeed, in the absence of reliable gradient information, the noise is used to explore the landscape, but it is unclear what type of noise is optimal in terms of exploration ability. In order to narrow this gap in our knowledge, we study a general type of continuous-time non-Markovian process, based on fractional Brownian motion, that allows for the increments of the process to be correlated. This generalizes processes based on Brownian motion, such as the Ornstein-Uhlenbeck process. We demonstrate how to discretize such processes which gives rise to the new algorithm fPGD. This method is a generalization of the known algorithms PGD and Anti-PGD. We study the properties of fPGD both theoretically and empirically, demonstrating that it possesses exploration abilities that, in some cases, are favorable over PGD and Anti-PGD. These results open the field to novel ways to exploit noise for training machine learning models