Heavy Tails in SGD and Compressibility of Overparametrized Neural Networks

Neural network compression techniques have become increasingly popular as they can drastically reduce the storage and computation requirements for very large networks. Recent empirical studies have illustrated that even simple pruning strategies can be surprisingly effective, and several theoretical studies have shown that compressible networks (in specific senses) should achieve a low generalization error. Yet, a theoretical characterization of the underlying cause that makes the networks amenable to such simple compression schemes is still missing. In this study, we address this fundamental question and reveal that the dynamics of the training algorithm has a key role in obtaining such compressible networks. Focusing our attention on stochastic gradient descent (SGD), our main contribution is to link compressibility to two recently established properties of SGD: (i) as the network size goes to infinity, the system can converge to a mean-field limit, where the network weights behave independently, (ii) for a large step-size/batch-size ratio, the SGD iterates can converge to a heavy-tailed stationary distribution. In the case where these two phenomena occur simultaneously, we prove that the networks are guaranteed to be '$\ell_p$-compressible', and the compression errors of different pruning techniques (magnitude, singular value, or node pruning) become arbitrarily small as the network size increases. We further prove generalization bounds adapted to our theoretical framework, which indeed confirm that the generalization error will be lower for more compressible networks. Our theory and numerical study on various neural networks show that large step-size/batch-size ratios introduce heavy-tails, which, in combination with overparametrization, result in compressibility.

Manipulating SGD with Data Ordering Attacks

Machine learning is vulnerable to a wide variety of different attacks. It is now well understood that by changing the underlying data distribution, an adversary can poison the model trained with it or introduce backdoors. In this paper we present a novel class of training-time attacks that require no changes to the underlying model dataset or architecture, but instead only change the order in which data are supplied to the model. In particular, an attacker can disrupt the integrity and availability of a model by simply reordering training batches, with no knowledge about either the model or the dataset. Indeed, the attacks presented here are not specific to the model or dataset, but rather target the stochastic nature of modern learning procedures. We extensively evaluate our attacks to find that the adversary can disrupt model training and even introduce backdoors. For integrity we find that the attacker can either stop the model from learning, or poison it to learn behaviours specified by the attacker. For availability we find that a single adversarially-ordered epoch can be enough to slow down model learning, or even to reset all of the learning progress. Such attacks have a long-term impact in that they decrease model performance hundreds of epochs after the attack took place. Reordering is a very powerful adversarial paradigm in that it removes the assumption that an adversary must inject adversarial data points or perturbations to perform training-time attacks. It reminds us that stochastic gradient descent relies on the assumption that data are sampled at random. If this randomness is compromised, then all bets are off.

Convergence Rates of Stochastic Gradient Descent under Infinite Noise Variance

Recent studies have provided both empirical and theoretical evidence illustrating that heavy tails can emerge in stochastic gradient descent (SGD) in various scenarios. Such heavy tails potentially result in iterates with diverging variance, which hinders the use of conventional convergence analysis techniques that rely on the existence of the second-order moments. In this paper, we provide convergence guarantees for SGD under a state-dependent and heavy-tailed noise with a potentially infinite variance, for a class of strongly convex objectives. In the case where the $p$-th moment of the noise exists for some $p\in [1,2)$, we first identify a condition on the Hessian, coined '$p$-positive (semi-)definiteness', that leads to an interesting interpolation between positive semi-definite matrices ($p=2$) and diagonally dominant matrices with non-negative diagonal entries ($p=1$). Under this condition, we then provide a convergence rate for the distance to the global optimum in $L^p$. Furthermore, we provide a generalized central limit theorem, which shows that the properly scaled Polyak-Ruppert averaging converges weakly to a multivariate $\alpha$-stable random vector. Our results indicate that even under heavy-tailed noise with infinite variance, SGD can converge to the global optimum without necessitating any modification neither to the loss function or to the algorithm itself, as typically required in robust statistics. We demonstrate the implications of our results to applications such as linear regression and generalized linear models subject to heavy-tailed data.

On the Ergodicity, Bias and Asymptotic Normality of Randomized Midpoint Sampling Method

The randomized midpoint method, proposed by [SL19], has emerged as an optimal discretization procedure for simulating the continuous time Langevin diffusions. Focusing on the case of strong-convex and smooth potentials, in this paper, we analyze several probabilistic properties of the randomized midpoint discretization method for both overdamped and underdamped Langevin diffusions. We first characterize the stationary distribution of the discrete chain obtained with constant step-size discretization and show that it is biased away from the target distribution. Notably, the step-size needs to go to zero to obtain asymptotic unbiasedness. Next, we establish the asymptotic normality for numerical integration using the randomized midpoint method and highlight the relative advantages and disadvantages over other discretizations. Our results collectively provide several insights into the behavior of the randomized midpoint discretization method, including obtaining confidence intervals for numerical integrations.

Riemannian Langevin Algorithm for Solving Semidefinite Programs

We propose a Langevin diffusion-based algorithm for non-convex optimization and sampling on a product manifold of spheres. Under a logarithmic Sobolev inequality, we establish a guarantee for finite iteration convergence to the Gibbs distribution in terms of Kullback-Leibler divergence. We show that with an appropriate temperature choice, the suboptimality gap to the global minimum is guaranteed to be arbitrarily small with high probability. As an application, we analyze the proposed Langevin algorithm for solving the Burer-Monteiro relaxation of a semidefinite program (SDP). In particular, we establish a logarithmic Sobolev inequality for the Burer-Monteiro problem when there are no spurious local minima; hence implying a fast escape from saddle points. Combining the results, we then provide a global optimality guarantee for the SDP and the Max-Cut problem. More precisely, we show the Langevin algorithm achieves $\epsilon$-multiplicative accuracy with high probability in $\widetilde{\Omega}( n^2 \epsilon^{-3} )$ iterations, where $n$ is the size of the cost matrix.

A Brief Note on the Convergence of Langevin Monte Carlo in Chi-Square Divergence

We study sampling from a target distribution $\nu_* \propto e^{-f}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm when the target $\nu_*$ satisfies the Poincar\'e inequality, and the potential $f$ is first-order smooth and dissipative. Under an opaque uniform warmness condition on the LMC iterates, we establish that $\widetilde{\mathcal{O}}(\epsilon^{-1})$ steps are sufficient for LMC to reach $\epsilon$ neighborhood of the target in Chi-square divergence. We hope that this note serves as a step towards establishing a complete convergence analysis of LMC under Chi-square divergence.

Convergence Analysis of Langevin Monte Carlo in Chi-Square Divergence

We study sampling from a target distribution $\nu_* \propto e^{-f}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm when the target $\nu_*$ satisfies the Poincar\'e inequality and the potential $f$ is weakly smooth, i.e., $\nabla f$ is $\beta$-H\"older continuous. We prove that $\widetilde{\mathcal{O}}(\epsilon^{-1/\beta})$ steps are sufficient for LMC to reach $\epsilon$ neighborhood of the target in Chi-square divergence. We derive the dimension dependency of the convergence rate under various scenarios, where the effects of initialization and the Poincar\'e constant are particularly taken into consideration. For convex and first-order smooth potentials, if we assume the Kannan-Lov\'asz-Simonovits (KLS) conjecture, then LMC with warm-start achieves the best-known rate $\widetilde{\mathcal{O}}(d\epsilon^{-1})$ which was previously established for strongly convex potentials. In the pessimistic case when the KLS conjecture is not true, using the results of Lee and Vempala, and initializing LMC with a Gaussian, we obtain the rate $\widetilde{\mathcal{O}}(d^{3}\epsilon^{-1})$ for all smooth potentials that are convex up to finite perturbations. Translating this rate to KL divergence improves upon the best-known rate for smooth potentials that have linear tail growth. For weakly smooth potentials whose tails behave like $\|x\|^\alpha$, the regime of improvement becomes the interval $\alpha \in (1,10/7]$. Finally, as we rely on the Poincar\'e inequality, our framework covers a wide range of non-convex potentials that are weakly smooth, and have at least linear tail growth.

Hausdorff Dimension, Stochastic Differential Equations, and Generalization in Neural Networks

Despite its success in a wide range of applications, characterizing the generalization properties of stochastic gradient descent (SGD) in non-convex deep learning problems is still an important challenge. While modeling the trajectories of SGD via stochastic differential equations (SDE) under heavy-tailed gradient noise has recently shed light over several peculiar characteristics of SGD, a rigorous treatment of the generalization properties of such SDEs in a learning theoretical framework is still missing. Aiming to bridge this gap, in this paper, we prove generalization bounds for SGD under the assumption that its trajectories can be well-approximated by a Feller process, which defines a rich class of Markov processes that include several recent SDE representations (both Brownian or heavy-tailed) as its special case. We show that the generalization error can be controlled by the Hausdorff dimension of the trajectories, which is intimately linked to the tail behavior of the driving process. Our results imply that heavier-tailed processes should achieve better generalization; hence, the tail-index of the process can be used as a notion of ``capacity metric''. We support our theory with experiments on deep neural networks illustrating that the proposed capacity metric accurately estimates the generalization error, and it does not necessarily grow with the number of parameters unlike the existing capacity metrics in the literature.

An Analysis of Constant Step Size SGD in the Non-convex Regime: Asymptotic Normality and Bias

Structured non-convex learning problems, for which critical points have favorable statistical properties, arise frequently in statistical machine learning. Algorithmic convergence and statistical estimation rates are well-understood for such problems. However, quantifying the uncertainty associated with the underlying training algorithm is not well-studied in the non-convex setting. In order to address this shortcoming, in this work, we establish an asymptotic normality result for the constant step size stochastic gradient descent (SGD) algorithm--a widely used algorithm in practice. Specifically, based on the relationship between SGD and Markov Chains [DDB19], we show that the average of SGD iterates is asymptotically normally distributed around the expected value of their unique invariant distribution, as long as the non-convex and non-smooth objective function satisfies a dissipativity property. We also characterize the bias between this expected value and the critical points of the objective function under various local regularity conditions. Together, the above two results could be leveraged to construct confidence intervals for non-convex problems that are trained using the SGD algorithm.