Differential Privacy of Noisy (S)GD under Heavy-Tailed Perturbations

Injecting heavy-tailed noise to the iterates of stochastic gradient descent (SGD) has received increasing attention over the past few years. While various theoretical properties of the resulting algorithm have been analyzed mainly from learning theory and optimization perspectives, their privacy preservation properties have not yet been established. Aiming to bridge this gap, we provide differential privacy (DP) guarantees for noisy SGD, when the injected noise follows an $\alpha$-stable distribution, which includes a spectrum of heavy-tailed distributions (with infinite variance) as well as the Gaussian distribution. Considering the $(\epsilon, \delta)$-DP framework, we show that SGD with heavy-tailed perturbations achieves $(0, \tilde{\mathcal{O}}(1/n))$-DP for a broad class of loss functions which can be non-convex, where $n$ is the number of data points. As a remarkable byproduct, contrary to prior work that necessitates bounded sensitivity for the gradients or clipping the iterates, our theory reveals that under mild assumptions, such a projection step is not actually necessary. We illustrate that the heavy-tailed noising mechanism achieves similar DP guarantees compared to the Gaussian case, which suggests that it can be a viable alternative to its light-tailed counterparts.

Generalization Bounds for Heavy-Tailed SDEs through the Fractional Fokker-Planck Equation

Understanding the generalization properties of heavy-tailed stochastic optimization algorithms has attracted increasing attention over the past years. While illuminating interesting aspects of stochastic optimizers by using heavy-tailed stochastic differential equations as proxies, prior works either provided expected generalization bounds, or introduced non-computable information theoretic terms. Addressing these drawbacks, in this work, we prove high-probability generalization bounds for heavy-tailed SDEs which do not contain any nontrivial information theoretic terms. To achieve this goal, we develop new proof techniques based on estimating the entropy flows associated with the so-called fractional Fokker-Planck equation (a partial differential equation that governs the evolution of the distribution of the corresponding heavy-tailed SDE). In addition to obtaining high-probability bounds, we show that our bounds have a better dependence on the dimension of parameters as compared to prior art. Our results further identify a phase transition phenomenon, which suggests that heavy tails can be either beneficial or harmful depending on the problem structure. We support our theory with experiments conducted in a variety of settings.

Tighter Generalisation Bounds via Interpolation

This paper contains a recipe for deriving new PAC-Bayes generalisation bounds based on the $(f, \Gamma)$-divergence, and, in addition, presents PAC-Bayes generalisation bounds where we interpolate between a series of probability divergences (including but not limited to KL, Wasserstein, and total variation), making the best out of many worlds depending on the posterior distributions properties. We explore the tightness of these bounds and connect them to earlier results from statistical learning, which are specific cases. We also instantiate our bounds as training objectives, yielding non-trivial guarantees and practical performances.

Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models

This paper deals with the problem of efficient sampling from a stochastic differential equation, given the drift function and the diffusion matrix. The proposed approach leverages a recent model for probabilities \citep{rudi2021psd} (the positive semi-definite -- PSD model) from which it is possible to obtain independent and identically distributed (i.i.d.) samples at precision $\varepsilon$ with a cost that is $m^2 d \log(1/\varepsilon)$ where $m$ is the dimension of the model, $d$ the dimension of the space. The proposed approach consists in: first, computing the PSD model that satisfies the Fokker-Planck equation (or its fractional variant) associated with the SDE, up to error $\varepsilon$, and then sampling from the resulting PSD model. Assuming some regularity of the Fokker-Planck solution (i.e. $\beta$-times differentiability plus some geometric condition on its zeros) We obtain an algorithm that: (a) in the preparatory phase obtains a PSD model with L2 distance $\varepsilon$ from the solution of the equation, with a model of dimension $m = \varepsilon^{-(d+1)/(\beta-2s)} (\log(1/\varepsilon))^{d+1}$ where $0<s\leq1$ is the fractional power to the Laplacian, and total computational complexity of $O(m^{3.5} \log(1/\varepsilon))$ and then (b) for Fokker-Planck equation, it is able to produce i.i.d.\ samples with error $\varepsilon$ in Wasserstein-1 distance, with a cost that is $O(d \varepsilon^{-2(d+1)/\beta-2} \log(1/\varepsilon)^{2d+3})$ per sample. This means that, if the probability associated with the SDE is somewhat regular, i.e. $\beta \geq 4d+2$, then the algorithm requires $O(\varepsilon^{-0.88} \log(1/\varepsilon)^{4.5d})$ in the preparatory phase, and $O(\varepsilon^{-1/2}\log(1/\varepsilon)^{2d+2})$ for each sample. Our results suggest that as the true solution gets smoother, we can circumvent the curse of dimensionality without requiring any sort of convexity.

Cyclic and Randomized Stepsizes Invoke Heavier Tails in SGD

Cyclic and randomized stepsizes are widely used in the deep learning practice and can often outperform standard stepsize choices such as constant stepsize in SGD. Despite their empirical success, not much is currently known about when and why they can theoretically improve the generalization performance. We consider a general class of Markovian stepsizes for learning, which contain i.i.d. random stepsize, cyclic stepsize as well as the constant stepsize as special cases, and motivated by the literature which shows that heaviness of the tails (measured by the so-called "tail-index") in the SGD iterates is correlated with generalization, we study tail-index and provide a number of theoretical results that demonstrate how the tail-index varies on the stepsize scheduling. Our results bring a new understanding of the benefits of cyclic and randomized stepsizes compared to constant stepsize in terms of the tail behavior. We illustrate our theory on linear regression experiments and show through deep learning experiments that Markovian stepsizes can achieve even a heavier tail and be a viable alternative to cyclic and i.i.d. randomized stepsize rules.

Generalization Bounds with Data-dependent Fractal Dimensions

Providing generalization guarantees for modern neural networks has been a crucial task in statistical learning. Recently, several studies have attempted to analyze the generalization error in such settings by using tools from fractal geometry. While these works have successfully introduced new mathematical tools to apprehend generalization, they heavily rely on a Lipschitz continuity assumption, which in general does not hold for neural networks and might make the bounds vacuous. In this work, we address this issue and prove fractal geometry-based generalization bounds without requiring any Lipschitz assumption. To achieve this goal, we build up on a classical covering argument in learning theory and introduce a data-dependent fractal dimension. Despite introducing a significant amount of technical complications, this new notion lets us control the generalization error (over either fixed or random hypothesis spaces) along with certain mutual information (MI) terms. To provide a clearer interpretation to the newly introduced MI terms, as a next step, we introduce a notion of "geometric stability" and link our bounds to the prior art. Finally, we make a rigorous connection between the proposed data-dependent dimension and topological data analysis tools, which then enables us to compute the dimension in a numerically efficient way. We support our theory with experiments conducted on various settings.

Algorithmic Stability of Heavy-Tailed SGD with General Loss Functions

Heavy-tail phenomena in stochastic gradient descent (SGD) have been reported in several empirical studies. Experimental evidence in previous works suggests a strong interplay between the heaviness of the tails and generalization behavior of SGD. To address this empirical phenomena theoretically, several works have made strong topological and statistical assumptions to link the generalization error to heavy tails. Very recently, new generalization bounds have been proven, indicating a non-monotonic relationship between the generalization error and heavy tails, which is more pertinent to the reported empirical observations. While these bounds do not require additional topological assumptions given that SGD can be modeled using a heavy-tailed stochastic differential equation (SDE), they can only apply to simple quadratic problems. In this paper, we build on this line of research and develop generalization bounds for a more general class of objective functions, which includes non-convex functions as well. Our approach is based on developing Wasserstein stability bounds for heavy-tailed SDEs and their discretizations, which we then convert to generalization bounds. Our results do not require any nontrivial assumptions; yet, they shed more light to the empirical observations, thanks to the generality of the loss functions.

Generalization Bounds for Stochastic Gradient Descent via Localized $\varepsilon$-Covers

In this paper, we propose a new covering technique localized for the trajectories of SGD. This localization provides an algorithm-specific complexity measured by the covering number, which can have dimension-independent cardinality in contrast to standard uniform covering arguments that result in exponential dimension dependency. Based on this localized construction, we show that if the objective function is a finite perturbation of a piecewise strongly convex and smooth function with $P$ pieces, i.e. non-convex and non-smooth in general, the generalization error can be upper bounded by $O(\sqrt{(\log n\log(nP))/n})$, where $n$ is the number of data samples. In particular, this rate is independent of dimension and does not require early stopping and decaying step size. Finally, we employ these results in various contexts and derive generalization bounds for multi-index linear models, multi-class support vector machines, and $K$-means clustering for both hard and soft label setups, improving the known state-of-the-art rates.

Algorithmic Stability of Heavy-Tailed Stochastic Gradient Descent on Least Squares

Recent studies have shown that heavy tails can emerge in stochastic optimization and that the heaviness of the tails has links to the generalization error. While these studies have shed light on interesting aspects of the generalization behavior in modern settings, they relied on strong topological and statistical regularity assumptions, which are hard to verify in practice. Furthermore, it has been empirically illustrated that the relation between heavy tails and generalization might not always be monotonic in practice, contrary to the conclusions of existing theory. In this study, we establish novel links between the tail behavior and generalization properties of stochastic gradient descent (SGD), through the lens of algorithmic stability. We consider a quadratic optimization problem and use a heavy-tailed stochastic differential equation as a proxy for modeling the heavy-tailed behavior emerging in SGD. We then prove uniform stability bounds, which reveal the following outcomes: (i) Without making any exotic assumptions, we show that SGD will not be stable if the stability is measured with the squared-loss $x\mapsto x^2$, whereas it in turn becomes stable if the stability is instead measured with a surrogate loss $x\mapsto |x|^p$ with some $p<2$. (ii) Depending on the variance of the data, there exists a \emph{`threshold of heavy-tailedness'} such that the generalization error decreases as the tails become heavier, as long as the tails are lighter than this threshold. This suggests that the relation between heavy tails and generalization is not globally monotonic. (iii) We prove matching lower-bounds on uniform stability, implying that our bounds are tight in terms of the heaviness of the tails. We support our theory with synthetic and real neural network experiments.

Chaotic Regularization and Heavy-Tailed Limits for Deterministic Gradient Descent

Recent studies have shown that gradient descent (GD) can achieve improved generalization when its dynamics exhibits a chaotic behavior. However, to obtain the desired effect, the step-size should be chosen sufficiently large, a task which is problem dependent and can be difficult in practice. In this study, we incorporate a chaotic component to GD in a controlled manner, and introduce multiscale perturbed GD (MPGD), a novel optimization framework where the GD recursion is augmented with chaotic perturbations that evolve via an independent dynamical system. We analyze MPGD from three different angles: (i) By building up on recent advances in rough paths theory, we show that, under appropriate assumptions, as the step-size decreases, the MPGD recursion converges weakly to a stochastic differential equation (SDE) driven by a heavy-tailed L\'evy-stable process. (ii) By making connections to recently developed generalization bounds for heavy-tailed processes, we derive a generalization bound for the limiting SDE and relate the worst-case generalization error over the trajectories of the process to the parameters of MPGD. (iii) We analyze the implicit regularization effect brought by the dynamical regularization and show that, in the weak perturbation regime, MPGD introduces terms that penalize the Hessian of the loss function. Empirical results are provided to demonstrate the advantages of MPGD.