Private Covariance Approximation and Eigenvalue-Gap Bounds for Complex Gaussian Perturbations

We consider the problem of approximating a $d \times d$ covariance matrix $M$ with a rank-$k$ matrix under $(\varepsilon,\delta)$-differential privacy. We present and analyze a complex variant of the Gaussian mechanism and show that the Frobenius norm of the difference between the matrix output by this mechanism and the best rank-$k$ approximation to $M$ is bounded by roughly $\tilde{O}(\sqrt{kd})$, whenever there is an appropriately large gap between the $k$'th and the $k+1$'th eigenvalues of $M$. This improves on previous work that requires that the gap between every pair of top-$k$ eigenvalues of $M$ is at least $\sqrt{d}$ for a similar bound. Our analysis leverages the fact that the eigenvalues of complex matrix Brownian motion repel more than in the real case, and uses Dyson's stochastic differential equations governing the evolution of its eigenvalues to show that the eigenvalues of the matrix $M$ perturbed by complex Gaussian noise have large gaps with high probability. Our results contribute to the analysis of low-rank approximations under average-case perturbations and to an understanding of eigenvalue gaps for random matrices, which may be of independent interest.

Re-Analyze Gauss: Bounds for Private Matrix Approximation via Dyson Brownian Motion

Given a symmetric matrix $M$ and a vector $\lambda$, we present new bounds on the Frobenius-distance utility of the Gaussian mechanism for approximating $M$ by a matrix whose spectrum is $\lambda$, under $(\varepsilon,\delta)$-differential privacy. Our bounds depend on both $\lambda$ and the gaps in the eigenvalues of $M$, and hold whenever the top $k+1$ eigenvalues of $M$ have sufficiently large gaps. When applied to the problems of private rank-$k$ covariance matrix approximation and subspace recovery, our bounds yield improvements over previous bounds. Our bounds are obtained by viewing the addition of Gaussian noise as a continuous-time matrix Brownian motion. This viewpoint allows us to track the evolution of eigenvalues and eigenvectors of the matrix, which are governed by stochastic differential equations discovered by Dyson. These equations allow us to bound the utility as the square-root of a sum-of-squares of perturbations to the eigenvectors, as opposed to a sum of perturbation bounds obtained via Davis-Kahan-type theorems.

Private Matrix Approximation and Geometry of Unitary Orbits

Consider the following optimization problem: Given $n \times n$ matrices $A$ and $\Lambda$, maximize $\langle A, U\Lambda U^*\rangle$ where $U$ varies over the unitary group $\mathrm{U}(n)$. This problem seeks to approximate $A$ by a matrix whose spectrum is the same as $\Lambda$ and, by setting $\Lambda$ to be appropriate diagonal matrices, one can recover matrix approximation problems such as PCA and rank-$k$ approximation. We study the problem of designing differentially private algorithms for this optimization problem in settings where the matrix $A$ is constructed using users' private data. We give efficient and private algorithms that come with upper and lower bounds on the approximation error. Our results unify and improve upon several prior works on private matrix approximation problems. They rely on extensions of packing/covering number bounds for Grassmannians to unitary orbits which should be of independent interest.

Faster Sampling from Log-Concave Distributions over Polytopes via a Soft-Threshold Dikin Walk

We consider the problem of sampling from a $d$-dimensional log-concave distribution $\pi(\theta) \propto e^{-f(\theta)}$ constrained to a polytope $K$ defined by $m$ inequalities. Our main result is a "soft-threshold'' variant of the Dikin walk Markov chain that requires at most $O((md + d L^2 R^2) \times md^{\omega-1}) \log(\frac{w}{\delta}))$ arithmetic operations to sample from $\pi$ within error $\delta>0$ in the total variation distance from a $w$-warm start, where $L$ is the Lipschitz-constant of $f$, $K$ is contained in a ball of radius $R$ and contains a ball of smaller radius $r$, and $\omega$ is the matrix-multiplication constant. When a warm start is not available, it implies an improvement of $\tilde{O}(d^{3.5-\omega})$ arithmetic operations on the previous best bound for sampling from $\pi$ within total variation error $\delta$, which was obtained with the hit-and-run algorithm, in the setting where $K$ is a polytope given by $m=O(d)$ inequalities and $LR = O(\sqrt{d})$. When a warm start is available, our algorithm improves by a factor of $d^2$ arithmetic operations on the best previous bound in this setting, which was obtained for a different version of the Dikin walk algorithm. Plugging our Dikin walk Markov chain into the post-processing algorithm of Mangoubi and Vishnoi (2021), we achieve further improvements in the dependence of the running time for the problem of generating samples from $\pi$ with infinity distance bounds in the special case when $K$ is a polytope.

Sampling from Log-Concave Distributions with Infinity-Distance Guarantees and Applications to Differentially Private Optimization

For a $d$-dimensional log-concave distribution $\pi(\theta)\propto e^{-f(\theta)}$ on a polytope $K$, we consider the problem of outputting samples from a distribution $\nu$ which is $O(\varepsilon)$-close in infinity-distance $\sup_{\theta\in K}|\log\frac{\nu(\theta)}{\pi(\theta)}|$ to $\pi$. Such samplers with infinity-distance guarantees are specifically desired for differentially private optimization as traditional sampling algorithms which come with total-variation distance or KL divergence bounds are insufficient to guarantee differential privacy. Our main result is an algorithm that outputs a point from a distribution $O(\varepsilon)$-close to $\pi$ in infinity-distance and requires $O((md+dL^2R^2)\times(LR+d\log(\frac{Rd+LRd}{\varepsilon r}))\times md^{\omega-1})$ arithmetic operations, where $f$ is $L$-Lipschitz, $K$ is defined by $m$ inequalities, is contained in a ball of radius $R$ and contains a ball of smaller radius $r$, and $\omega$ is the matrix-multiplication constant. In particular this runtime is logarithmic in $\frac{1}{\varepsilon}$ and significantly improves on prior works. Technically, we depart from the prior works that construct Markov chains on a $\frac{1}{\varepsilon^2}$-discretization of $K$ to achieve a sample with $O(\varepsilon)$ infinity-distance error, and present a method to convert continuous samples from $K$ with total-variation bounds to samples with infinity bounds. To achieve improved dependence on $d$, we present a "soft-threshold" version of the Dikin walk which may be of independent interest. Plugging our algorithm into the framework of the exponential mechanism yields similar improvements in the running time of $\varepsilon$-pure differentially private algorithms for optimization problems such as empirical risk minimization of Lipschitz-convex functions and low-rank approximation, while still achieving the tightest known utility bounds.

Sync-Switch: Hybrid Parameter Synchronization for Distributed Deep Learning

Stochastic Gradient Descent (SGD) has become the de facto way to train deep neural networks in distributed clusters. A critical factor in determining the training throughput and model accuracy is the choice of the parameter synchronization protocol. For example, while Bulk Synchronous Parallel (BSP) often achieves better converged accuracy, the corresponding training throughput can be negatively impacted by stragglers. In contrast, Asynchronous Parallel (ASP) can have higher throughput, but its convergence and accuracy can be impacted by stale gradients. To improve the performance of synchronization protocol, recent work often focuses on designing new protocols with a heavy reliance on hard-to-tune hyper-parameters. In this paper, we design a hybrid synchronization approach that exploits the benefits of both BSP and ASP, i.e., reducing training time while simultaneously maintaining the converged accuracy. Based on extensive empirical profiling, we devise a collection of adaptive policies that determine how and when to switch between synchronization protocols. Our policies include both offline ones that target recurring jobs and online ones for handling transient stragglers. We implement the proposed policies in a prototype system, called Sync-Switch, on top of TensorFlow, and evaluate the training performance with popular deep learning models and datasets. Our experiments show that Sync-Switch achieves up to 5.13X throughput speedup and similar converged accuracy when comparing to BSP. Further, we observe that Sync-Switch achieves 3.8% higher converged accuracy with just 1.23X the training time compared to training with ASP. Moreover, Sync-Switch can be used in settings when training with ASP leads to divergence errors. Sync-Switch achieves all of these benefits with very low overhead, e.g., the framework overhead can be as low as 1.7% of the total training time.

A Provably Convergent and Practical Algorithm for Min-max Optimization with Applications to GANs

We present a new algorithm for optimizing min-max loss functions that arise in training GANs. We prove that our algorithm converges to an equilibrium point in time polynomial in the dimension, and smoothness parameters of the loss function. The point our algorithm converges to is stable when the maximizing player can respond using any sequence of steps which increase the loss at each step, and the minimizing player is empowered to simulate the maximizing player's response for arbitrarily many steps but is restricted to move according to updates sampled from a stochastic gradient oracle. We apply our algorithm to train GANs on Gaussian mixtures, MNIST and CIFAR-10. We observe that our algorithm trains stably and avoids mode collapse, while achieving a training time per iteration and memory requirement similar to gradient descent-ascent.

A Second-order Equilibrium in Nonconvex-Nonconcave Min-max Optimization: Existence and Algorithm

Min-max optimization, with a nonconvex-nonconcave objective function $f: \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$, arises in many areas, including optimization, economics, and deep learning. The nonconvexity-nonconcavity of $f$ means that the problem of finding a global $\varepsilon$-min-max point cannot be solved in $\mathrm{poly}(d, \frac{1}{\varepsilon})$ evaluations of $f$. Thus, most algorithms seek to obtain a certain notion of local min-max point where, roughly speaking, each player optimizes her payoff in a local sense. However, the classes of local min-max solutions which prior algorithms seek are only guaranteed to exist under very strong assumptions on $f$, such as convexity or monotonicity. We propose a notion of a greedy equilibrium point for min-max optimization and prove the existence of such a point for any function such that it and its first three derivatives are bounded. Informally, we say that a point $(x^\star, y^\star)$ is an $\varepsilon$-greedy min-max equilibrium point of a function $f: \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ if $y^\star$ is a second-order local maximum for $f(x^\star,\cdot)$ and, roughly, $x^\star$ is a local minimum for a greedy optimization version of the function $\max_y f(x,y)$ which can be efficiently estimated using greedy algorithms. The existence follows from an algorithm that converges from any starting point to such a point in a number of gradient and function evaluations that is polynomial in $\frac{1}{\varepsilon}$, the dimension $d$, and the bounds on $f$ and its first three derivatives. Our results do not require convexity, monotonicity, or special starting points.

Faster algorithms for polytope rounding, sampling, and volume computation via a sublinear "Ball Walk''

We study the problem of "isotropically rounding" a polytope $K\subseteq\mathbb{R}^n$, that is, computing a linear transformation which makes the uniform distribution on the polytope have roughly identity covariance matrix. We assume that $K$ is defined by $m$ linear inequalities, with guarantee that $rB\subseteq K\subseteq RB$, where $B$ is the unit ball. We introduce a new variant of the ball walk Markov chain and show that, roughly, the expected number of arithmetic operations per-step of this Markov chain is $O(m)$ that is sublinear in the input size $mn$--the per-step time of all prior Markov chains. Subsequently, we give a rounding algorithm that succeeds with probability $1-\varepsilon$ in $\tilde{O}(mn^{4.5}\mathrm{polylog}(\frac{1}{\varepsilon},\frac{R}{r}))$ arithmetic operations. This gives a factor of $\sqrt{n}$ improvement on the previous bound of $\tilde{O}(mn^{5} \mathrm{polylog}(\frac{1}{\varepsilon},\frac{R}{r}))$ for rounding, which uses the hit-and-run algorithm. Since the cost of the rounding preprocessing step is in many cases the bottleneck in improving sampling or volume computation, our results imply these tasks can also be achieved in roughly $\tilde{O}(mn^{4.5}\mathrm{polylog}(\frac{1}{\varepsilon},\frac{R}{r})+mn^4\delta^{-2})$ operations for computing the volume of $K$ up to a factor $1+\delta$ and $\tilde{O}(m n^{4.5}\mathrm{polylog}(\frac{1}{\varepsilon},\frac{R}{r})))$ for uniformly sampling on $K$ with TV error $\varepsilon$. This improves on the previous bounds of $\tilde{O}(mn^{5}\mathrm{polylog}(\frac{1}{\varepsilon},\frac{R}{r})+mn^4\delta^{-2})$ for volume computation and $\tilde{O}(mn^{5}\mathrm{polylog}(\frac{1}{\varepsilon},\frac{R}{r}))$ for sampling. We achieve this improvement by a novel method of computing polytope membership, where one avoids checking inequalities which are estimated to have a very low probability of being violated.

Nonconvex sampling with the Metropolis-adjusted Langevin algorithm

The Langevin Markov chain algorithms are widely deployed methods to sample from distributions in challenging high-dimensional and non-convex statistics and machine learning applications. Despite this, current bounds for the Langevin algorithms are slower than those of competing algorithms in many important situations, for instance when sampling from weakly log-concave distributions, or when sampling or optimizing non-convex log-densities. In this paper, we obtain improved bounds in many of these situations, showing that the Metropolis-adjusted Langevin algorithm (MALA) is faster than the best bounds for its competitor algorithms when the target distribution satisfies weak third- and fourth- order regularity properties associated with the input data. In many settings, our regularity conditions are weaker than the usual Euclidean operator norm regularity properties, allowing us to show faster bounds for a much larger class of distributions than would be possible with the usual Euclidean operator norm approach, including in statistics and machine learning applications where the data satisfy a certain incoherence condition. In particular, we show that using our regularity conditions one can obtain faster bounds for applications which include sampling problems in Bayesian logistic regression with weakly convex priors, and the nonconvex optimization problem of learning linear classifiers with zero-one loss functions. Our main technical contribution in this paper is our analysis of the Metropolis acceptance probability of MALA in terms of its "energy-conservation error," and our bound for this error in terms of third- and fourth- order regularity conditions. Our combination of this higher-order analysis of the energy conservation error with the conductance method is key to obtaining bounds which have a sub-linear dependence on the dimension $d$ in the non-strongly logconcave setting.