arXiv subjects

Jason M. Altschuler

Publications and source records attributed to Jason M. Altschuler.

31 records · Page 2Linked to original sources

arXiv2023

Privacy of Noisy Stochastic Gradient Descent: More Iterations without More Privacy Loss

A central issue in machine learning is how to train models on sensitive user data. Industry has widely adopted a simple algorithm: Stochastic Gradient Descent with noise (a.k.a. Stochastic Gradient Langevin Dynamics). However, foundational theoretical questions about this algorithm's privacy loss remain open -- even in the seemingly simple setting of smooth convex losses over a bounded domain. Our main result resolves these questions: for a large range of parameters, we characterize the differential privacy up to a constant factor. This result reveals that all previous analyses for this setting have the wrong qualitative behavior. Specifically, while previous privacy analyses increase ad infinitum in the number of iterations, we show that after a small burn-in period, running SGD longer leaks no further privacy. Our analysis departs from previous approaches based on fast mixing, instead using techniques based on optimal transport (namely, Privacy Amplification by Iteration) and the Sampled Gaussian Mechanism (namely, Privacy Amplification by Sampling). Our techniques readily extend to other settings, e.g., strongly convex losses, non-uniform stepsizes, arbitrary batch sizes, and random or cyclic choice of batches.

cs.LG↗

arXiv2023

Faster high-accuracy log-concave sampling via algorithmic warm starts

Understanding the complexity of sampling from a strongly log-concave and log-smooth distribution $π$ on $\mathbb{R}^d$ to high accuracy is a fundamental problem, both from a practical and theoretical standpoint. In practice, high-accuracy samplers such as the classical Metropolis-adjusted Langevin algorithm (MALA) remain the de facto gold standard; and in theory, via the proximal sampler reduction, it is understood that such samplers are key for sampling even beyond log-concavity (in particular, for distributions satisfying isoperimetric assumptions). In this work, we improve the dimension dependence of this sampling problem to $\tilde{O}(d^{1/2})$, whereas the previous best result for MALA was $\tilde{O}(d)$. This closes the long line of work on the complexity of MALA, and moreover leads to state-of-the-art guarantees for high-accuracy sampling under strong log-concavity and beyond (thanks to the aforementioned reduction). Our starting point is that the complexity of MALA improves to $\tilde{O}(d^{1/2})$, but only under a warm start (an initialization with constant Rényi divergence w.r.t. $π$). Previous algorithms took much longer to find a warm start than to use it, and closing this gap has remained an important open problem in the field. Our main technical contribution settles this problem by establishing the first $\tilde{O}(d^{1/2})$ Rényi mixing rates for the discretized underdamped Langevin diffusion. For this, we develop new differential-privacy-inspired techniques based on Rényi divergences with Orlicz--Wasserstein shifts, which allow us to sidestep longstanding challenges for proving fast convergence of hypocoercive differential equations.

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arXiv2022

Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling

Sampling from a high-dimensional distribution is a fundamental task in statistics, engineering, and the sciences. A canonical approach is the Langevin Algorithm, i.e., the Markov chain for the discretized Langevin Diffusion. This is the sampling analog of Gradient Descent. Despite being studied for several decades in multiple communities, tight mixing bounds for this algorithm remain unresolved even in the seemingly simple setting of log-concave distributions over a bounded domain. This paper completely characterizes the mixing time of the Langevin Algorithm to its stationary distribution in this setting (and others). This mixing result can be combined with any bound on the discretization bias in order to sample from the stationary distribution of the continuous Langevin Diffusion. In this way, we disentangle the study of the mixing and bias of the Langevin Algorithm. Our key insight is to introduce a technique from the differential privacy literature to the sampling literature. This technique, called Privacy Amplification by Iteration, uses as a potential a variant of Rényi divergence that is made geometrically aware via Optimal Transport smoothing. This gives a short, simple proof of optimal mixing bounds and has several additional appealing properties. First, our approach removes all unnecessary assumptions required by other sampling analyses. Second, our approach unifies many settings: it extends unchanged if the Langevin Algorithm uses projections, stochastic mini-batch gradients, or strongly convex potentials (whereby our mixing time improves exponentially). Third, our approach exploits convexity only through the contractivity of a gradient step -- reminiscent of how convexity is used in textbook proofs of Gradient Descent. In this way, we offer a new approach towards further unifying the analyses of optimization and sampling algorithms.

math.ST↗

arXiv2022

Flows, Scaling, and Entropy Revisited: a Unified Perspective via Optimizing Joint Distributions

In this short expository note, we describe a unified algorithmic perspective on several classical problems which have traditionally been studied in different communities. This perspective views the main characters -- the problems of Optimal Transport, Minimum Mean Cycle, Matrix Scaling, and Matrix Balancing -- through the same lens of optimization problems over joint probability distributions P(x,y) with constrained marginals. While this is how Optimal Transport is typically introduced, this lens is markedly less conventional for the other three problems. This perspective leads to a simple and unified framework spanning problem formulation, algorithm development, and runtime analysis.

math.OC↗

arXiv2022

Polynomial-time algorithms for Multimarginal Optimal Transport problems with structure

Multimarginal Optimal Transport (MOT) has attracted significant interest due to applications in machine learning, statistics, and the sciences. However, in most applications, the success of MOT is severely limited by a lack of efficient algorithms. Indeed, MOT in general requires exponential time in the number of marginals k and their support sizes n. This paper develops a general theory about what "structure" makes MOT solvable in poly(n,k) time. We develop a unified algorithmic framework for solving MOT in poly(n,k) time by characterizing the "structure" that different algorithms require in terms of simple variants of the dual feasibility oracle. This framework has several benefits. First, it enables us to show that the Sinkhorn algorithm, which is currently the most popular MOT algorithm, requires strictly more structure than other algorithms do to solve MOT in poly(n,k) time. Second, our framework makes it much simpler to develop poly(n,k) time algorithms for a given MOT problem. In particular, it is necessary and sufficient to (approximately) solve the dual feasibility oracle -- which is much more amenable to standard algorithmic techniques. We illustrate this ease-of-use by developing poly(n,k) time algorithms for three general classes of MOT cost structures: (1) graphical structure; (2) set-optimization structure; and (3) low-rank plus sparse structure. For structure (1), we recover the known result that Sinkhorn has poly(n,k) runtime; moreover, we provide the first poly(n,k) time algorithms for computing solutions that are exact and sparse. For structures (2)-(3), we give the first poly(n,k) time algorithms, even for approximate computation. Together, these three structures encompass many -- if not most -- current applications of MOT.

math.OC↗

arXiv2021

Wasserstein barycenters are NP-hard to compute

Computing Wasserstein barycenters (a.k.a. Optimal Transport barycenters) is a fundamental problem in geometry which has recently attracted considerable attention due to many applications in data science. While there exist polynomial-time algorithms in any fixed dimension, all known running times suffer exponentially in the dimension. It is an open question whether this exponential dependence is improvable to a polynomial dependence. This paper proves that unless P=NP, the answer is no. This uncovers a "curse of dimensionality" for Wasserstein barycenter computation which does not occur for Optimal Transport computation. Moreover, our hardness results for computing Wasserstein barycenters extend to approximate computation, to seemingly simple cases of the problem, and to averaging probability distributions in other Optimal Transport metrics.

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arXiv2021

Approximating Min-Mean-Cycle for low-diameter graphs in near-optimal time and memory

We revisit Min-Mean-Cycle, the classical problem of finding a cycle in a weighted directed graph with minimum mean weight. Despite an extensive algorithmic literature, previous work falls short of a near-linear runtime in the number of edges $m$. We propose an approximation algorithm that, for graphs with polylogarithmic diameter, achieves a near-linear runtime. In particular, this is the first algorithm whose runtime scales in the number of vertices $n$ as $\tilde{O}(n^2)$ for the complete graph. Moreover, unconditionally on the diameter, the algorithm uses only $O(n)$ memory beyond reading the input, making it "memory-optimal". Our approach is based on solving a linear programming relaxation using entropic regularization, which reduces the problem to Matrix Balancing -- á la the popular reduction of Optimal Transport to Matrix Scaling. The algorithm is practical and simple to implement.

cs.DS↗

arXiv2021

Near-linear convergence of the Random Osborne algorithm for Matrix Balancing

We revisit Matrix Balancing, a pre-conditioning task used ubiquitously for computing eigenvalues and matrix exponentials. Since 1960, Osborne's algorithm has been the practitioners' algorithm of choice and is now implemented in most numerical software packages. However, its theoretical properties are not well understood. Here, we show that a simple random variant of Osborne's algorithm converges in near-linear time in the input sparsity. Specifically, it balances $K\in\mathbb{R}_{\geq 0}^{n\times n}$ after $O(mε^{-2}\logκ)$ arithmetic operations, where $m$ is the number of nonzeros in $K$, $ε$ is the $\ell_1$ accuracy, and $κ=\sum_{ij}K_{ij}/(\min_{ij:K_{ij}\neq 0}K_{ij})$ measures the conditioning of $K$. Previous work had established near-linear runtimes either only for $\ell_2$ accuracy (a weaker criterion which is less relevant for applications), or through an entirely different algorithm based on (currently) impractical Laplacian solvers. We further show that if the graph with adjacency matrix $K$ is moderately connected--e.g., if $K$ has at least one positive row/column pair--then Osborne's algorithm initially converges exponentially fast, yielding an improved runtime $O(mε^{-1}\logκ)$. We also address numerical precision by showing that these runtime bounds still hold when using $O(\log(nκ/ε))$-bit numbers. Our results are established through an intuitive potential argument that leverages a convex optimization perspective of Osborne's algorithm, and relates the per-iteration progress to the current imbalance as measured in Hellinger distance. Unlike previous analyses, we critically exploit log-convexity of the potential. Our analysis extends to other variants of Osborne's algorithm: along the way, we establish significantly improved runtime bounds for cyclic, greedy, and parallelized variants.

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arXiv2021

Asymptotics for semi-discrete entropic optimal transport

We compute exact second-order asymptotics for the cost of an optimal solution to the entropic optimal transport problem in the continuous-to-discrete, or semi-discrete, setting. In contrast to the discrete-discrete or continuous-continuous case, we show that the first-order term in this expansion vanishes but the second-order term does not, so that in the semi-discrete setting the difference in cost between the unregularized and regularized solution is quadratic in the inverse regularization parameter, with a leading constant that depends explicitly on the value of the density at the points of discontinuity of the optimal unregularized map between the measures. We develop these results by proving new pointwise convergence rates of the solutions to the dual problem, which may be of independent interest.

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arXiv2021

Kernel approximation on algebraic varieties

Low-rank approximation of kernels is a fundamental mathematical problem with widespread algorithmic applications. Often the kernel is restricted to an algebraic variety, e.g., in problems involving sparse or low-rank data. We show that significantly better approximations are obtainable in this setting: the rank required to achieve a given error depends on the variety's dimension rather than the ambient dimension, which is typically much larger. This is true in both high-precision and high-dimensional regimes. Our results are presented for smooth isotropic kernels, the predominant class of kernels used in applications. Our main technical insight is to approximate smooth kernels by polynomial kernels, and leverage two key properties of polynomial kernels that hold when they are restricted to a variety. First, their ranks decrease exponentially in the variety's co-dimension. Second, their maximum values are governed by their values over a small set of points. Together, our results provide a general approach for exploiting (approximate) "algebraic structure" in datasets in order to efficiently solve large-scale data science problems.

cs.LG↗

arXiv2020

Hardness results for Multimarginal Optimal Transport problems

Multimarginal Optimal Transport (MOT) is the problem of linear programming over joint probability distributions with fixed marginals. A key issue in many applications is the complexity of solving MOT: the linear program has exponential size in the number of marginals k and their support sizes n. A recent line of work has shown that MOT is poly(n,k)-time solvable for certain families of costs that have poly(n,k)-size implicit representations. However, it is unclear what further families of costs this line of algorithmic research can encompass. In order to understand these fundamental limitations, this paper initiates the study of intractability results for MOT. Our main technical contribution is developing a toolkit for proving NP-hardness and inapproximability results for MOT problems. We demonstrate this toolkit by using it to establish the intractability of a number of MOT problems studied in the literature that have resisted previous algorithmic efforts. For instance, we provide evidence that repulsive costs make MOT intractable by showing that several such problems of interest are NP-hard to solve--even approximately.

math.OC↗

arXiv2020

Wasserstein barycenters can be computed in polynomial time in fixed dimension

Computing Wasserstein barycenters is a fundamental geometric problem with widespread applications in machine learning, statistics, and computer graphics. However, it is unknown whether Wasserstein barycenters can be computed in polynomial time, either exactly or to high precision (i.e., with $\textrm{polylog}(1/\varepsilon)$ runtime dependence). This paper answers these questions in the affirmative for any fixed dimension. Our approach is to solve an exponential-size linear programming formulation by efficiently implementing the corresponding separation oracle using techniques from computational geometry.

math.OC↗

arXiv2019

Lyapunov Exponent of Rank One Matrices: Ergodic Formula and Inapproximability of the Optimal Distribution

The Lyapunov exponent corresponding to a set of square matrices $\mathcal{A} = \{A_1, \dots, A_n \}$ and a probability distribution $p$ over $\{1, \dots, n\}$ is $λ(\mathcal{A},p) := \lim_{k \to \infty} \frac{1}{k} \,\mathbb{E} \log \|A_{σ_k} \cdots A_{σ_2}A_{σ_1}\|$, where $σ_i$ are i.i.d. according to $p$. This quantity is of fundamental importance to control theory since it determines the asymptotic convergence rate $e^{λ(\mathcal{A},p)}$ of the stochastic linear dynamical system $x_{k+1} = A_{σ_k} x_k$. This paper investigates the following "design problem": given $\mathcal{A}$, compute the distribution $p$ minimizing $λ(\mathcal{A},p)$. Our main result is that it is NP-hard to decide whether there exists a distribution $p$ for which $λ(\mathcal{A},p)< 0$, i.e. it is NP-hard to decide whether this dynamical system can be stabilized. This hardness result holds even in the "simple"' case where $\mathcal{A}$ contains only rank-one matrices. Somewhat surprisingly, this is in stark contrast to the Joint Spectral Radius -- the deterministic kindred of the Lyapunov exponent -- for which the analogous optimization problem for rank-one matrices is known to be exactly computable in polynomial time. To prove this hardness result, we first observe via Birkhoff's Ergodic Theorem that the Lyapunov exponent of rank-one matrices admits a simple formula and in fact is a quadratic form in $p$. Hardness of the design problem is shown through a reduction from the Independent Set problem. Along the way, simple examples are given illustrating that $p \mapsto λ(\mathcal{A},p)$ is neither convex nor concave in general. We conclude with extensions to continuous distributions, exchangeable processes, Markov processes, and stationary ergodic processes.

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