Measuring Sample Quality with Diffusions

Gorham, Jackson; Duncan, Anthony; Vollmer, Sebastian J.; Mackey, Lester

doi:10.48550/arxiv.1611.06972

Cited by 20 publications

(29 citation statements)

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“…The bound M i (f ) on the i-th order derivative of f are termed as the i-th Stein factor. The main important property of Lemma 7 is that it connects the third Stein factor to the smoothness of the test function h. In the one-dimensional case, one can obtain a bound on third Stein factor in terms of the Lipschitz constant of the test function [Ste86,Röl18] without any smoothness assumptions; however, at least polynomial smoothness is required in the higher dimensions [GDVM16,EMS18]. The proof for the above result is given below for reader's convenience.…”

Section: A Proofs Of the Martingale Clt Resultsmentioning

confidence: 99%

Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT

Anastasiou,

Balasubramanian,

Erdogdu

2019

Preprint

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We provide non-asymptotic convergence rates of the Polyak-Ruppert averaged stochastic gradient descent (SGD) to a normal random vector for a class of twice-differentiable test functions. A crucial intermediate step is proving a non-asymptotic martingale central limit theorem (CLT), i.e., establishing the rates of convergence of a multivariate martingale difference sequence to a normal random vector, which might be of independent interest. We obtain the explicit rates for the multivariate martingale CLT using a combination of Stein's method and Lindeberg's argument, which is then used in conjunction with a non-asymptotic analysis of averaged SGD proposed in [PJ92]. Our results have potentially interesting consequences for computing confidence intervals for parameter estimation with SGD and constructing hypothesis tests with SGD that are valid in a non-asymptotic sense. * Authors are listed in alphabetical order. This paper provides a solution for an open problem formulated at the AIM workshop on Stein's method and Applications in High-dimensional Statistics, 2018 (problem (4) in page 2 of https://aimath.org/pastworkshops/steinhdrep.pdf by KB). We thank Jay Bartroff, Larry Goldstein, Stanislav Minsker, and Gesine Reinert for organizing a stimulating workshop, and the participants for several discussions. MAE is partially funded by CIFAR AI Chairs program at the Vector Institute.

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Section: A Proofs Of the Martingale Clt Resultsmentioning

confidence: 99%

Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT

Anastasiou,

Balasubramanian,

Erdogdu

2019

Preprint

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“…One of our principal contributions is a careful enumeration of the dependencies of these Stein factors and Markov chain moments on the objective f and the candidate diffusion. Our convergence analysis builds on the arguments of [15,23], and our Stein factor bounds rely on distant and uniform dissipativity conditions for L 1 -Wasserstein rate decay [12,15] and the smoothing effect of the Markov semigroup [4,15]. Our Stein factor results significantly generalize the existing bounds of [15] by accommodating pseudo-Lipschitz objectives f and quadratic growth in the covariance coefficient and deriving the first four Stein factors explicitly.…”

Section: Related Workmentioning

confidence: 96%

“…The diffusion Z z t starts at a point z ∈ R d and, under the conditions of Section 3, admits a limiting invariant distribution P with (Lebesgue) density p. To encourage sampling near the minima of f , we would like to choose p so that the maximizers of p correspond to minimizers of f . Fortunately, under mild conditions, one can construct a diffusion with target invariant distribution P (see, e.g., [15,20,Thm. 2]), by selecting the drift coefficient…”

Section: Optimization With Discretized Diffusions: Preliminariesmentioning

confidence: 99%

“…and bound each term on the right-hand side separately. The integration error-which captures both the short-term non-stationarity of the chain and the long-term bias due to discretization-is the subject of Section 3; we develop explicit bounds using techniques that build upon [15,23]. The expected suboptimality quantifies how well exact samples from p minimize f on average.…”

Section: Optimization With Discretized Diffusions: Preliminariesmentioning

confidence: 99%

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Global Non-convex Optimization with Discretized Diffusions

Erdogdu¹,

Mackey²,

Shamir³

2018

Preprint

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An Euler discretization of the Langevin diffusion is known to converge to the global minimizers of certain convex and non-convex optimization problems. We show that this property holds for any suitably smooth diffusion and that different diffusions are suitable for optimizing different classes of convex and non-convex functions. This allows us to design diffusions suitable for globally optimizing convex and non-convex functions not covered by the existing Langevin theory. Our non-asymptotic analysis delivers computable optimization and integration error bounds based on easily accessed properties of the objective and chosen diffusion. Central to our approach are new explicit Stein factor bounds on the solutions of Poisson equations. We complement these results with improved optimization guarantees for targets other than the standard Gibbs measure.

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“…There now exist several excellent books and reviews on Stein's method and its consequences in various settings, such as [68,9,10,57,21]. There also exist several non-equivalent general frameworks for the theory covering to large swaths of probability distributions, of which we single out the works [26,70] for univariate distributions under analytical assumptions, [6,7] for infinitely divisible distributions and [55,35,37] as well as [29] for multivariate densities under diffusive assumptions. A "canonical" differential Stein operator theory is also presented in [49,50,63].…”

Section: Introductionmentioning

confidence: 99%

Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions

Ernst,

Reinert,

Swan

2018

Preprint

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In this paper, following on from [49,50,63] we present a minimal formalism for Stein operators which leads to different probabilistic representations of solutions to Stein equations. These in turn provide a wide family of Stein-Covariance identities which we put to use for revisiting the very classical topic of bounding the variance of functionals of random variables. Applying the Cauchy-Schwarz inequality yields first order upper and lower Klaassen [45]-type variance bounds. A probabilistic representation of Lagrange's identity (i.e. Cauchy-Schwarz with remainder) leads to Papathanasiou [60]-type variance expansions of arbitrary order. A matrix Cauchy-Schwarz inequality leads to Olkin-Shepp [59] type covariance bounds. All results hold for univariate target distribution under very weak assumptions (in particular they hold for continuous and discrete distributions alike). Many concrete illustrations are provided.

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Measuring Sample Quality with Diffusions

Cited by 20 publications

References 0 publications

Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT

Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT

Global Non-convex Optimization with Discretized Diffusions

Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions

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