29 research outputs found
Concentration of the Langevin Algorithm's Stationary Distribution
A canonical algorithm for log-concave sampling is the Langevin Algorithm, aka
the Langevin Diffusion run with some discretization stepsize . This
discretization leads the Langevin Algorithm to have a stationary distribution
which differs from the stationary distribution of the
Langevin Diffusion, and it is an important challenge to understand whether the
well-known properties of extend to . In particular, while
concentration properties such as isoperimetry and rapidly decaying tails are
classically known for , the analogous properties for are open
questions with direct algorithmic implications. This note provides a first step
in this direction by establishing concentration results for that
mirror classical results for . Specifically, we show that for any
nontrivial stepsize , is sub-exponential (respectively,
sub-Gaussian) when the potential is convex (respectively, strongly convex).
Moreover, the concentration bounds we show are essentially tight.
Key to our analysis is the use of a rotation-invariant moment generating
function (aka Bessel function) to study the stationary dynamics of the Langevin
Algorithm. This technique may be of independent interest because it enables
directly analyzing the discrete-time stationary distribution
without going through the continuous-time stationary distribution as an
intermediary
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
after
arithmetic operations, where is the number of nonzeros in ,
is the accuracy, and measures the conditioning of . Previous work had established
near-linear runtimes either only for 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 is moderately
connected--e.g., if has at least one positive row/column pair--then
Osborne's algorithm initially converges exponentially fast, yielding an
improved runtime . We also address numerical
precision by showing that these runtime bounds still hold when using
-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.Comment: v2: Fixed minor typos. Modified title for clarity. Corrected
statement of Thm 6.1; this does not affect our main result
Acceleration by Stepsize Hedging I: Multi-Step Descent and the Silver Stepsize Schedule
Can we accelerate convergence of gradient descent without changing the
algorithm -- just by carefully choosing stepsizes? Surprisingly, we show that
the answer is yes. Our proposed Silver Stepsize Schedule optimizes strongly
convex functions in iterations, where
is the silver ratio and is the condition number. This is
intermediate between the textbook unaccelerated rate and the accelerated
rate due to Nesterov in 1983. The non-strongly convex setting is
conceptually identical, and standard black-box reductions imply an analogous
accelerated rate .
We conjecture and provide partial evidence that these rates are optimal among
all possible stepsize schedules.
The Silver Stepsize Schedule is constructed recursively in a fully explicit
way. It is non-monotonic, fractal-like, and approximately periodic of period
. This leads to a phase transition in the convergence rate:
initially super-exponential (acceleration regime), then exponential (saturation
regime).Comment: 7 figure
Acceleration by Stepsize Hedging II: Silver Stepsize Schedule for Smooth Convex Optimization
We provide a concise, self-contained proof that the Silver Stepsize Schedule
proposed in Part I directly applies to smooth (non-strongly) convex
optimization. Specifically, we show that with these stepsizes, gradient descent
computes an -minimizer in iterations, where is the silver
ratio. This is intermediate between the textbook unaccelerated rate
and the accelerated rate due to
Nesterov in 1983. The Silver Stepsize Schedule is a simple explicit fractal:
the -th stepsize is where is the -adic valuation
of . The design and analysis are conceptually identical to the strongly
convex setting in Part I, but simplify remarkably in this specific setting.Comment: 10 pages, 3 figure
Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent
We study first-order optimization algorithms for computing the barycenter of
Gaussian distributions with respect to the optimal transport metric. Although
the objective is geodesically non-convex, Riemannian GD empirically converges
rapidly, in fact faster than off-the-shelf methods such as Euclidean GD and SDP
solvers. This stands in stark contrast to the best-known theoretical results
for Riemannian GD, which depend exponentially on the dimension. In this work,
we prove new geodesic convexity results which provide stronger control of the
iterates, yielding a dimension-free convergence rate. Our techniques also
enable the analysis of two related notions of averaging, the
entropically-regularized barycenter and the geometric median, providing the
first convergence guarantees for Riemannian GD for these problems.Comment: 48 pages, 8 figure
Development and implementation of a prescription opioid registry across diverse health systems
Objective: Develop and implement a prescription opioid registry in 10 diverse health systems across the US and describe trends in prescribed opioids between 2012 and 2018.
Materials and Methods: Using electronic health record and claims data, we identified patients who had an outpatient fill for any prescription opioid, and/or an opioid use disorder diagnosis, between January 1, 2012 and December 31, 2018. The registry contains distributed files of prescription opioids, benzodiazepines and other select medications, opioid antagonists, clinical diagnoses, procedures, health services utilization, and health plan membership. Rates of outpatient opioid fills over the study period, standardized to health system demographic distributions, are described by age, gender, and race/ethnicity among members without cancer.
Results: The registry includes 6 249 710 patients and over 40 million outpatient opioid fills. For the combined registry population, opioid fills declined from a high of 0.718 per member-year in 2013 to 0.478 in 2018, and morphine milligram equivalents (MMEs) per fill declined from 985 MMEs per fill in 2012 to 758 MMEs in 2018. MMEs per member declined from 692 MMEs per member in 2012 to 362 MMEs per member in 2018.
Conclusion: This study established a population-based opioid registry across 10 diverse health systems that can be used to address questions related to opioid use. Initial analyses showed large reductions in overall opioid use per member among the combined health systems. The registry will be used in future studies to answer a broad range of other critical public health issues relating to prescription opioid use