Non-asymptotic Convergence Research Articles

Finite-time and fixed-time input-to-state stability: Explicit and implicit approaches

Abstract The present article gathers the analysis of non-asymptotic convergence rates (finite-time and fixed-time) with the property of input-to-state stability. Theoretical tools to determine this joint property are presented for the case where an explicit ISS Lyapunov function is known, and when it remains in implicit form (e.g. as a solution of an algebraic equation). For the case of finite-time input-to-state stability, necessary and sufficient conditions are given whereas for the fixed-time case only a sufficient condition is obtained. Academic examples and numerical simulations support the obtained results.

The CLT in high dimensions: Quantitative bounds via martingale embedding

We introduce a new method for obtaining quantitative convergence rates for the central limit theorem (CLT) in a high-dimensional setting. Using our method, we obtain several new bounds for convergence in transportation distance and entropy, and in particular: (a) We improve the best known bound, obtained by the third named author (Probab. Theory Related Fields 170 (2018) 821–845), for convergence in quadratic Wasserstein transportation distance for bounded random vectors; (b) we derive the first nonasymptotic convergence rate for the entropic CLT in arbitrary dimension, for general log-concave random vectors (this adds to (Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019) 777–790), where a finite Fisher information is assumed); (c) we give an improved bound for convergence in transportation distance under a log-concavity assumption and improvements for both metrics under the assumption of strong log-concavity. Our method is based on martingale embeddings and specifically on the Skorokhod embedding constructed in (Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 1259–1280).

New nonasymptotic convergence rates of stochastic proximal point algorithm for stochastic convex optimization

Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first-order schemes for constrained convex models under progressively relaxed assumptions. Stochastic proximal point is an iterative scheme born from the adaptation of proximal point algorithm to noisy stochastic optimization, with a resulting iteration related to stochastic alternating projections. Inspired by the scalability of alternating projection methods, we start from the (linear) regularity assumption, typically used in convex feasiblity problems to guarantee the linear convergence of stochastic alternating projection methods, and analyze a general weak linear regularity condition which facilitates convergence rate boosts in stochastic proximal point schemes. Our applications include many non-strongly convex functions classes often used in machine learning and statistics. Moreover, under weak linear regularity assumption we guarantee convergence rate for SPP, in terms of the distance to the optimal set, using only projections onto a simple component set. Linear convergence is obtained for interpolation setting, when the optimal set of the expected cost is included into the optimal sets of each functional component.

Fast state tomography with optimal error bounds

Projected least squares is an intuitive and numerically cheap technique for quantum state tomography: compute the least-squares estimator and project it onto the space of states. The main result of this paper equips this point estimator with rigorous, non-asymptotic convergence guarantees expressed in terms of the trace distance. The estimator’s sample complexity is comparable to the strongest convergence guarantees available in the literature and—in the case of the uniform POVM—saturates fundamental lower bounds. Numerical simulations support these competitive features.

Aggregated Gradient Langevin Dynamics

In this paper, we explore a general Aggregated Gradient Langevin Dynamics framework (AGLD) for the Markov Chain Monte Carlo (MCMC) sampling. We investigate the nonasymptotic convergence of AGLD with a unified analysis for different data accessing (e.g. random access, cyclic access and random reshuffle) and snapshot updating strategies, under convex and nonconvex settings respectively. It is the first time that bounds for I/O friendly strategies such as cyclic access and random reshuffle have been established in the MCMC literature. The theoretic results also indicate that methods in AGLD possess the merits of both the low per-iteration computational complexity and the short mixture time. Empirical studies demonstrate that our framework allows to derive novel schemes to generate high-quality samples for large-scale Bayesian posterior learning tasks.

Confidence Region of Singular Subspaces for Low-Rank Matrix Regression

Low-rank matrix regression refers to the instances of recovering a low-rank matrix based on specially designed measurements and the corresponding noisy outcomes. Numerous statistical methods have been developed over the recent decade for efficiently reconstructing the unknown low-rank matrices. It is often interesting, in certain applications, to estimate the unknown singular subspaces. In this paper, we revisit the low-rank matrix regression model and introduce a two-step procedure to construct confidence regions of the singular subspaces. We investigate distributions of the joint projection distance between the empirical singular subspaces and the unknown true singular subspaces. We prove asymptotical normality of the joint projection distance with data-dependent centering and normalization when $r^{3/2}(m_{1}+m_{2})^{3/2}=o(n/\log n)$ where $m_{1}, m_{2}$ denote the matrix row and column sizes, $r$ is the rank and $n$ is the number of independent random measurements. Consequently, data-dependent confidence regions of the true singular subspaces are established which attain pre-determined confidence levels asymptotically. Additionally, non-asymptotic convergence rates are also established. Numerical results are presented to show the merits of our methods.

Non-asymptotic rate for high-dimensional covariance estimation with non-independent missing observations

In this paper, we study non-asymptotic convergence rate of the inverse probability weight estimator of covariance matrix when some values of the data are missing completely at random.

An alternative to EM for Gaussian mixture models: batch and stochastic Riemannian optimization

We consider maximum likelihood estimation for Gaussian Mixture Models (Gmm s). This task is almost invariably solved (in theory and practice) via the Expectation Maximization (EM) algorithm. EM owes its success to various factors, of which is its ability to fulfill positive definiteness constraints in closed form is of key importance. We propose an alternative to EM grounded in the Riemannian geometry of positive definite matrices, using which we cast Gmm parameter estimation as a Riemannian optimization problem. Surprisingly, such an out-of-the-box Riemannian formulation completely fails and proves much inferior to EM. This motivates us to take a closer look at the problem geometry, and derive a better formulation that is much more amenable to Riemannian optimization. We then develop Riemannian batch and stochastic gradient algorithms that outperform EM, often substantially. We provide a non-asymptotic convergence analysis for our stochastic method, which is also the first (to our knowledge) such global analysis for Riemannian stochastic gradient. Numerous empirical results are included to demonstrate the effectiveness of our methods.

Convergence Rate Analysis of Inertial Krasnoselskii–Mann Type Iteration with Applications

It is well known that the Krasnoselskii–Mann iteration of nonexpansive operators find applications in many areas of mathematics and known to be weakly convergent in the infinite dimensional setting. In this present paper, we provide a nonasymptotic convergence rate result for a Krasnoselskii–Mann iteration with inertial extrapolation step in real Hilbert spaces. We give some applications of our results to the Douglas–Rachford splitting method and the alternating projection method by John von Neumann. Our result serves as supplement to many existing results on convergence rate of Krasnoselskii–Mann iteration in the literature.

Streaming PCA and Subspace Tracking: The Missing Data Case.

For many modern applications in science and engineering, data are collected in a streaming fashion carrying time-varying information, and practitioners need to process them with a limited amount of memory and computational resources in a timely manner for decision making. This often is coupled with the missing data problem, such that only a small fraction of data attributes are observed. These complications impose significant, and unconventional, constraints on the problem of streaming Principal Component Analysis (PCA) and subspace tracking, which is an essential building block for many inference tasks in signal processing and machine learning. This survey article reviews a variety of classical and recent algorithms for solving this problem with low computational and memory complexities, particularly those applicable in the big data regime with missing data. We illustrate that streaming PCA and subspace tracking algorithms can be understood through algebraic and geometric perspectives, and they need to be adjusted carefully to handle missing data. Both asymptotic and non-asymptotic convergence guarantees are reviewed. Finally, we benchmark the performance of several competitive algorithms in the presence of missing data for both well-conditioned and ill-conditioned systems.

Pointwise and Ergodic Convergence Rates of a Variable Metric Proximal Alternating Direction Method of Multipliers

In this paper, we obtain global pointwise and ergodic convergence rates for a variable metric proximal alternating direction method of multipliers for solving linearly constrained convex optimization problems. We first propose and study nonasymptotic convergence rates of a variable metric hybrid proximal extragradient framework for solving monotone inclusions. Then, the convergence rates for the former method are obtained essentially by showing that it falls within the latter framework. To the best of our knowledge, this is the first time that global pointwise (resp. pointwise and ergodic) convergence rates are obtained for the variable metric proximal alternating direction method of multipliers (resp. variable metric hybrid proximal extragradient framework).

Non-asymptotic convergence analysis of inexact gradient methods for machine learning without strong convexity

Many recent applications in machine learning and data fitting call for the algorithmic solution of structured smooth convex optimization problems. Although the gradient descent method is a natural choice for this task, it requires exact gradient computations and hence can be inefficient when the problem size is large or the gradient is difficult to evaluate. Therefore, there has been much interest in inexact gradient methods (IGMs), in which an efficiently computable approximate gradient is used to perform the update in each iteration. Currently, non-asymptotic linear convergence results for IGMs are typically established under the assumption that the objective function is strongly convex, which is not satisfied in many applications of interest; while linear convergence results that do not require the strong convexity assumption are usually asymptotic in nature. In this paper, we combine the best of these two types of results by developing a framework for analysing the non-asymptotic convergence rates of IGMs when they are applied to a class of structured convex optimization problems that includes least squares regression and logistic regression. We then demonstrate the power of our framework by proving, in a unified manner, new linear convergence results for three recently proposed algorithms—the incremental gradient method with increasing sample size [R.H. Byrd, G.M. Chin, J. Nocedal, and Y. Wu, Sample size selection in optimization methods for machine learning, Math. Program. Ser. B 134 (2012), pp. 127–155; M.P. Friedlander and M. Schmidt, Hybrid deterministic–stochastic methods for data fitting, SIAM J. Sci. Comput. 34 (2012), pp. A1380–A1405], the stochastic variance-reduced gradient (SVRG) method [R. Johnson and T. Zhang, Accelerating stochastic gradient descent using predictive variance reduction, Advances in Neural Information Processing Systems 26: Proceedings of the 2013 Conference, 2013, pp. 315–323], and the incremental aggregated gradient (IAG) method [D. Blatt, A.O. Hero, and H. Gauchman, A convergent incremental gradient method with a constant step size, SIAM J. Optim. 18 (2007), pp. 29–51]. We believe that our techniques will find further applications in the non-asymptotic convergence analysis of other first-order methods.

Nonasymptotic convergence analysis for the unadjusted Langevin algorithm

In this paper, we study a method to sample from a target distribution $\pi$ over $\mathbb{R}^d$ having a positive density with respect to the Lebesgue measure, known up to a normalisation factor. This method is based on the Euler discretization of the overdamped Langevin stochastic differential equation associated with $\pi$. For both constant and decreasing step sizes in the Euler discretization, we obtain non-asymptotic bounds for the convergence to the target distribution $\pi$ in total variation distance. A particular attention is paid to the dependency on the dimension $d$, to demonstrate the applicability of this method in the high dimensional setting. These bounds improve and extend the results of (Dalalyan 2014).

A pseudo-stable structure in a completely invertible bouncer system

It is shown that a pseudo-stable structure of non-asymptotic convergence may exist in a completely invertible bouncing ball model. Visualization of the pattern of H-ranks helps to identify this structure. It appears that this structure is similar to the stable manifold of non-invertible nonlinear maps which govern the non-asymptotic convergence to unstable periodic orbits. But this convergence to the unstable repeller of the bouncing ball problem is only temporary since non-asymptotic convergence cannot exist in completely invertible maps. This nonlinear effect is exploited for temporary stabilization of unstable periodic orbits in completely reversible nonlinear maps.

Estimation in the partially observed stochastic Morris–Lecar neuronal model with particle filter and stochastic approximation methods

Parameter estimation in multidimensional diffusion models with only one coordinate observed is highly relevant in many biological applications, but a statistically difficult problem. In neuroscience, the membrane potential evolution in single neurons can be measured at high frequency, but biophysical realistic models have to include the unobserved dynamics of ion channels. One such model is the stochastic Morris-Lecar model, defined by a nonlinear two-dimensional stochastic differential equation. The coordinates are coupled, that is, the unobserved coordinate is nonautonomous, the model exhibits oscillations to mimic the spiking behavior, which means it is not of gradient-type, and the measurement noise from intracellular recordings is typically negligible. Therefore, the hidden Markov model framework is degenerate, and available methods break down. The main contributions of this paper are an approach to estimate in this ill-posed situation and nonasymptotic convergence results for the method. Specifically, we propose a sequential Monte Carlo particle filter algorithm to impute the unobserved coordinate, and then estimate parameters maximizing a pseudo-likelihood through a stochastic version of the Expectation-Maximization algorithm. It turns out that even the rate scaling parameter governing the opening and closing of ion channels of the unobserved coordinate can be reasonably estimated. An experimental data set of intracellular recordings of the membrane potential of a spinal motoneuron of a red-eared turtle is analyzed, and the performance is further evaluated in a simulation study.

On the Nonasymptotic Convergence of Cyclic Coordinate Descent Methods

Cyclic coordinate descent is a classic optimization method that has witnessed a resurgence of interest in signal processing, statistics, and machine learning. Reasons for this renewed interest include the simplicity, speed, and stability of the method, as well as its competitive performance on $\ell_1$ regularized smooth optimization problems. Surprisingly, very little is known about its nonasymptotic convergence behavior on these problems. Most existing results either just prove convergence or provide asymptotic rates. We fill this gap in the literature by proving $O(1/k)$ convergence rates (where $k$ is the iteration count) for two variants of cyclic coordinate descent under an isotonicity assumption. Our analysis proceeds by comparing the objective values attained by the two variants with each other, as well as with the gradient descent algorithm. We show that the iterates generated by the cyclic coordinate descent methods remain better than those of gradient descent uniformly over time.

Clocking convergence to Arnold tongues – the circle map revisited

Computational techniques based on ranks of Hankel matrices is used to study the convergence to Arnold tongues. It appears that the process of convergence to the phaselocked mode is far from being trivial. The stable, the unstable and the manifold of nonasymptotic convergence intertwine in the parameter plane of the circle map. Pseudoranks of Hankel matrices carry important physical information about transient processes taking place in discrete nonlinear iterative maps. These pictures in the parameter plane are also beautiful from the aesthetical point of view.

Clocking convergence to a stable limit cycle of a periodically driven nonlinear pendulum

Convergence to a stable limit cycle of a periodically driven nonlinear pendulum is analyzed in this paper. The concept of the H-rank of a scalar sequence is used for the assessment of transient processes of the system. The circle map is used to illustrate the complex structure of the manifold of non-asymptotic convergence to a fixed point. It is demonstrated that the manifold of non-asymptotic convergence to a stable limit cycle also exists in the stroboscopic representation of the transient data of the periodically driven nonlinear pendulum. A simple method based on a short external impulse is proposed for the control of transient processes when the transition time to stable limit cycles must be minimized.

A Geometric Convergence Theory for the Preconditioned Steepest Descent Iteration

Preconditioned gradient iterations for very large eigenvalue problems are efficient solvers with growing popularity. However, only for the simplest preconditioned eigensolver, namely the preconditioned gradient iteration (or preconditioned inverse iteration) with fixed step size, are sharp nonasymptotic convergence estimates known. These estimates require a properly scaled preconditioner. In this paper a new sharp convergence estimate is derived for the preconditioned steepest descent iteration which combines the preconditioned gradient iteration with the Rayleigh--Ritz procedure for optimal line search convergence acceleration. The new estimate always improves that of the fixed-step-size iteration. The practical importance of this new estimate is that arbitrarily scaled preconditioners can be used. The Rayleigh--Ritz procedure implicitly computes the optimal scaling constant.

On the Robustness of the Snell Envelope

We analyze the robustness properties of the Snell envelope backward evolution equation for the discrete time optimal stopping problem. We consider a series of approximation schemes, including cut–off-type approximations, Euler discretization schemes, interpolation models, quantization tree models, and the stochastic mesh method of Broadie and Glasserman. In each situation, we provide nonasymptotic convergence estimates, including $\mathbb{L}_p$-mean error bounds and exponential concentration inequalities. We deduce these estimates from a single and general robustness property of Snell envelope semigroups. In particular, this analysis allows us to recover existing convergence results for the quantization tree method and to improve significantly the rates of convergence obtained for the stochastic mesh estimator of Broadie and Glasserman. In the second part of the article, we propose a new approach based on a genealogical tree approximation model of the reference Markov process in terms of a neutral-type genetic model. In contrast to Broadie–Glasserman Monte Carlo models, the computational cost of this new stochastic approximation is linear in the number of particles. Some simulation results are provided and confirm the interest of this new algorithm.