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Daniel Paulin

Publications and source records attributed to Daniel Paulin.

35 records · Page 2Linked to original sources
arXiv2020

Randomized Hamiltonian Monte Carlo as Scaling Limit of the Bouncy Particle Sampler and Dimension-Free Convergence Rates

The Bouncy Particle Sampler is a Markov chain Monte Carlo method based on a nonreversible piecewise deterministic Markov process. In this scheme, a particle explores the state space of interest by evolving according to a linear dynamics which is altered by bouncing on the hyperplane tangent to the gradient of the negative log-target density at the arrival times of an inhomogeneous Poisson Process (PP) and by randomly perturbing its velocity at the arrival times of an homogeneous PP. Under regularity conditions, we show here that the process corresponding to the first component of the particle and its corresponding velocity converges weakly towards a Randomized Hamiltonian Monte Carlo (RHMC) process as the dimension of the ambient space goes to infinity. RHMC is another piecewise deterministic non-reversible Markov process where a Hamiltonian dynamics is altered at the arrival times of a homogeneous PP by randomly perturbing the momentum component. We then establish dimension-free convergence rates for RHMC for strongly log-concave targets with bounded Hessians using coupling ideas and hypocoercivity techniques.

stat.CO↗
arXiv2020

Dual Space Preconditioning for Gradient Descent

The conditions of relative smoothness and relative strong convexity were recently introduced for the analysis of Bregman gradient methods for convex optimization. We introduce a generalized left-preconditioning method for gradient descent, and show that its convergence on an essentially smooth convex objective function can be guaranteed via an application of relative smoothness in the dual space. Our relative smoothness assumption is between the designed preconditioner and the convex conjugate of the objective, and it generalizes the typical Lipschitz gradient assumption. Under dual relative strong convexity, we obtain linear convergence with a generalized condition number that is invariant under horizontal translations, distinguishing it from Bregman gradient methods. Thus, in principle our method is capable of improving the conditioning of gradient descent on problems with non-Lipschitz gradient or non-strongly convex structure. We demonstrate our method on p-norm regression and exponential penalty function minimization.

math.OC↗
arXiv2018

Concentration inequalities for Markov chains by Marton couplings and spectral methods

We prove a version of McDiarmid's bounded differences inequality for Markov chains, with constants proportional to the mixing time of the chain. We also show variance bounds and Bernstein-type inequalities for empirical averages of Markov chains. In the case of non-reversible chains, we introduce a new quantity called the "pseudo spectral gap", and show that it plays a similar role for non-reversible chains as the spectral gap plays for reversible chains. Our techniques for proving these results are based on a coupling construction of Katalin Marton, and on spectral techniques due to Pascal Lezaud. The pseudo spectral gap generalises the multiplicative reversiblication approach of Jim Fill.

math.PR↗
arXiv2018

Hamiltonian Descent Methods

We propose a family of optimization methods that achieve linear convergence using first-order gradient information and constant step sizes on a class of convex functions much larger than the smooth and strongly convex ones. This larger class includes functions whose second derivatives may be singular or unbounded at their minima. Our methods are discretizations of conformal Hamiltonian dynamics, which generalize the classical momentum method to model the motion of a particle with non-standard kinetic energy exposed to a dissipative force and the gradient field of the function of interest. They are first-order in the sense that they require only gradient computation. Yet, crucially the kinetic gradient map can be designed to incorporate information about the convex conjugate in a fashion that allows for linear convergence on convex functions that may be non-smooth or non-strongly convex. We study in detail one implicit and two explicit methods. For one explicit method, we provide conditions under which it converges to stationary points of non-convex functions. For all, we provide conditions on the convex function and kinetic energy pair that guarantee linear convergence, and show that these conditions can be satisfied by functions with power growth. In sum, these methods expand the class of convex functions on which linear convergence is possible with first-order computation.

math.OC↗
arXiv2018

Optimization Based Methods for Partially Observed Chaotic Systems

In this paper we consider filtering and smoothing of partially observed chaotic dynamical systems that are discretely observed, with an additive Gaussian noise in the observation. These models are found in a wide variety of real applications and include the Lorenz 96' model. In the context of a fixed observation interval $T$, observation time step $h$ and Gaussian observation variance $σ_Z^2$, we show under assumptions that the filter and smoother are well approximated by a Gaussian with high probability when $h$ and $σ^2_Z h$ are sufficiently small. Based on this result we show that the Maximum-a-posteriori (MAP) estimators are asymptotically optimal in mean square error as $σ^2_Z h$ tends to $0$. Given these results, we provide a batch algorithm for the smoother and filter, based on Newton's method, to obtain the MAP. In particular, we show that if the initial point is close enough to the MAP, then Newton's method converges to it at a fast rate. We also provide a method for computing such an initial point. These results contribute to the theoretical understanding of widely used 4D-Var data assimilation method. Our approach is illustrated numerically on the Lorenz 96' model with state vector up to 1 million dimensions, with code running in the order of minutes. To our knowledge the results in this paper are the first of their type for this class of models.

stat.ME↗
arXiv2018

On Concentration Properties of Partially Observed Chaotic Systems

This article presents results on the concentration properties of the smoothing and filtering distributions of some partially observed chaotic dynamical systems. We show that, rather surprisingly, for the geometric model of the Lorenz equations, as well as some other chaotic dynamical systems, the smoothing and filtering distributions do not concentrate around the true position of the signal, as the number of observations tends to infinity. Instead, under various assumptions on the observation noise, we show that the expected value of the diameter of the support of the smoothing and filtering distributions remains lower bounded by a constant times the standard deviation of the noise, independently of the number of observations. Conversely, under rather general conditions, the diameter of the support of the smoothing and filtering distributions are upper bounded by a constant times the standard deviation of the noise. To some extent, applications to the three dimensional Lorenz 63' model and to the Lorenz 96' model of arbitrarily large dimension are considered.

math.ST↗
arXiv2018

Error Bounds for Sequential Monte Carlo Samplers for Multimodal Distributions

In this paper, we provide bounds on the asymptotic variance for a class of sequential Monte Carlo (SMC) samplers designed for approximating multimodal distributions. Such methods combine standard SMC methods and Markov chain Monte Carlo (MCMC) kernels. Our bounds improve upon previous results, and unlike some earlier work, they also apply in the case when the MCMC kernels can move between the modes. We apply our results to the Potts model from statistical physics. In this case, the problem of sharp peaks is encountered. Earlier methods, such as parallel tempering, are only able to sample from it at an exponential (in an important parameter of the model) cost. We propose a sequence of interpolating distributions called interpolation to independence, and show that the SMC sampler based on it is able to sample from this target distribution at a polynomial cost. We believe that our method is generally applicable to many other distributions as well.

math.PR↗
arXiv2015

Mixing and Concentration by Ricci Curvature

We generalise the coarse Ricci curvature method of Ollivier by considering the coarse Ricci curvature of multiple steps in the Markov chain. This implies new spectral bounds and concentration inequalities. We also extend this approach to the bounds for MCMC empirical averages obtained by Joulin and Ollivier. We prove a recursive lower bound on the coarse Ricci curvature of multiple steps in the Markov chain, making our method broadly applicable. Applications include the split-merge random walk on partitions, Glauber dynamics with random scan and systemic scan for statistical physical spin models, and random walk on a binary cube with a forbidden region.

math.PR↗
arXiv2015

Non-asymptotic confidence intervals for MCMC in practice

Using concentration inequalities, we give non-asymptotic confidence intervals for estimates obtained by Markov chain Monte Carlo (MCMC) simulations, when using the approximation $\mathbb{E}_π f\approx (1/(N-t_0))\cdot \sum_{i=t_0+1}^N f(X_i)$. To allow the application of non-asymptotic error bounds in practice, here we state bounds formulated in terms of the spectral properties of the chain and the properties of $f$ and propose estimators of the parameters appearing in the bounds, including the spectral gap, mixing time, and asymptotic variance. We introduce a method for setting the burn-in time and the initial distribution that is theoretically well-founded and yet is relatively simple to apply. We also investigate the estimation of $\mathbb{E}_πf$ via subsampling and by using parallel runs instead of a single run. Our results are applicable to both reversible and non-reversible Markov chains on discrete as well as general state spaces. We illustrate our methods by simulations for three examples of Bayesian inference in the context of risk models and clinical trials.

math.PR↗
arXiv2015

Hypothesis testing for Markov chain Monte Carlo

Testing between hypotheses, when independent sampling is possible, is a well developed subject. In this paper, we propose hypothesis tests that are applicable when the samples are obtained using Markov chain Monte Carlo. These tests are useful when one is interested in deciding whether the expected value of a certain quantity is above or below a given threshold. We show non-asymptotic error bounds and bounds on the expected number of samples for three types of tests, a fixed sample size test, a sequential test with indifference region, and a sequential test without indifference region. Our tests can lead to significant savings in sample size. We illustrate our results on an example of Bayesian parameter inference involving an ODE model of a biochemical pathway.

stat.ME↗
arXiv2015

Probabilistic verification of partially observable dynamical systems

The construction and formal verification of dynamical models is important in engineering, biology and other disciplines. We focus on non-linear models containing a set of parameters governing their dynamics. The value of these parameters is often unknown and not directly observable through measurements, which are themselves noisy. When treating parameters as random variables, one can constrain their distribution by conditioning on observations and thereby constructing a posterior probability distribution. We aim to perform model verification with respect to this posterior. The main difficulty in performing verification on a model under the posterior distribution is that in general, it is difficult to obtain \emph{independent} samples from the posterior, especially for non-linear dynamical models. Standard statistical model checking methods require independent realizations of the system and are therefore not applicable in this context. We propose a Markov chain Monte Carlo based statistical model checking framework, which produces a sequence of dependent random realizations of the model dynamics over the parameter posterior. Using this sequence of samples, we use statistical hypothesis tests to verify whether the model satisfies a bounded temporal logic property with a certain probability. We use sample size bounds tailored to the setting of dependent samples for fixed sample size and sequential tests. We apply our method to a case-study from the domain of systems biology, to a model of the JAK-STAT biochemical pathway. The pathway is modeled as a system of non-linear ODEs containing a set of unknown parameters. Noisy, indirect observations of the system state are available from an experiment. The results show that the proposed method enables probabilistic verification with respect to the parameter posterior with specified error bounds.

eess.SY↗
arXiv2014

The convex distance inequality for dependent random variables, with applications to the stochastic travelling salesman and other problems

We prove concentration inequalities for general functions of weakly dependent random variables satisfying the Dobrushin condition. In particular, we show Talagrand's convex distance inequality for this type of dependence. We apply our bounds to a version of the stochastic salesman problem, the Steiner tree problem, the total magnetisation of the Curie-Weiss model with external field, and exponential random graph models. Our proof uses the exchangeable pair method for proving concentration inequalities introduced by Chatterjee (2005). Another key ingredient of the proof is a subclass of $(a,b)$-self-bounding functions, introduced by Boucheron, Lugosi and Massart (2009).

math.PR↗
arXiv2014

Efron-Stein Inequalities for Random Matrices

This paper establishes new concentration inequalities for random matrices constructed from independent random variables. These results are analogous with the generalized Efron-Stein inequalities developed by Boucheron et al. The proofs rely on the method of exchangeable pairs.

math.PR↗
arXiv2013

Deriving Matrix Concentration Inequalities from Kernel Couplings

This paper derives exponential tail bounds and polynomial moment inequalities for the spectral norm deviation of a random matrix from its mean value. The argument depends on a matrix extension of Stein's method of exchangeable pairs for concentration of measure, as introduced by Chatterjee. Recent work of Mackey et al. uses these techniques to analyze random matrices with additive structure, while the enhancements in this paper cover a wider class of matrix-valued random elements. In particular, these ideas lead to a bounded differences inequality that applies to random matrices constructed from weakly dependent random variables. The proofs require novel trace inequalities that may be of independent interest.

math.PR↗
arXiv2012

Concentration Inequalities in Locally Dependent Spaces

This paper studies concentration inequalities for functions of locally dependent random variables. We show that the usual definition of local dependence does not imply concentration for general Hamming Lipschitz functions. We define hypergraph dependence, which is a special case of local dependence, and show that it implies concentration if the maximal neighborhood size is small. We prove concentration in Hamming distance, Talagrand distance, and for self-bounding functions of a particular type under this dependence structure.

math.PR↗
arXiv2010

Locally Perturbed Random Walks with Unbounded Jumps

In \cite{SzT}, D. Szász and A. Telcs have shown that for the diffusively scaled, simple symmetric random walk, weak convergence to the Brownian motion holds even in the case of local impurities if $d \ge 2$. The extension of their result to finite range random walks is straightforward. Here, however, we are interested in the situation when the random walk has unbounded range. Concretely we generalize the statement of \cite{SzT} to unbounded random walks whose jump distribution belongs to the domain of attraction of the normal law. We do this first: for diffusively scaled random walks on $\mathbf Z^d$ $(d \ge 2)$ having finite variance; and second: for random walks with distribution belonging to the non-normal domain of attraction of the normal law. This result can be applied to random walks with tail behavior analogous to that of the infinite horizon Lorentz-process; these, in particular, have infinite variance, and convergence to Brownian motion holds with the superdiffusive $\sqrt{n \log n}$ scaling.

math.PR↗