Optimization is akeycomponent for training machine learning models and has a strong impact on their generalization. In this paper, we consider a particular optimization method--the stochastic gradient Langevin dynamics (SGLD) algorithm--and investigate the generalization of models trained by SGLD.

Optimization is akeycomponent for training machine learning models and has a strong impact on their generalization. In this paper, we consider a particular optimization method--the stochastic gradient Langevin dynamics (SGLD) algorithm--and investigate the generalization of models trained by SGLD.

This paper provides a finite-sample analysis of a passive stochastic gradient Langevin dynamics algorithm (PSGLD) designed to achieve adaptive inverse reinforcement learning (IRL). By passive, we mean that the noisy gradients available to the PSGLD algorithm (inverse learning process) are evaluated at randomly chosen points by an external stochastic gradient algorithm (forward learner) that aims to optimize a cost function. The PSGLD algorithm acts as a randomized sampler to achieve adaptive IRL by reconstructing this cost function nonparametrically from the stationary measure of a Langevin diffusion. Previous work has analyzed the asymptotic performance of this passive algorithm using weak convergence techniques. This paper analyzes the non-asymptotic (finite-sample) performance using a logarithmic-Sobolev inequality and the Otto-Villani Theorem. We obtain finite-sample bounds on the 2-Wasserstein distance between the estimates generated by the PSGLD algorithm and the cost function. Apart from achieving finite-sample guarantees for adaptive IRL, this work extends a line of research in analysis of passive stochastic gradient algorithms to the finite-sample regime for Langevin dynamics.

We revisit processes generated by iterated random functions driven by a stationary and ergodic sequence. Such a process is called strongly stable if a random initialization exists, for which the process is stationary and ergodic, and for any other initialization, the difference of the two processes converges to zero almost surely. Under some mild conditions on the corresponding recursive map, without any condition on the driving sequence, we show the strong stability of iterations. Several applications are surveyed such as stochastic approximation and queuing. Furthermore, new results are deduced for Langevin-type iterations with dependent noise and for multitype branching processes.

Stochastic Gradient Langevin Dynamics (SGLD) ensures strong guarantees with regards to convergence in measure for sampling log-concave posterior distributions by adding noise to stochastic gradient iterates. Given the size of many practical problems, parallelizing across several asynchronously running processors is a popular strategy for reducing the end-to-end computation time of stochastic optimization algorithms. In this paper, we are the first to investigate the effect of asynchronous computation, in particular, the evaluation of stochastic Langevin gradients at delayed iterates, on the convergence in measure. For this, we exploit recent results modeling Langevin dynamics as solving a convex optimization problem on the space of measures. We show that the rate of convergence in measure is not significantly affected by the error caused by the delayed gradient information used for computation, suggesting significant potential for speedup in wall clock time. We confirm our theoretical results with numerical experiments on some practical problems.

Stochastic gradient Markov chain Monte Carlo (MCMC) algorithms have received much attention in Bayesian computing for big data problems, but they are only applicable to a small class of problems for which the parameter space has a fixed dimension and the log-posterior density is differentiable with respect to the parameters. This paper proposes an extended stochastic gradient MCMC lgoriathm which, by introducing appropriate latent variables, can be applied to more general large-scale Bayesian computing problems, such as those involving dimension jumping and missing data. Numerical studies show that the proposed algorithm is highly scalable and much more efficient than traditional MCMC algorithms. The proposed algorithms have much alleviated the pain of Bayesian methods in big data computing.

Applying standard Markov chain Monte Carlo (MCMC) algorithms to large data sets is computationally expensive. Both the calculation of the acceptance probability and the creation of informed proposals usually require an iteration through the whole data set. The recently proposed stochastic gradient Langevin dynamics (SGLD) method circumvents this problem by generating proposals which are only based on a subset of the data, by skipping the accept-reject step and by using decreasing step-sizes sequence $(\delta_m)_{m \geq 0}$. %Under appropriate Lyapunov conditions, We provide in this article a rigorous mathematical framework for analysing this algorithm. We prove that, under verifiable assumptions, the algorithm is consistent, satisfies a central limit theorem (CLT) and its asymptotic bias-variance decomposition can be characterized by an explicit functional of the step-sizes sequence $(\delta_m)_{m \geq 0}$. We leverage this analysis to give practical recommendations for the notoriously difficult tuning of this algorithm: it is asymptotically optimal to use a step-size sequence of the type $\delta_m \asymp m^{-1/3}$, leading to an algorithm whose mean squared error (MSE) decreases at rate $\mathcal{O}(m^{-1/3})$