Motivated by applications to distributed asynchronous optimization and large-scale data processing, we analyze the incremental aggregated gradient method for minimizing a sum of strongly convex functions from a novel perspective, simplifying the global convergence proofs considerably and proving a linear rate result. We also consider an aggregated method with momentum and show its linear conver...

We analyze online [5] and mini-batch [16] k-means variants. Both scale up the widely used k-means algorithm via stochastic approximation, and have become popular for large-scale clustering and unsupervised feature learning. We show, for the first time, that starting with any initial solution, they converge to a “local optimum” at rateO( 1t ) (in terms of the k-means objective) under general con...

where X is the decision set and Q0, the distribution of ξ, is a probability distribution supported on [0, 1]. This kind of problem can be solved numerically or (in special cases) in closed form if we know the exact distribution of Q0. Unfortunately, in practice one rarely knows the exact distribution Q0. Instead, one often has only partial information about Q0, which may include limited distrib...

In this paper we show that for discounted MDPs with discount factor, > 1/2 the asymptotic rate of convergence of Q-Iearning is O(1/tR(1-1') if R(1 ,) < 1/2 and O( Jlog log tit) otherwise provided that the state-action pairs are sampled from a fixed probability distribution. Here R = Pmin/Pmax is the ratio of the minimum and maximum state-action occupation frequencies. The results extend to conv...

Convergence rate results are derived for a stochastic optimization problem where a performance measure is minimized with respect to a vector parameter θ. Assuming that a gradient estimator is available and that both the bias and the variance of the estimator are (known) functions of the budget devoted to its computation, the gradient estimator is employed in conjunction with a stochastic approx...

— An aiiempt to set up a method for estimating convergence of itérative processes using inequalities of the type where (Ù is a "small" fonction as deflned by the author. This method is compared with the classical notion of a rate of convergence ; for convex fonctions (ù inequalities using the distance of xn from the solution and those using the distance of two consécutive steps are shown to be ...

In this paper we develop tools for analyzing the rate at which a reversible Markov chain converges to stationarity. Our techniques are useful when the Markov chain can be decomposed into pieces which are themselves easier to analyze. The main theorems relate the spectral gap of the original Markov chains to the spectral gap of the pieces. In the first case the pieces are restrictions of the Mar...

Godunov type schemes form a special class of transport projection methods for the approximate solution of nonlinear hyperbolic conservation laws. We study the convergence rate of such schemes in the context of scalar conservation laws. We show how the question of consistency for Godunov type schemes can be answered solely in terms of the behavior of the associated projection operator. Namely, w...

A number of mortgage prepayment models require a specification of the mortgage rate process. In 2006, Citigroup published its MOATS model where the prepayment model was based on the endogenously defined mortgage rate process. MOATS was based on the “End of Finance” (EOF) assumption, which stated that the term for a new mortgage is 30 years or the rest of time until EOF (assumed to be in sixty y...

This paper is concerned with the convergence rate of the solutions of nonlinear switched systems. We first consider a switched system which is asymptotically stable for a class of inputs but not for all inputs. We show that solutions corresponding to that class of inputs converge arbitrarily slowly to the origin. Then we consider analytic switched systems for which a common weak quadratic Lyapu...