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We study the problem of estimating the parameters of a regression model froma set of observations, each consisting of a response and a predictor. Theresponse is assumed to be related to the predictor via a regression model ofunknown parameters. Often, in such models the parameters to be estimated areassumed to be constant. Here we consider the more general scenario where theparameters are allowed to evolve over time, a more natural assumption for manyapplications. We model these dynamics via a linear update equation withadditive noise that is often used in a wide range of engineering applications,particularly in the well-known and widely used Kalman filter (where the systemstate it seeks to estimate maps to the parameter values here). We derive anapproximate algorithm to estimate both the mean and the variance of theparameter estimates in an online fashion for a generic regression model. Thisalgorithm turns out to be equivalent to the extended Kalman filter. Wespecialize our algorithm to the multivariate exponential family distribution toobtain a generalization of the generalized linear model (GLM). Because thecommon regression models encountered in practice such as logistic, exponentialand multinomial all have observations modeled through an exponential familydistribution, our results are used to easily obtain algorithms for online meanand variance parameter estimation for all these regression models in thecontext of time-dependent parameters. Lastly, we propose to use thesealgorithms in the contextual multi-armed bandit scenario, where so far modelparameters are assumed static and observations univariate and Gaussian orBernoulli. Both of these restrictions can be relaxed using the algorithmsdescribed here, which we combine with Thompson sampling to show the resultingperformance on a simulation.

Online Algorithms For Parameter Mean And Variance Estimation In Dynamic Regression Models