Stochastic Tracking of Drifting Affine Fixed Points

Sajad Khodadadian

Linear stochastic approximation (SA) is widely used to estimate fixed points from noisy observations, but most finite-time analyses assume that the underlying operator remains fixed. We study linear SA with time-varying affine operators, where the fixed points change over time and the observation noise may be correlated across rounds. For constant-stepsize updates, we establish a bound on the cumulative expected squared distance between the iterates and the corresponding fixed points. The bound accounts for the initial error, observation noise, and changes in the fixed points over time. We also establish lower bounds for scalar Gaussian models, showing that, under specified conditions, a suitably chosen constant stepsize achieves the smallest possible worst-case cumulative error up to constant factors, even compared with estimators that use future observations. These results further quantify the effect of temporal correlation on estimation accuracy. Since choosing an appropriate stepsize generally requires knowledge of the noise level and how much the fixed points change, we develop an adaptive method that uses an additional bounded, unbiased observation of the fixed point to…