System identification with binary observations by stochastic approximation and active learning

We investigate the problem of estimating a constant based on noisy observations via a binary sensor. This problem is well-studied for the case when the noise characteristics are known, for example, the noise is i.i.d. and we have access to its cumulative distribution function (CDF). Here, we try to reduce the assumptions on the noise to a minimum and, for example, assume only that the noise is symmetrically distributed about zero in each time step, but otherwise the CDF is unknown. We neither assume that the noise variables are independent nor that they are stationary. They may also not have densities. We do assume, however, that the threshold of the binary sensor can be controlled. Based on the setting that the threshold can be set to any value or only to some predefined ones, we suggest solutions based on stochastic approximation (SA) and active learning (AL). In the former case, we provide a strongly consistent estimator, while in the latter case we give a probably approximately correct (PAC) algorithm. Finally, we present numerical experiments to support the results.