From PAC bounds on the generalization error to PAC bounds on the parameter estimation error

Mihaly Petreczky, John Leth, Emma Lukacs, Balint Daroczy

Probably Approximately Correct (PAC) bounds on the generalization error (the difference between the empirical prediction error computed on the training data and its expectation) are an established tool in statistical machine learning, and there have been several extensions of these bounds to dynamical systems. In parallel, in recent years, there has been growing interest in finite-sample bounds for parameter estimation error in system identification. The latter bounds tend to be algorithm-specific, relying on reformulating the identification algorithm as a least-squares regression problem. This is in contrast to classical PAC bounds, which are algorithm-agnostic and can be applied to any identification algorithm based on minimizing a cost function involving the empirical loss. In this paper, we show how oracle bounds derived from PAC bounds can be used to derive finite-sample bounds for dynamical systems. As an illustration of this approach, we present finite-sample parameter-estimation bounds for linear and bilinear systems with bounded inputs and noise. These bounds apply to identification algorithms based on minimizing a regularized empirical loss function. The key ingredient…