Bounds on the Excess Minimum Risk via Generalized Information Divergence Measures

Given finite-dimensional random vectors <i>Y</i>, <i>X</i>, and <i>Z</i> that form a Markov chain in that order (Y→X→Z), we derive the upper bounds on the excess minimum risk using generalized information divergence measures. Here, <i>Y</i> is a target vector to be estimated from an observed feature vector <i>X</i> or its stochastically degraded version <i>Z</i>. The excess minimum risk is defined as the difference between the minimum expected loss in estimating <i>Y</i> from <i>X</i> and from <i>Z</i>. We present a family of bounds that generalize a prior bound based on mutual information, using the Rényi and α-Jensen-Shannon divergences, as well as Sibson's mutual information. Our bounds are similar to recently developed bounds for the generalization error of learning algorithms. However, unlike these works, our bounds do not require the sub-Gaussian parameter to be constant, and therefore, apply to a broader class of joint distributions over <i>Y</i>, <i>X</i>, and <i>Z</i>. We also provide numerical examples under both constant and non-constant sub-Gaussianity assumptions, illustrating that our generalized divergence-based bounds can be tighter than the ones based on mutual information for certain regimes of the parameter α.