Improving the union bound: A distribution dependent approach

Statistical Learning Theory deals with the problem of estimating the performance of a learning procedure. Any learning procedure implies making choices and this choices imply a risk. When the number of choices is finite, the state-of-the-art tool for evaluating the total risk of all the choice made is the Union Bound. The problem of the Union Bound is that it is very loose in practice if no a-priori information is available. In fact, the Union Bound considers all choices equally plausible while, as a matter of fact, a learning procedure targets just particular choices disregarding the others. In this work we will show that it is possible to improve the Union Bound based results using a distribution dependent weighting strategy of the true risks associated to each choice. Then we will prove that our proposal outperforms or, in the worst case, it degenerate in the Union Bound.