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Abstract:
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A number of problems in Statistical Learning Theory, such as regression and pattern classification, can be formulated as penalized empirical risk minimization over a linear span of a very large dictionary of functions. The complexity penalties are often based on $\ell_p$-norm of the vector of coefficients, most often, with $p=1.$ The central problem that will be discussed in this talk is to show that if the true solution of the problem is "approximately sparse," then its empirical solution is also "approximately sparse." Several "sparsity bounds" that provide mathematical description of this phenomenon will be considered and the influence of the sparsity on excess risk bounds and oracle inequalities will be also discussed.
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