Iterative regularization for convex regularizers

Machine learning often reduces to estimating some model parameters. This approach raises at least two orders of questions: first, multiple solutions may exist, amongst which a specific one must be selected; second, potential instabilities with respect to noise and sampling must be controlled. A classical way to achieve both goals is to consider explicitly penalized or constrained objective functions. In machine learning, this leads to regularized empirical risk minimization (Shalev-Shwartz and Ben-David, 2014). A more recent approach is based on directly exploiting an iterative optimization procedure for an unconstrained/unpenalized problem. This approach is shared by several related ideas. One is implicit regularization (Mahoney, 2012; Gunasekar et al., 2017), stemming from the observation that the bias is controlled increasing the number of iterations, just like in penalized methods it is controlled decreasing the penalty parameter. Another one is early stopping (Yao et al., 2007; Raskutti et al., 2014), putting emphasis on the fact that running the iterates to convergence might lead to instabilities in the presence of noise. Yet another, and more classical, idea is iterative regularization, where both aspects (convergence and stability) are considered to be relevant (Engl et al., 1996; Kaltenbacher et al., 2008).