# SCH

## Seminar

### Stability - based regularisation

Seminar Room 1, Newton Institute

#### Abstract

The properties of L1-penalized regression have been examined in detail in recent years. I will review some of the developments for sparse high-dimensional data, where the number of variables p is potentially very much larger than sample size n. The necessary conditions for convergence are less restrictive if looking for convergence in L2-norm than if looking for convergence in L0-quasi-norm. I will discuss some implications of these results. These promising theoretical developments notwithstanding, it is unfortunately often observed in practice that solutions are highly unstable. If running the same model selection procedure on a new set of samples, or indeed a subsample, results can change drastically. The choice of the proper regularization parameter is also not obvious in practice, especially if one is primarily interested in structure estimation and only secondarily in prediction. Some preliminary results suggest, though, that the stability or instability of results is informative when looking for suitable data-adaptive regularization.

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