Enumerating Multiple Equivalent Lasso Solutions
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Predictive modelling is a data-analysis task common in many scientific fields. However, it is rather unknown that multiple predictive models can be equally well-performing for the same problem. This multiplicity often leads to poor reproducibility when searching for a unique solution in datasets with low number of samples, high dimensional feature space and/or high levels of noise, a common scenario in biology and medicine. The Lasso regression is one of the most powerful and popular regularization methods, yet it also produces a single, sparse solution. In this paper, we show that nearly-optimal Lasso solutions, whose out-of-sample statistical error is practically indistinguishable from the optimal one, exist. We formalize various notions of equivalence between Lasso solutions, and we devise an algorithm to enumerate the ones that are equivalent in a statistical sense: we define a tolerance on the root mean square error (RMSE) which creates a RMSE-equivalent Lasso solution space. Results in both regression and classification tasks reveal that the out-of-sample error due to the RMSE relaxation is within the range of the statistical error due to the sampling size.
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