Fig. 3.

Empirical evidence for consistency. The figure shows the mean squared estimation error $\mathrm{E}[||{\widehat{\mathit{\theta }}}_n-{\mathit{\theta }}^{\circ }|{|}^2]$ for the examples in Fig. 2 as a function of the sample size n (solid lines, circles). The mean was computed as an average over 100 outcomes. The dashed lines depict the mean ±2 standard errors. The linear trend on the log–log scale suggests convergence in quadratic mean, and hence consistency of the estimator ${\widehat{\mathit{\theta }}}_n$. The results are for $L_1$-regularized logistic regression, see Supplementary material 3 for the other classification methods