Recently, there has been a marked increase in interest regarding the generalization capabilities of unregularized gradient-based learning methods.

Recently, there has been a marked increase in interest regarding the generalization capabilities of unregularized gradient-based learning methods.

D () has lower space complexity than the original problem or can be minimized more efficiently.

Fenchel-Y oung losses are a family of loss functions, encompassing the squared, logistic and sparsemax losses, among others. They are convex w.r.t. the model

In this Appendix, we will derive the fixed-point equations for the order parameters presented in the main text, following and generalising the analysis in Ref. [ Saddle-point equations The saddle-point equations are derived straightforwardly from the obtained free energy functionally extremising with respect to all parameters. The zero-regularisation limit of the logistic loss can help us study the separability transition. N 5 + \ 1 p 0, 1 d 5. (66) As a result, given that \ 2( 0, 1 ], the smaller value for which E is finite is U This result has been generalised immediately afterwards by Pesce et al. Ref. [ 59 ] for the Gaussian case, we can obtain the following fixed-point equations, 8 > > > > > >< > > > > > >: E = Mean universality Following Ref. [ In our case, this condition is simpler than in Ref. [ We see that mean-independence in this setting is indeed verified. Numerical experiments Numerical experiments regarding the quadratic loss with ridge regularisation were performed by computing the Moore-Penrose pseudoinverse solution.

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