Add unpenalized logistic regression

[lorentzenchr]

The problem sets of logistic regression all have a penalty. It would be very interesting, at least to me, to add the zero penalty case.

Note: For n_features > n_samples, like the 20 news dataset, this is real fun (from an optimization point of view).

[mathurinm]

If n_features > n_samples Then it's likely that the data is perfectly separable and so the minimizer does not even exist, no? all algorithms go to infinity (at a log log speed)

@tomMoral what's our policy between 3 separate repos (L1, L2, no pen) or a single one with some solvers skipping configs?

[lorentzenchr]

If n_features > n_samples Then it's likely that the data is perfectly separable and so the minimizer does not even exist, no? all algorithms go to infinity (at a log log speed)

Not quite, the solution is just not unique. Then, a desirable property of the coefficients is to have minimum L2 norm among all solutions. But that is hardly ever promised by any solver.

[mathurinm]

• If there exists a vector w which perfectly classifies the training data, then t * w with scalar t going to infinity make the Logistic loss go to 0, because - t y_i * x_i^\top w goes to - infty for all training samples.
• Since log(1 + exp(u)) > 0, a loss value of exactly 0 is unreachable.

hence there is no minimizer in that case (see discussion above equation 2 in https://arxiv.org/pdf/1710.10345.pdf).

Edit: it's still a very interesting problem

[lorentzenchr]

Thanks for the linked literature.

Since log(1 + exp(u)) > 0, a loss value of exactly 0 is unreachable.

A good implementation will reach a loss very close to zero, e.g. I got about n_samples * 1e-10 for the 20 news dataset.

[mathurinm]

Yes, the objective converges to 0, but the iterates are diverging (there is no minimizer, ie the argmin is empty and the min is an inf)

[Badr-MOUFAD]

I'm transferring the issue to benchmark_logreg_l2 as it seems to be less directly linked to benchopt.

[agramfort]

PR welcome :)

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