AFT and PO share the PH optimiser's two omissions — measure optimise_nm_tnc before changing it

[derrynknife]

Split out of #328, which fixed the same two omissions for PH only.

What is shared

optimise_nm_tnc in surpyval/univariate/regression/_fit_skeleton.py serves the AFT and PO families:

def optimise_nm_tnc(fun: Callable, init_t):
    """AFT/PO's historical ladder: Nelder-Mead, then TNC kept only on
    success."""
    res = minimize(
        fun, init_t, method="Nelder-Mead", options={"maxiter": 1000}
    )
    res2 = minimize(fun, res.x, method="TNC")
    return res2 if res2.success else res

Like the old optimise_ph, it passes no jac even though fun closes over regression_neg_ll, which is written in autograd.numpy and is fully traceable. And like the old optimise_ph, the search is not preconditioned, so it inherits the scale sensitivity fixed in #323 and #328: the gradient shrinks like 1/theta with the data magnitude and grows like n with the sample size, and no absolute gtol serves every fit.

What is not shared, and why this is not a copy of #328

It does already guard the TNC rung on success — that was the third defect in optimise_ph and it does not apply here.

More importantly the first rung is Nelder-Mead, which is derivative-free. The specific failure in #328 was BFGS meeting scipy's absolute gtol well short of the optimum on large-magnitude data and reporting success; a derivative-free method does not stop for that reason. So the plausible outcomes here are different:

• it may be slow but correct, where PH was fast and wrong — Nelder-Mead pays for its robustness in function evaluations, and every evaluation is O(n);
• it may fail somewhere else entirely, since TNC still polishes the answer and TNC is gradient-based;
• or the missing gradient may cost little because Nelder-Mead never asks for one.

I have not measured any of this. That is the point of this issue.

Do this first

Run the same A/B I ran for PH before touching the code:

• Grid over n and p, and a separate sweep over data magnitude (1e-3 to 1e9), which is where PH broke.
• Compare against lifelines' WeibullAFTFitter and LogNormalAFTFitter. This is a direct comparison — no reparameterisation needed, unlike the PH case which had to go through beta_aft = -gamma_ph / shape.
• Score both packages' answers on an independently written AFT log-likelihood rather than trusting either one's reported figure. Scoring on surpyval's own objective is what caught the Truncation correction manufactures likelihood when the survival probability underflows #326 misdiagnosis.
• Proportional odds has no lifelines counterpart, so it needs a different oracle — a fit at unit scale transported to large scale by the exact change of units, as in test_weibull_ph_at_large_scale_reaches_the_optimum.

Only then decide the change. If it looks like PH did, the fix is the same: preconditioned_bfgs (now in fitters/__init__.py) on the autograd gradient, then TNC, then Nelder-Mead as the derivative-free fallback, stopping at the first rung that converges.

Reference

For PH the equivalent change gave 1.4–6x on wall clock and turned a 1.468-nat shortfall at data scale 1e6 into 1.1e-08. There is no reason yet to expect the same numbers here — the ladder is differently shaped.

Test pattern to reuse: test_weibull_ph_fit_is_invariant_to_the_units_of_x and test_weibull_ph_at_large_scale_reaches_the_optimum in surpyval/tests/univariate/regression/test_proportional_hazards.py. Both fail on the pre-#328 PH ladder, so they are known to discriminate.