docs(searches): add multi-start gradient searches to the MLE example

[Jammy2211]

Overview

Phase 2 (redefined) of the multi-start gradient MAP search promotion (PyAutoFit#1369; library + JAX validation shipped in Fit#1370 + autofit_workspace_test#43). The original "config + packaged defaults" phase was found to be empty — PyAutoFit searches have no per-search parameter config and the new searches add zero config keys. The real remaining work is the user-facing workspace example.

Plan

• Add a Search: MultiStartAdam section to scripts/searches/mle.py (after LBFGS), running the new search on the shared 1D Gaussian model and plotting the max-log-likelihood fit, mirroring the Drawer/LBFGS sections.
• Give that section its own JAX-traceable analysis (the searches are JAX-native): enable_pytrees() + register_model(model) + af.ex.Analysis(use_jax=True); leave the shared use_jax=False analysis for Drawer/LBFGS untouched.
• Note MultiStartADABelief / MultiStartLion as drop-in alternatives, and what multi-start buys (escaping wrong basins).
• Update the module docstring bullets, __Contents__, and Relevant links.
• Regenerate the paired notebooks/searches/mle.ipynb.

Detailed implementation plan

Affected Repositories

• autofit_workspace (primary)

Branch Survey

Repository	Current Branch	Dirty?
./autofit_workspace	main	clean

Suggested branch: feature/multi-start-gradient-examples

Implementation Steps

1. scripts/searches/mle.py: extend header (docstring bullets + __Contents__ + Relevant links); add a MultiStartAdam section with enable_pytrees()/register_model(model), af.ex.Analysis(use_jax=True), af.MultiStartAdam(path_prefix="searches", name="MultiStartAdam", n_starts=…, n_steps=…), search.fit, and the standard fit plot; add a short note on ADABelief/Lion.
2. Regenerate notebooks/searches/mle.ipynb from the script (keep the mirror in sync).
3. Run the script end-to-end to confirm it executes and recovers the truth basin.

Key Files

• autofit_workspace/scripts/searches/mle.py — the MLE searches example
• autofit_workspace/notebooks/searches/mle.ipynb — generated mirror
• Reference: autofit_workspace_test scripts/searches/Nautilus_jax.py (JAX analysis + pytree registration pattern)

Autonomy

• --auto, effective level safe (docs cap; difficulty small). Plan written here at start, unmodified since.

Original Prompt

Click to expand

Document af.MultiStartAdam / MultiStartADABelief / MultiStartLion in autofit_workspace scripts/searches/mle.py (JAX-native → use_jax=True analysis + pytree registration), mirroring Drawer/LBFGS; regenerate the paired notebook; update header + links. Opus-authored prose. Cheap run.

[Jammy2211]

Shipped + merged ✅

PR #95 squash-merged (f23254bd). scripts/searches/mle.py now documents af.MultiStartAdam (with MultiStartADABelief / MultiStartLion); paired notebook regenerated.

Gate: tests n/a (workspace) · example validated end-to-end (MultiStartAdam recovers 50/25/10 under JAX) · review self-CLEAN · Heart YELLOW, full reason set human-acked (incl. the benign "release validation stale: source moved since rehearsal" from the Phase-1 merge).

This completes the multi-start gradient search promotion (PyAutoFit#1369): library + JAX validation + user-facing example all landed. Note during review: the searches need no JAX pytree registration in user scripts (they build instances inside the traced objective), unlike Nautilus_jax.