Median branch assignment

[josh-lee-MIE2t5]

Summary

Speeds up AbsoluteError (median / MAE) branch assignment during coordinate descent by replacing full partition re-sorts with dynamic structures.

Previously, AbsoluteError branch assignment either skipped CD or re-sorted each partition’s samples on every bin move (O(n log n) per trial). This PR keeps sorted structure across leave/join so moves cost closer to O(n) or O(k log n).

Algorithmic approaches

1. Merge/filter (default, production)

   • Each bin stores a pre-sorted (y, w) array.
   • Each partition stores a sorted (y, w) array plus source-bin ids.
   • Join: mergesort-style merge of the bin into the partition — O(n + k).
   • Leave: scan/filter the partition by bin id — O(n).
   • MAE uses a presorted pinball/median pass (no re-sort).

2. BST (deprecated, A/B)

   • Augmented AVL (WeightedMAETree) per partition/output with subtree (Σw, Σw·y).
   • Batch insert/erase of a bin — O(k log N); median/MAE — O(log N).

3. Sort (deprecated, reference)

   • Original path: collect partition samples and std::sort on every add/remove — O(n log n).

Backends are hot-swappable via SGTLEARN_MAE_BACKEND=merge|bst|sort. MAE CD remains gated by SGTLEARN_MAE_CD (off by default for sklearn CART parity). Implementations are split into separate files: AbsoluteErrorBranchAssignment (merge), …Bst, …Sort, plus shared helpers.

Benchmarks

Timing scripts and CSVs are kept out of this PR and live on the fork:

josh-lee-MIE2t5/sgtlearn-MAE-CD-SpeedUp @ median-branch-assignment

• CD microbench CSV — sort vs BST vs merge (branch assignment only)
• End-to-end fit CSV — SGTRegressor(absolute_error), MAE CD on, tao_n_runs=0
• How to re-run

CD-only (branch assignment)

case	sort	bst	merge	bst×	merge×
small_64×20	1119 ms	77 ms	364 ms	14.6×	3.1×
medium_128×50	10356 ms	382 ms	2201 ms	27.1×	4.7×
large_256×100	57983 ms	2859 ms	13313 ms	20.3×	4.4×
multiout_128×40×3	13717 ms	747 ms	3840 ms	18.4×	3.6×

Objectives matched across backends.

End-to-end shape-tree fit (median predictor)

case	sort	bst	merge	bst×	merge×
2k × 8	15.7 s	7.0 s	5.2 s	2.24×	2.99×
5k × 12	81.3 s	39.3 s	31.2 s	2.07×	2.61×
8k × 16	259 s	149 s	101 s	1.74×	2.57×

Leaf counts identical across backends. Merge wins end-to-end fit; BST wins the CD-only microbench.

[upadhyan]

looks good!