From Transfer Probes to Expert Grouping: Evidence from LoRA Post-Training

Xueyang Wu

cs.LGOct 10, 2026

From Transfer Probes to Expert Grouping: Evidence from LoRA Post-Training

Authors: Xueyang Wu

Abstract

A platform that post-trains one base model for many business lines often has more lines than experts it can maintain, so lines must share experts, yet lines seldom come with a taxonomy saying which should share. We ask whether transfer probes (train on one source, measure paired changes on all sources) can discover a grouping when the standard one is unknown. To test probes against a reference, we use seven public sources that do have a label tree (five mathematics subfields and two kinds of programming, standing in for business lines) and treat the tree's grouping as a label reference that the evaluator holds and the method does not. With four LoRA experts under fixed per-source exposure and per-expert budgets, we compare a purely probe-derived grouping and an uninformed uniform default (whose tied routing serves every line from one expert) against this reference. Trained once for one interval from a shared warm-up state and routed by a rule fixed by each allocation, neither reached the prespecified practical gain of 2 percentage points, and both had lower overall utility than the reference ( $-2.43$ and $-2.54$ pp; 95% intervals below zero). In a post hoc comparison, the probe-derived grouping was indistinguishable from the uniform default ( $+0.11$ pp, $[-1.25,+1.54]$ ). In fixed-budget LoRA post-training, a grouping built from initial-state single-source probes thus did not deliver the advantage predicted from the probes once carried to a later training state, a finite budget, and specified routing.

Figures & tables

Platform aspect	Simulated	Simplified	Not covered
Lines and labels	Every request carries its line label; seven sources act as lines	Lines are public mathematics subfields and programming types with automatic verifiers	Enterprise data; noisy or changing labels; lines without automatic scoring
Expert budget	Fewer experts than lines ( $M=4<K=7$ ); LoRA experts on one shared base	Equal expert budgets and equal line exposure in every interval (Eq. 1 )	Choosing $M$ ; unequal data volumes, budgets, or hosting costs
Knowledge of the grouping	The default and probe-derived candidates use no labels; the tree’s grouping is held by the evaluator as the label reference	A tree exists, and at $M=4$ it has a single four-group cut	Platforms with no tree at all, which motivate the study but are not tested
Traffic	Per-line traffic weights in the utility (Eq. 3 )	Equal weights; other static mixes examined after the fact through linearity (Table 5 )	Drifting traffic; new or retired lines
Training	One allocation interval from a shared warm-up state W	One starting state; one training run per candidate	Repeated reallocation over many intervals; training-seed variation; later states and longer budgets
Serving	Each line served by a single expert chosen by a fixed rule	Route read off the block matrix (Eq. 2 )	Competence-calibrated routing; latency and throughput of multi-adapter serving

Table 1: The testbed against the motivating multi-line platform.

Candidate	Blocks $n$	Route $h_{\rm blk}(n)$	Role and information used
C0	$\left(\begin{smallmatrix}0&0&7&7&0&0&0\\ 7&7&0&0&0&0&0\\ 0&0&0&0&0&7&7\\ 1&1&1&1&8&1&1\end{smallmatrix}\right)$	$(1,1,0,0,3,2,2)$	Label reference: the tree’s only four-group cut, balanced onto the margins. Uses the tree; not available to a method in the label-unknown setting.
C1	$\left(\begin{smallmatrix}2&2&2&2&2&2&2\\ 2&2&2&2&2&2&2\\ 2&2&2&2&2&2&2\\ 2&2&2&2&2&2&2\end{smallmatrix}\right)$	$(0,0,0,0,0,0,0)$	Uninformed default: uses no information. Every source is a tie, so expert 0 serves all sources.
C2	$\left(\begin{smallmatrix}0&0&0&6&8&0&0\\ 0&8&2&2&0&0&2\\ 0&0&0&0&0&8&6\\ 8&0&6&0&0&0&0\end{smallmatrix}\right)$	$(3,1,3,0,0,2,2)$	Probe-derived: transportation-LP vertex for the additive proxy built from the s17 probe table. Uses the probes and the margins, not the tree.

Table 2: Candidates in the main comparison. Block matrices list experts e0–e3 as rows and sources (Integral, Differential, Abstract, Linear, Statistics, algorithmic, generic) as columns; every row sums to 14 and every column to 8. Routes are block-matrix routes (Eq. 2 ) in the same source order.

Link	State	Data	What it answers, and what it does not
1. Probes	$S_{0}$ : base model with a fresh adapter ( $B=0$ )	Training split, one source, 16 blocks; audit split, 80 problems per source (OpenCodeInstruct for both code sources); shuffles s17 (7/7 rows) and s23 (6/7 rows), sharing problems and initial predictions	Whether training on source $k$ moves accuracy on source $j$ at $S_{0}$ , as paired flips. Not: effects at W, under mixed-source training, or at other doses.
2. Candidate construction (development)	No training; uses the $S_{0}$ probe table	C0: label tree and margins. C1: margins only. C2: s17 table, margins, and the additive proxy with competence routing. No test data and no data from W	Which allocations to train. Not: how they perform. C3, an edit of C0, is described in Appendix D .
3. Training at W	W: shared warm-up, one uniform interval (two blocks per expert per source)	Training split; one interval of 56 blocks per candidate, with identical tokens per source across candidates; one run per candidate	What each allocation produces from W in one interval. Not: training-seed variation, other starting states, or longer budgets.
4. Fixed-routing test	W plus one candidate interval	Test split, first 400 problems per source (algorithmic: 376 MBPP+ and 24 OpenCodeInstruct problems), two samples each (seeds 2001 and 2002); route by Eq. 2	$U_{\rm all}$ , $U_{\rm LS}$ , $U_{\rm other5}$ , and per-source accuracy of allocation and route together, with intervals for test sampling. Not: performance under competence-calibrated routing; for algorithmic programming, performance on the benchmark that was probed.

Table 3: State and evidence along the chain from probes to scores.

	When specified	$U_{\rm all}$	$U_{\rm LS}$	$U_{\rm other5}$
C0 (label reference)	—	0.5036	0.5019	0.5042
C1 (uninformed default)	—	0.4782	0.4888	0.4740
C2 (probe-derived)	—	0.4793	0.4906	0.4747
		$\Delta U_{\rm all}$	$\Delta U_{\rm LS}$	$\Delta U_{\rm other5}$
C1 $-$ C0	before the run	$-2.54$ $[-4.04,-1.12]$	$-1.31$ $[-4.19,+1.50]$	$-3.02$ $[-4.70,-1.42]$
C2 $-$ C0	before the run	$-2.43$ $[-3.84,-1.14]$	$-1.13$ $[-3.88,+1.38]$	$-2.95$ $[-4.50,-1.40]$

Table 4: Candidate comparison at W. Top: routed utilities (mean accuracy; 400 test problems per source, two samples each). Bottom: paired differences in percentage points with 95% joint paired bootstrap intervals.

U_{\rm LS}

averages Linear Algebra and Statistics;

U_{\rm other5}

averages the other five sources.

	C1 $-$ C0		C2 $-$ C0
Source	$\Delta_{j}$	Simultaneous 95%	$\Delta_{j}$	Simultaneous 95%
Integral Calculus	$-4.500$	$[-10.42,+1.42]$	$-3.250$	$[-8.87,+2.37]$
Differential Calculus	$-4.250$	$[-10.02,+1.52]$	$-4.625$	$[-10.14,+0.89]$
Abstract Algebra	$+0.875$	$[-4.87,+6.62]$	$-0.125$	$[-5.73,+5.48]$
Linear Algebra	$-1.250$	$[-7.57,+5.07]$	$-3.125$	$[-8.85,+2.60]$
Statistics	$-1.375$	$[-7.10,+4.35]$	$+0.875$	$[-4.70,+6.45]$

Table 5: Other traffic mixes and single sources, relative to the label reference (percentage points).

\Delta_{j}

is the per-source accuracy difference. Because utility is linear in the traffic weights,

\Delta U_{w}=\sum_{j}w_{j}\Delta_{j}

lies in

[\min_{j}\Delta_{j},\max_{j}\Delta_{j}]

for every

w

. Intervals are simultaneous 95% intervals (max-

|t|

bootstrap, critical value 3.025).

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Cell	$+g/-l$	$p$	$+g/-l$	$p$
	s17 (7 rows)		s23 (6 rows)
Linear $\to$ Statistics	$+16/-3$	0.004	$+14/-4$	0.031
Statistics $\to$ Statistics	$+18/-4$	0.004	$+13/-6$	0.167
Linear $\to$ Integral	$+21/-6$	0.006	$+17/-10$	0.248
Statistics $\to$ Linear	$+22/-8$	0.016	$+19/-8$	0.052
Abstract $\to$ Abstract	$+21/-9$	0.043	$+20/-15$	0.500

Appendix

Table 6: Probe cells with nominal

p<0.05

in either shuffle, plus the boundary case Integral

\to

Abstract. Two-sided exact McNemar

p

; the Bonferroni-adjusted value is

p_{\rm Bonf}=\min(1,49p)

over the 49-cell family; † marks the only cell with

p_{\rm Bonf}<0.05

(

p_{\rm Bonf}=0.00103

). The two shuffles share audit items and initial predictions.

Reading	Tensor	C1 $-$ C0	C2 $-$ C0	C3 $-$ C0	C3 $\Delta U_{\rm LS}$
Block-matrix routing	s17	$-2.23$	$+2.01$	$+1.22$	$+4.57$
	s23*	$-1.92$	$+0.67$	$+0.74$	$+2.27$
Competence routing	s17	$-3.90$	$+0.90$	$+0.58$	$+0.94$
	s23*	$-3.67$	$+1.06$	$+0.40$	$+1.64$
Zero transfer	s17	$-3.73$	$+0.02$	$-0.02$	$-0.55$
	s23*	$-2.98$	$-0.10$	$-0.02$	$-0.27$

Appendix

Table 7: Model-predicted and measured differences from C0 (pp); post hoc. The two rightmost columns concern the candidate outside the setting (Appendix D ). Model rows use the

S_{0}

audit tensors; the measured row uses state W, 400 test items per source, and two samples per item.

		C1 $-$ C0		C2 $-$ C0
Reading	Tensor	$\Delta U_{\rm LS}$	$\Delta U_{\rm other5}$	$\Delta U_{\rm LS}$	$\Delta U_{\rm other5}$
Block-matrix routing	s17	$+0.23$	$-3.22$	$+4.92$	$+0.84$
	s23*	$-1.48$	$-2.09$	$+1.95$	$+0.16$
Competence routing	s17	$-3.40$	$-4.09$	$+1.29$	$+0.75$
	s23*	$-3.75$	$-3.64$	$+3.12$	$+0.23$
Zero transfer	s17	$-4.45$	$-3.44$	$-0.23$	$+0.13$

Appendix

Table 8: Model-predicted and measured differences from C0 on the Linear–Statistics lines (

\Delta U_{\rm LS}

) and the other five lines (

\Delta U_{\rm other5}

), pp; post hoc.

Source	Disagreeing items	$\sigma_{d}^{2}$	$\sigma_{t}^{2}$	$\sigma_{b}^{2}$	Decoding share
Integral	22/80	0.138	0.252	0.115	54%
Differential	27/80	0.169	0.253	0.084	67%
Abstract	25/80	0.156	0.248	0.092	63%
Linear	23/80	0.144	0.252	0.108	57%
Statistics	20/80	0.125	0.237	0.112	53%
algorithmic	17/80	0.106	0.250	0.144	42%

Appendix

Table 9: Decoding-variance decomposition used to choose

k

: two independent samples of one model on 80 items per source. The 80 algorithmic items are MBPP+ problems.

	$\Delta U_{\rm all}$	$\Delta U_{\rm LS}$	$\Delta U_{\rm other5}$
C3 $-$ C0	$-1.54$ $[-2.89,-0.20]$	$-3.56$ $[-6.25,-0.88]$	$-0.72$ $[-2.30,+0.83]$
C3 $-$ C1	$+1.00$ $[-0.37,+2.45]$	$-2.25$ $[-5.00,+0.50]$	$+2.30$ $[+0.67,+4.02]$

Appendix

Table 10: The candidate outside the setting, C3: differences in pp with 95% jointly paired bootstrap intervals. The comparison with C0 was specified before the run; the comparison with C1 was added after observing results and is not adjusted for multiplicity.

Line	Difference	Marginal 95%	Simultaneous 95%
Integral	$-3.625$	$[-7.500,+0.375]$	$[-9.43,+2.18]$
Differential	$-0.875$	$[-4.625,+2.875]$	$[-6.78,+5.03]$
Abstract	$+1.500$	$[-2.125,+5.000]$	$[-4.15,+7.15]$
Linear	$-4.750$	$[-8.625,-0.875]$	$[-10.64,+1.14]$
Statistics	$-2.375$	$[-6.125,+1.125]$	$[-7.80,+3.05]$
algorithmic	$+0.875$	$[-2.125,+3.875]$	$[-3.75,+5.50]$

Appendix

Table 11: Per-line differences C3

-

C0 (pp). Point estimates are exact multiples of 0.125 pp. Marginal 95% intervals come from the analysis specified before the run; simultaneous 95% intervals (max-

|t|

bootstrap over all 21 executed per-line comparisons,

q=3.025

; Appendix A ) were added after observing results.

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