Routing Should Pay for Itself: Sparse Supervision for Economical LLM Routing

Guannan Lai; Gelin Bian; Hao-Xuan Ma; Jun-Peng Jiang; Long Chen; Jian-Dong Liu; Zhi-Hao Tan; Han-Jia Ye

cs.AISep 29, 2026

Routing Should Pay for Itself: Sparse Supervision for Economical LLM Routing

Authors: Guannan Lai, Gelin Bian, Hao-Xuan Ma, Jun-Peng Jiang, Long Chen, Jian-Dong Liu, Zhi-Hao Tan, Han-Jia Ye

Organizations: School of Artificial Intelligence, Nanjing University · National Key Laboratory for Novel Software Technology, Nanjing University · SinapisAI · The Hong Kong University of Science and Technology

Abstract

Large language model (LLM) routing reduces serving cost by assigning each query to an appropriate model while preserving response quality. Learning such a router, however, often requires executing multiple candidate models on historical queries to collect query--model quality feedback, creating a nontrivial supervision cost before deployment. Existing work largely focuses on serving-time efficiency, overlooking whether the resulting savings are sufficient to recover this upfront expenditure. We further observe that routing quality often saturates well before all query--model feedback is collected, suggesting that dense supervision can be economically over-provisioned. We propose SaveRouter, a sparse-supervision routing framework that selectively acquires informative model feedback and shares capability information across related queries, while retaining query-level refinement for fine-grained routing. We evaluate routing by jointly accounting for supervision expenditure and subsequent serving-time savings. Across four routing benchmarks, the main setting uses only about 33--41% of available training feedback while maintaining competitive or better routing quality, and reduces the break-even deployment volume by approximately 1.9--9.5 times compared with the fastest conventional router. Further analysis shows that acquiring more supervision is not always economically preferable: the supervision level that minimizes serving cost can differ from the one that achieves the earliest payback. Our code is publicly available at https://github.com/LAMDA-Model-Reuse/SaveRouter.

Figures & tables

Panel of Figure 1 — Figure 1: Dense supervision can be economically over-provisioned. Left: supervision expenditure can substantially delay the break-even point of routers with lower serving-time cost. Right: routing quality often saturates well before the full query–model matrix is observed; EmbedLLM and kNN recover 99% of their fully supervised accuracy with only 30% and 60% supervision, respectively.

Figure 2: Overview of SaveRouter . SaveRouter first groups related training queries and adaptively acquires sparse query–model feedback using capability and uncertainty. It then estimates model capability through a structured group–model prior, evidence-based correction from the acquired observations, and a query-specific residual predictor. At deployment time, the resulting quality estimates are combined with estimated serving costs for cost-aware routing.

	LLMRouterBench				Mixinstruct
Method	$P_{s}$ $\uparrow$	CR $\downarrow$	SA-BEP $\downarrow$	SA-CR@1M $\downarrow$	$P_{s}$ $\uparrow$	CR $\downarrow$	SA-BEP $\downarrow$	SA-CR@1M $\downarrow$
EmbedLLM	0.6094	0.5246	15.6K	0.5321	0.7488	0.9924	28.60M	1.2088
kNN	0.6230	0.3770	11.9K	0.3845	0.7484	0.9814	11.63M	1.1977
OmniRouter	0.6218	0.3564	11.5K	0.3638	0.7435	$\infty$	$\infty$	$\infty$
RMSoftmax	0.6126	0.6328	20.2K	0.6403	0.7495	0.9893	20.19M	1.2056
TRouter	0.6145	0.5221	15.6K	0.5295	0.7488	1.0000	$\infty$	$\infty$

Table 1: Main results on four routing benchmarks. Higher

P_{s}

is better; lower CR, SA-BEP, and SA-CR@1M are better.

\infty

denotes an unreachable target operating point; for supervision-amortized metrics, it also indicates non-positive per-query saving. For sparse- or partial-feedback methods, every model invocation used to acquire feedback is charged separately.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameter	Symbol	Value
Global prior parameter	$\alpha_{0}$	1
Global prior parameter	$\beta_{0}$	1
Acquisition prior strength	$\tau_{0}$	10
UCB exploration coefficient	$\beta_{\mathrm{ucb}}$	0.35
Additive-prior Ridge weight	$\lambda_{\mathrm{prior}}$	10
Final shrinkage strength	$\tau$	40

Appendix

Table A1: Core hyperparameters of SaveRouter .

Benchmark	Training groups	New-query assignment
LLMRouterBench	10 task groups	Logistic classifier
Mixinstruct	Single global group	Single group
MMRBench	7 dataset groups	Logistic classifier
RouterBench	85 task groups	Logistic classifier

Appendix

Table A2: Query grouping used in the main experiments.

Benchmark	#Queries	#Train	#Test	#Models	#Groups	Train Pairs
LLMRouterBench	12,446	2,489	9,957	12	10	29,868 / 29,868
Mixinstruct	110,000	22,000	88,000	12	1	264,000 / 264,000
MMRBench	10,370	2,074	8,296	10	7	20,149 / 20,740
RouterBench	36,497	7,299	29,198	11	85	80,289 / 80,289

Appendix

Table A3: Statistics of the routing benchmarks used in our experiments. Available train pairs count query–model pairs for which both quality and cost fields are available.

Method	Configuration
EmbedLLM	$\alpha=0.001$ ; 30 epochs; learning rate $10^{-3}$ .
kNN	$k=300$ .
OmniRouter	top- $k=50$ ; $\gamma=\delta=0.6$ ; 50 epochs.
RMSoftmax	30 linearly spaced cost weights from $0$ to $1000$ ; 300 epochs.
TRouter	hidden dimension 256; dropout 0.1; temperature 0.07; 20 epochs.
UniRoute	10 clusters; mapping network trained for 5 epochs.

Appendix

Table A4: Key configurations of the compared routing methods.

Benchmark	$K$	Sup. (%)	$P_{s}$ $\uparrow$	CR $\downarrow$	$C_{0}$ $\downarrow$	SA-BEP $\downarrow$	SA-CR@1M $\downarrow$
LLMRouterBench	1	8.33	0.621874	0.2395	44.64	1,311	0.2405
	2	16.67	0.631415	0.2357	85.21	2,489	0.2376
	3	25.00	0.630160	0.2228	146.69	4,213	0.2261
	4	33.33	0.633825	0.2320	181.09	5.3K	0.2360
	6	50.00	0.627699	0.2180	241.15	6,884	0.2234
	8	66.67	0.627147	0.2260	279.86	8,071	0.2322

Appendix

Table A5: Effect of supervision budget

K

across four routing benchmarks. Higher

P_{s}

is better; lower CR,

C_{0}

, SA-BEP, and SA-CR@1M are better. “–” indicates that the router does not reach the best-single-model quality target, so the corresponding supervision-amortized metrics are undefined.

Method	$P_{s}$ $\uparrow$	CR $\downarrow$	SA-BEP $\downarrow$	SA-CR@1M $\downarrow$
Dense-supervision routers
EmbedLLM	0.776135	$\infty$	$\infty$	$\infty$
kNN	0.795418	0.816906	4,772,514	1.6907
MLP	0.776705	$\infty$	$\infty$	$\infty$
SVM	0.718650	$\infty$	$\infty$	$\infty$
RouteLLM-MF	0.780365	$\infty$	$\infty$	$\infty$

Appendix

Table A6: Results on RouterEval. SaveRouter uses

K=108

. Higher

P_{s}

is better; lower CR, SA-BEP, and SA-CR@1M are better.

\infty

denotes an unreachable target operating point.

Train/Test	$P_{s}$ $\uparrow$			CR $\downarrow$
	SaveRouter	Best Dense	$\Delta P_{s}$ (pp)	SaveRouter	Best Dense	$\Delta$ CR
20/80	0.6338	0.6230	+1.08	0.2320	0.3564	34.9%
40/60	0.634842	0.624665	+1.018	0.1888	0.3275	42.4%
60/40	0.633461	0.626029	+0.743	0.2137	0.2965	27.9%
80/20	0.641566	0.636145	+0.542	0.1412	0.2275	38.0%

Appendix

Table A7: Sensitivity to the training-set fraction on LLMRouterBench. “Best dense” denotes the strongest dense baseline for the corresponding metric among EmbedLLM, kNN, OmniRouter, TRouter, and InferenceDynamics.

\Delta P_{s}

is the absolute improvement of SaveRouter in percentage points, and

\Delta

CR is the relative reduction in CR.

Configuration	$P_{s}$ $\uparrow$	CR $\downarrow$	SA-BEP $\downarrow$	SA-CR@1M $\downarrow$
Main: $\gamma=2,\tau=40,\lambda_{\mathrm{ctx}}=200$	0.633825	0.2320	5,263	0.2360
$\lambda_{\mathrm{ctx}}=100$	0.631365	0.2182	5,171	0.2223
$\lambda_{\mathrm{ctx}}=50$	0.625791	0.2208	5,188	0.2248
$\gamma=1$	0.633373	0.2517	5,402	0.2558
$\gamma=4$	0.624385	0.2430	5,340	0.2471
$\tau=20$	0.628955	0.2024	5,068	0.2065

Appendix

Table A8: Hyperparameter robustness on LLMRouterBench with fixed

K=4

supervision. Higher

P_{s}

is better; lower CR, SA-BEP, and SA-CR@1M are better.

Panel of Figure 13 — Figure A1: Full quality–cost frontiers across four routing benchmarks. Each solid curve shows all evaluated SaveRouter operating points, while markers denote the peak-quality endpoint of each baseline method. The horizontal axis reports serving cost normalized by the best single model within each benchmark; lower is better, while higher quality is better.

Method	$0.90Q_{b}$	$0.925Q_{b}$	$0.95Q_{b}$	$0.975Q_{b}$	$1.00Q_{b}$
EmbedLLM	0.0286 / 7.7K / 0.0360	0.0953 / 8.2K / 0.1027	0.5246 / 15.6K / 0.5321	0.5246 / 15.6K / 0.5321	0.5246 / 15.6K / 0.5321
kNN	0.0863 / 8.1K / 0.0938	0.1107 / 8.4K / 0.1182	0.1530 / 8.8K / 0.1605	0.2490 / 9.9K / 0.2564	0.3770 / 11.9K / 0.3845
OmniRouter	0.1419 / 8.7K / 0.1493	0.1618 / 8.9K / 0.1693	0.2037 / 9.3K / 0.2112	0.2579 / 10.0K / 0.2653	0.3564 / 11.5K / 0.3638
RMSoftmax	0.0285 / 7.7K / 0.0359	0.6328 / 20.2K / 0.6403	0.6328 / 20.2K / 0.6403	0.6328 / 20.2K / 0.6403	0.6328 / 20.2K / 0.6403
TRouter	0.0328 / 7.7K / 0.0402	0.1500 / 8.7K / 0.1575	0.2447 / 9.8K / 0.2521	0.2995 / 10.6K / 0.3069	0.5221 / 15.6K / 0.5295
UniRoute	0.0285 / 7.7K / 0.0359	0.1962 / 9.2K / 0.2036	0.5583 / 16.8K / 0.5657	0.5583 / 16.8K / 0.5657	0.5583 / 16.8K / 0.5657

Appendix

Table A9: Performance across quality targets on LLMRouterBench. Each entry reports CR / SA-BEP / SA-CR@1M; lower values are better.

Method	$0.90Q_{b}$	$0.925Q_{b}$	$0.95Q_{b}$	$0.975Q_{b}$	$1.00Q_{b}$
EmbedLLM	0.5000 / 432.7K / 0.7163	0.5000 / 432.7K / 0.7163	0.9924 / 28.60M / 1.2088	0.9924 / 28.60M / 1.2088	0.9924 / 28.60M / 1.2088
kNN	0.5000 / 432.7K / 0.7163	0.5000 / 432.7K / 0.7163	0.8561 / 1.50M / 1.0725	0.8561 / 1.50M / 1.0725	0.9814 / 11.63M / 1.1977
OmniRouter	0.5596 / 491.2K / 0.7759	0.5596 / 491.2K / 0.7759	0.6292 / 583.4K / 0.8455	0.8097 / 1.14M / 1.0261	$\infty$ / $\infty$ / $\infty$
RMSoftmax	0.5000 / 432.7K / 0.7163	0.5000 / 432.7K / 0.7163	0.9893 / 20.19M / 1.2056	0.9893 / 20.19M / 1.2056	0.9893 / 20.19M / 1.2056
TRouter	0.5000 / 432.7K / 0.7163	0.5000 / 432.7K / 0.7163	0.6441 / 607.9K / 0.8604	0.9879 / 17.95M / 1.2043	1.0000 / $\infty$ / $\infty$
UniRoute	0.5000 / 432.7K / 0.7163	0.5000 / 432.7K / 0.7163	1.0000 / $\infty$ / $\infty$	1.0000 / $\infty$ / $\infty$	1.0000 / $\infty$ / $\infty$

Appendix

Table A10: Performance across quality targets on Mixinstruct. Each entry reports CR / SA-BEP / SA-CR@1M; lower values are better.

Method	$0.90Q_{b}$	$0.925Q_{b}$	$0.95Q_{b}$	$0.975Q_{b}$	$1.00Q_{b}$
EmbedLLM	1.0000 / $\infty$ / $\infty$	1.0000 / $\infty$ / $\infty$	1.0000 / $\infty$ / $\infty$	1.0000 / $\infty$ / $\infty$	1.0000 / $\infty$ / $\infty$
kNN	0.5503 / 12.2K / 0.5557	0.7515 / 22.0K / 0.7569	0.9274 / 75.3K / 0.9329	1.0000 / $\infty$ / $\infty$	1.0000 / $\infty$ / $\infty$
OmniRouter	0.2911 / 7.7K / 0.2965	0.4700 / 10.3K / 0.4755	0.6581 / 16.0K / 0.6635	0.8215 / 30.6K / 0.8269	$\infty$ / $\infty$ / $\infty$
RMSoftmax	1.0350 / $\infty$ / $\infty$	1.0350 / $\infty$ / $\infty$	1.0350 / $\infty$ / $\infty$	1.0350 / $\infty$ / $\infty$	1.0350 / $\infty$ / $\infty$
TRouter	0.2103 / 6.9K / 0.2158	0.6730 / 16.7K / 0.6784	0.6730 / 16.7K / 0.6784	0.6730 / 16.7K / 0.6784	0.8396 / 34.1K / 0.8451
UniRoute	0.8173 / 29.9K / 0.8227	0.8173 / 29.9K / 0.8227	0.8173 / 29.9K / 0.8227	0.8173 / 29.9K / 0.8227	$\infty$ / $\infty$ / $\infty$

Appendix

Table A11: Performance across quality targets on MMRBench. Each entry reports CR / SA-BEP / SA-CR@1M; lower values are better.

Method	$0.90Q_{b}$	$0.925Q_{b}$	$0.95Q_{b}$	$0.975Q_{b}$	$1.00Q_{b}$
EmbedLLM	0.0742 / 24.8K / 0.0972	0.2648 / 31.3K / 0.2878	0.9296 / 326.4K / 0.9526	0.9296 / 326.4K / 0.9526	0.9296 / 326.4K / 0.9526
kNN	0.0655 / 24.6K / 0.0885	0.1443 / 26.9K / 0.1672	0.4101 / 39.0K / 0.4331	0.7455 / 90.3K / 0.7684	$\infty$ / $\infty$ / $\infty$
OmniRouter	0.1617 / 27.4K / 0.1847	0.1811 / 28.1K / 0.2040	0.3522 / 35.5K / 0.3752	0.6616 / 67.9K / 0.6846	$\infty$ / $\infty$ / $\infty$
RMSoftmax	0.9865 / 1.70M / 1.0095	0.9865 / 1.70M / 1.0095	0.9865 / 1.70M / 1.0095	0.9865 / 1.70M / 1.0095	0.9865 / 1.70M / 1.0095
TRouter	0.0901 / 25.3K / 0.1131	0.3151 / 33.5K / 0.3381	0.4450 / 41.4K / 0.4680	0.5987 / 57.3K / 0.6217	$\infty$ / $\infty$ / $\infty$
UniRoute	0.0741 / 24.8K / 0.0970	0.9327 / 341.3K / 0.9557	0.9327 / 341.3K / 0.9557	0.9327 / 341.3K / 0.9557	0.9327 / 341.3K / 0.9557

Appendix

Table A12: Performance across quality targets on RouterBench. Each entry reports CR / SA-BEP / SA-CR@1M; lower values are better.

	$P_{s}$ $\uparrow$	CR $\downarrow$	SA-BEP $\downarrow$		SA-CR@1M $\downarrow$
Benchmark			Cached	Fresh	Cached	Fresh
LLMRouterBench	0.6181	0.3161	10.8K	114.0K	0.3235	0.3941
Mixinstruct	0.7482	0.9695	6.13M	44.66M	1.1565	2.3329
MMRBench	0.7448	0.9052	57.6K	640.8K	0.9107	0.9660
RouterBench	0.8053	0.9355	329.2K	3.25M	0.9567	1.1449

Appendix

Table A13: Sensitivity of BaRP to fresh versus cached feedback accounting.

P_{s}

and CR are unchanged because caching affects only supervision expenditure. Lower SA-BEP and SA-CR@1M are better.

Benchmark	Arrival	$\|\Omega^{+}\|$	$\rho_{C,+}$	$P_{s,10}$	$P_{s,100}$	$P_{s,\mathrm{dense}}$	$\mathrm{CR}_{10}$	$\mathrm{CR}_{100}$	$\mathrm{CR}_{\mathrm{dense}}$	SA-BEP 10	SA-CR@1M 10
LLMRouterBench	Cheap	254	11.92%	0.6276	0.6273	0.6230	0.2296	0.2261	0.3564	4,921	0.2334
	Median	254	9.35%	0.6263	0.6340	0.6230	0.2650	0.2287	0.3564	5,610	0.2691
	Expensive	254	10.97%	0.6339	0.6305	0.6230	0.2606	0.2412	0.3564	3,887	0.2635
Mixinstruct	Cheap	2,200	10.00%	0.7486	0.7496	0.7495	0.9847	0.9516	0.9814	6,088,547	1.0779
	Median	2,200	10.00%	0.7498	0.7498	0.7495	0.9354	0.9354	0.9814	1,257,581	1.0166
	Expensive	2,200	10.00%	0.7498	0.7497	0.7495	0.9368	0.9569	0.9814	1,287,017	1.0181

Appendix

Table A14: Sparse model-pool expansion with

K=4

supervision for the existing model pool. Subscripts 10 and 100 denote using approximately 10% and 100% of the available training feedback for the arriving model.

\rho_{C,+}

is the acquisition cost of the 10% feedback relative to fully labeling the arriving model. Dense

P_{s}

and CR are independent per-metric optima over the fully supervised baselines and need not correspond to the same router. Higher

P_{s}

is better; lower

\rho_{C,+}

, CR, SA-BEP, and SA-CR@1M are better.

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