When Less Compute Is More: Adaptive Early Exit Improves Pretrained Outlier Detection

Tianyang Zhou; Leman Akoglu

cs.LGSep 26, 2026

When Less Compute Is More: Adaptive Early Exit Improves Pretrained Outlier Detection

Authors: Tianyang Zhou, Leman Akoglu

Organizations: Carnegie Mellon University

Abstract

Pretrained tabular foundation models process every dataset at a fixed depth, with inference costs growing with dataset size. To address this, we present the first study of depth-adaptive early-exit for pretrained outlier detection models. While early-exit is typically motivated by efficiency, we uncover a surprising benefit: exiting at the optimal intermediate layer can also improve detection performance on diverse real-world benchmarks by 4.7-7.3% on average, consistent across three distinct foundation models. First, we investigate the factors driving these gains, and identify a key mechanism: context pollution, i.e., the presence of outliers among in-context samples. Our analysis reveals that nearby in-context samples exert increasing influence on query predictions at greater depths, consistent with a retrieval-based view of these models. In effect, early-exit alleviates the adverse effects of retrieving accurate-yet-polluted neighbors, with gains of 13-21% when context pollution matches the natural outlier rate. Motivated by these findings, we pretrain a plug-in router to select a dataset-specific exit layer, using query outlier labels as privileged information available only during router training. The router operates post hoc, leaving the base model parameters and prediction head unchanged. Experiments on three large real-world benchmarks show that, on clean context, the router recovers up to 45% of the oracle gain with up to 1.8x speedup across three pretrained backbones, with larger gains as context pollution increases.

Figures & tables

Figure 1: OD performance when exiting at a fixed layer under clean (–) and polluted (- -) context; context pollution hinders performance across different pretrained models and benchmarks. Per-dataset oracle early-exit boosts performance for both clean (

\bigstar

) and especially polluted ( P ) context.

Figure 2: Hidden pollution in two official real-world splits: (left) t-SNE of the model input, polluting context rows circled; (right) layer-wise AUROC ( OutFormer ) with and without them.

Figure 3: Oracle gain over full depth, grouped by which third of the depth the best exit layer falls in (square-root scaled y-axis); bars give the share of datasets per group.

Backbone	OddBench (690)		OvRBench (755)		ADBench (47)
OutFormer	$+8.3$ $\to$	$+24.0\;(3{\times}10^{-48})$	$+6.6$ $\to$	$+18.1\;(2{\times}10^{-56})$	$+4.0$ $\to$	$+21.4\;(2{\times}10^{-8})$
TACTIC	$+8.1$ $\to$	$+18.5\;(3{\times}10^{-48})$	$+6.2$ $\to$	$+15.0\;(2{\times}10^{-44})$	$+4.6$ $\to$	$+7.0\;(0.086)$
ICLAD	$+5.2$ $\to$	$+15.4\;(4{\times}10^{-60})$	$+4.4$ $\to$	$+11.6\;(2{\times}10^{-48})$	$+2.0$ $\to$	$+6.4\;(10^{-5})$

Table 1: Oracle performance gains (%): Clean

\to

Polluted;

p

-values in parentheses.

Figure 4: Fixed-layer and oracle performance of OutFormer as held-out context pollution increases;

\bigstar

depicts the oracle on clean context and P the oracle under pollution.

Figure 5: Effect of one planted sibling per targeted outlier ( OutFormer ): (left) Detection of targeted outliers declines with depth; (middle) Outlier scores increasingly rely on the sibling with depth; (right) Sibling influence and AUROC loss rise together, showing a strong correlation of 0.99.

Figure 6: Left: rOUT reads the frozen backbone layer by layer, stops at layer

k

, and returns the best computed layer

j\leq k

. Right: deeper layers can lean on polluted context that masks query outliers; thus, rOUT stops early and looks back; query labels are training-only privileged information.

		OddBench				OvRBench				ADBench
Backbone	Method	PFLOPs	Time (s)	Speedup ( $\uparrow$ )	AUROC gain ( $\uparrow$ )	PFLOPs	Time (s)	Speedup ( $\uparrow$ )	AUROC gain ( $\uparrow$ )	PFLOPs	Time (s)	Speedup ( $\uparrow$ )	AUROC gain ( $\uparrow$ )
OutFormer ( $L{=}10$ )	Full depth	2.29	111	–	–	3.07	146	–	–	0.21	9.9	–	–
OutFormer ( $L{=}10$ )	rOUT	1.21	69.0	1.61 $\times$	+2.8%	1.48	82.4	1.77 $\times$	+3.0%	0.11	6.0	1.64 $\times$	+0.9%
ICLAD ( $L{=}12$ )	Full depth	2.04	70.6	–	–	2.64	89.9	–	–	0.18	6.4	–	–
ICLAD ( $L{=}12$ )	rOUT	1.25	53.1	1.33 $\times$	+0.8%	1.36	56.7	1.59 $\times$	+1.4%	0.087	3.8	1.67 $\times$	+0.0%
TACTIC ( $L{=}12$ )	Full depth	2.71	183	–	–	3.64	246	–	–	0.25	17.0	–	–

Table 3: Total compute and inference time (s) for each benchmark on one H100 GPU, full depth vs. rOUT (router calls and extra features included).

Figure 7: rOUT under near-duplicate pollution improves over Full depth while exiting early ( OutFormer ).

Held-out Pollution	OddBench				OvRBench				ADBench
Method	Clean	1:4	1:2	1:1	Clean	1:4	1:2	1:1	Clean	1:4	1:2	1:1
Oracle	.802	.757	.741	.719	.756	.706	.684	.657	.887	.841	.813	.773
Best fixed layer	.746	.666	.637	.606	.713	.634	.601	.563	.854	.771	.726	.666
Full depth	.740	.646	.616	.580	.709	.625	.594	.556	.853	.760	.704	.636
Half depth	.745	.651	.622	.587	.707	.623	.594	.562	.843	.747	.694	.626
LA-Entropy	.709	.657	.639	.612	.680	.627	.606	.579	.770	.723	.706	.689

Table 4: rOUT under held-out pollution of the context ( OutFormer ). Columns give the context-to-query outlier ratio; at a 1:1 ratio, the context pollution rate is close to the dataset’s natural outlier rate (median 0.8–1.2

\times

). The row under rOUT gives its relative AUROC gain over full depth.

OutFormer	Odd	OvR	AD
rOUT (full)	.761	.730	.860
w/o $\mathcal{L}_{\mathrm{layer}}$	$-$ 1.5 ∗∗∗	$-$ 2.0 ∗∗∗	$-$ 0.5
w/o $\mathcal{L}_{\mathrm{seq}}$	$-$ 0.6 ∗∗∗	$-$ 0.6 ∗∗	$-$ 0.4 ∗
w/o PI	$-$ 2.0 ∗∗∗	$-$ 2.4 ∗∗∗	$-$ 1.2 ∗
w/o layer attention	$-$ 0.8 ∗∗	$-$ 0.7 ∗∗∗	$-$ 1.2 ∗
w/o additional feat.	$-$ 1.7 ∗∗∗	$-$ 1.0 ∗∗∗	$-$ 1.5 ∗

Table 5: Ablations of rOUT : relative AUROC change (%) from the full router at matched depth; Full depth is the backbone without routing; one-sided Wilcoxon

{}^{*}p{<}.05

{}^{**}p{<}.01

{}^{***}p{<}.001

Appendix figures & tables21 assets

Supplementary material from the paper’s appendix.

Appendix

Feature	Count	Definition
Neighbor distances	21	For each $k\in\{1,2,5,10,20,50,100\}$ : log mean distance to the $k$ nearest context rows, its percentile among context rows, and log mean distance to the $k$ nearest other query rows
Distance to center	2	Standardized distance from the context mean, as a log value and as a percentile
Local density	4	For each $k\in\{5,20\}$ : local outlier factor and reverse-neighbor count
Boundary and duplicate	2	Fraction of features at the boundary of the quantile transform; whether an exact copy exists in the context

Appendix

Table 6: Geometric features of each row. Distances are Euclidean in the preprocessed input space; context rows are scored leave-one-out.

	Stage 1	Stage 2
	query labels shown	privileged inputs withdrawn
Epochs	12	8
Batch size (datasets)	64	32
Peak learning rate	$5\times 10^{-4}$	$2\times 10^{-4}$
Warmup steps	300	100
Training time (hours, one H100)

Appendix

Table 7: Training recipe of rOUT .

Benchmark	Tables	Datasets	rows	columns	anomalies
			Median per dataset
OddBench	real	690	7,504	6	9.2%
OvRBench	real	755	5,610	10	18.7%
ADBench	real	47	4,207	19	18.2%
SynBench	synthetic	800	3,202	49	25.8%

Appendix

Table 8: Benchmarks used in this study.

Prior family	Mechanisms	Context	Train	Val	Held out
OutFormer priors	GMM, Copula-Dependence, Copula-Probability, SCM-Structure, SCM-Measurement	clean polluted	17,298 51,878	1,000 1,000	999 1,000
TabPFN priors	class relabeling: One-vs-Rest, One-vs-One, Many-vs-Rest	clean polluted	71,376 70,943	2,037 2,016	2,081 2,048
		Total	211,495	6,053	6,128

Appendix

Table 9: Synthetic training corpus of rOUT .

Hyperparameter	Values	Description
Each dataset
$d$	$\mathrm{LogUniform}(2,100)$ , rounded	Dimensionality of data
$N$	$5{,}000$	Rows per dataset, context and query together
$n_{\mathrm{ctx}}$	$U\{500,750,\ldots,4{,}750\}$	Context rows, a uniform draw from the grid with step 250; inliers only in the clean pools
$n_{\mathrm{query}}$	$N-n_{\mathrm{ctx}}$	Query rows, 250 to 4,500, all of them scored
$r$	$\mathrm{Beta}(1,4)$ , capped at $0.5$	Outlier rate among the query rows

Appendix

Table 10: Sampling parameters of the training corpus.

Figure 8: Distribution of the oracle exit layer over datasets for every backbone and benchmark; the dashed line marks the mean.

Figure 9: Per-layer performance of ICLAD under held-out pollution .

Figure 10: Per-layer performance of TACTIC under held-out pollution .

Figure 11: Per-layer performance of OutFormer under near-duplicate pollution .

Figure 12: Per-layer performance of ICLAD under near-duplicate pollution .

Figure 13: Per-layer performance of TACTIC under near-duplicate pollution .

Figure 14: One planted sibling per targeted outlier ( ICLAD ).

Figure 15: One planted sibling per targeted outlier ( TACTIC , clean-context checkpoint).

Figure 16: LA-Entropy over its full threshold sweep, with exit heads averaged over three seeds, compared with fixed exit layers and rOUT ; its layer-specific heads lift TACTIC ’s early layers well above its fixed-layer curve.

Figure 17: Mean AUROC of fixed exit layers, Full depth, the Oracle , and rOUT for every backbone and benchmark (Table 2 ).

		OddBench		OvRBench		ADBench
Backbone		PFLOPs	Time (s)	PFLOPs	Time (s)	PFLOPs	Time (s)
OutFormer ( $L{=}10$ )	Full depth	2.29	111	3.07	146	0.21	9.9
	rOUT total	1.21	69.0	1.48	82.4	0.11	6.0
	backbone to exit	1.15	57.8	1.42	70.5	0.11	5.1
	router calls	0.051	7.6	0.055	8.1	0.003	0.5
	extra features	0.006	3.6	0.006	3.7	0.000	0.4

Appendix

Table 11: Full cost breakdown behind Table 3 : totals over each benchmark on one H100.

Figure 18: rOUT under held-out pollution improves over Full depth while exiting early ( OutFormer ).

Held-out Pollution		OddBench				OvRBench				ADBench
Backbone	Method	Clean	1:4	1:2	1:1	Clean	1:4	1:2	1:1	Clean	1:4	1:2	1:1
OutFormer ( $L{=}10$ )	Oracle	.802	.757	.741	.719	.756	.706	.684	.657	.887	.841	.813	.773
	Best fixed layer	.746	.666	.637	.606	.713	.634	.601	.563	.854	.771	.726	.666
	Full depth	.740	.646	.616	.580	.709	.625	.594	.556	.853	.760	.704	.636
	Half depth	.745	.651	.622	.587	.707	.623	.594	.562	.843	.747	.694	.626
	LA-Entropy	.709	.657	.639	.612	.680	.627	.606	.579	.770	.723	.706	.689

Appendix

Table 12: Held-out pollution on all three backbones. Columns give the context-to-query outlier ratio; the row under rOUT gives its relative AUROC gain over full depth.

Near-duplicate Pollution		OddBench				OvRBench				ADBench
Backbone	Method	Clean	1:4	1:2	1:1	Clean	1:4	1:2	1:1	Clean	1:4	1:2	1:1
OutFormer ( $L{=}10$ )	Oracle	.802	.764	.745	.720	.756	.708	.686	.656	.887	.851	.823	.785
	Best fixed layer	.746	.669	.639	.608	.713	.628	.593	.568	.854	.794	.745	.675
	Full depth	.740	.658	.628	.582	.709	.617	.581	.521	.853	.778	.723	.655
	Half depth	.745	.663	.633	.586	.707	.616	.581	.531	.843	.773	.729	.654
	LA-Entropy	.709	.658	.635	.600	.680	.623	.601	.565	.770	.737	.717	.688

Appendix

Table 13: Near-duplicate pollution on all three backbones. Columns give the context-to-query outlier ratio; the row under rOUT gives its relative AUROC gain over full depth.

Backbone: OutFormer		OvR	OvR	Syn
			Poll.
	rOUT (full)	.730	.580	.967
Data recipe	w/o polluted ctx.	+0.4%	$-$ 4.4% ∗∗∗	0.0%
	w/o TabPFN priors	$-$ 2.6% ∗∗∗	$-$ 0.5%	+0.1%
	w/o OutFormer priors	0.0%	+0.5%	$-$ 2.3% ∗∗∗

Appendix

Table 14: Training-data ablations of rOUT on OutFormer : relative AUROC change from the full router at matched depth. Poll. OvR averages both pollution modes at 1:1; shading marks each source’s largest loss; stars as in Table 5 .

		OddBench		OvRBench		ADBench		SynBench
Backbone	$\lambda$	AUROC	Layers	AUROC	Layers	AUROC	Layers	AUROC	Layers
OutFormer ( $L$ =10)	0.0025	.761	5.5	.730	5.4	.860	5.5	.967	5.7
OutFormer ( $L$ =10)	0	.761	8.8	.731	8.7	.856	9.2	.973	10.0
ICLAD ( $L$ =12)	0.0025	.776	6.8	.755	6.0	.891	6.0	.938	4.4
ICLAD ( $L$ =12)	0	.777	11.1	.757	11.0	.896	11.3	.939	11.7
TACTIC ( $L$ =12)	0.0025	.747	10.2	.729	10.0	.849	10.1	.838	9.0

Appendix

Table 15: Depth penalty ablations : rOUT with depth cost

\lambda=0.0025

(default) vs.

\lambda=0

: AUROC and mean exit layer.

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