Out of Support, Not Out of Reach: In-Context Learning with Unsupported Instructions and Composed Tasks

Xuhan Tong; Yuchen Zeng; Jiawei Zhang

cs.LGOct 16, 2026

Out of Support, Not Out of Reach: In-Context Learning with Unsupported Instructions and Composed Tasks

Authors: Xuhan Tong, Yuchen Zeng, Jiawei Zhang

Organizations: Department of Computer Sciences, University of Wisconsin–Madison. · Microsoft Research.

Abstract

In-Context Learning (ICL) enables pretrained LLMs to adapt to downstream tasks from a small set of input-output demonstrations, without parameter updates. Yet existing theory primarily explains ICL as identifying a latent task already represented in pretraining, leaving unclear whether ICL can succeed when either the instruction or the target task falls outside the pretraining support. We develop a theoretical framework under mild assumptions that separates these two forms of distribution shift. For target tasks represented in pretraining, we show that demonstrations can identify the task and reduce sensitivity to pretrain-supported prompt templates as their number increases. For an unsupported instruction, we show that under a fixed model-induced reference extension, demonstration evidence can dominate an $N$ -independent instruction shift while preserving the exponential identification rate. We then use the same ICL framework as an interpretive lens for long-horizon target tasks that are not represented in pretraining but admit a plan that decomposes them into pretraining-supported subtasks. Applying the supported-task analysis step by step yields a standard accumulated-error bound and clarifies how invoking familiar subtasks along an intermediate-state trajectory can avoid directly selecting the unsupported complete task. Together, our results characterize when demonstrations, instructions, and task decomposition can support ICL beyond the corresponding pretraining supports.

Figures & tables

#	Support	Appearance	Prompt format	Example
1	Supported	Single	${\bm{z}}=(i,(x_{n},y_{n})_{n=1}^{N},x_{\text{query}})$	A standard instruction stated once: “Classify ticket priority.”
2	Unsupported	Single	${\bm{z}}=(\tilde{i},(x_{n},y_{n})_{n=1}^{N},x_{\text{query}})$	Indian-English revert used once to mean “reply.”
3	Supported	Repeated	${\bm{z}}=\bigl((i,x_{n},y_{n})_{n=1}^{N},(i,x_{\text{query}})\bigr)$	A standard schema key, e.g. task: sentiment , repeated in every record.
4	Unsupported	Repeated	${\bm{z}}=\bigl((\tilde{i},x_{n},y_{n})_{n=1}^{N},(\tilde{i},x_{\text{query}})\bigr)$	One private API label, e.g. operation: Atlas , reused in every record.
5	Supported	Varying	${\bm{z}}=\bigl((i_{n},x_{n},y_{n})_{n=1}^{N},(i_{\text{query}},x_{\text{query}})\bigr)$	Retrieved examples using familiar paraphrases such as “sentiment” and “tone.”
6	Unsupported	Varying	${\bm{z}}=\bigl((\tilde{i}_{n},x_{n},y_{n})_{n=1}^{N},(\tilde{i_{\text{query}}},x_{\text{query}})\bigr)$	Multi-tenant examples carrying distinct private labels such as triage_acme and route_7G .

Table 1: Seven prompt formats classified by support and instruction placement. Supported means that the instruction was paired with the intended task with positive probability during pretraining; unsupported means that this probability is zero. Appendix D.2 gives complete examples.

Panel of Figure 1 — Figure 1: Empirical analogues of (a) consistent and (b) varying instruction mismatch, together with (c) the role of supported, stable subtasks in Theorem 4.2 and Section 4.2 . In (a)–(b), the fixed correct answer is 1345 ; Char “1” , Char “3” , Char “4” , and Char “5” denote the score-gradient traces for these correct digits at the first through fourth output positions, respectively.

Demonstration	Qwen3-30B-A3B	Qwen3-235B-A22B
Ambiguous / Identifying	$0.164\pm 0.039/\textbf{0.559$ \pm $0.074}$	$0.202\pm 0.037/\textbf{0.602$ \pm $0.043}$

Table 2: Test accuracy (

\uparrow

) of ICL under ambiguous and identifying demonstrations (

k=8

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

$N$	$r_{0}$	$d_{N}^{\mathrm{path}}$ ( $\delta_{\mathrm{emb}}=.25$ )	$d_{N}^{\mathrm{path}}$ ( $\delta_{\mathrm{emb}}=.50$ )
1	$0.088\pm 0.046$	$0.125\pm 0.081$	$0.253\pm 0.170$
2	$0.056\pm 0.078$	$0.069\pm 0.087$	$0.146\pm 0.151$
4	$0.064\pm 0.082$	$0.071\pm 0.080$	$0.139\pm 0.113$
8	$0.066\pm 0.059$	$0.099\pm 0.084$	$0.316\pm 0.175$
16	$0.035\pm 0.024$	$0.058\pm 0.038$	$0.182\pm 0.127$
32	$0.090\pm 0.102$	$0.115\pm 0.127$	$0.206\pm 0.179$

Appendix

Table 3: Model–reference discrepancy on local paths. Mean

\pm

standard deviation over five independently trained models. The empirical supported-anchor gap (denoted

r_{0}

in this table) is common to the endpoint-distance conditions.

$\delta_{\mathrm{emb}}$	$d_{1}^{\mathrm{path}}$	$d_{64}^{\mathrm{path}}$	$d_{64}^{\mathrm{path}}/d_{1}^{\mathrm{path}}$
$0.25$	$0.125\pm 0.081$	$0.099\pm 0.086$	$0.799$
$0.50$	$0.253\pm 0.170$	$0.192\pm 0.157$	$0.757$
$1.00$	$0.323\pm 0.270$	$0.459\pm 0.189$	$1.421$
$2.00$	$0.413\pm 0.229$	$0.543\pm 0.081$	$1.315$
$4.00$	$0.525\pm 0.057$	$0.562\pm 0.057$	$1.072$

Appendix

Table 4: Dependence on normalized endpoint distance. Pathwise model–reference gap at the smallest and largest tested demonstration counts. Mean

\pm

standard deviation over five independently trained models; the last column is the ratio of the two means.

Demo number	Ambiguous Acc (235B)	Identifying Acc (235B)	Ambiguous Acc (30B)	Identifying Acc (30B)
2	$0.265\pm 0.073$	$0.421\pm 0.080$	$0.122\pm 0.029$	$0.176\pm 0.062$
4	$0.267\pm 0.084$	$0.542\pm 0.048$	$0.151\pm 0.078$	$0.410\pm 0.059$
8	$0.202\pm 0.037$	$0.602\pm 0.043$	$0.164\pm 0.039$	$0.559\pm 0.074$

Appendix

Table 5: Improved Demonstration Experiment. Task: sports, repeated 5 times with different random seeds.

Demo number	Ambiguous Acc (235B)	Identifying Acc (235B)	Ambiguous Acc (30B)	Identifying Acc (30B)
2	$0.713\pm 0.074$	$0.940\pm 0.049$	$0.735\pm 0.083$	$0.908\pm 0.059$
4	$0.708\pm 0.096$	$0.931\pm 0.027$	$0.775\pm 0.037$	$0.950\pm 0.058$
8	$0.648\pm 0.008$	$0.845\pm 0.044$	$0.820\pm 0.027$	$0.990\pm 0.001$

Appendix

Table 6: Improved Demonstration Experiment. Task: People, repeated 5 times with different random seeds.

$\lambda$	Exact $\alpha(\lambda)$	Subtraction competitor	Multiplication competitor
$0.25$	$1.3533$	$1.3428\ [1.3298,1.3554]$	$1.3432\ [1.3298,1.3551]$
$0.50$	$2.7067$	$2.7144\ [2.6981,2.7292]$	$2.7134\ [2.6988,2.7315]$
$0.75$	$4.0600$	$4.0681\ [4.0533,4.0830]$	$4.0679\ [4.0531,4.0847]$
$1.00$	$5.4133$	$5.4141\ [5.4070,5.4212]$	$5.4127\ [5.4051,5.4198]$

Appendix

Table 7: Controlled evidence-rate recovery. Fitted slopes of cumulative evidence versus

N

; intervals are 95% bootstrap confidence intervals over prompts.

$\lambda$	$N=1$	$2$	$4$	$8$	$16$	$32$	$64$
$0.25$	$.5033$	$.3742$	$.2205$	$.07523$	$.01243$	$2.61\!\times\!10^{-4}$	$1.08\!\times\!10^{-8}$
$0.50$	$.3541$	$.1892$	$.04979$	$.003138$	$3.40\!\times\!10^{-4}$	$3.90\!\times\!10^{-16}$	$<10^{-16}$
$0.75$	$.1998$	$.05201$	$.005197$	$2.52\!\times\!10^{-4}$	$1.18\!\times\!10^{-13}$	$<10^{-16}$	$<10^{-16}$
$1.00$	$.04103$	$.006837$	$2.56\!\times\!10^{-4}$	$2.99\!\times\!10^{-13}$	$<10^{-16}$	$<10^{-16}$	$<10^{-16}$

Appendix

Table 8: Reference task-posterior error. Mean posterior mass on the two competing tasks over 2,000 prompts.

$\lambda$	$N=1$	$2$	$4$	$8$	$16$	$32$	$64$
$0.25$	$.6598$	$.6285$	$.5576$	$.4520$	$.2868$	$.1321$	$.0483$
$0.50$	$.6265$	$.5662$	$.4375$	$.2531$	$.1067$	$.0293$	$.0086$
$0.75$	$.5625$	$.4458$	$.2952$	$.1444$	$.0485$	$.0150$	$.0093$
$1.00$	$.4994$	$.3471$	$.1828$	$.0630$	$.0139$	$.0031$	$.0028$

Appendix

Table 9: Learned-Transformer task error. Means over five model seeds and 200 evaluation prompts per model and condition.

Training regime	$b_{N}$	95% CI	$b_{K}$	95% CI
Shared-template	$-0.000197$	$[-0.000794,\ 0.000395]$	$0.000030$	$[-0.000002,\ 0.000062]$
Hierarchical	$0.001244$	$[-0.001079,\ 0.003472]$	$0.000328$	$[0.000207,\ 0.000447]$
Independent-occurrence	$-0.001733$	$[-0.007019,\ 0.003765]$	$0.001230$	$[0.000960,\ 0.001518]$

Appendix

Table 10: Effective instruction sample size. Coefficients in

C_{\theta}(N,K)=a+b_{N}N+b_{K}K

for posterior-distilled models. Intervals are 95% cluster-bootstrap confidence intervals.

Panel of Figure 2 — Figure 2: Posterior confidence under instruction variation. ( Qwen-235B-A22B-Instruct ) (a) Equivalent instructions yield stable, high-confidence predictions. (b) Incorrect but consistent instructions allow confidence to recover as more demonstrations are added. (c) Incorrect and inconsistent instructions lead to persistently low and unstable confidence despite increasing context length.

Panel of Figure 3 — Figure 3: Posterior confidence under instruction variation. ( Qwen-30B-A3B-Instruct ) (a) Equivalent instructions yield stable, high-confidence predictions. (b) Incorrect but consistent instructions allow confidence to recover as more demonstrations are added. (c) Incorrect and inconsistent instructions lead to persistently low and unstable confidence despite increasing context length.

Panel of Figure 4 — Figure 4: Discrete gradients of log probability. ( Qwen-235B-A22B-Instruct ) (a) Equivalent instructions yield small and stable gradients. (b) Incorrect but consistent instructions show gradual stabilization. (c) Incorrect and inconsistent instructions produce large and irregular gradients.

Panel of Figure 5 — Figure 5: Discrete gradients of log probability. ( Qwen-30B-A3B-Instruct ) (a) Equivalent instructions yield small and stable gradients. (b) Incorrect but consistent instructions show gradual stabilization.(c) Incorrect and inconsistent instructions produce large and irregular gradients.

Panel of Figure 6 — Figure 6: Agent transition accuracy. Left: ID plan; right: OOD plan.

Panel of Figure 7 — Figure 7: Step-wise agent log-probability gradients. Left: ID plan; right: OOD plan.

$N$	Direct error	Supported-agent rollout	OOD-agent rollout	$\sum_{k=0}^{2}q_{k}(N)$
1	$0.964\pm 0.023$	$0.456\pm 0.302$	$0.877\pm 0.103$	$0.531\pm 0.385$
2	$0.973\pm 0.020$	$0.185\pm 0.170$	$0.865\pm 0.098$	$0.194\pm 0.178$
4	$0.974\pm 0.021$	$0.072\pm 0.091$	$0.877\pm 0.113$	$0.074\pm 0.093$
8	$0.966\pm 0.022$	$0.038\pm 0.048$	$0.900\pm 0.078$	$0.038\pm 0.048$
16	$0.963\pm 0.023$	$0.016\pm 0.018$	$0.904\pm 0.054$	$0.016\pm 0.018$
32	$0.966\pm 0.022$	$0.0088\pm 0.0035$	$0.892\pm 0.061$	$0.0088\pm 0.0035$

Appendix

Table 11: Controlled validation of supported agent composition. Probabilistic error under direct prediction and stochastic agent rollout, together with the sum of the supported-agent conditional transition errors. Values are mean

\pm

standard deviation over five models and twelve prompt seeds; every measurement includes all

17

query states.

Path	Region	$\widehat{L}^{\log p}$	$\widehat{L}^{\mathrm{err}}$	$\widehat{L}^{\mathrm{sub}}$
Symbolic mismatch	Anchor $[0,.25]$	$3.668$	$0.166$	$3.708$
	Middle $[.25,.75]$	$20.674$	$0.911$	$26.458$
	Endpoint $[.75,1]$	$7.518$	$0.076$	$7.431$
Synthetic soft-OOD	Anchor $[0,.25]$	$1.221$	$0.053$	$1.198$
	Middle $[.25,.75]$	$6.805$	$0.249$	$6.143$
	Endpoint $[.75,1]$	$3.447$	$0.112$	$2.962$

Appendix

Table 12: Empirical sensitivity along genuine Qwen3 soft-prompt paths. Values average the finite-difference sensitivities within each path region and then across the six demonstration counts. Both paths show a substantially smoother neighborhood around the supported anchor than the intermediate task-transition region.

In-Context Learning

In-context learning studies how models adapt to new tasks from examples or instructions supplied at inference time, without changing their weights.

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Mar 20, 2026J/K move · Enter open · S save

Demonstrations, CoT, and Prompting: A Theoretical Analysis of ICL

Xuhan Tong, Yuchen Zeng, Jiawei Zhang

Department of Computer Sciences, University of Wisconsin-Madison · Microsoft Research

In-Context Learning CoT Reasoning

Open paper AskPDF arXiv

In-Context Learning (ICL) enables pretrained LLMs to adapt to downstream tasks by conditioning on a small set of input-output demonstrations, without any parameter updates. Although there have been many theoretical efforts to explain how ICL works, most either rely on strong architectural or data assumptions, or fail to capture the impact of key practical factors such as demonstration selection, Chain-of-Thought (CoT) prompting, the number of demonstrations, and prompt templates. We address this gap by establishing a theoretical analysis of ICL under mild assumptions that links these design choices to generalization behavior. We derive an upper bound on the ICL test loss, showing that performance is governed by (i) the quality of selected demonstrations, quantified by Lipschitz constants of the ICL loss along paths connecting test prompts to pretraining samples, (ii) an intrinsic ICL capability of the pretrained model, and (iii) the degree of distribution shift. Within the same framework, we analyze CoT prompting as inducing a task decomposition and show that it is beneficial when demonstrations are well chosen at each substep and the resulting subtasks are easier to learn. Finally, we characterize how ICL performance sensitivity to prompt templates varies with the number of demonstrations. Together, our study shows that pretraining equips the model with the ability to generalize beyond observed tasks, while CoT enables the model to compose simpler subtasks into more complex ones, and demonstrations and instructions enable it to retrieve similar or complex tasks, including those that can be composed into more complex ones, jointly supporting generalization to unseen tasks. All theoretical insights are corroborated by experiments.

Abstract

In-Context Learning

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