Universality and Generalization of Causal Transformers Across Context Lengths

Takashi Furuya; Maarten V. de Hoop; Gabriel Peyré

cs.LGSep 28, 2026

Universality and Generalization of Causal Transformers Across Context Lengths

Authors: Takashi Furuya, Maarten V. de Hoop, Gabriel Peyré

Organizations: Doshisha University, RIKEN AIP · Rice University · CNRS, ENS, PSL Université

Abstract

Long contexts are central to modern transformer systems, but most expressivity results choose a different network for each fixed sequence length. We study whether one masked transformer can approximate causal token-to-token maps uniformly over sequences of arbitrary length sampling a fixed normalized horizon. To relate sampling resolutions, we model tokens by $α$ -Hölder sequences or, more generally, a common modulus of continuity. Our notion of continuity across resolutions characterizes the causal families admitting uniform approximation on these compact input classes by a single transformer with length-independent parameters. The result extends to the infinite-length mean-field limit, where tokens form continuous curves and masked attention becomes a causal time integral. For bounded regression with target maps satisfying a $β$ -smooth stability condition defined using regular test functions, quantitative approximation yields a generalization bound: exact empirical risk minimization over suitably sized bounded-weight transformers gives root mean-square prediction error $O((\log\log N/\log N)^{β/(d+2)})$ from $N$ iid labeled sequences. The bound holds at fixed confidence on the same sampling distribution, with $d$ the token dimension and no maximum-length factor. Finally, experiments on physical time series support the Hölder-regular token model at observed scales, with dataset-dependent fitted exponents, whereas text input embeddings provide a contrasting case. Native and dense sampling, shuffled controls, and refinement checks delimit this empirical regularity regime.

Figures & tables

Panel of Figure 1 — Figure 1: Representation-dependent finite-scale regularity. Worst-increment curves against normalized lag $h$ (horizontal axis): physical time in (a)–(c), token position in (d). Curves are normalized per window by the finest-lag increment before averaging; shading is one sample SD. This preserves each window’s slope and fixes the first point at one with zero SD. Raw-patch and shuffled controls probe patch geometry and temporal organization; L a T e X and Python are technical text controls. Dashed lines mark the fit cutoff; all panels have lower ordinate $0.85$ .

Domain	$m$	Input/dense	Native	Shuffled
Jena weather	8	$.335{\pm}.037$	$.167{\pm}.040$	$.005{\pm}.003$
Beijing PM 2.5	6	$.257{\pm}.085$	–	$.023{\pm}.016$
Appliances	4	$.087{\pm}.010$	$.020{\pm}.017$	$.015{\pm}.012$
Road traffic	3	$.303{\pm}.009$	–	$.035{\pm}.011$

Table 1: Pretrained content comparison. Fine-band

\widehat{\alpha}_{\infty}

, mean

\pm

sample SD across

m

windows. Physical rows use Bolt content; WikiText-103 uses BigBird input. Shuffling precedes patching. Native dashes denote insufficient scales; the text shuffled control is unmeasured.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

data type	representation	$d$	embedding map
Scalar series	Chronos-Bolt-tiny	256	Standardized 16-value patch plus 16 observation-mask entries, pretrained residual ReLU MLP; content only, evaluated at stride 16 and diagnostic stride one.
NLP	BigBird-RoBERTa	768	Learned word and token-type embeddings, linearly interpolated learned absolute position, pretrained input LayerNorm.

Appendix

Table 2: Pretrained front ends. Every row uses Equations ( 23 )– ( 24 );

d

is the increment dimension.

domain	dense stride 1	native stride 16	lags
Jena weather	$.157\pm.039$	$.167\pm.040$	7
Beijing PM2.5	–	–	2
Building appliances	$.020\pm.022$	$.020\pm.017$	4
Road traffic	–	–	2
Transformer oil temp.	$.133\pm.044$	$.175\pm.041$	4
Transformer load	$.025\pm.027$	$.043\pm.055$	4

Appendix

Table 3: Matched physical fine band for the same Bolt map. Mean

\widehat{\alpha}_{\infty}\pm

sample SD; the last column counts dyadic lags. Native cadence uses a subset of the dense starting positions.

domain	$N$	$d$	$m$	$\widehat{\alpha}_{\infty}$	$R^{2}$	$\alpha_{.99}$
Jena weather	32,769	1	8	$.253\pm.040$	.89	.421
Beijing PM2.5	1,025	1	6	$.139\pm.061$	.73	.365
Building appliances	4,097	1	4	$.041\pm.020$	.61	.101
Road traffic	1,025	1	3	$.172\pm.019$	.60	.178
Transformer oil temp.	4,097	1	4	$.134\pm.027$	.85	.291
Transformer load	4,097	1	4	$.129\pm.025$	.77	.176

Appendix

Table 4: Raw physical-signal worst-increment regressions.

N

is waveform length,

d

the number of measured coordinates, and

m

the number of windows. Slopes are mean

\pm

sample SD on

h\leq 1/32

;

R^{2}

is the mean fit quality, and

\alpha_{.99}

the mean 99th-percentile sensitivity slope.

Figure 2: Calibration of finite-band slopes. Twelve realizations per scenario, with mean

\pm

sample SD. The ordering of fractional-Brownian roughness is visible, but the positive slope of the jump control after patch encoding prevents interpreting the regression as an asymptotic certificate.

Figure 3: Exact candidate Hölder prefactors on nested restrictions. Each thin curve is a domain’s mean ratio

A_{n}(\alpha)/A_{65}(\alpha)

, for raw endpoints or dense Bolt content. Domains are not pooled as independent samples. The finite cadence and central subinterval limit the scope of this stability check.

Source	$\widehat{\alpha}_{\infty}$	$\widehat{\alpha}_{2}$	Shuffled $\widehat{\alpha}_{\infty}$
WikiText-103	$0.0122\pm 0.0012$	$0.0188\pm 0.0006$	–
AG News	$0.0126\pm 0.0018$	$0.0191\pm 0.0003$	–
IMDb	$0.0130\pm 0.0008$	$0.0180\pm 0.0001$	–
L a T e X control	$0.0119\pm 0.0012$	$0.0148\pm 0.0004$	–
Shakespeare drama	$0.0106\pm 0.0018$	$0.0173\pm 0.0002$	$0.0109\pm 0.0017$
KJV Bible	$0.0115\pm 0.0005$	$0.0178\pm 0.0001$	$0.0117\pm 0.0024$

Appendix

Table 5: The text screen does not reveal a more regular input regime. Mean

\pm

sample SD on the fixed fine band, using identical full-input BigBird embeddings. Shuffled controls are measured for all three new sources; dashes denote unmeasured controls at this long context. The original L a T e X stream is a repeated technical control, as described above.

Figure 4: BigBird’s multiscale geometry varies with depth. All depths are overlaid on the same log–log axes, using the same eight windows per source. Curves are ratios of mean worst increments as in ( 242 ); purple and yellow mark the input and final block. The dotted line is the fixed fine-band cutoff $h=1/32$ . Intermediate blocks have steeper finite-band curves, but the gain does not persist to the final block.

BigBird (12 bidirectional blocks)
Source	Input	Block 8	Final
WikiText-103	$.0130\pm.0016$	$.0308\pm.0027$	$.0023\pm.0060$
AG News	$.0141\pm.0015$	$.0306\pm.0042$	$.0039\pm.0082$
IMDb	$.0143\pm.0014$	$.0279\pm.0037$	$.0106\pm.0082$
L a T e X control	$.0143\pm.0019$	$.0320\pm.0043$	$.0140\pm.0086$
DistilGPT2 (6 causal blocks)

Appendix

Table 6: Effective maximum-increment slopes across depth. Mean

\pm

sample SD across eight matched natural-order windows per source, on

h\leq 1/32

. BigBird’s final state is block 12; DistilGPT2’s is the output after the separate final LayerNorm. SD is descriptive, not a confidence interval: windows may share documents or repeated content.

Figure 5: Depth, token order, and the choice of increment statistic. Mean fine-band slopes at every block for BigBird and causal DistilGPT2. Solid lines use natural order, dashed lines use shuffled inputs; shading is one sample SD for natural-order windows. Top: maximum increments. Bottom: RMS increments, a weaker statistic. DistilGPT2’s

+\mathrm{LN}

point applies the final normalization to block six. Architectures, tokenizers, and context lengths differ, so cross-model differences do not isolate an effect of masking.

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Shi Chen, Zhengjiang Lin, Kaizhao Liu, Philippe Rigollet

Transformer Architectures Flow Map Wasserstein Distance Chaos Propagation

Open paper AskPDF arXiv

We develop a quantitative statistical theory of transformers in the large-context regime by adopting the abstraction of contextual flow maps (CFMs): dynamical systems that evolve a distinguished token in the presence of a contextual measure across a stack of attention blocks. Within this framework, the finite-context model approximates an idealized infinite-context system in which the contextual measure is replaced by its underlying population, so that the context length

n

becomes a statistical resource. Exploiting the McKean--Vlasov structure of the dynamics and the classical machinery of propagation of chaos, we establish a forward bound controlling the deviation between the finite- and infinite-context CFMs uniformly along depth, and a backward bound controlling the deviation between the corresponding training trajectories uniformly across iterations of online gradient descent. Both bounds achieve the optimal Wasserstein rate

n^{-1/d}

for general CFMs and parametric rate

n^{-1/2}

for a restricted class of CFMs that includes transformers as a special case. The analysis rests on a new Eulerian adjoint formulation of the loss gradient and stability estimates for the resulting forward--adjoint system, both of which may be of independent interest.

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