Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Authors: Yijun Dong
Yicheng Li
Yunai Li
Jason D. Lee
Qi Lei
Venue: ICML 2025 poster (2025)
Keywords: Weak-to-Strong generalization
Intrinsic dimension
Finetuning
Primary area: theory->learning_theory
TL;DR: A theoretical analysis of W2S from an intrinsic dimension perspective, unveiling the role of teacher-student discrepancies and the sample complexities.

Abstract: Weak-to-strong (W2S) generalization is a type of finetuning (FT) where a strong (large) student model is trained on pseudo-labels generated by a weak teacher. Surprisingly, W2S FT often outperforms the weak teacher. We seek to understand this phenomenon through the observation that FT often occurs in intrinsically low-dimensional spaces. Leveraging the low intrinsic dimensionality of FT, we analyze W2S in the ridgeless regression setting from a variance reduction perspective. For a strong student-weak teacher pair with sufficiently expressive low-dimensional feature subspaces $\mathcal{V}_s, \mathcal{V}_w$, we provide an exact characterization of the variance that dominates the generalization error of W2S. This unveils a virtue of discrepancy between the strong and weak models in W2S: the variance of the weak teacher is inherited by the strong student in $\mathcal{V}_s \cap \mathcal{V}_w$, while reduced by a factor of $\mathrm{dim}(\mathcal{V}_s)/N$ in the subspace of discrepancy $\mathcal{V}_w \setminus \mathcal{V}_s$ with $N$ pseudo-labels for W2S. Our analysis further casts light on the sample complexities and the scaling of performance gap recovery in W2S. The analysis is supported by experiments on synthetic regression problems, as well as real vision and NLP tasks.

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Yijun Dong'
Yicheng Li
Yunai Li
Jason D. Lee Qi
Abstract
of a weak teacher model. W2S is related to various learning paradigms like co-training (Blum & Mitchell, 1998), self- training (Scudder, 1965), knowledge distillation (Hinton, 201 and self-distillation (Zhang et al., 2019; 2021), yet being critically dissimilar.
Weak-to-strong (W2S) generalization is a type of finetuning (FT) where strong (large) student model is trained on pseudo-labels generated by a weak teacher. Surprisingly. W2S FT often outper- forms the weak teacher. We seek to understand this phenomenon through the observation that FT often occurs in intrinsically low-dimensional spaces. Leveraging the low intrinsic dimensional- ity of FT, we analyze W2S in the ridgeless regres- sion setting from a variance reduction perspec- tive. For a strong student-weak teacher pair with sufficiently expressive low-dimensional feature subspaces Vs: Vw. we provide an exact character- ization of the variance that dominates the gener- alization eπror of W2S. This unveils virtue of discrepancy between the strong and weak mod- els in W2S: the variance of the weak teacher is inherited by the strong student in Va O V.r. while reduced by a factor of
Formalizing the discrepancy between the student and the teacher in their model capacities is essential for understand- ing W2S. Most existing theories for W2S treat model ca- pacity as an absolute notion with respect to the downstream task, e.g. the weak teacher lacks the robustness to perturba- tion (Lang et al., 2024; Shin et al., 2024) or the ability to fit the target function (Ildiz et al., 2024; Wu Sahai, 2024). Nevertheless, empirical observations suggest W2S models also surpass weak models' performance less challenging tasks (Burns et al., 2024), where the weak teacher has suf- ficient capacity to achieve good performance. This gap of understanding motivates some natural questions:
Why W2S happens when both the teacher and student have sufficient capacities for the downstream task?
in the subspace of discrepancy V \ V. with N pseudo-labels for W2S. Our analysis further casts light the sam- ple complexities and the scaling of performance gap recovery in W2S. The analysis is supported by experiments on synthetic regression problems, as well as real vision and NLP tasks.
What affects W2S generalization beyond the absolute notion
of model capacity?
To answer the above questions, we analyze W2S general- ization through the lens of intrinsic dimension beyond the absolute notion of model capacity. We develop theoretical framework that incorporates student-teacher correlation, pro- viding a more nuanced explanation of when and why W2S model surpasses the weak teacher's performance.
1. Introduction
As the capabilities of modern machine learning models grow and exceed human performance in many domains, an emerging problem is whether it would be possible to align the strong superhuman models with weaker supervi- sors such as human feedback. The weak-to-strong (W2S) framework introduced in (Burns et al., 2024) feasible analogy for this problem, inquiring how much capacity of strong student model can be evoked under the supervision
Our framework is built finetuning (FT): (i) FT tends to fall in the kernel regime (Ja- cot et al., 2018; Wei et al., 2022; Malladi et al., 2023); and (ii) for downstream task. relevant features in a stronger pretrained model tend to concentrate a subspace of lower dimension, known as the intrinsic dimension, even when FT is highly overparametrized (Aghajanyan al., 2021). Leveraging these properties, we cast FT as a ridgeless re- gression problem
inspiring observations on
'Nww York University *Shanghai Jiaotong University 'Princeton University. Correspondence to: Yijun Dong @yd1319@nyu.cduu Qi Lci <q1518@nyu.coe
subgaussian features. In particular, we consider two subspaces V., V,r C Rdd of low dimen- sions da dw < d that encapsulate relevant features in the strong student and weak teacher, respectively. The "absolute" model capacities are measured from two aspects: (i) the in- trinsic dimensions d, dn that quantify the representation
Proceedings of the nnnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
"complexity" and (ii) the approximation errors Ps < Pw that quantify the representation "accuracy" of the strong and weak models, respectively. In addition, the student-teacher correlation is measured by alignment between the strong and weak feature subspaces through their canonical angles (see Appendix D), dsyw = Σ cos( (Vv,, Vw)) such that darw E [0, minn,,, J.
Our main contributions are summarized as follows:
We introduce theoretical framework for W2S based on the low intrinsic dimensions of FT, where we characterize model capacities from three aspects: approximation errors for "accuracy". intrinsic dimensions for "complexity", and student-teacher correlation for "alignment" (Section 2).
We provide a generalization analysis for W2S with an exact characterization of the variance under Gaussian feature assumption, unveiling the virtue of discrepancy between the student and teacher in W2S (Section 3.1).
This framework reveals the roles of low intrinsic dimensions and student- teacchr correlation in W2S. Decomposing the W2S generalization error into variance and bias, the bias is due to the approximation errors, Pa: Pw, which are low when both student and teacher have sufficient capabilities; whereas the variance comes from noise in the labeled sam- ples for finetuning the weak teacher. When finetuning the strong student with N Z de pseudo-labels generated by a weak teacher supervisedly finetuned with 2 dw noisy labels, the variance of W2S is proportional to:
We investigate the relative W2S performance in terms of performance gap recovery (PGR) (Burns et al., 2024) and outperforming ratio (OPR) compared to the strong baseline model supervisedly finetuned with labels. A case study provides insights into the scaling of PGR and OPR with respect to the sample sizes n, N and sample complexities in W2S (Section 3.2).
dghww
d
d danw
N
1.1. Related works
Var. in V. n. V
W2S
Var. in Va V/
In this section, review literature directly related to W2S and intrinsic dimension, while deferring detailed discussions on other related topics to Appendix A.
Specifically, the student mimics variance of the weak teacher in the overlapped feature subspace V. 7 Vw but reduces the variance by a factor of d,/N in the discrepancy be- tween Va and V., Compared to the weak teacher variance that scales as d,,//,, W2S happens (i.e. the student outper- forms its weak teacher) with sufficient sample sizes n, NN when: (i) the strong student has a lower intrinsic dimension, de < du (as empirically observed in (Aghajanyan et al., 2021) NLP tasks), or (ii) the student-teacher correlation is low, daww < d This unveils the benefit of discrepancy between the teacher and student features for W2S:
W2S alignment: emergence & growing influence. W2S generalization was first introduced by (Burns et al., 2024), offering a promising avenue for aligning superhuman mod- els. A rapidly expanding body of work has empirically validated this phenomenon across diverse tasks in vision and language models since then. Guo al. (2024); Liu & Alahi (2024) propose loss functions and multi-teacher algorithms. Guo & Yang (2024); Yang et al. (2024b) re- fine training data to improve W2S alignment, while Li et al. (2024); Sun et al. (2024) use weak models for data filtering and reranking. In contrast, Yang et al. (2024a) highlight the issue of W2S deception, where strong models superficially align with weak teachers but fail in new or conflicting cases. This calls for theoretical understanding of the mechanism behind W2S generalization and better strategies to mitigate misalignment. Coinciding with our theoretical findings, the strong negative correlation between model similarity and W2S performance is empirically observed in the concurrent work (Goel al., 2025) in extensive experiments.
In the variance-donen dominated regime, W2S comes from variance reduction in the discrepancy of weak teacher features from strong student features.
To provide intuitions for such variance reduction, let's con- sider V. and V., with large discrepancy as two distinct as- pects of downstream task that both provide sufficient in- formation. For example, to classify the brand of car in image, one can use either the simple information in the logo (strong features V. with a lower intrinsic dimension d,) or the complex information in the design (weak features V., with a higher intrinsic dimension ). In a high--dimensioal feature space, V₂ and V.,, that encode irrelevant information are likely almost orthogonal, leading to a small danu- Then, the error of weak teacher induced by noise in the n labeled samples is only correlated to the design features in V. but almost independent of the logo features in V., Therefore, the error in weak supervision can be viewed as independent label noise for the student. With an intrinsic dimension of dg. the generalization error of student induced by such independent noise vanishes at a rate of (dg/N).
Theoretical perspectives on W2S generalization. Exist- ing theories on W2S interpret the difference between strong and weak models in terms of the quality of their represen- tations (from the bias perspective in our context). Lang et al. (2024) study W2S in classification through the lens of neighborhood expansion (Wei et al., 2020; Cai et al., 2021) where model capacity is interpreted as the robustness to perturbation. Within this framework, Shin et al. (2024) highlights the importance of data selection in W2S while
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Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
proposing metrics and algorithms for data selection in W2S. In the same classification setting, Somerstep et al. (2024) takes transfer learning perspective and highlights the limi- tation of naive FT in W2s. Wu & Sahai (2024) take a benign overfitting (Bartlett et al., 2020; Muthukumar et al., 2021) perspective and show the asymptotic transition between W2S generalization and random guessing. For regression tasks, Charikar al. (2024) reveals the connection between W2S gain and misfit error of the strong student weak pseudo-labels. Ildiz et al. (2024) treats W2S as special case of knowledge distillation, showing its limitation in terms of improving the data scaling law (Spigler et al., 2020; Bahri et al., 2024). We consider a similar ridgeless regres- sion setting as Ildiz et al. (2024) but from a fundamentally different aspect- variance reduction. This offers a fresh take on the roles of intrinsic dimension and student-teacher correlation in W2S. Parallel to this work, Medvedev et al. (2025); Yao et al. (2025) analyze W2S in the context of early stopping and loss functions, respectively.
if - (g) and f ( Also, we denote / = o(g) or f/9 (1) if lim f(z)//()) = 0.
2. Problem setup
In this section, we cast FT' as a ridgeless regression problem. The setup is introduced in three parts: model capacity, FT algorithms, and metrics for W2S performance.
Consider the problem of learning an unknown data distribu- tion Dff+)= X x y - [0. 1] (where X' is set and ) C R) associated with downstream task characterized by an un- known ground truth function fa : X + R. Every sample (x, y) ~ D(f=) satisfies y = f=(x + where 2 ~ (0 o²) is an independent Gaussian label noise. Let D: : X - [0, 1] be the marginal distribution over X. We assume that f. is bounded: [x)] < for x D almost surely (under normalization without loss of generality).
2.1. Measures for model capacity
Model capacity is a key notion in W2S that distinguishes the weak and strong models. Intuitively, a stronger model is capable of representing downstream task D(f=) more ac- curately and efficiently. We formalize such "accuracy" and "complexity" through the notions of intrinsic dimensions and FT approximation errors, as introduced below.
Intrinsic dimension. There has been prevailing empiri- cal and theoretical evidence that natural high-dimensional systems often exhibit low-dimensional structures (Udell & Townsend, 2019). The concept of intrinsic dimension has been widely studied in manifold learning (Tenenbaum et al., 2000), dimensionality reduction (Van der Maaten & Hinton, 2008), and representation learning (Bengio et al., 2013). In the context of neural network training. Li et al. (2018)
Consider two pretrained models, a weak model @n. and a strong model e.. that output features X' Rd.
propose a method to measure the intrinsic dimension of the objective landscape based on the type transforms (Johnson, 1984). This offers a structural perspective on task complexity. which is largely absent from prior W2S studies. Aghajanyan et al. (2021) investigate the intrinsic dimensions of FT, showing that FT over large mod- els usually has surprisingly low intrinsic dimensions, while good pretraining tends to reduce the intrinsic dimension. Our work extends these insights by linking the intrinsic dimension to W2S, decomposing generalization error into bias and variance, and building upon findings from Yang et al. (2020); Amari et al. (2020) on variance-dominated risks in learning from noisy labels.
Assumption 2.1 "Sub-gaussing features). For D, as- sume both (x) and (,, are zero-mean sub-gaussian random vectors with Bl (x)] = [x) = Oz. and (x) (x) = Ew.. El (x) Σ.
Approximation errors measure the model capacity from the "accuracy" perspective: how accurately can the downstream task D(f=) be represented by the pretrained features of Φ and &w over the population.
Definition 2.2 (FT approximation error). Given D(J+), let the FT approximation errors of o, and 6.. be
Ps min Ex-D [[0. (x) T e fx
BER
1.2. Notations
Pvo min Ex~D [lo (x) To))oo-00 fx)))
Given any n E Z4, we denote [n] = {1, [1,...,,]] Let be the n-th canonical basis of conformable dimension; I, is the n x n identity matrix; and O,, 1n E Rn are vectors with all zeroes and ones. For any distribution p and n E Z4. let p" & fi p as the n-fold product distribution of p. sam- pling which yields i.i.d. samples from p. For any matrix pseudoinverse. We adapt the standard asymptotic notations: for any func-
nn.
OER
such that Ps. Pru E (0, 1] (given PixwD(lf.(x)) < 1] = 1 by assumption). We assume both Ps and Pio are small compared to label noise: Ps + Pie < g²; while the stronger model o. has lower FT approximation error: Ps < Pw-
A E Raxd, let Af be the Moore-
Notice that FT approximation error is different from approx- imation error of the full model. Precisely, Definition 2.2 quantifies the approximation error of finetuning the pre- trained model, whose dynamics (Wei et al., 2022; Malladi et al., 2023) fall in the kernel regime (Jacot et al., 2018).
tions f.g: : R. + R.,. we write f = O (g) or f 2 g if there exists some constant C > 0 such that f(x) & Cg(x) for all x E R f = (g) or f 2 g if = 0 (f): =
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Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Since feature learning is limited in the kernel regime (Wood- worth et al., 2020), low FT approximation error requires the pretrained features o, and dw to provide an expressive set of features for the downstream task (ff.).
2.2. W2S and supervised finetuning
With the task D(f.) and models ds. Pio specified, we are ready to formalize the data and algorithms for FT.
We consider two sample sets drawn i.i.d. from D(f=): small labeled set S [x, Wili E [n]] ~ D(J+)" n and large sample set S [x,, Dx)l [N]] ~ D(S.)* where the labels y4 are inaccessible, denoting the unlabeled part as S₂ = fxa]i i E [N]] The goal is to learn function f :x R using S and Sr that generalizes well to D(f.).
In addition to "accuracy", a strong model ought to be able to represent a downstream task concisely. We quantity such "complexity" through intrinsic dimension the mini- mum dimension of a feature subspace that can represent the downstream task D(f.) accurately. In light of the ubiqui- tous observations on low intrinsic dimensions of FT (Agha- janyan et al., 2021), we introduce a common assumption for FT (Xia et al., 2024; Dong et al., 2024c) that the pretrained features of Φe, &w are concentrated in diweeniiiiilll sub- spaces, as formalized below.
For 8, let $ w
[ow (S.)..... Pu (x,) )T $
(x,( ... (X,)) E RRuxd be the weak and strong fea- tures with associated labels ÿ [T,.... F..] E R. Anal- ogously for S, let t ~ [ ) ...., Hn (xN)] D. [x1)...... (xN)) E Raxd be the weak and strong features with unknown labels y = l. UNN E R.,, For conciseness of notations, we introduce a mild regularity assumption on the ranks of these feature matrices.
Definition 2.3 (Intrinsic dimensions). Let rank (E,)
and dw rank(E.,) be the intrinsic dimensions of os and 6w. Assume low intrinsic dimensions': dg, dw < d.
Moreover, Aghajanyan et al. (2021) observed that the stronger pretrained models tend to have lower intrinsic di- mensions, i.e. we often have d, < d in practice.
Assumption 2.6 (Sufficient finetuning data). Assume $ and S are sufficiently large such that rank ( w) rank (P..) du and rank ).) = rank 2 = d, almost surely?.
Beyond the absolute notion of model capacity in terms of intrinsic dimensions and FT approximation errors, we intro- duce a relative measure for the similarity between weak and strong models- - the correlation dimension characterizing the overlap between feature subspaces of d, and ow.
Given regularization hyperparameters aws aw2s agg ag
0, we consider the following FT algorithms:
(a) Weak teacher model fur (x) &u (x) o,. is super- visedly finetuned over S:
Definition 2.4 (Correlation dimension). Consider spectral decompositions Σ, V,A, VT and E,, A. V where A₂ ddxx xd and A Re are diagonal ma- trices with positive eigenvalues in decreasing order; while V. E Rdxd, and V E [gxx consist of the correspond- ing orthonormal eigenvectors. Let dy V V. F be the correlation dimension between o. and &. w such that
1 n1
x - 31/ + aw 10122 (I)
0 E = argmin BER2
(b) W2S model fx2s(x) = (x(x) Ow is finetuned over the strong feature o, through S. and their pseudo- labels generated by the weak teacher model:
0 & darw < min {d,, ).
Ow2e argmin dERU
1|0.00 0 00 + aw |00
N
Remark 2.5 (Extension to general FT). While we focus on learning D(f+) via linear probing over @w and d., since the finetuning dynamics fall approximately in the kernel regime (Wei et al., 2022; Malladi et al., 2023), the linear probing analysis naturally extends to general FT. Precisely, let (-(aa)))) : X + R and (-10,) : X - R be the pre- trained weak and strong models, where d is the number of finetunable parameters. By denoting die (x)= = (x(xxx)(0x) and d. (x) = Vo (x(O). the general FT process effec- tively reduces to linear probing over dhe and ds.
(2)
(c) Strong SFT model fa(x) = (x) is a strong baseline where the strong feature @g is supervisedly finetuned over the small labeled set directly:
1 5500 1::0-ylo
-51 as 110
0. = argmin - OER
(3)
n
T (d) Strong ceiling model fe(x) = xx( 0g is a refer- ence for the ceiling performance where &, is super- visedly finetuned over S U.S assuming access to the
1 In practice, Σ. and E.. usually admit fast-decaying eigen- values, but exactly low-rank. In this more realistic case, ridge regression with suitable choices of regularization hyperparameters intuitively performs "soft" truncation of the small singular values, effectively leading to low intrinsic dimensions da, d_ & d. For conciseness of the main message, we focus on the ideal case of exactly low-rank E, and E. in the main text, while deferring the ridge regression analysis for general E, and Eng to Appendix C.
AAssuming distributions of dx((x) and (x) are absolutely continuous with respect to the Lebesgue measure, for any n ? d and n > dg. rank (..) rank(# de and rank( rank( ( .) = d, almost surely (Vershynin, 2018 $3.3.1).
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Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
unknown labels y = [V1..... :1.....; yN]
W2S model sw2s and the strong ceiling model fe:
0- aargmmn
AERd
72
1
N
& + 0
$
2
+ a.
1
PGR ER(fw) ER(fw22)
(R( - (R))
(5)
(4)
In practice, ER(Sw2s) typically falls between ER(S.) and ( (Burns et al., 2024). Therefore, it usually holds that 0 & PGR & 1. higher PGR indicates better W2S generalization: the W2S model fw2s can recover more of the excess risk gap between the weak teacher fu and the strong ceiling model fe.
For any f with randomness from its training samples S ((f.))5r], let X be the unlabeled part of Sy, and let S, D(f.))ss) be test set. We measure the generalization error via the expected excess risk of f over Sn and St:
1 SS
(b) Outperforming ratio (OPR) compares excess risks of the strong baseline S& and the W2S model sw2s:
{f( = S.
(f(x) - fx(x))2
XES,
Notice that ER( = Var(f) + Bfas(f) can be decom-
OPR E (f((f
(6)
posed into variance and bias, where
A higher OPR implies better W2S generalization: fw2s outperforms f. when OPR > 1. This metric could be of interest in practice when the labeled sam- ples S are limited if OPR < 1, SFT the strong model over S would be better choice than W2S both in terms of generalization and computational efficiency.
1
Var(f) = S.
Σ(f(x) - Ix,, 1f(x)))
|st|
xES.
(fas( = 5.,
1
Σ XES.
(Es,ix, - [x)))
|
For clarity of the main message, we set S. S fa in (1), fw2s in (2), Sa in (3) for fair comparison, and S. = S US for fo in (4) for simplicity?.
3. Main results
In this section, we first analyze the generalization errors of W2S and its reference models in Section 3.1. Then in Sec- tion 3.2, we conduct case study on the W2S performance in terms of the metrics introduced in Section 2.3.
Remark 2.7 (Regularization prevents W2S from overfitting).
As pointed out in (Burns et al., 2024), suitable regularization is crucial to prevent W2S from overfitting the weak teacher. For overparametrized problems*, even without explicit reg- ularization, gradient descent implicitly biases toward the minimum 22-norm solutions in the kernel regime (Wood- worth et al., 2020), equivalent to solving Equations (1) to (4) with Ow Ww2s ag,, a. - O. Therefore, we focus on ridge- less regression here under the idealized intrinsic dimension assumption in Definition 2.3. In Appendix C, we extend our analysis to the more general scenario: when E,, Σn: are not exactly low-rank, a careful choice of Ow, awws > 0 brings a W2S generalization bound, Theorem C.4, that conveys the same message as Theorem 3.1 the ridgeless case.
3.1. Generalization errors
We start with the W2S model fw22(x) = (x()) Ow2s fine- tuned as in (1), (2) with both Ow Oww2 + O. For demon- stration purposes, we consider an idealized Gaussian feature case in the main text, where the variance of fw2s can be exactly characterized (instead of upper bounded)
Theorem 3.1 (W2S model (formally in B.1)). Assuming Assumptions 2.1 and 2.6 and (x) ~ ~N(Os,, ).). for > du + 1, (R(222) Var Var(fw2)) + (Bias(fwz)) satisfies
Var(fwa)) = n. du - donnu + de (du - donue) N
2.3. Metrics for W2S performance
Bias(fwas) & Bias(fw) + Ps & <((p)) + Ps.
In addition to the absolute generalization error of W2S, [f(fw)) we quantify the W2S performance of fw2s rela- tive to Jmi sa. and fa through the following metrics:
where the inequality for Bias(fwzs) is strict if (iasssf) O and d, < dny-
(a) Performance gap recovery (PGR) introduced in (Burns et al., 2024) measures the ratio between cess risk reductions from the weak teacher fu of the
Remark 3.2 (Discrepancy is virtue). Notice that Var(swas) consists of two terms. (a) In the overlapped subspace Range Σ, )rRannRRnee., with correlation dimension danans the variance o² d (n d 1) mimics that of the weak teacher, where more pseudo-labels N fail to re- duce the variance. (b) Whereas variance in the subspace
'The strong ceiling performance [R(ff) only serves a ref- erence in (5). irrelevant the rest of the analysis.
"While the feature dimension d can be either larger (over- parametrized) or smaller (underparametrizee)) than the sample sizes n, N, + N, with the low intrinsic dimensions de, d & d, Equations (1) to (4) are always underdetermined.
TThe analogous generalization bound holds up constants for sub-gaussian features Assumption 2.1, see Theorem B.2.
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Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
of discrepancy Range(Eu, \ Range(E, takes the form (d,//N) ddwww)/( du 1), reduced by a factor of d,/N and vanishing as N grows.
Proposition 3.5 (PGR and OPR lower bounds (B.3)). Given fw, Jwass Scs and Is in Theorem 3.1, Proposi- tion 3.3, and Corollary 3.4, under Assumptions 2.1 and 2.6, assuming ow (x) ~ (Os,, Ew) and (x(x) ~ (Os E,); with n = dw + 9 for some constant E N, we have
As reference, we also look into the weak teacher model fu (x) (x) Ow in (1) with aw - 0:
PGR 7 1 danw
d, du - -aaww
9. O(pw) + Ps
Proposition 3.3 (Weak teacher (B.2)). Under the setting of Theorem 3.1, ER(Sw)= = (Vr(sw)+ + Bias(sw) satisfies
dw
N
du
dw
o²
and OPR >
o²dn
Var(fw) dw
(iws(fw) S Pw.
(
n O(ww) + [ darw + (dh - danw)da/M q
dg
d r²
when (x) ~ (0a,, Iw)); Var(fw) S o'dw otherwise.
We recall from Section 2.3 that the larger PGR and OPR imply better W2S generalization. Then, a natural question hinted by Proposition 3.5 is how do the sample sizes n, N affect the W2S performance? ? The concrete answers to this question depend on the relative magnitude of the FT approx- imation errors and label noise, (Pw + ps)/
To measure the W2S performance in a relative sense, an- other two necessary references are the strong SFT base- line fx) (x) Te, in (3) and strong ceiling model fa(x) = (x) 0g in (4), with both as; Ole 0:
Corollary 3.4 (Strong SFT and ceiling). Under the setting of Theorem 3.1, ER(f.) Var(fa) + (Baas(f)) satisfies
Case I: negligible FT approximation error. In the ideal case where the FT approximation errors are negligible com- pared to label noise. (pw + p=)/o² - 0. Proposition 3.5 suggests better lower bounds for PGR. OPR as n, N in- crease:
d²d,
Var(fa) n1 d, Bfas(fa) & Ps:
when (x) ~ NOs,, E.); and VJr(J.) S g'de otherwise. ER(fe)= = Var(fe) + Bias(fe) satisfies
n
PGR > 1 darw + (dw denw )
d²d,
Var(Ja)= N + n² Bias(Ja) & Ps-
dry
OPR > n dho -1
d, denw + (dw darw)d Jde/
W2S in variance. Assuming P. Pr < o² (Defini- tion 2.2), variance dominates the generalization error. Theo- rem 3.1 and Proposition 3.3 suggest that W2S generalization generally occurs in variance, i.e. Var(Swa)) < Var(Jw), as long as the W2S FT sample size is reasonably large, N > ds. Mcanwhile, with a low correlation dimension dgww, W2S in variance is more pronounced, especially when N is much larger than d,.
Depending dwww, we have the following cases:
(a) When duaw > O, with sample sizes n. Z du and N 2 (dw/danw 1)d,, PGR > 1 - Oddsnw/ (ddnwr/ww)) and OPR > M(d./dam imply good W2S performance if dgww < min {dz, d.
(b) When daww 0, a labeled sample size of n 2 du leads PGR > 1 1- Odd/NN and OPR 2 ((N/dw), implying good w2S performance when N > max (dhe, d.).
W2S in bias. When the strong student has zero FT ap- proximation error, Ps = 0, as long as Bias(sw) > 0, and the strong student has a lower intrinsic dimension, d, < du. Theorem 3.1 and Proposition 3.3 further suggest that W2s also enjoys strictly lower bias than the weak teacher: (ias(fw)) < (
Case II: small non-negligible FT approximation error. In a more realistic scenario where 0 < (p. + pww/o² < 1 is small but non-negligible, the trade-off between variance and bias brings about a non-monotonic scaling of the PGR and OPR lower bounds with respect to n::
3.2. W2S performance: a case study
With the generalization analysis, we are ready to take a closer look at the W2S performance in terms of PGR and OPR defined in Section 2.3.
Corollary 3.6 (Non-monotonic scaling w.r.t. n. (B.4)). For conciseness, denote (wzz(()) = derw + (dw - (daww)da/NN and e (O(pw) + 0=) Under the same setting as Proposition 3.5, with n = dw + q + 1 for some q E N, the lower bound of PGR is maximized when g = O, where
Quantifying the advantage of W2S in bias requires further assumptions on the downstream task D(f.) and the covariance matrices E.. Σ., analogous to the settings in (Ildiz et al., 2024; Wu & Sahai, 2024), which deviating from focus variance but could be an interesting future direction.
PGR > 1 dwwa((N)
6

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Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
while the lower bound of OPR is maximized when q = v [dw + 1) (N)) e where
ance over bias (characterized by Pas (ey).). Each error bar reflects the standard deviation over 40 runs.
OPR 2dg (Viwaa(N) + e + +1)
-2
dy - 200, d, 100, d,s. 10, o² 02, Pr. 2.00
04, A 5.50 5.50
- 40000
n-000
10-2.
Such non-monotonic scaling for PGR with respect to n coincides with some empirical observations in (Burns et al., 2024) on NLP tasks. While the variance of fw2e in Theo- rem 3.1 decreases monotonically as n grows, so do those of the reference models fw. fa. and fc- With non-negligible FT approximation errors, as n increases, the PGR and OPR lower bounds decrease with the improvements in bias but increase with the improvements in variance. Therefore, the optimal for the lower bounds of PGR and OPR is de- termined by the trade-off between variance and bias.
10-3
Excoss rick
10
Xx10-
3x100
1000
2:00
3000
4000
1000
2000
3000
4000
W25 Bound
Bound
W25
SBsssllnee
Weak Bound
Bound
Weak
S-Celing
Figure I. Scaling for excess risks the synthetic regression task in variance-dominated regime with a low correlation dimension.
Assuming + Pv: < o² in Definition 2.2, we have < 1. Again, consider two cases depending on dgΛw:
du 200, 100, d..- 90, of le pv. 1.60 04, 5.5e
= 4000
n= 2000
(a) If dxyw > O, we have Aw2s(N), X daaw when N (dw//daww - 1) 1).s. implying large PGR and OPR when derw < min {da, d.
10-2
10"2
Exposs rikk
5:330
(b) If danu = O, we have (N) d./,/N, implying large PGR and OPR when N > max {dhn, da) ).
10
Kx100-
2000
4. Experiments
W25 Bound
Saaas Bound
W25
-+-
S-aasdine
Weak Bound
5: Bound
Weak
Salling
We conduct experiments to validate the theoretical findings on both synthetic and real tasks. In this section, we focus two illustrative settings: synthetic regression (Section 4.1) and real-worl image regression (Section 4.2). For brevity, we defer more experiments on image and sentiment classifi- cation tasks to Appendices E.2 and E.3, respectively.
Figure 2. Scaling for excess risks on the synthetic regression task in variance-domin dominated regime with a high correlation dimension.
dy 200, d, 100, d. = 90, of de 04, Pv 1.6e- 04, A 5.55 50555-05 - 4000
n- 2000
2
4.1. Synthetic regression
Excoss FSk
We start by grounding the theoretical framework introduced in Section 2 with synthetic regression tasks.
10-1
Setup. We concretize the downstream task D(Ja) as a re- gression problem over Gaussian features. Let f. Ed - R be linear function in high-dimensional feature space = 20, 000 of form fx( = x A 9. where A.- = (Xi, ... ,X)) is a diagonal matrix with a low rank d. 300 such that A; = i-1 for i & d, and X; otherwise; and O. E I&d is random unit vector. Every sample (x, y) D(f.) is generated by
1x10-
1000
2200
3000
3000
W25 Bound
Salace in Bound
W75
S-ieeiine SCcel no
What Bound
5-5e ling Bound
Week
Figure 3. Scaling for excess risks on the synthetic regression task when the variance is dominant, o² pa + pu.
NV(O,, La) and y (x) + with 2 ~ (0, o²). Given x, the as- sociated strong and weak features in Assumption 2.1 are generated by (x) E!/² X and &e (x) E1/2x, x, with in- trinsic dimensions d, = 100 and dnp = 200 such that Σ, = + duam
Scaling for generalization errors. Figures 1 to 3 show scaling for ER(/w2s) (W2S), ER(Sw) (Weak), ER(Sa (S- (S-Baseline), and ER(fe) (S-Ceiling) with respect to the sample sizes n, N. The dashes show theoretical predictions in Theorem 3.1, Proposition 3.3, and Corollary 3.4, consis- tent with the empirical measurements shown in the solid lines. In particular, we consider three cases:
Di=1 Tee and Σ.,- Lied. -d +1 eee.eTT T. For all synthetic experiments, we have Pa + Pw < 0.0004.
In the experiments, we vary danw to control the student- teacher correlation and o² to control the dominance of vari-
Figure 1: When variance dominates (02 = 0.01 >
Pw + Ps), with a low correlation dimension daww 10,
7

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Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Jws outperforms both Sw and fs for moderate and a large enough N. However, larger sample sizes do not necessarily lead to better W2S generalization in relative sense. For example, when keeps increasing, the strong baseline/ fa eventually outperforms fw2s-
UTKFace dataset (Zhang al., 2017). Each error bar in this section reflects standard deviation of 10 runs.
Dataset. UTKFaaee (Aligned & Cropped) (Zhang et al., 2017) consists of 23, 708 face images with age labels rang- ing from 0 to 116. We preprocess the images to 224 x 224 pixels and split the dataset into training and testing sets of sizes 20, 000 and 3, 708. Generalization errors are estimated with the mean squared error (MSE) over the test set.
Figure 2: When variance dominates, with high correla- tion dimension da = 90, fw2s still generalizes better than fw but fails outperform the strong baseline fa.
Figure 3: When the variance is low (not dominant, e.g. a² = 0.0004 R Ps + Pw), swzs can fail to outperform fw- This suggests that variance reduction is a key ad advantage of W2S over supervised FT.
Linear probing over pretrained features. We fix the strong student as CLIP ViT-B/32 (Radford et al., 2021) (CIIP--332 and vary the weak teacher among the ResNet series (He et al., 2016) (ResNet18, ResNet ResNet50, ResNet 101, ResNet152). We treat the backbones of these models (excluding the classification lay- ers) as Φs. &u and finetune them via linear probing. We use ridge regression with a small fixed regularization hyper- parameter Ow, Ow Ogg, ae = 10-6 6 close to the machine epsilon of single precision floating point numbers.
2000 d, 100, o²= 1e 02, 90,50, 10, Pa 2.6e P 5.5e 05
N- 4000
n= 2000
1.
0.8
0.6
0.2
1000
2000
3000
4000
1000
2030
1000
4000
Intrinsic dimension. The intrinsic dimensions dwo d, are measured based the empirical covariance matrices Ew, E, of the weak and strong features over the entire dataset (including training and testing). As mentioned in Footnote I, these covariances generally have fast-decaying eigenvalues (but not exactly low-rank) in practice, effec- tively leading to low intrinsic dimensions under ridge regres- sion. We estimate such low intrinsic dimensions as the min- imum rank for the best low-rank approximation of Ew, Σ, with a relative error in trace less than r 0.01.
1.50
1.25
1.00
0.75
2000
2000
3000
3000
M
90.0 Bound
10.0 Bound
Figure 4. Scaling for PGR and OPR under different d on the synthetic regression task in variance dominnttd regime.
Correlation dimension. The pretrained feature dimen- sions (or the finetunable parameter counts) of the weak and strong models can be different in practice (see Ap- pendix E. Table 1). We introduce an estimation for dsaw in this case. Consider the (truncated) spectral decompo- sitions [E = V.A, V and [Ewld = V. V of two empirical covariances with different feature dimen- sions D., D., such that V, E D. and V E P xd... consists of the top d,, dw orthonormal eigenvectors, respec- tively. We estimate the correlation dimension darw under different feature dimensions D. * Dw by matching the dimensions through a random unitary matrix (Vershynin, 2018) r E DRD,D.,; dyAw V rv w: 1| This pro- vides a good estimation for daaw because with low intrinsic dimensions max(d,, d < D., Du in practice, mild di- mension reduction through I well preserves the essential information in Vs, V
Scaling for PGR and OPR. Figure 4 show the scaling for PGR and OPR with respect to sample sizes n, N in the variance-dominated regime (with small non-negligible FT approximation errors), at three different correlation di- mensions da = 90, 50, 10. The solid and dashed lines represent the empirical measurements and lower bounds in (11), (12), respectively.
Coinciding with the theoretical predictions in Corol- lary 3.6 and the performance gaps between W2S and the references in Figure 1, we observe that the relative W2S performance in terms of PGR and OPR can degenerate as increases, while the larger N generally leads to better W2S generalization in the relative sense.
The lower correlation dimension deaw leads to higher PGR and OPR, i.e. larger discrepancy between the strong and weak features improves W2S generalization.
Discrepancies lead to better W2s. Figure 5 shows the scaling of PGR and OPR with respect to the sample sizes n, N for different weak teachers in the ResNet series with respect to fixed student, CL.P-B32. We first observe that
4.2. UTKFace regression
Beyond the synthetic regression, we investigate W2S on a real-world image regression task - age estimation on the
8

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Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
d_
199 (Restetti R0, 150 (Res 615 (Restie1101), 589 (Reslee112)).c m 643 (CUP-B321
level. In Figure 6, we show the scaling for PGR and OPR with respect the sample sizes n, N under different noise levels We observe that the relative W2S performance in terms of PGR and OPR improves as the noise level < increases. This provides empirical evidence that variance reduction is a key advantage of W2S over supervised FT, highlighting the importance of understanding the mecha- nisms of W2S in the variance-dominated regime.
0.75
#- 10000
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5. Limitations and future directions
2000
In this work, we introduce a theoretical framework for un- derstanding the mechanism of weak-to-strong (W2S) gener- alization in the -ominntdd regime where both the student and teacher have sufficient capacities for the down- stream task. Leveraging the low intrinsic dimensionality of finetuning (FT), we characterize model capacities from three perspectives: FT approximation errors for "accuracy". intrinsic dimensions for "complexity". and student-teacher correlation for "alignment". Our analysis shows that W2S generalization is driven by variance reduction in the discrep- ancy between the weak teacher and strong student features. This generalization analysis is followed by case study on the relative W2S performance in terms of performance gap recovery (PGR) and outperforming ratio (OPR). We show that while larger sample sizes imply better W2S generaliza- tion in an absolute sense, the relative W2S performance can degenerate as the sample size increases. Our results provide thcoretical insights into the choice of weak teachers and sample sizes in W2S pipelines.
Figure 5. Scaling for PGR and OPR of different weak teachers with a fixed strong student on UTKFace. The legends show the comparison between d, and ch..
the relative W2S performance in terms of PGR and OPR is closely related to the correlation dimension danw and the intrinsic dimensions d3, deg
When the strong student has lower intrinsic dimen- sion than the weak teacher (as widely observed in prac- tice (Aghajanyan et al., 2021)), i.e. d, < du, the relative W2S performance tends to be better than when d, > dw-
The relative W2S performance tends to be better when deww/da is lower, i.e. the larger discrepancy between weak and strong features leads to better W2S generaliza- tion.
Meanwhile, both PGR and OPR scale inversely with the labeled sample size n and exhibit diminishing return with respect to the increasing pscudolabel size N, consistent with the theoretical predictions in Corollary 3.6 and the synthetic experiments in Figure 4.
An interesting implication of our analysis is that the mech- anism of W2S may differ as the balance between variance and bias shifts. In the iitaace--domnntee regime studied in this work, W2S can benefit from a lower intrinsic dimension of the strong student due to the resulting variance reduction in the subspace of discrepancy from the weak teacher. In contrast, in the bias-dominated regime, the lower approxi- mation error of the strong student is generally brought by the larger "capacity" of the strong model corresponding to higher intrinsic dimension (Ildiz et al., 2024; Wu & Sahai, 2024). This calls for future studies on unified views and transitions between the two regimes, which will provide a more comprehensive understanding of W2S. Toward this goal, a limitation of our analysis is the quantification of the advantage of W2S in bias (see Footnote 6), which could be promising next step.
d, 522 (Rsseet5)) 4 - 443 [CUP 832), 9, 301.06
0.5
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3.70
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Injected lsbel nalen Eo
0.05
Ireected netie ç- 5-
Injected notel 30.5
0.0
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Reccted
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veected
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Figure 6. Scaling for PGR and OPR on UTKeaae with injected label noise: ys y4 + 6, where & ~ (O. i.d.d
A cknowledgements
The authors would like to thank Denny Wu for insightful discussions. The experiments are supported by the PLI com- puting cluster. YD acknowledges support of NYU Courant Instructorship. JDL acknowledges support of Open Philan- thropy, NSF IIS 2107304, NSF CCF 22 NSF CA-
Variance reduction is a key advantage of W2S. To in- vestigate the impact of variance on W2S generalization, we inject noise to the training label by y: + 3/ + S, where S ~ (0, s²) ii.dd, and 5 controls the injected labels noise

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Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
REER Award 21 2144994, and NSF CCF 2019844. This mate- rial is based upon work supported by the U.S. Department of Energy, Office of Science Energy Earthshot Initiative as part of the project "Learning reduced models under extreme data conditions for design and rapid decision-making in complex systems" under Award #DE-SC0024721.
sian process regression and classification. arXiv preprint arXiv:2304.0244, 2023.
Burns, C., Izmailov, P., Kirchner, J. H., Baker, B., Gao, L., Aschenbrenner, I.., Chen, Y., Ecoffet, A., Joglekar, M., Leike, J., et al. Weak-to-strong generalization: eliciting strong capabilities with weak supervision. In Proceed- ings of the 41st International Conference on Machine Learning, pp. 497 1 5012, 2024.
Impact Statement
This paper presents work whose goal is advance the field of machine learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
Cai, T., Gao, R., Lee, J., and Lei, Q. A theory of label propagation for subpopulation shift. In International Con- ference on Machine Learning, pp. 1170-1182. PMLR, 2021.
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Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Appendix
Contents
Additional related works
14
Proofs in Section 3
14
B.1
Proof of Theorem 3.1
15
B.2 Proof of Proposition 3.3 and Corollary 3.4
20
B.3 Proof of Proposition 3.5
21
B.4 Proof of Corollary 3.6
22
Ridge regression analysis
22
Canonical angles
28
Additional experiments
29
E.1
Additional experiments and details on UTKFace regression
29
E.2
Experiments on image classification
31
E.3
Experiments on sentiment classification
31
A. Additional related works
Knowledge distillation. Knowledge distillation (KD) (Hinton, 2015; Gou al., 2021) closely connected W2S generalization regarding the teacher-student setup, while W2S reverts the capacities of teacher and student in KD, In KD, strong teacher model guides weak student model to learn the teacher's 's knowledge. In contrast, W2S generalization occurs when strong student model surpasses weak teacher model under weak supervision. Phuong & Lampert (2019); Stanton et al. (2021); Ojha et al. (2023); Nagarajan et al. (2023); Dong et al. (2024b); Ildiz et al. (2024) conducted rigorous statistical analyses for the student's generalization from knowledge distillation. From the analysis perspective, a key difference between KD and W2S is that W2S is usually analyzed in the context finetuning since the notions of "weak" and "strong" are built upon pretraining. This finetuning perspective introduces distinct angles from KD for examining intrinsic dimension al., 2018) and student-teache correlation W2S.
Self-distillation and self-training. In contrast to W2S, which considers distinet student and teacher models, self- distillation (Zhang et al., 2019; 2021) and related paradigms such as Born-Again Networks (Furlanello al., 2018) use the same progressively refined architectures to iteratively distill knowledge from a "previous version" of the model. There have been extensive theoretical analyses toward understanding the mechanism behind self-distillation (Mobahi et al., 2020; Das Sanghavi, 2023; Borup & Andersen, 2023; Pareck et al., 2024).
Self-training (Scudder, 1965; Lee et al., 2013) a closely related method to self-distillation that takes a single model's confident predictions create pseudo-labels for unlabeled data and refines that model iteratively. Wei al. (2020); Oymak & Gulcu (2021); Frei al. (2022) provide theoretical insights into the generalization of self-training. In particular, Wei al. (2020) introduced a theoretical framework based on neighborhood expansion, which was later extended to various settings of weakly supervised learning, including domain adaptation (Cai et al., 2021), contrastive learning (Shen et al., 2022; Huang et al., 2021), consistency regularization (Yang et al., 2023; Dong et al., 2023), and recently weak-to-strong generalization (Lang et al., 2024; Shin et al., 2024).
B. Proofs in Section 3
With respect any sample size N, let
(n( = Ex-D= (1oo, (x) fxxx)
(n) Ex<D=Illow (X)o (X) fx(X) fx(X)13)
14

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
where db (X) and (x()) () n feature matrices; and [x( R is vector of the noiseless ground truth labels.
Lemma B.1. Given the FT approximation errors ps and pm in Definition 2.2, we have
(a( & nps and (n) & npw Vn E.
Proof of Lemma B.I. Let argmingeRt E F (x) T [x such that
E PxbDl((brx)) 0. - [.(x))²) = Pw-
Then, by observing that conditioned on X,
Φ J.(X) argmin
f.(X)
BER2
we have
(n) = EXND" [1 ()) - (x)]
Ex-D" [1 (x)
Ex~Dr [X]0. f. (X)11
n Ex-D [x0m(() [-(())
n Pw
The proof for (a(n) follows analogously.
B.1. Proof Theorem 3.1
Theorem B.2 (Formal restatement of Theorem 3.1). Consider fw2s(x) = xx(x)) Ow2s finetuned as in (1), (2) with both
Omny Ol22s + Under Assumptions 2.1 and 2.6, when n 2 M(ww), the excess risk ER(Sw2s) = Var(Jwas) + Bias(Swz) satisfies
0² Var(fw2)) Z (deree n1 donnw + de (dw danve))
o²
Bias(fwaa) < Bias(fw) + Ps O(pw) + Ps.
Moreover, when (x) (Os. Ew), for any n > du + 1, we have
Var(Jwze) dw (denu
d²
dg N ( denw))
+
Proof of Theorem 3.1 and Theorem B.2. We first observe that the solution of (1) Oig 0 is given by
θ = t T w (f. + 2):
where 7 ~ N (Nn,, Meanwhile, the solution of (2) as O'22s
+ is given by
ww + n. ', $ क5 2).
Then, the excess risk of fw2s be decomposed into variance and bias follows:
1
ER(ww)) E
Es..5
o 0
- f.
N
1
2
Ess, $
[b 1 m TI..
f.) +
$.
$
2
N
1
E
5
T. $
+ T T
B
Tb.
Φ T
T f.
f.
NN
N
2
Var(Swz)
Blas Swaz)
15

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Recall the spectral decomposition En w V Since (x [] E for each D, write (x) E// Y, where E I2d is an independent random vector that is zero-mean and isotropic (i.e. E[y]] 0a and E La). The same holds for >, V.A,V V and (x(x) >/// 7
Then, for S and S, there exist independent random matrices r ( [ E R.Nxd and r= (.,.... 7n T E R consisting of i.i.d. zero-mean isotropic rows such that
1/ 8
1 1/2
rE
rV V mn 1
DE TV. V T V $ rΣ - -/ rv.V
1/2
(7)
$,Σ, 1/2
Let T = IV RN and F IV E Rxdd throughout the proof.
Bias.
For the bias term, by observing that P. = o Φ is an N x N orthogonal projection, we can decompose the bias
term as
Bias = E S.
P.
@ f. 11
N Bs.
[I(TN -P.)
N
where Es, [I(TN P.) f.) Pa(NN by Definition 2.2, and
Pa(NN
Es, [IIN f.)
& Ps.
For the first term, since P. is an orthogonal projection, we have
p f. - -..))
$ w DI.. f. -.113]
s. S
P.
<
Bias(fw).
N
Notice that when Bias fm) > 0 and dg dny such that tb D f. & Range( (.)) almost surely, the inequality is strict, i.e., Bias(fw2s) < (wa)(f)) + Overall, we have
Bias & Bias(fw) t & ((ww) + Ps.
where the second inequality follows from Proposition 3.3.
Variance. For the variance term, we observe that
Var Var(fw2)
1
$1
S.S
2 26 (tt) }
E
$
tr
N
E
s,.s
$
tr
which implies
0²
Eg (x= Th D 20112) (22) Σ- 1/21/2 T.P.P. 4 1-12
Var(Swad N
tr
Es.
(8)
We observe that
Eg
1/
Eg
(v....o V.ll'
(T )/
V
Now, consider the following two cases for the feature distribution of ow (x)), corresponding the distribution of I and
16

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Gaussian features: In Theorem 3.1, assuming (w) (Od: Ew) such that F consists of i.i.d. Gaussian rows, have % ~ ; Notice that under the assumption n > du + 1, rank(T w) = du almost surely, and therefore T w is invertible.
Meanwhile, with f. ~ (0 lag) for all i E [m). (T) ~ W(Id;; follows the Wishart distribution (Wishart, 1928, Definition 3.4.1) with n degrees of freedom and scale matrix Ld_. Therefore, (w) W- (Id., n) follows the inverse Wishart distribution (Mardia al., 2024, $3.8), whose mean takes the form (Mardia et al., 2024, (3.8.3))
Eg
[T
Idw.
10
n1 dw
Then, have
3
Σ 1/2
2.
2 112)
1 dw
V w w.
V
Therefore, (8) implies
0²
Var (f22s]
tr V V...S Dw 1/2 x 2 2. 1/2 V(2
N n
dny
0²
tr [Es, V V.rr V
(9)
N
thys
Ir (Es. (.P.))
N
dw
Recall that P. = - Let r, rV, E Nxd, and we can write
P. ( n 1/2 q D. 1/2 (T.V.. r.r!..
Therefore, with r,, = rV., m and r, rV we can decompose
tr (Bs, [ = Es, [tr (T. (...T
Es. tr (V V. V V w...rrr
Es, [tr (V (Ia V V.
For the first term, since Iw. VVV. rV.. V. and r, TV.. the range of I w V V. is subspace of that of r, and therefore,
Es, [tr (V. V. r.rrt E.s. [ir
V... Es. tr I I V v.)] = =t [V] V Es. T V.V.).
Since Es, [TT... m
NIa., we have
Es, [tr V.V. VV N V.V.V.
- V 2
NN
For the second term, first observe that the space of V V.V.) is orthogonal to that of T, rV.. and therefore, P,, VT (Ia V.V. and I, are independent, which implies
Es. [ur (V (Ia V.V
-.l..v..V. V.V. (B
V.. r.r.))
17

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Since I' consists of independent isotropic rows, do rV, E BBxx and T rV RRxxe, which implies
dg N
In
and
[ m = Ia..
Then, we have
Es, [rr (V (Ia V. V.V.)
= tr
( -. B(.T)
dg
(1.
ds
(V (I. V.V. V. (TT.))
N
de
N
-la
N
d. (dho dsnw).
Combining the two terms, we have
tr (Es, [T.P. Ndanw + do(dw derw).
Then, by (9), the variance is exactly characterized by
Var(Sw2s) = 0²
Ndsww + dywdw danw))
N
dw -1
dw -1 (onn + dg
N (dw danue)
(b) Sub-gaussian features: Relaxing the Gaussian feature assumption, when I consists of i.idd sub-gaussian random vectors that are zero-mean and isotropic (i.e. [6 Od... and [[(T:T]] = Ia with 2 Rd(d))) Lemma B.3 implies that
(T.Tw)] & x0 Iaw.
E3
and therefore,
tl 1 x0(1) V V
Eş
Σ 1/-1/22 | T E 1/2)
1112)
Then, via an analogous argument as (9), (8) implies that
0 ( (Es, [.P.K.))
Var Var(fw2e)
(10)
We observe that in the analysis of the Gaussian feature case, the characterization
tr (Es, [.P..W]] (N - da)daaw + dydw
does not involve the Gaussianity of T and therefore holds for general subgaussian features. This leads to an upper bound on the variance:
0 1 (Ndara + do (dw, danw))
Var(fw2s)
N
0²
darw + de de (dro denu))
N
18

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Lemma B.3 (Adapting (Vershynin, 2010) Theorem 5.39). Let F ",..., In consist i.i.d. sub- gaussian random vectors that zero-mean and isotropic (i.e. Blai]] Od,, and B17 L When n 2 (dd we have
: be an dy matrix whose rows
E (FTT.) <o (()
where (2(-) and O(-) suppresses constants that depend only the sub-gaussian
1Rlll
supve8ta-1 pupp31 (Bl1% /rrl /p. independent of n, dw-
w- Leveraging (Vershynin, 2010) Theorem 5.39, we notice that for n > ())dw,,, there exist constants 61, 02 > 0 that depend only on the sub-gaussian norm || lly. such that
Proof of Lemma B.3. Let miin (T (T be the smallest singular value of ff
Pr m C wT) < - - -1) & exp (-czt²).
Therefore, we have
Vī In-cavda - (vi-
Pr
< exp
-02
mmin
I w
Notice that for any non-negative random variable Z with a cumulative density function Fz(z).
E 7 / zdFz(=) =dFz(z)=--
-- f 2d Fz(2))
= [12(1- F2(2))10 + =/(0 (1- (-F2(z)) de
10
1/0 PriZ 2/
Therefore, have
es ddn - V21 2)
//-
0
<
exp
-02
vn
de.
Omin
T w
)
0
Let to 1/ (Vn- ) such that vn - 6i Vi
and
f
t0
Vvv the vt)
exp
-02
Vn
& to
19

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Then, have
-02 Vn - vi - vi)
exp
mmiul (TTw)
Vn 6i the Vt) -it)) de
<
to
exp
12
l0
/dw
(Vn c v ()- -3
to
exp (-c21?)
du
2
-02 Vi ai v vdw 2 ui² (1 u)- du
lo + (VT-- - 0I dw
exp
2
V% exp (-2(n²) (1 u) du
+
(Vn vvvdw) 1
(Vn- - -vvddw
(Vn (Vn--avdu) chv
When 3 M(dw), we have vn c vdy 12( (l())), and therefore.
/
<0
E
< E
min
B.2. Proof Proposition 3.3 and Corollary 3.4
Proof of Proposition 3.3 and Corollary 3.4. The excess risk of the finetuned weak teacher fm. = (x) (x) can be expressed as
S 1 N
ER(fw) = Es
fw - f-
where f, = (xI)..... [xx)] E N; and we recall that ow = 1o (x11....... ow (XN)). Notice that the randomness of i. comes from the SFT samples S ~ D(f
Observe that the solution of (1) aw - 0 is given by O., T where ÿ F. + 2 is the noisy label vector with 2 ~ ~N,0 I2)) Therefore, with the randomness over S D(f.) )", we have
ER(S2) E
w
f.
2
1
f..) 1.)113]
E
N
JE
E
w L
f.
Var fm)
Bias fw.)
For bias, leveraging Lemma B.3 and by the definition of finetuning capacity (see Definition 2.2), we have
Bias (fw)
=E
N
m
-Eg
ExnD
(dxx(x)
$1..
fx(x))
I]]
Peo pwr (n)
<E3
$
f.
F
n
2
20

---

o²ds Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension P=(n)
We observe that (wa I For variance, observe that
" < Pw by Lemma B.1.
Var(Sw)
R
= ir En $ 22 ($) (8))))
N
=o² E
ir ((
1125 $b,, D 12)1) = =0²1B |ir
(T
Assuming (x) ~ (N,, En,, we then have E ir (-)
du and n-dw-1
0² ddd
Var(fw)
n2 dw
Meanwhile, assuming (x) is sub-gaussian, Lemma B.3 implies that E tr (ITTTw)-)) 2 0 (dw) and therefore,
Var(fw) S dddu
The analogous results hold for the strong SFT model: ER(fs) = Bias(f)) + Var(f.) satisfies
4 I - f. x Px(n)
Bias(
< Ps:
and
d²ds Var(Js) S
Var(Js) n1 dg
n
when (x (O,, E.) or (x) is sub-gaussian, respectively.
For the strong ceiling model fe, the variance, Var(fe), follows from the standard generalization analysis for fixed design linear regression, and the bias, Bias(fe), follows directly from the definition of p=(n + N).
B.3. Proof of Proposition 3.5
Proof of Proposition 3.5. Noticing that with rank($ = du and rank( = rank (b) = d, almost surely, the excess risks of fw, Js. Jc are characterized exactly in Proposition 3.3 and Corollary 3.4, and ER( (w2s) is upper bounded by Theorem 3.1. Therefore, by directly plugging in the excess risks to the definitions of PGR and OPR, we have
PGR ER(Sw) (R(Sw2s)
ER(Sw) ER(fo) j2d
daww (dw danm)) (Bias(fw) + pa)
7 dn Bias(fw)
n dn - de N + +
(11)
dd
1 dnn Bias(fw)
Ps)
dd danw (di danw)da//N
dd dw -
ps)
+ +0(pw)
11 dey
n - dw 1
n2
and
OPR ER(S.)
o2d, /
(22dhnu (dhw danu )dg/N
+ O(pw) + +p.)
>
(12)
ER(Sw2))
71 d2
n - dy -
21

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
When taking dw q + 1 for small constant o N, observe that
ddd
Maaw + (dw - dedwwdd//
/
dddu
PGR
Ps
+ 0 (pw)
dw
n dwy
n
dw
dy
dyho
dg du - danw Ps
du
+0
N
x2
dw
dynw
dg due darw
Ps
0
Pw
du
N
q
dena
de dw
denw
O(pw) ) + Pa
dw
N
dw
duy
and
o²da
22 @2dern + (dw danw)da/N + (pww) Ps)
OPR
d2 -1
dw
dg
daww + (dw )anw)d//NN O(pw) t
n
0²
nt
darw + (dwr - damw)da/N
((pw) +
-1
q
d2
d
σ²
B.4. Proof Corollary 3.6 Proof of Corollary 3.6. Recall the notations introduced for conciseness:
(N2 danw + (dw darw) d2
O(pw) + P=
N
0
o²
Then, the lower bounds for PGR and OPR in Proposition 3.5 can be expressed in terms of d22((N) and as
(22 du du
PGR >
and
(wzz((N) dg
du +
(N) dw +1 q e dg )
OPR
+
dg e
dg
Both lower bound of PGR. is maximized when the last term is minimized, i.e., q = 0, which yields PGR > de) Meanwhile, the lower bound of OPR is maximized when the last two terms in the expressions that involve are minimized, which is achieved when = V(dw + 1) (N)/o. Substituting the optimal q back into the expression yields the best lower bound for OPR:
dy
(N2 d,
dn + d,
dn + 1 dwza (N) 2
OPR
+
d,
d
-2
(N)
du +
+ Ve
dg
dg
C. Ridge regression analysis
In this section, we investigate the more realistic scenario where the weak and strong feature covariances are not exactly low-rank but admit small number of dominating eigenvalues.
22

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Concretely, consider the same data distribution (x,y) D({.) with y fx)x for some independent Gaussian label noise (O, d²) and unknown ground truth function f. X R in Section 2. Under the same sub-gaussian feature assumption in Assumption 2.1, we adapt Definitions 2.3 and 2.4 to the ridge regression setting follows.
Assumption C.1 (Data distribution). Let d, X' + Rd and ow : X + Rd be the strong and weak pretrained models that take x D and output pretrained features o. (x),, (x) E R respectively.
Ground truth: Assume f. can expressed as a linear function over an unknown ground truth feature Φ. : X + Rd such that (+) (-) 0., for some fixed 0. RR
111 Sub-gaussian features (Assumption 2.1): Let ow (x), O (x). (x)) be zero-mean sub- gabb-gausiin random vectors with E 1 (x)] = B (x) = Eió (x) = Od.
(x) 1 E
(x), Es, (x) = E.,
and Σ; 1/2 (x(x satisfies for canonical basis vectors e; E I2 that
E
e Σ, 02)x(
= =3
(e.g., when o. (x) is Gaussian random vector)?. We assume without loss of generality that these features are roughly normalized, i..., |well2 x 1, |_ = 1, and ||>||| > 1.
(ii) Low intrinsic dimension: Let E, and Ew both be positive-detinite with spectral decompositions Σ, V.A.V. and E,,- = -V.,A.. VT, lo where A,, E Rdxd are diagonal matrices with positive eigenvalues in decreasing order; while V, E dddd and V E Rddd are orthogonal matrices consisting of the corresponding orthonormal eigenvectors. The low intrinsic dimension of FT implies that A, = dia ... .A1) and A. diag(>". .... A consist of few dominating eigenvalues, while the rest of the eigenvalues are negligible, i.e., there exist da, du d such that Eisd, X (E,) and Eisde X tr w. Here,
tr (.) S de and tr(Ew) S dw
effectively measure the intrinsic dimensions of o, and ow.
Remark C.2 (Weak-strong similarity). In place of correlation dimension (Definition 2.4) in the ridgeless setting, for the ridge regression analysis, we measure the similarity between the weak and strong models directly through tr trC,,ww). Notice that
E.Ew) & min (tr((,)) |E tr( |E, 12 & min (tr(E,) tr(Ew))
In particular, when Σ, and Σ.r admit low intrinsic dimensions, tr Σ,,.) can be much smaller than min (tr(E,), tr ir(E.))) if their eigenvectors corresponding the dominating eigenvalues almost orthogonal.
Remark C.3 (FT approximation errors). assumptions Assumption Ci1 iii). the FT approximation errors in Definition 2.2 satisfy
worth noting that under the ground truth and positive-defintt covariance
min Ex-D [x) T fx())) 0 (when = E.,'..0.),
OERd
(13)
Pv: min EX~D [( (x) [x() = (when 0 = ==
OCRd
In place of Definition 2.2, with positive-definite covariances in Assumption C.1, we measure the alignment between the ground truth feature d, and the weak/strong feature du, d. through
2. エ
1/ 0 2w | 1/ /
"e make this assumption only for the conciseness of final results. The assumption on the fourth moment can be relaxed to any El(eT 1-1/2 (x))) x (N E R) at the cost of an additional term in the upper bound for variance:
Var(fwz) ((amn)(aa
(T.E... N
tt((diag( (()ddiiggg...))) E.)E..))
which has negligible impact on the sample complexity when large.
23

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Intuitively, for Σ, and E. with few dominating eigenvalues (Assumption C.I(ii)), 2a and ew are small if the eigensubspace associated with non-negligible eigenvalues of Σ, is fully covered by the eigensubspaces associated with the dominating eigenvalues of E, and Ew. respectively.
The W2S FT under ridge regression consists of two steps.
(a) First, the weak teacher (x) = Dw (x) ( On
supervisedly finetuned over S:
@w - 5/ $11 + aw 10112 Ow > 0.
v = argmin OER
(14)
(b) Then, the W2S model (x) (x) Ow2e is finetuned over the strong feature d, through the unlabeled samples Sg and their pseudo-labels generated by the weak teacher model:
Bw2s argmin BCR
2.0 & N
aw2s |10122
aw2s 0.
(15)
1/ 0.
|| and Os
|
-/
Theorem C.4 (W2S under ridge regression). Let Du
Under
Assumption C.I, the generalization error of W2S FT via ridge regression with fixed awr aw2s Var(Sw2)) + Bias([w2s), is upper bounded by
0. ER
(R(fw2))
((++N) N tr (E,Ew)) + N tr tr(2,) tr(2w))
Var(fu2)) < (a
(fas() & aw tio + Ow2alls
In particular, when taking
( (E,
En
2s
1/3
(E,t E.:) Ow
1/3
Ow
Ow2s
4nN
Li
4nN
the excess risk of W2S FT is upper bounded by
( o²
1 (1+
1
)) 1/3
ER(fw2)) < 4nN AnNe lalw
tr (E,Ew) + N tr tr(E.) tr tr(Enn)
N
Notice that for large enough N
tr (tr()) tr.
Theorem C.4 implies that
302 4nN lsww tr (E, Ew)
1/3
ER(fw2)) <
Theorem C.4 conveys a similar high-level intuition as in Theorem 3.1 regarding the effect of the weak-strong similarity on the generalization error of W2S FT. In particular, the larger discrepancy between o, and dw (corresponding to the smaller tr (Σ,E.)) leads to lower variance and therefore better W2S generalization.
Meanwhile, a key difference in W2S between the ridge and ridgeless settings (Theorem C.4 versus Theorem 3.1) is that the FT approximation errors in Theorem C.4, reflected by e. 1/25 and Lu ||_"" 1/2 compensated by larger sample sizes N and directly affect the sample complexity:
1// 2 113 can be
nN x 0² tr (E,E)) 2slw.
Such difference is result of optimizing the regularization hyperparameters aw, awss in ridge regression that control the variance-binc tradeoff.
Proof of Theorem C.4. We first formalize some useful facts on the features and labels as in (7). In particular, the sub- sub-gaussian assumption in Assumption Cii implies that for each D. the corresponding strong/weak feature Φ2 (x),, d (x) Td..
24

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
and the ground truth (x) with E(y] Oa and |8] y Id, Lee,
simultaneously characterized by an independent subgaussian random vector 7 Rd
Φ (x) T (x(x)= 7, fx(x) (x)) 0, =77
Then, for S and S, there exist independent random matrices r m,.... YN] E RRxxd and r , E B consisting of i..dd. zero-mean isotropic rows such that
i. = rE TAA.V
i rE 2 I A 1
= f. + f, = -E, 1/2 0.,
~~ NO,, IIIN),
(16)
y f. b,
f, r>''' 0., 2 7~ W(,,,
where r, rV,, r, rV mi and r,, rV
Varianee-bias decomposition. With aw (14) yields a weak teacher model far(x) (x) On with
-
Ow o 2w + a.d
& fs +
Then, the W2S model (x) = o, (x)
w2s is given by (15) with aw2s > 0:
ww2
(Φb ob NIs) & 2h
$ + + Now..NI.
Φ o u
② + + awdld)
T P. 8))
which implies
E s.. ww2s]] =00_ 4. + NIa)
$ D..
T + aw
1 T
Then, we can concretize the variance and bias terms as:
1
Var(fw2)) = E
Φ. (Owzs - ss (0w222)))
N
(17)
1
Eg
o
T + o t + awnld o. + Nw2
$ T
N
and
-002))) -Fa((]]
Bias(Sw2s)
R.s..
t E. Fss.s(002z -fa
(18)
12
E.S..
+ ( + aw2 NIz) -1 $ o0
$ t + awnld -1
$
TT -
Now, we are ready upper-bound the variance and bias terms separately.
Variance. Denote Ç = A
1/2 1/2
VT
T Th + aw
$ L. Z Rd, whose randomness comes from S only, independent
of S,. Then, the variance term (17) can be expressed
Var(fw2)) = Es..5
D Φ + +
NI.)
Φ &
N
1 N [Es.
TT T. ( Φb + a2 NL.)
Φ t. ( $. Nw2,
$
Eg [c
25

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Notice that
(b_ d. + NI
$ D. (D. Φb + -Ww2
N ② Φ. + +a2oti
$ Φ.
$ 4, + awzto
=
20 Id.
N
N
Therefore, we have
1
Var (ar()
N2 lis.
T T. B3 V Cc
20rw2n
1
Es
T FTrrrTT
B3 V VV.
2aw22
N2
Since Assumption C.I (ii) implies that
1 N2
(1+11
E. + 1 N Lr
Es.
TE
we then have
Var(fwa))
& 2erwer
(((++N) N
E. N (((x))aa) E5 V Sc [
1
Meanwhile, we observe that
Eş V
=Eg
1/2 T ~ + aw
$ 2
$ T v + aw
T
E!/²
1//2 Φ Eu + ad) =o
d v
1 u
d + awlls)
11/2
where
-1
$ T
+ aw
T T + aw T..
3
Id-
o1
2a
Therefore, we have
0² E...
Eg Vc VV]
11/2
la
11/
20
20
Overall, the variance of fwis can be upper bounded as
Var(fwz))
(( ((++
a²
NV tr tr,EEw) + tr (E(E)) br(Ew))
1
(19)
N
1/2 Recall from (18), the bias term (18) can be decomposed
Bias. Let g
w tr + n
f. Rd, whose randomness comes from S only, independent of S,.
-1
27
Bias(fw2s) = =E,,
p. ( TTT
2. + a12)
- 2w T + aw
$ $T..
= E...
2.
o v + NL)
2 T.
DIf. 2
+ +I1[IN - 2. E. E1112)
N
where by Lemma B.1 and (13)
(N(N)
Es,
[1 2.
1.
P
N
26

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Therefore, with 5 21/ T 1 + awnld)
+ TT we have
Bias( Bias(fw2s) E8 S
$ $ p. + VIa
-1 4 rg - F.
N
Recall that f. = T 0, and Φ
rE,
1/2
r s 1/2 V.. Then, we can express the bias term as
1
Bias fw2a) -1E s
r
[
r + N
re r
N
s
1
r
N
1
1!!20 -E) +r La - awT++a r & $1/
= E3 S
3
N
r
By the Woodbury matrix identity, we have
Ia T +
aw2,
N
- r [Ia
1 E,TTT)
(20)
aw22 N
Therefore, we have
Bias Bias(fw2s) = Ls,,
r
/20 0. -E) +T
La
aw2. N Ex r
E
(21)
N
Term
Term B
For Term A, notice that & = 11/ $ x + (ig
T implies
$ T.f. Σ!/20. 0, & -Z!/20 0. - E!/² 1 $ aannIs
=2.220 0, [ T +
aw a
r
TTTE!
122 0.
[ + (
aw
T E!
Ia +
E
where the last equality follows from Woodbury matrix identity as (20). Therefore,
1 r
2 - $ >>]
E 1/20. 0. - -5) 1/
E
S.
5
N
E
n
r
-F3
r
La
1 E
-1
E./2 0-
27
n
awn
Since
La
I
T
La
E
r
Rw >m
we have
1//2
27
Qz
x- 12-1g 1/2 0.0 x!/-
E Es..
3
r
-
tr
2
n
2
(22)
C'u
Σ
1/2 E 1/2 0
2
27

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
For Term B, leveraging Woodbury matrix identity in (20), we notice that
1
O'22s
E
sss..
La
E
&
S...
EE
O'
N
Olw2
E
Σg 1/2 1/
T annlo
S
Ow E S
E/ [ (TT a
r
0
Since T awn
T 3 Ldl we know that
E S.
3
N
la +
N
Ow22 N
Ow2s 2
Σ 1/2 1/2
1/1//2//22 0.
(23)
Combining (21), (22), and (23), we can upper bound the bias term as
Bias(sw2s) = Eg
S
N
T
Σ 12020 0.
+
Ia
+
aw2. N
E, T
E
Term
Term B
12
(24)
<2 Su
N r 02'20 -5) -2) 15
21E S..S
r
Ia
ET
N
Oi N
E01/2 1-1/2 12.
0. ||- + Ow22x
1タ 1 1/2
!200
&0w
Variance-hias tradeoff. Recall es = |E 1//2 0. 112 and Lw and (24), we have
1 1//22,1100000 0..13 from Remark C.3. Overall, by (19)
Var(fwz)) & a) (1+
tr [.) tr(Eww) +
1
N tr (E
N
?
Bias fw2s) & Ouw
E. 1/2
1/2 0.
+ aw2s
Σ 1-1/2 !
0
< awlw + aw2sls-
The upper bound the excess risk ER(fw2s
Var Var(fw2s) + Bias ) is minimized by taking
a² 2s
1/3 ( (1+ (E tr(E.) tr (Z..)
Oww
AnN
2
0² @u
((++))
1
1/3
(+/N) tr (E,E tr tr(E,)
tr(,tt((ww))) tr(Eww)
Ow2s
4nN
o3
which leads the optimal upper bound for the excess risk:
0²
(1+
1 111 N
1 tr ( + N Ir ir(E.) ((( 1/3
ER(Sw2s) <
eow
D. Canonical angles
In this section, review the concept of canonical angles between two subspaces that connect the formal definition of the correlation dimension d V V= I| in Definition 2.4 to the intuitive notion of the alignment between the corresponding subspaces V, and VI,. in the introduction: Σ cos( ((V,,, V..)) = |
28

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
Definition D.1 (Canonical angles (Golub Van Loan, 2013), adapting from (Dong et al., 2024a)). Let Ve, Vu Rd be two subspaces with dimensions dim (V,) dy and dim (V.)) d (assuming du 3 d, without loss of generality). The canonical angles 2 (Ve, Vw) diag (.1,..., Ma,) are d, angles that jointly measure the alignment between V, and Vw. defined recursively as follows:
u, v, - argmax u,v; Vi
s.t. u E (v. span {u. ) n ngo-1
E (V \ span (v, [v.ki=] n
cos(v: v. -i= 1.......
such that 0 < M .... < < 2/2.
Given two subspaces Ve: V. S Rd, let V, Raxd, and V. E Rdxdw be the matrices whose columns form orthonormal bases for Vs and Va., respectively. Then, the canonical angles <(v,, Vv)) are determined by the singular values of V.V. w (Björck & Golub, 1973, $3):
cos (os(,, Vv)) =i= Li.,,...,d
where (V. Vw) denotes the i-th singular value of V V.....
In particular, since V,, V. consist of orthonormal columns, the singular values of V V fall in [0, 1]. and therefore,
Σ cos( cos(< v.)) = [ V.n I| E [0, min {d, d
dara =
E. Additional experiments
E.1. Additional experiments and details on UTKFaae regression
This section provides some additional details and results for the UTKFaaee regression experiments in Section 4.2.
d w 194 (ResNet18), d, 443 (CLIP-B32), dsaw 167.64
N 10000
1000
W2S
10
103
Weak
S-Baseline
S-Ceiling
MSE
102
10
1000
1500
2000
2500
3000
5000
10000
15000
N
Figure 7. Scaling for MSE on UTKFace with CLIP-B32 as the strong student and ResNet 1 8 as the weak teacher
We summarize the relevant dimensionality in Table We observe the following:
The intrinsic dimensions of the pretrained features are significantly smaller than the ambiance feature dimensions, which is consistent with our theoretical analysis and the empirical observations in (Aghajanyan et al., 2021).
The correlation dimensions danw are considerably smaller than the corresponding intrinsic dimensions, indicating that the subspaces spanned by the weak and strong features are not aligned in practice. As shown Section 4.2. such discrepancies the weak and strong features facilitate W2S generalization.
29

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
d 522 (ResNet50), d, 443 (CLIP-B32), Aaww 301.06
N 10000
1000
W25
103
10
Weak
S-Baseline
S-Ceiling
MSE
102
102
1000
1500
2000
2500
3000
5000
10000
15000
N
Figure 8. Scaling for MSE on UTKFace with CLIP-B32 as the strong student and ResNet 50 as the weak teacher
dw 589 (ResNet152), d, 443 (CLIP-B32), ds AW 339.90
N = 10000
1000
103
W25
3
Weak
S-Baseline
S-Ceiling
MSE
102
102
1000
1500
2000
2500
3000
5000
10000
15000
N
Figure 9. Scaling for MSE on UTKFaae with CLIP-B32 as the strong student and ResNet 152 as the weak teacher
Table 1. Summmary of the pretrained feature dimensions, along with the intrinsic dimensions d, du and correlation dimensions deww (with respect to the strong student CLIP-B32) computed over the entire UTKFace dataset (including training and testing).
Pretrained Model
Feature Dimension
Intrinsic Dimension (T 0.01) Correlation Dimension
ResNet18
512
194
167.64
34
512
150
129.97
ResNet.50
2048
522
301.06
ResNet101
2048
615
354.52
ResNet152
2048
589
339.90
CLIP-B33
768
443
X
For reference, we provide the scaling for MSE losses of three representative teacher-student pairs in Figures 7 to 9.
is worth highlighting that while the MSE loss fw2s monotonically decreases with respect to both sample sizes n, N. the different rates convergence compared to fws fa, and fe lead to the distinct scaling behavior of the relative W2S performance (PGR and OPR) with respect versus N in Figures and 6.
30

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
When the strong student has lower intrinsic dimension than the weak teacher (cf. Figure versus Figures 8 and 9), dg dw, the W2S model sw2s tends to achieve better generalization in terms of the test MSE. This is consistent with analysis in Section 3.1.
When d, < dw. the W2S model fw2s tends to achieve (slightly) better generalization for (slightly) smaller correlation dimension derw (cf. Figure 8 versus Figure 9), again coinciding with our analysis in Section 3.1.
W2S generalization generally happens (i.e. Sw2s is able to outperform (w) with sufficiently large sample sizes n, N. However, as the labeled sample size increases, the test MSE of fw2s converges slower than that of the strong baseline and ceiling models, Ja and fe, leading to the inverse scaling for PGR and OPR with respect in Figures 5 and 6. When is too large, the W2S model fw2s may not be able to achieve better generalization than the strong baseline fs.
E.2. Experiments image classification
Dataset. ColoredMNIST (Arjovsky et al., 2019) consists of groups of different colors and reassign the label be binary (digits 0-4 vs 5-9). We pool together the groups to form dataset. The choice is to bring diversity the feature space with additional color features and thus potential feature discrepancies. We hold out test set of 7000 samples and use the rest 63000 samples for training.
Linear probing over pretrained features. We fix strong student DINOv2-s14 (Oquab al., 2024) and vary the weak teacher among the ResNet-d series and ResNet series (ResNet18D, ResNet34D, ResNet101, ResNet152) (He et al., 2019; 2016). We replace ResNet18 and ResNet34 used in Section 4.2 to experiment on weak models with similar intrinsic dimensions but different correlation dimensions. We treat the backbone of the models (excluding the classification layer) as os and &w and finetune them via linear probing. We train the models with cross-entropy loss and Adam W optimizer. tune the training hyperparameters of weak and strong models using a validation set and train them for 800 steps with a learning rate 1e-3 and weight decay 1e-6.
Table Summary of the pretrained feature dimensions, along with the intrinsic dimensions da dw and correlation dimensions daww (with respect the strong student DINOv2-S14 computed over the entire ColoredMNIST dataset (including training and testing).
Pretrained Model
Feature Dimension
Intrinsic Dimension (r = 0.01) Correlation Dimension
ResNet-18-D
512
117
6.23
ReSNet-34-D
512
127
7.07
ResNet101
2048
121
1.74
ResNet152
2048
128
1.88
DINOV2-S14
384
28
Discrepancies lead better W2S.
Figure 10 shows the scaling of PGR and OPR with respect to the sample sizes n, N for different weak teachers in the ResNet series with respect to fixed student, CLIP-B32. As in Section 4.2, observe that with similar intrinsic dimensions de, dw. W2S achieves better relative performance in terms of PGR and OPR when the correlation dimension d' is smaller.
Variance reduction is a key advantage of W2S. We inject noise to the labels of the original ColoredMNIST training samples by randomly flipping the ground truth labels with probability < E [0, 1] (following (Arjovsky et al., 2019)). Figure 11 shows the scaling of PGR and OPR with respect n and N when taking DINOv2-S14 the strong student and ResNet101 as the weak teacher. We observe that the larger artificial label noise ş leads to higher PGR and OPR.
E.3. Experiments on sentiment classification
Dataset. The Stanford Sentiment Treebank (Socher et al., 2013) is a corpus with fully labeled parse trees that allows for complete analysis of the compositional effects of sentiment in language. The corpus is based on the dataset introduced by Pang & Lee (2005) and consists of 1 11,855 single sentences extracted from movie reviews. was parsed with the Stanford parser and includes a total 215.154 unique phrases from those parse trees, each annotated by human judges. We conduct binary classification experiments on full sentences (negative or somewhat negative versus somewhat positive or positive,
31

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
dw 117 (ResNet18D), 127 (ResNet34D) 121 (ResNet101), 128 (ResNet152), d; = 28 (DINOv2-S14)
N = 40000
n 5000
0.6
0.6
0.5
PGR
0.4
0.4
0.2
0.3
0.2
0.0 0
2000
4000
6000
8000
10000
12000
10000
20000
30000
40000
50000
deww/dw
6.23/117 (R)sNet18D)
1.05
dexwldy
7.07/127 (ResNe(34D)
0.98
dexw/dw = 1.74/121 (ResNet101)
dsaw/d 1.88/128 (ResNeti52)
0.96
1.00
0.94
0.95
0.92
2000
4000
6000
8000
10000
12000
10000
20000
30000
40000
50000
N
Figure 10. Scaling PGR OPR of different weak teachers with a fixed strong student on ColoredMNIST.
with neutral sentences discarded), the so-called SST-2 dataset, and split the dataset into training and testing sets of sizes 63000 and 4349. Generalization errors are estimated with the 0-1 loss over the test set.
Full finetuning. We fix the strong student as Eleett--aas-ddicriinnato (Clark et al., 2020) and vary the weak teacher among the Bert series (Ture et al., 2019) (Bert-Tiny, Bert-Mini, Bert-Small, Bert-Medium). With manageable model sizes, conduct full finetuning experiments following the setup in (Burns et al., 2024). We the standard cross-entropy loss for supervised finetuning. When training strong students weak labels (W2S), by (Burns et al., 2024), which is suggested to be able to improve weak-to-strong generalization many NLP tasks. All training is conducted via Adam optimizers (Kingma & Ba, 2014) with a learning rate of 5e-5, a cosine learning rate schedule, and 40 warmup steps. We train for epochs, which sufficient for the train and validation losses to stabilize.
the confidence- confidence-weighteed loss proposed
The intrinsic dimensions den, de are measured based on the Structure-Awar Intrinsic Dimension (SAID) method proposed by (Aghajanyan et al., 2021). We first train the full models on the whole training set, and then train the models with only d trainable parameters based on SAID transformation. The du or dy are the smallest number of parameters under SAID that is necessary retain 90% accuracy of the full models. Here, the 90% accuracy is common threshold used to estimate intrinsic dimensions in the literature (Li et al., 2018).
Intrinsic dimension.
Correlation Dimension. Let D,, Das be the finetunable parameter counts of the strong and weak models, respectively, For full FT whose dynamics fall in the kernel regime, explained in Remark 2.5, the strong and weak "features" become the gradients + Vefa ( ( x)) E NDD, and o Vel ( (X)) Nx of the respective models at the
pretrained initialization, o( E DDD and (L!) E D....
A practical challenge is that Da, Dw, N are all huge for full FT on most NLP tasks, making it infeasible to compute the D. x D, and Dw x Dw, Gram matrices and their spectral decompositions. As a remedy, we leverage the significantly lower
*Notice that fa, fw are scalar-valued functions for binary classification tasks like SST-2, and thus the gradients VeJa and Veff are row vectors. For multi-class classification tasks where fs, fo output vectors of logits, common heuristic to keep T., 4... as matrices of manageable sizes (in constrast tensors) is to replace gradients of the models, Vofs and Vofw. with gradients of MSE losses at the pretrained initialization. The gradients of MSE can viewed as weighted sum of the model gradients for each class.
32

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
dw 110 (1es, d5 25 (DINOv2-S14), Asa 1.4
N = 40000
n= 5000
0.8
0.8
0.6
0.6
PGR
0.4
0.4
0.2
0.2
0.0
0
2000
4000
6000
8000
10000
12000
10000
20000
30000
40000
50000
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0
2000
4000
6000
8000
10000
12000
10000
20000
30000
40000
50000
N
Injected label noise 5
Injected label noise ç 0.2
Injected label noise 5 0.1
Injected label noise = 0.3
Figure 11. Scaling for PGR and OPR of W2S ColoredMNIST with injected label noise.
intrinsic dimensions d, < Da, dhy < Da (see Table to accelerate estimation of daww via sketching (Halko al., 2011; Woodruff et al., 2014).
(i) We first reduce both D,, D.. to the same lower dimension D = 0.01 (D., D (with D > max ) by subsampling columns of 4.,, P.. (uniformly for efficiency, or adaptively via sketching-basee interpolative decomposi- tion (Dong & Martinsson, 2023) when affordable) to obtain " T', E ND
(ii) Then, we use randomized rangefinder (Halko al., 2011, Algorithm 4. to approximate the first d, du right singular vectors, V, E Dxd, and V w Dxd.. of &' & Taking the evaluation of V. an example, draw Gaussian random matrix G, x
and compute the orthornormalization V. ortho Φ G,) via QR decomposition.
(iii) Finally, we compute the correlation dimension de = | V
Table 3. Summary of finetunable parameter counts D,, D.. intrinsic dimensions dy dw and correlation dimensions darw (with respect to the strong student Elect ra) computed over the entire SST-2 dataset (including training and testing).
Pretrained Model
D., D
Intrinsic Dimension (r 0.01) Correlation Dimension
Bert -Tny
4.4M
7000
81.13
Bert-Mini
11.2M
8500
38.67
Bert-Smale
28.8M
8000
26.19
Bert-Medium
41.4M
4000
8.52
Electra
109.5M
700
X
Discrepancies lead better W2S. Figure 12 shows the scaling of PGR and OPR with respect to n and N for different dyra- As Section 4.2 and Appendix E.2, we observe the better relative W2S performance in terms of PGR and OPR
33

---

Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
d₂ 7000 (Bert-liny), 8500 (Berl-mini), 8000 (Berl-small), 4000 (Berl-medium)) dg 700 (Elecraa-vaxe-diccciinaaat))))))
N = 10000
n7 = 2500
0.4
0.2
0.3
PGR
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
1000
1500
2000
2500
3000
3500
4000
4000
5000
6000
7000
8000
9000
10000
1.0
0.6
0.9
0.5
OPR
0.8
0.4
0.4
0.3
0.2
0.2
1000
1500
2000
2500
3000
3500
4000
4000
5000
6000
7000
8000
9000
10000
N
ds dww 8.52/4000 (Bert-medium)
d w/d 26.19/8000 (Bert-small)
ww/dd 38.67/8500 (Bert-mini)
ds w/d, 81.13/7000 (Bert-liny)
Figure 12. Scaling for PGR and OPR of different weak teachers with fixed strong student on SST-2.
when deww dw is smaller.
Variance reduction is key advantage of W2S. We inject noise the labels of training samples by randomly flipping
labels with probability 0, 0.1, 0.2, 0.3. Figure 13 shows the scaling of PGR and OPR with respect to and N when as the weak teacher. We observe that the larger artificial label
taking Electra as the strong student and noise ç leads to higher PGR and OPR.
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Discrepancies are Virtue: Weak-to-Strong Generalization through Lens of Intrinsic Dimension
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Reviews:
[{"review_id": "bz7ptX7739", "reviewer": "Reviewer_5dnE", "summary": "This paper presents a theoretical analysis of weak-to-strong (W2S) generalization, a recently observed phenomenon where a strong student model outperforms a weak teacher model when trained on the teacher's pseudo-labels. The authors provide a variance reduction perspective by leveraging the concept of intrinsic dimensionality by analyzing kernel models.  The analysis is technically sound, and the findings offer some insights into the conditions under which W2S generalization may occur.  The paper is well-written and organized, making it accessible to readers familiar with machine learning theory.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and suggestions. We are glad that they found this work well-presented and provided a good understanding of W2S. Since we could not include figures in the OpenReview rebuttal, we will synopsize our new experiments in text and present the formal results in an anonymous URL: https://anonymous.4open.science/r/W2S_IntrinsicDim_demo-FEAF/demo.pdf.\n\n## Key clarifications\n1. **Additional experiments on image regression and LLMs**: We appreciate the suggestion on strengthening the experimental validation. In the URL, we included additional real experiments on image regression (UTKFace) based on CLIP and ResNet embeddings, as well as NLP task (SST-2) based on finetuning LLMs. Please see **R#4 (W6Uk), Key#1** for detailed discussions.\n\n2. **Empirical estimation of intrinsic and correlation dimensions**: We first highlight that with small finetunable parameter counts (e.g., linear probing or finetuning last few layers), we estimate intrinsic dimensions based on traces of data covariances (see Line 413-419, i.e., the minimum rank for the best low-rank approximation of $\\Sigma_w, \\Sigma_s$ with a relative error in trace less than $\\tau=0.01$). \n    1. For correlation dimension, a practical challenge is that **the feature dimensions of the weak and strong model can be different**--$V_s, V_w$ of sizes $D_s \\times d_s$ and $D_w \\times d_w$ can have $D_s \\ne D_w$. In this case, we gauge the correlation dimension by matching $V_s, V_w$ through a random unitary matrix $G \\in \\mathbb{R}^{D_s \\times D_w}$ s.t. $d_{s \\wedge w} = ||V_s^\\top G V_w||^2_F$. This provides a good estimation for $d_{s \\wedge w}$ because with low intrinsic dimensions $\\max(d_s, d_w) \\ll D_s, D_w$ in practice, mild dimension reduction through $G$ well preserves the essential information in $V_s, V_w$.\n    2. We appreciate the question on **empirical estimations of intrinsic and correlation dimensions for finetuning large models**. When finetuning LLMs, $D_s, D_w$ will be on the order of millions, making the covariance-based estimation infeasible. In this case, we use the SAID method proposed by Aghajanyan et al. (2020) to estimate intrinsic dimensions. Following Remark 2.5, for full FT, we take $\\Phi_s, \\Phi_w$ as the gradients of the strong and weak models at pretrained initializations. We use randomized rangefinder based on sparse JLTs and the random unitary projection trick above to estimate $d_{s \\wedge w}$ efficiently (see Appendix C in the above URL for detailed procedures).\n\n3. **Our assumptions are reasonable and generalizable**. We kindly highlight that footnote 4 and Theorem 3.1 provide explicit pointers to Theorem A.1, the formal version of Theorem 3.1 that rigorously extends the results to sub-Gaussian features. As explained in footnote 4, both theorems convey the same message. Due to page limit, we present the Gaussian results in the main text for clarity. Meanwhile, in the introduction, we provide strong motivations for studying FT in the kernel regime. The choice of subgaussian features is also well-justified by literature (Wei et al., 2022) and related works (Wu & Sahai, 2024; Ildiz et al., 2024) on W2S. Most importantly, the empirical evidence in our new experiments demonstrates the generalizability of our assumptions and results to real-world scenarios.\n\n4. **Ridge regression analysis**: Our analysis can be extended to the ridge regression setting. As mentioned in footnote 1, when $\\Sigma_s, \\Sigma_w$ admit full ranks with fast decaying eigenvalues, ridge regression effectively brings low intrinsic dimensions by filtering out the small eigenvalues. The result for ridge regression again conveys the same message as Theorem 3.1, with the correlation dimension replaced by $tr(\\Sigma_s \\Sigma_w)$. Intuitively, a large discrepancy corresponds to a $\\Sigma_s, \\Sigma_w$ pair with approximately orthogonal leading eigenvectors associated with the large eigenvalues, bringing a small $tr(\\Sigma_s \\Sigma_w)$. (See Appendix A in the above URL for detailed statements and proofs.)\n\n## Detailed responses\n* The **practical implications** of our theoretical insights on the choice of weak v.s. strong models and sample sizes in W2S are self-evident and discussed in detail in Sections 3.2 and 5. We will try to further emphasize them in the revision.\n\n* Assuming both weak and strong models have sufficient capacities to achieve **low FT approximation errors on the downstream task**, our theory and experiments show that the larger discrepancy between weak and strong models brings better W2S, with **no potential downsides**. \n\nWe are happy to answer any further questions you may have. If our responses above have addressed your concerns, we would truly appreciate a re-evaluation accordingly.", "claims_and_evidence": "The authors make several key claims, including:\n\n- W2S generalization can be explained through the lens of intrinsic dimension.    \n- The discrepancy between strong and weak models in W2S has a positive effect on reducing variance.    \n- W2S generalization occurs in variance, with the student model inheriting the teacher's variance in the overlapped feature subspaces and reducing it in the discrepancy subspace.    \n- The student-teacher correlation influences W2S, with lower correlation benefiting W2S.\n\nThe authors somewhat support these claims:\n\n- **Theoretical Framework**: The authors develop a theoretical framework using ridgeless regression and the concept of intrinsic dimension. They provide mathematical formulations and theorems (like Theorem 3.1) to characterize the variance and bias in W2S generalization.  The framework is built on established observations on fine-tuning and the concept of intrinsic dimension.    \n\n- **Variance Reduction Analysis**: The authors provide a detailed analysis of how variance is reduced in the discrepancy subspace, supported by Theorem 3.1 and related discussions.  They use the analogy of car logo vs. design to provide intuition.    \n\n- **Role of Student-Teacher Correlation**: The authors define and incorporate student-teacher correlation (using correlation dimension) into their framework.  They explain how lower correlation (greater discrepancy) can lead to better W2S generalization.    \n\n- **Experimental Validation**: The authors present experiments on synthetic regression tasks and real image classification tasks.  These experiments aim to validate their theoretical findings.", "methods_and_evaluation_criteria": "**Theoretical Framework**\n\n- The authors propose a theoretical framework based on ridgeless regression and the concept of intrinsic dimension.   \n- They use mathematical tools and theorems to analyze the variance and bias components of W2S generalization.\n- This theoretical approach is appropriate for gaining a deeper understanding of the underlying mechanisms driving W2S. Ridgeless regression, while simplified, allows for tractable analysis and can provide valuable insights.\n\n\n**Experimental Validation**\n\n- The authors use both synthetic regression tasks and real image classification tasks for experimental validation.\n\n- Synthetic data allows them to control specific parameters and test the theoretical predictions in a controlled setting.\n\n- Real image classification tasks (using CIFAR-10) demonstrate the relevance of their findings to practical applications.\n\n---\n**Suggestion**: While the current methods and evaluation criteria are generally sound, the experimental validation could be expanded. For example, exploring a wider range of datasets, model architectures, and training paradigms would provide a more comprehensive evaluation of the theory. Additional experiments that directly measure or manipulate the intrinsic dimension could further strengthen the connection between the theory and the empirical results.", "theoretical_claims": "I did not check the correctness of the theoretical proofs.", "experimental_designs_or_analyses": "Please see above.", "relation_to_broader_scientific_literature": "That aspect seems fine.", "essential_references_not_discussed": "It seems like most key contributions have been discussed.", "other_strengths_and_weaknesses": "**Weaknesses**\n\n- **Idealized Assumptions**: The theoretical analysis relies on strong assumptions, such as the Gaussian feature assumption. While the authors mention that the results hold for sub-Gaussian features, the analysis in the main text is limited to the Gaussian case. It is unclear how sensitive the results are to these assumptions and how well they generalize to more realistic scenarios.\n\n- **Practical Implications**: The theoretical results provide a good understanding of W2S, but their practical implications are not fully explored. The paper could benefit from a more detailed discussion on how these findings can be used to improve W2S training or model design in real-world applications.\n\n- **Experimental Validation**: While the experiments support the theory, they are somewhat limited. Additional experiments on more diverse datasets and model architectures such as language models would strengthen the empirical validation of the proposed framework.\n\n**Strengths**\n\n- **Clearly written**: The paper is well-motivated, clearly written, and easy to follow.\n\n- **New perspective on W2S**: The theoretical framework based on intrinsic dimension provides a novel perspective on W2S generalization. The analysis of variance reduction and the role of student-teacher discrepancy is insightful.", "other_comments_or_suggestions": "The paper has merit in its theoretical analysis and the use of intrinsic dimension to explain W2S generalization. However, the strong assumptions, limited novelty, and the need for more extensive experimental validation and discussion of practical implications suggest that the paper is not yet ready for publication. I encourage the authors to address these concerns and resubmit the work in the future.", "questions_for_authors": "- **Gaussian Feature Assumption**: The theoretical analysis relies heavily on the assumption of Gaussian features. While you mention that the results can be extended to sub-Gaussian features, could you provide more detail on how this extension is achieved and what the key differences in the results would be? Are there specific types of sub-Gaussian distributions where the results would be significantly weaker or not applicable.\n\n- **Alternative distributions**: How does the Gaussian assumption influence the specific forms of the generalization error bounds derived in Theorem 3.1? Are there alternative distributional assumptions that would allow for a similar analysis, and what would be the trade-offs?\n\n- **Estimate intrinsic dimension**: The concept of intrinsic dimension is central to your analysis. How do you propose to estimate or approximate the intrinsic dimension of a model in practice, especially for deep learning models where it's not straightforward? Are there practical methods or heuristics that can be used to guide the choice of student and teacher models based on their estimated intrinsic dimensions?\n\n- **Potential downsides**: Your analysis suggests that a discrepancy in intrinsic dimensions between the student and teacher is beneficial for W2S. Are there any potential downsides to having a very large discrepancy? Could there be a point where the discrepancy becomes too large, and W2S is negatively affected\n\n- **Correlation Mesaure**: The correlation dimension is used to quantify the similarity between the student and teacher models. How sensitive are your results to the specific way this correlation is measured? Are there other measures of student-teacher similarity that could be used, and would they lead to qualitatively different results?\n\n- **Ridge case**: The paper focuses on ridgeless regression. How do you anticipate the results might change with the introduction of regularization, which is commonly used in practice? What are the key challenges in extending your analysis to the regularized setting?\n\n- **Practical Implications**: What are the most important practical implications of your findings? How can practitioners use your theoretical insights to improve W2S training or design more effective student and teacher models?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MkvNxB8J2y", "reviewer": "Reviewer_WLsf", "summary": "This paper theoretically investigates the weak-to-strong (W2S) generalization phenomenon in the setting of ridgeless regression. From a bias-variance decomposition perspective, the authors utilize the intrinsic dimensionality of fine-tuned models to analyze the generalization performance of the weak teacher model, the W2S student model, the strong model, and the strong ceiling model.\nWhen variance dominates the generalization error, the paper finds that the W2S student model behaves similarly to the weak teacher model within their overlapping feature space, but reduces variance in the other parts of weak teacher's feature subspace. This reduction in variance creates an opportunity for the W2S student to outperform the weak teacher. Additionally, the authors characterize the performance gap recovery metric and the outperforming ratio metric based on their bias-variance decomposition analysis.\nFurthermore, the theoretical findings are empirically verified in both the ridgeless regression setting and a binary image classification task.\n\n## update after rebuttal\n\nI would like to thank the authors for their further responses. However, I still do not see how the explicit effect of early stopping is reflected in the results. As I understand it, in the main context of the paper, the parameter $\\alpha_{w2s} \\to 0$ is assumed, and all main results are derived under this setting. It is unclear to me (quantitatively) how choosing a suitable (non-zero) value of $\\alpha_{w2s}$ would impact the theoretical results presented in the main text. At this point, I intend to maintain my original evaluation.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the constructive suggestions from the reviewer. We are glad that they found our paper well-presented and our perspective novel. Since we could not include figures in OpenReview rebuttal, we will synopsize our new experiments in text and present the formal results in an anonymous URL: https://anonymous.4open.science/r/W2S_IntrinsicDim_demo-FEAF/demo.pdf.\n\n## Key clarifications\n1. **Additional experiments on image regression and LLMs**: In the URL, we included additional real experiments on image regression (UTKFace) based on CLIP and ResNet embeddings, as well as NLP task (SST-2) based on finetuning LLMs. Please see **R#4 (W6Uk), Key#1** for details.\n\n2. **Empirical estimation of intrinsic and correlation dimensions**: Please see **R#1 (5dnE), Key#2**.\n\n3. **Our analysis extends to weak-strong pairs with different feature dimensions**: Notice that since the intrinsic dimensions are far lower than the feature dimensions, the larger feature dimension (e.g., $D_s$) can always be reduced to the smaller one ($D_w < D_s$) via random projection (Johnson-Lindenstrauss transforms) with a negligible information loss $\\epsilon \\asymp d_s / D_w$. This is the main idea of the empirical estimation in Key#2. Since the high feature dimensions are not essential in our setting, we consider $D_s = D_w = d$ for a clean analysis. We will clarify this in the revision.\n\n4. **Variance-dominated regime is crucial for understanding W2S, focusing on which does not compromise our contributions**: We respectfully disagree with the reviewer and emphasize that our focus on variance-dominated regime is a strength rather than a weakness. This setting is empirically motivated and fills an important gap in existing theoretical understanding. As our motivation in Lines 23-26 right said, empirical evidence in (Burns et al. 2023) suggests that larger W2S gain tends to occur on easier tasks (i.e., variance-dominated tasks). This can also be observed in our synthetic experiments (cf. Fig. 2&3) and our new vision and NLP experiments (see **R#4 (W6Uk), Key#1.2**). **Despite the appealing empirical performance of W2S in variance-dominated regimes, to the best of our knowledge, this regime has never been rigorously studied in prior theories on W2S. Our work fills this gap by providing a descriptive yet clean theoretical underpinning for W2S in the variance-dominated regime under finite samples**.\n\n## Detailed responses\n* Concern in \"Theoretical Claims\": Notice that taking expectation over the training dataset is a standard way to connect the excess risks (ER) of random and fixed designs: $$\\text{random design ER}=\\mathbb{E}\\_{S_x \\sim \\mathcal{D}^N}[\\text{fixed design ER over }S_x]$$ Thanks to the reviewer's question, we realized that the inner expectation $\\mathbb{E}\\_{\\theta_{w2s}}$ with shared randomness between $S_x$ and $\\theta_{w2s}$ could be misleading (although it is technically correct as $\\theta_{w2s}$ inside is conditioned on $S_x$). A better expression for the right-hand side may be $\\mathbb{E}\\_{S_x,\\widetilde{S}}[||\\Phi_s\\theta - f_*||^2/N]$. We will revise these notations in the proofs.\n\n* \"Questions For Authors\" QFA#3,5,6 are addressed in the Key clarifications.\n\n* QFA#1: The variance-bias decomposition in Lines 595-604 is equivalent to directly deriving $Var(f)$ and $Bias(f)$ in Lines 121-123, both following the standard variance-bias decomposition for linear regression (see e.g. Liang, (2016), Statistical learning theory, Sec 2.9). The key observation here is that when opening the square in Line 600, the cross term vanishes because $\\widetilde{z}$ is an independent random vector with zero mean.\n\n* QFA#2: First, as the analysis in Sec 3.2 shows, even if the weak teacher lacks capacity (FT approximation error $\\rho_w$ is large), as long as the label noise $\\sigma^2$ dominates $\\rho_w$ (intuitively, harder tasks are more likely to have noisy labels), the problem still falls in the variance-dominated regime. Second, beyond the variance-dominated regime, our theory suggests a degeneration of W2S performance due to the vanishing advantage in variance reduction. This is empirically confirmed in both synthetic and real experiments (see discussions in Key#4).\n\n* QFA#4: Our analysis is not directly related to early stopping, but there are shared insights. Early stopping has a known intuitive connection with weight decay, which translates to ridge regression in our setting. As footnote 3 explained, ridge regression effectively brings low intrinsic dimensions by filtering out small eigenvalues. In the revision, we formally extend our analysis to ridge regression (see **R#1 (5dnE), Key#4** or the URL). We show that a suitable choice of ridge parameter (or early stopping intuitively) can bring better W2S performance by revealing the underlying low intrinsic dimensions.\n\nWe are happy to answer any further questions you may have. If our responses above help address your concerns, we would truly appreciate a re-evaluation accordingly.", "claims_and_evidence": "The theoretical results seem to be well supported by the empirical findings. However, I have some concerns regarding certain parts of the proofs, see the Theoretical Claims section for details.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria seem reasonable.", "theoretical_claims": "I checked the proofs in the appendix and have some concerns that need to be addressed. Specifically, in the proof of Theorem 3.1 (or Theorem A.2), the authors use the following equality in Lines 595–597:\n$$\n\\mathbb{E}\\_{\\mathbf{x}\\sim\\mathcal{D}}[\\mathbb{E}\\_{f\\_{w2s}}[(f\\_{w2s}(x)-f\\_{\\*}(x))^2]]=\\mathbb{E}\\_{\\mathcal{S}\\_x}[\\mathbb{E}\\_{\\theta\\_{w2s}}[\\frac{1}{N}||\\Phi\\_s\\theta\\_{w2s}(x)-f\\_{\\*}||^2]].\n$$\n\nOn the LHS, the expression represents the definition of excess risk, where $\\mathbf{x}\\sim\\mathcal{D}$ is an independently drawn test sample, which is independent of the random variable $f\\_{w2s}$. However, on the RHS, the expectation over the test sample $\\mathbf{x}\\sim\\mathcal{D}$  is replaced by an average over the training sample $\\mathcal{S}\\_x$, which is **not independent** of $\\theta_{w2s}$. This suggests that the RHS might be the training error rather than the excess risk. This issue seems to exist in other parts of the proofs in the appendix. Could you clarify why this equality holds and whether it correctly represents the excess risk rather than the training error?", "experimental_designs_or_analyses": "The ridgeless regression experiments align with the theoretical analysis and appear to be sound. However, using a binary image classification task with MSE loss may not substantially strengthen the paper in terms of supporting the claims in a more realistic setting. Would similar observations hold for multi-class image classification with a cross-entropy loss? Exploring this could provide stronger empirical support for the theoretical results.", "relation_to_broader_scientific_literature": "This paper contributes to the broader literature on learning theory, with a particular focus on the weak-to-strong generalization phenomenon, first identified by Burns et al. (2023).", "essential_references_not_discussed": "The relevant works seem to be properly cited.", "other_strengths_and_weaknesses": "*Other Strengths:* \n\n1. The paper is well written and easy to follow. \n\n2. It introduces the use of intrinsic dimensionality to quantitatively characterize model capacity, an aspect that has not been explored in previous works on W2S generalization.\n\n*Other Weaknesses:* \n\n1. This paper mainly focuses on a setting where both the weak and strong models perform well, that is, both the weak teacher and the W2S student have relatively high model capacity and achieve low approximation error. This may differ from many scenarios in previous W2S generalization studies, where the weak teacher typically has limited capacity and performs poorly.\n\n2. The analysis is largely restricted to the variance-dominated regime, which limits its general applicability. However, the authors explicitly acknowledge this limitation in the paper.\n\n3. Another limitation is that the paper assumes both the weak model features and the strong model features exist in the same feature space (i.e. $\\mathbb{R}^d$), which may not always hold in practical scenarios.", "other_comments_or_suggestions": "Typo in Line 66 (left column): \"both student and teach\" ---> \"both student and teacher\"", "questions_for_authors": "1. In addition to the question raised in Theoretical Claims, could you clarify how the obtained variance and bias terms in Line 603 correspond to the terms defined in Lines 121–123 (right column)? Can these terms be directly derived from $\\mathrm{Var}(f)$ and $\\mathrm{Bias}(f)$ as formulated in Lines 121–123?\n\n2. If the weak teacher model is not sufficiently strong (e.g., if $\\rho_w$ is no longer small), will W2S generalization still occur in your analysis?\n\n3. Will your analysis remain valid if  $\\phi_w(x)\\in\\mathbb{R}^{d_1}$ and $\\phi_s(x)\\in\\mathbb{R}^{d_2}$ with $d_1\\leq d_2$? This would correspond to a scenario where the strong model has more parameters than the weak model.\n\n4. Does your analysis provide any insights into why early stopping often benefits W2S generalization?\n\n5. Based on Proposition 3.5, it seems that large label noise (i.e., large  $\\sigma^2$) can facilitate W2S generalization. Does this suggest that artificially injecting independent noise into the labels when training the weak teacher model could make W2S generalization more likely?\n\n6. Could you provide more details on how the intrinsic dimension was estimated in your image classification experiments?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tylky7dmci", "reviewer": "Reviewer_299q", "summary": "The paper studies weak-to-strong generalization in ridgeless regression with (sub-)Gaussian features. It reveals that weak-to-strong generalization arises from the discrepancy between the weak model's features and the strong model's features.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and suggestions. We are glad that they found our paper well-written and provides novel understanding for W2S. Since we could not include figures in OpenReview rebuttal, we will synopsize our new experiments in text and present the formal results in an anonymous URL: https://anonymous.4open.science/r/W2S_IntrinsicDim_demo-FEAF/demo.pdf.\n\n## Key clarifications\n1. **The key intuitions on how discrepancies lead to better W2S are unrolled in two steps in the introduction**:\n    1. In Lines 68-79, we first break down our main theoretical result $$Var(f_{w2s}) \\asymp \\frac{d_{s \\wedge w}}{n} + \\frac{d_s}{N}\\frac{d_w - d_{s \\wedge w}}{n}$$ to explain **why a larger discrepancy (smaller $d_{s \\wedge w}$) leads to better W2S**. That is, the strong student mimics variance of the weak teacher in the overlapped subspace of weak and strong features (with dimension $d_{s \\wedge w}$). In contrast, **variance in the discrepancy subspace between weak and strong features (with dimension $d_w - d_{s \\wedge w}$) is reduced by a factor of $\\frac{d_s}{N}$ in W2S**. Recall that $d_s$ is the low intrinsic dimension, and the unlabeled W2S sample size $N$ is generally large in practice. Therefore, a multiplicative factor of $\\frac{d_s}{N}$ significantly reduces the variance in the discrepancy subspace, leading to better W2S performance.\n    2. Then, in Lines 91-107, we use an example to provide high-level intuition for **how variance reduction in the discrepancy subspace happens**. In particular, we consider a downstream task on classifying the brands of vehicles based on their images. \n        - The weak features, spanning a low-dimensional subspace $\\mathcal{V}_w$, capture the designs of vehicles that tend to be more complicated (with a higher intrinsic dimension $d_w$) but contain irrelevant/spurious information that makes the model weak for the downstream task. \n        - The strong features, spanning a low-dimensional subspace $\\mathcal{V}_s$, capture the logos of vehicles that are often simpler (with a lower intrinsic dimension $d_s$) and more relevant for the downstream task. \n    \n        Since the design and logo of a vehicle are typically irrelevant, $\\mathcal{V}\\_w$ and $\\mathcal{V}\\_s$ are likely to be almost orthogonal in a high-dimensional feature space, leading to a small $d_{s \\wedge w}$. Then, **the weak teacher $f_w$ brought by noisy SFT labels will make mistakes that only correlate to the design features in $\\mathcal{V}_w$, independent of the logo features in $\\mathcal{V}_s$. Such errors in weak supervision can be viewed as independent label noise with respect to the strong features in $\\mathcal{V}_s$**. With an intrinsic dimension $d_s$, the generalization error of strong student induced by such independent label noise vanishes at a rate of $\\frac{d_s}{N}$ (following the intuition of classical regression analysis). \n    \n    We will further clarify these key intuitions in the revision.\n\n2. **Empirical estimation of intrinsic and correlation dimensions**: Please see **R#1 (5dnE), Key#2**.\n\n3. **Additional experiments on image regression and LLMs**: In the URL, we included additional real experiments on image regression (UTKFace) based on CLIP and ResNet embeddings, as well as NLP task (SST-2) based on finetuning LLMs. In particular, our new experiments provide strong empirical evidence for our theoretical results by showing that (i) **model discrepancies lead to better W2S**; and (ii) **variance reduction is a key advantage of W2S**. Please see **R#4 (W6Uk), Key#1** for details.\n\n## Detailed responses\n* Discussion on concurrent works: We are keeping track of relevant concurrent works on W2S and will include discussions on them in the revision. \n* We appreciate all the detailed suggestions in \"Other Comments Or Suggestions\" and will revise accordingly in the next version.\n\nWe are happy to answer any further questions you may have. If our responses above help address your concerns, we would truly appreciate a re-evaluation accordingly.", "claims_and_evidence": "The main theorem statement is clearly presented, and the proof in the appendix  appear to be well-structured and understandable.", "methods_and_evaluation_criteria": "In their main results and experimental evaluations, the authors use PGR and OPR, which are standard evaluation metrics in the weak-to-strong generalization literature.", "theoretical_claims": "I have checked the proof of the main results, and it appears to be correct.", "experimental_designs_or_analyses": "The authors provide experimental results on synthetic regression and an image classification task. However, in the image classification task, I have concerns about the choice of intrinsic dimension. Why do the authors define the intrinsic dimension as 90% of the trace of the feature matrix? I believe it is essential to analyze the spectrum of the feature matrix, as the intrinsic dimension and trace can differ between the strong and weak models. Additionally, I am confused about the meaning of \"threshold dimension $k$\" in line 416 (left).", "relation_to_broader_scientific_literature": "This provides novel insights into understanding the weak-to-strong generalization phenomenon, particularly in relation to intrinsic dimension. This contribution enhances the broader understanding of weak-to-strong generalization.", "essential_references_not_discussed": "I believe most related works are cited in the manuscript, at least regarding the theory of weak-to-strong generalization, which I am familiar with. However, several relevant references have been released after the ICML submission deadline, and I hope the authors will cite and discuss them in the next revision. Here are some of them (but not limited to these):\n\n[1] Medvedev, Marko, et al. \"Weak-to-Strong Generalization Even in Random Feature Networks, Provably.\" *arXiv preprint arXiv:2503.02877* (2025).\n\n[2] Yao, Wei, et al. \"Understanding the Capabilities and Limitations of Weak-to-Strong Generalization.\" *arXiv preprint arXiv:2502.01458* (2025).", "other_strengths_and_weaknesses": "One weakness of the paper is its simplified problem setting, specifically ridgeless regression and (sub-)Gaussian feature maps. However, the authors provide a strong motivation for this choice, and I believe that results derived from a well-motivated simple setting can also be a strength of the work.", "other_comments_or_suggestions": "Here are some minor comments on the manuscript:\n\n- Typo: Line 142 (left): $p \\in [0,1]^b$ → $p \\in [0,1]^n$\n- Clarification: Line 158, 162 (left): $\\mathcal{D} : \\mathcal{X} \\times \\mathcal{Y} \\rightarrow [0,1]$. I believe it is not appropriate to represent the data distribution as a function mapping to [0,1].\n- Clarification: Line 155–156 (right): Does $\\dagger$ refer to the Moore–Penrose inverse? I think it would be better to clarify this.\n- Figures: The figures in the paper have low resolution. I suggest the authors provide high-resolution versions (e.g., in .pdf format).", "questions_for_authors": "Can you provide a high-level intuition on why discrepancies between the features of the weak and strong models lead to weak-to-strong generalization? I believe the current version of the paper lacks this high-level intuition, even though the theoretical results are interesting. If the authors can provide a clearer high-level explanation of their findings and promise to emphasize this in the next revision, I am open to increasing my score.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "H8kmQqdz1I", "reviewer": "Reviewer_W6Uk", "summary": "The paper considers the problem of weak-to-strong generalization, which is the study of how the performance of a strong model trained on pseudo labels of a weaker model generalize. More formally the following setup is consider. Two features transforms, $\\phi_s$ the strong model and $\\phi_w$ the weak model, are given where it is a assumed that the feature transform $\\phi_s$ and $\\phi_w$ has output in some spaces with dimension respectively $d_s$ and $d_w$. \nNow a linear layer $\\theta_w$ ontop of the weak model $\\phi_w$ is trained on a dataset \n\n$\\tilde{S}=((\\tilde{x}_1,\\tilde{y}_1),\\ldots,(\\tilde{x}_n,\\tilde{y}_n))$\n\n of i.i.d. examples where $\\tilde{x}_1\\sim \\mathcal{D}$\n\n and ${\\tilde{y}}_1=f^*(\\tilde{x}_1)+z_1$,\n\nwhere $z_1 \\sim  N(0,\\sigma^2)$, $f^*$ being uniformly bounded by 1. Now given a second dataset $S=((x_1,y_1),\\ldots,(x_N,y_N))$, drawn i.i.d. in the same manner as $\\tilde{S}$, where the learner is only given the unlabelled data $x_1,\\ldots,x_N$ a linear layer $\\theta_{w2s}$ ontop of the strong model $\\phi_s$ is trained with the pseudo labels given by $\\theta_w$, i.e. the data set $(x_1,x_1\\theta_w),\\ldots,(x_N,x_N\\theta_w).$ The paper then want to show how the strong model generalize on new data examples $(x,y)$ drawn in the above manner. To this end the paper studies the expected generlization error of $\\theta_{w2s},$ i.e. $err(\\theta_{w2s})=E_x[E_f[(\\theta_{w2s}\\phi_s(x)-f*(x))^2]]$ via the bias variance decomposition. Under sufficient conditions, the paper shows that the bias can be bounded by the sum of the generalization error of the best fixed linear lay on top of $\\phi_s$, $\\rho_s= \\min_\\theta E_x[(\\theta\\phi_s(x)-f*(x))^2]$ and the best fixed linear lay on top of $\\phi_w$,  $\\rho_w= \\min_\\theta E_x[(\\theta\\phi_w(x)-f*(x))^2].$ The variance term of the strong model can be bounded by $\\sigma/(n-d_w-1) (d_{sw}+d_s(d_w-d_{sw}/N),$ where $d_{sw}$ denotes the intersection of the subspaces that $\\phi_s$ and $\\phi_w$ spans, here the term $\\sigma d_{sw}/(n-d_w-1)$ do not change with the number of pseudo labels where as the term $\\sigma/(n-d_w-1)d_s(d_w-d_{sw}/N),$ does. \nThus, this theorem suggests that the generalization error of the stronger model trained with more pseudolabels improves this variance term, where as the other terms do not change with more pseudolabels. This variance-term can be thought of the discrepancy of the subspaces span by $\\phi_s$ and $\\phi_w$. For comparison of the performance of $\\theta_s$, the paper also considers the model $\\theta_f$ a linear layer trained ontop of the strong model $\\phi_s$ with the dataset \n$\\tilde{S}$ and  the model $\\theta_c$ a linear layer trained ontop of the strong model $\\phi_s$ with the dataset $\\tilde{S}\\cup S.$\nFrom these quantities, the paper defines the Performance Gap recovery, shorten PGR\n$$PGR=\\frac{err(\\theta_w)-err(\\theta_{w2s}}{err(\\theta_w)-err(\\theta_{w2s}}$$\nand the Outperforming ratio (OPR)\n$$OPR=\\frac{err(\\theta_{s})}{err(\\theta_{w2s})}.$$\nThe paper then shows that under suitable conditions, which on a high levels says that the noise $\\sigma$ is much larger than the approximations errors $\\rho_s$ and $\\rho_w,$ $(\\rho_s+\\rho_w) / \\sigma \\rightarrow 1$ and $n$ and $N$ large enough, then the PGR becomes lower bounded by $1-(d_{sw}/d_w)$ where it is argued that it have been observed in some experiments that $d_{sw}/d_w$ is small, since the weak model as a \"large\" dimension and the intersection of subspace spanned by the strong and the weak models $d_{sw}$ being \"small\". Furthermore, under these assumptions the OPR becomes $d_s/d_{sw}$, again being large since $d_{sw}\\leq d_s.$ Furthermore when $(\\rho_s+\\rho_w)/\\sigma \\not\\rightarrow 0$, the paper observes that the PGR and OPR is non-monotonic in $n$. Solving for the optimal $n$, the paper show that when $(\\rho_s+\\rho_w)/\\sigma<<1$ and $N$ is large enough the $PGR$ and $OPR$ under this optimal $n$ becomes larger. \nIn general when the approximation errors $\\rho_w+\\rho_s$ is larger than $\\sigma$ the lower bounds on PGR and OPR becomes not so meaningful. \nThe paper then examine there findings on an synthetic datasets and a real-world datasets. On the synthetic datasets where $\\rho_w+\\rho_s$ is smaller than $\\sigma$, $\\theta_{w2s}$ outperformes $\\theta_s$ when $n<<N$, and in the case $\\rho_w+\\rho_s$ is close to $\\sigma$, $\\theta_{w2s}$'s performance is close to $\\theta_w$ so giving little or no weak to strong generalization. \nFor the synthetic datasets with $\\rho_w+\\rho_s$ smaller than $\\sigma$ the PGR as a function of $n$ shows a decreasing trend and as a function of $N$ shows a no monotonic trend, with an increase in the beginning and then turns and decreases, the increase is steeper than the decrease. The OPR as a function of $n$ shows a decreasing trend and the OPR as a function of N shows a non decreasing trend. For the real word dataset the picture is similar expect now PGR can go negative and PGR as a function of N is increasing.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their constructive suggestions and supportive feedback. We are glad that they found our paper well-written and provided novel perspectives on W2S. Since we could not include figures in OpenReview rebuttal, we will synopsize our new experiments in text and present the formal results in an anonymous URL: https://anonymous.4open.science/r/W2S_IntrinsicDim_demo-FEAF/demo.pdf.\n\n## Key clarifications\n1. **Additional experiments on image regression and LLMs**: We appreciate the suggestion on strengthening the experimental validation. In the revision, we will include additional real experiments on image regression (UTKFace) based on CLIP and ResNet embeddings, as well as NLP task (SST-2) based on finetuning LLMs (see the URL for concrete plots). In particular, our new experiments provide strong empirical evidence for our theoretical results:\n    1. **Discrepancies lead to better W2S**: As Figures 4 and 5 in the submission, we plot the scaling of PGR/OPR against sample sizes $n,N$ for weak-strong pairs with various correlation dimensions on real vision and NLP tasks. The results show that the lower $d_{s \\wedge w}/d_w$ (larger discrepancy) consistently brings better W2S.\n    2. **Variance reduction is a key advantage of W2S**: While the variance-dominated regime that we focus on is overlooked by most prior theoretical works on W2S, our new experiments confirm that this is a particularly important regime where strong W2S performance (in terms of PGR/OPR) occurs in practice. By injecting artificial label noise that leads to larger variance, we observe significant improvements in PGR and OPR, implying that variance reduction is a key advantage of W2S.\n\n2. **Our analysis extends to weak-strong pairs with different feature dimensions**: Please see **R#2 (WLsf), Key#3** for detailed explanations.\n\n3. **Empirical estimation of intrinsic and correlation dimensions**: Thanks for the question. In practice, the feature dimensions of the weak and strong models can be different. Please see **R#1 (5dnE), Key#2.1 and 2.2** for how we estimate $d_{s \\wedge w}$ in practice.\n\n4. **Our main technical contributions** are:\n    1. the theoretical framework based on low intrinsic dimensions of finetuning that provides **a theoretical underpinning for W2S in the variance-dominated regime**, along with\n    2. the insight on **how weak-strong discrepancy brings better W2S generalization**.\n    \n    As pointed out in **R#2 (WLsf), Key#4**, W2S has shown appealing empirical performance when variance dominates (e.g., see Key#1.2). However, the variance-dominated regime is often overlooked in prior analyses on W2S. (We kindly point out the confusion that Ildiz et al. (2024) explain W2S from the bias instead of variance perspective. We will further clarify this in the revision.) To the best of our knowledge, this is **the first theoretical explanation for W2S in the important variance-dominated regime**. While our analysis is built upon various fundamental tools in random matrix theory, high-dimensional statistics, and linear algebra, the unique structure of the learning problem posed by W2S makes the combination of these tools highly non-trivial. Moreover, our framework is flexible enough to be extended to finetuning problems with similar structures like knowledge distillation and model collapse. We will clarify this in the revision.\n\n## Detailed responses\nWe appreciate the detailed suggestions on notations and presentation. Here, we address questions in \"Other Comments Or Suggestions\" (O):\n* O#1: We assume $f_*$ being bounded for conciseness of the analysis. This can be trivially relaxed by adding a constant $C = \\max_x |f(x)|$ to the generalization error in our analysis.\n* O#2, 3: In Case I, $(\\rho_s + \\rho_w)/\\sigma^2 \\to 0$ essentially says that the FT approximation errors are small enough compared to the label noise so that we can treat them as zeros. In contrast, for Case II, we define a small constant $\\varrho = (\\rho_s + \\rho_w)/\\sigma^2 > 0$ to quantify the variance domination. This $\\varrho$ affects the PGR and OPR lower bounds in Coro. 3.6. While the quantitative conclusions are different, it's correct that the qualitative takeaways for both cases are similar.\n* O#4: We agree. This is exactly a simple explanation for how $f_s$ eventually outperforms $f_{w2s}$ as $n$ increases. We will rephrase the sentence to avoid ambiguity.\n* O#5: The reason why we did not report the spectra is that they do not contain much relevant information. See the spectra of ResNet embeddings in our new UTKFace experiments for examples: https://anonymous.4open.science/r/W2S_IntrinsicDim_demo-FEAF/fig/utkface_svd_cutoff0.99.pdf. \n\nWe are happy to answer any further questions you may have. If our responses above help address your concerns, we would truly appreciate a re-evaluation accordingly.", "claims_and_evidence": "There is no proofs in the main, which is deferred to the appendixes. The experiments are very small and I think should be seen as a pilot study giving some indications of what the theory showed.", "methods_and_evaluation_criteria": "As my comment above I think the experiments should be seen as a complement to the theoretical works of this papers and there is room to further study the theoretical findings of this paper on more datasets to give a stronger indication on to what extend the theoretical frame work explains strong to weak generalization.", "theoretical_claims": "There was no proofs in the main so i did not check the correctness of the proofs.", "experimental_designs_or_analyses": "I did not check the code for the experiments.", "relation_to_broader_scientific_literature": "The paper first examines the empirical literature, which is related to the paper in the sense that this was the ground for the interest in theoretically better understanding W2S, generalization that this paper studies. More specifically the W2S concept was introduced by Burns et. al. 2023. The theoretical work which the paper says is the closest is Ildiz et al, which also studies the ridgeless regressors, assumably also with the focus on variance reduction, but here the paper studies variance reduction from an intrinsic dimension perspective inspired by empirical observations by Aghajanyan et al. 2020.", "essential_references_not_discussed": "To the best of my knowledge no.", "other_strengths_and_weaknesses": "The paper is well written, and explains nicely how they see their findings.", "other_comments_or_suggestions": "Here are the notes that I have collected while reading your nice paper. The first questions/notes(the enumarted ones) would be nice to get and answer on but it is very much not required. The last bullets points are notes i took while reading.\n\n1. page 3 second column line 110, why can the boundedness over the whole distribution be assumed without loss of generality, i get it with normalization, but f* could be unbounded?\n\n3. page 6 first column line 308. what is meant with $(\\rho_s+\\rho_w)/\\sigma \\rightarrow 0$ my understanding of theses quanties it not being dependent on neither n and N, so what is meant by going to 0?\n\n4. page 6 second column line 308-312, i feel like the conclusion made in the paragraph is the same as in the paragraph page 6 first column line 318 to 276 in the second column, but no without any concrete lower bound, is this correct?\n\n5. page 7 second column line 338, Is it that intriguingly that f_s outperfroms f_w2s when n\\approx N? In this case f_ws was trained on N examples labelled by the weaker model, and f_s was trained on n examples from the true distribution?\n\n6. page 8 first column line 417, why not report the spectrum such that one can get an idea of changing the cut off would change $d_s$ and $d_w$?\n\n- Allot of assymptotical notation was defined what is meant by $<<$\n- page 4 first column line 206 what is the gradient of $\\nabla_\\theta$ taken with respect to?", "questions_for_authors": "1. It is a bit unclear to me what the technical contribution was? Do the proofs require new novel ideas, or is it similar to previous work of X, but the novelty is to put it in the context of weak to strong learning?\n\n2. page 3 second column line 132, the embedings goes $\\phi_s$ and $\\phi_w$ to the same dimension, which my guess is because one wants to define the intersection of their covariance matrices later. What could be done without this assumption in terms of intrinsic dimension? \n\nTo the end of the above question\n\n3. page 8 first column line 413-419, do ResNet50 and ResNet15 have the same output dimension?\n\nThanks for the response. I would like to keep my initial score. This decision is based on the response, which clarified my understanding of the paper and reading the other reviewers' comments and concerns.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper provides a theoretical study of the weak-to-strong (W2S) generalization phenomenon highlighting the role of the intrinsic dimension. The theoretical analysis is carried out in the setting of ridgeless regression (as common in related literature), and the main result is a decomposition of the variance into three terms: one depending on the variance in the intersection of the two subspaces and on the number of training samples for the teacher, one depending on the dimension of the strong subspace and on the number of training samples for the student, and the last one depending on the variance in the difference between the two subspaces and on the number of training samples for the teacher. This implies that W2S occurs when the student has low intrinsic dimension, or student-teacher correlation is low -- something in agreement with existing experimental evidence.

The paper is currently borderline: I have disregarded Review 5dnE as LLM-generated, and the remaining three reviews express borderline evaluations even after the rebuttal of the authors and the discussion between AC and reviewers. On the positive side, the reviewers have praised the insights coming the theoretical analysis and, specifically, the use of intrinsic dimensionality to quantitatively characterize model capacity. On the negative side, criticisms about the simplified setting and practical relevance have been raised. The authors have provided additional experiments that have partially addressed such criticisms. One remaining issue concerns the restriction to the ridgeless regression setting. To address that, the authors have shared an additional appendix via an anonymous link. I do think that carefully proof-reading that goes well beyond what is possible in the ICML reviewing timeline. At the same time, after my own reading of the manuscript, my opinion is that (1) the extension to ridge regression is interesting but the paper is already strong enough to meet the bar for acceptance, and (2) the extension to ridge regression is already mentioned as a footnote in the original version of the paper and, based on my knowledge of the area, I find it likely that it can be carried out using similar techniques. For these reasons, I am leaning towards accepting the paper. 

I do warmly encourage the authors to provide a careful revision to the camera ready incorporating the non-trivial amount of material that has been discussed during the rebuttal process.
Decision: Accept (poster)