Mitigating the Curse of Detail: Scaling Arguments for Feature Learning and Sample Complexity
Authors: Noa Rubin
Orit Davidovich
Zohar Ringel
Venue: ICLR 2026 Poster (2026)
Keywords: Deep learning theory
feature learning
sample complexity
scaling laws
Primary area: learning theory
TL;DR: We introduce a heuristic framework for scaling analysis that adopts variational methods from statistical field theory. It yields predictions of feature learning emergence in deep NNs, going beyond the state-of-the-art to hitherto intractable regimes.

Abstract: Two pressing topics in the theory of deep learning are the interpretation of feature learning (FL) mechanisms and the determination of implicit bias of networks in the rich regime. Current theories of rich FL often appear in the form of high-dimensional non-linear equations, which require computationally intensive numerical solutions. Given the many details that go into defining a deep learning problem, this analytical complexity is a significant and often unavoidable challenge. Here, we propose a powerful heuristic route for predicting the data and width scales at which various patterns of FL emerge. This form of scale analysis is considerably simpler than such exact theories and reproduces the scaling exponents of various known results. In addition, we make novel predictions on complex toy architectures, such as three-layer non-linear networks and attention heads, thus extending the scope of first-principle theories of deep learning.

Published as a conference paper ICLR 2026
MITIGATING THE CURSE OF DETAIL: SCALING ARGU- MENTS FOR FEATURE LEARNING AND SAMPLE COMPLEX- ITY
Noa Rubin', Orit Davidovich", and Zohar Ringel'
Racah Institute of Physics, Hebrew University, Jerusalem, Israel 2BBM Research, Haifa, Israel.
ABSTRACT
Two pressing topics in the theory of deep learning are the interpretation of feature learning (FL) mechanisms and the determination of implicit bias of networks in the rich regime. Current theories of rich FL often appear in the form of high-dimensional non-linear equa- tions, which require computationally intensive numerical solutions. Given the many de- tails that into defining a deep learning problem, this analytical complexity is a signif- icant and often unavoidable challenge. Here, we propose a powerful heuristic route for predicting the data and width scales at which various patterns of FL emerge. This form of scale analysis is considerably simpler than such exact theories and reproduces the scaling exponents of various known results. In addition, we make novel predictions complex toy architectures, such as three-layer non-linear networks and attention heads, thus extend- ing the scope of first seinntpricppleeee theories of deep learning.
INTRODUCTION
There is clear need for a better theoretical understanding of deep learning. However, efforts to construct such theories inevitably suffer from "curse of details".. Indeed, since any choice of architecture, activation, data measure, and training protocol affects performance, finding a theory with true predictive power that accurately accounts for all those details is unlikely. One workaround is to focus analytically tractable toy models, an approach that can often uncover interesting fundamental aspects. However, analytical tractability is a fragile, fine-tuned property; thus, large explainability gap remains between such toy models and more complex data/architecture settings.
An alternative approach focuses on scaling properties of neural networks, which appear more robust. Two well-established examples are empirically predicting network performance by extrapolating learning curves using power laws Kaplan al. Hestness et al. (2017), and providing theory-inspired suggestions for hyperparameter transfer techniques Yang et al. (2022): Bordelon et al. (2023). Indeed, it is often the case Cardy (1996) that predicting scaling exponents easier than predicting exact or approximate behaviors. As a simple toy model of this, consider the integral f_ dzg(zz P). While g(-) needs to be fine-tuned for exact computations, change of variable reveals robust linear scaling with P for any 9(-)-
This work focuses on scaling properties of feature learning (FL). This phenomenon, and, more generally, interpretability, has been studied extensively both from the practical and theoretical sides. On the practical side, mechanistic interpretability Bereska and Gavves (2024) has provided us with statistical explanations for why some predictions are made and the underlying decision mechanisms. On the theory side, kernel

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Hounds
Approximations
Hearisties
Learning cortificate: Ay (hounds MSE)
Upper bound on posterior m|A,
Prior A rale function: (e))
Lower on sample size P 2 B(a)
Variational B(a) = Bg (a)
Feature Leuriing
Kernel Fenture Propagation
Pallers
Figure| 1: Logical flow of sample complexity derivation. Bounds: (i) we lower bound the test MSE by (1 A, where A, is the alignment of output and target; (ii) we establish upper bound on the proba- bility to observe good learning (i.e., strong alignment A 2 Rs 1) in the posterior using the negative-log- probability of the rare event of good learning in the prior; (iii) we define the energy E(a)) the negative log-Chernoff bound on the prior probability of successful learning (rate function); (iv) we leverage E(a) to bound the minimal sample size necessary for good learning in the posterior, Approximations: (v) Since the exact bound ((a) is intractable, we derive a variational approximation using cornellldaptatioon techniques to provide an explicit formula (Sec. Heuristies: (vi)-(vii) we propose "feature learning patterns" as heuris- tic variational candidates, selecting the pattern 9. that minimizes the approximation and utilizing heuristic scaling relations to model how feature amplification propagates through downstream layers (Sec. 5
approaches Bordelon and Pehlevan (2022); Aitchison (2021) I Saad and Solla type approaches Saad and Solla (1995); Arnaboldi et al. (2023): ; Bietti et al. (2022) (and their Bayesian counterparts Cui (2025) allow us to solve simple non- linerr sucher-student networks the rich regime.
This work examines feature learning through a Bayesian lens, where it is often defined as a deviation from the Gaussian Process (GP) prior. This deviation has been shown to manifest in several distinct regime dependent ways, including: (i) updates to the kernel while the distribution remains Gaussian Li and Som- polinsky (202T); I) Rubin et al. (2025;; AAiistoo et al.
( (2022); [Seroussi et al. (2023b); Aitchison (209); (ii) the distribution changes from zero mnaa-Gaussiia to a mixture of Gaussians Rubin et al. (2024); and (iii) the specialization of specific neurons Barbier al. (2025); well as combinations of the above Ringel al. (2025); Meegen and Sompolinsky (2024). However, the frameworks capturing these FL effects are often mathematically cumbersome and restricted to shallow networks. Even when extended to deep architectures Barbier et al. (202)) Li and Sompolinsky (2020), their computational complexity prevents the derivation of intuitive scaling laws. Ultimately, this difficulty in computation compounds with the context-specificity of these mechanisms, complicating unified comparison of scaling laws across varied architectural settings.
this work, we introduce novel heuristic framework addressing the challenging task of making first- principles predictions on sample complexity and feature learning effects in Bayesian NNs. We consider Bayesian NNs as these are a standard proxy for SGD-trained networks at equilibrium Mingard et al. (2020); Mandt et al. (2017). The heuristic nature of our approach (see Fig. [ for schematic overview), allows to bypass otherwise highly complex analysis and extract insights using simple pen-and- calculations, resulting in non-trivial predictions across broad range of architectures and settings. We indeed validate framework for two- and three-layer fully connected networks (using ReLU and Erf activations), attention heads, and convolutional neural networks, covering both teacher-student, regression, and classification tasks. Across these diverse settings, our main results are:
Predicting feature learning patterns: We predict which of the FL phenomena from the literature dis- cussed above will emerge in different architectures and across varying scales of architectural variables, such as input dimension, layer width, regularization, and parametrization choices such as mean-fiee mean-field versus standard scaling. We map transitions between these patterns and derive the explicit scaling laws governing their strength.
2. Predicting sample complexity scale:
We determine the scaling behavior of P., the threshold sample size at which learning is possible, as a function of the architectural variables and parametrization choices.
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2 SETUP
We consider here several types of feedforward networks, but, for the sake of clarity, we illustrate the main derivation deep fully connected networks (FCNs) and, later, when we analyze specific problems, augment it for convolutional neural networks (CNNs) as needed. Our FCNs are defined by
NL
Ni
d
f(x) - Σ of (h4 where 11 - Σ w Wijor(d ==1
1)(()) - W Jk
Σ k=1
(I)
==1
where can be any activation function, and refer to ht's 's pre-acitvations. We consider Bayesian neural networks, as Bayesian descriptions are a commonly used proxy for network behavior after long-time stochastic training |Wilson and Izmailov (2020); [Wilson| (2020): Naveh al. (202T). Alternatively, they represent an exact solution to Langevin dynamics with weight decay Welling and Teh (201 We denote the target function by y and the training sample size by P. and assume training with Mean Squared Error (MSE) loss. The quadratic weight decay for each layer / is set to KN/11/o where k is the ridge parameter and No = d is the input dimension. This choice of weight decay results in a Gaussian prior distribution for the weights given by W i.j (N0 of (1 for i = 1..., Ni J = I,.., Ni-1. The possible outputs f of such a network, given y and P, are then distributed according to the posterior:
(f v. Z exp
1 2x
-
If(a))) - v(2)))) po(f),
(2)
where Z is the normalization constant, { )== is the training dataset of size P. and po( is the prior defined as po( = d0p (0) 8 - determined by the weight decay. Here, is the collection of all network weights, and (8) corresponds to the density of the prior weight distribution, which take to be Gaussian with a diagonal covariance, representing quadratic weight decay, and fe is the network architecture with weights O. We further set p(O) such that antaaiiivsso are all (11) under the prior He let al. (2015). For classification tasks, see App. A.2 As a measure for learning, we consider an observable, which we refer as alignment, given by
A, (/.))/yy. y):
(3)
where (g. h1) = f ( (x(x)) is the functional inner product, and dus is some test measure, which, con- veniently, does not need to be the measure from which the training set was drawn. We similarly define (g, K, h) = ʃ du K() (x, x' e²)h(z') for any kernel K. Alignment represents the extent to which the network learns a function that proportional to the target. bounds the test MSE via the Cauchy-Schwarr inequality f (f(x((()) - r(z))² 2 (v. (A, - 1)2. Having As = is thus a necessary condition for successful learning.
3 ALIGNMENT AND SAMPLE COMPLEXITY
We turn to analyze sample complexity via an upper bound on the probability of finding A 2 a for R 1. We begin with a theoretical bound on the posterior that mainly depends on the chance that random network, chosen from the prior, produces an alignment of at least a. We denote the prior and posterior alignment probabilities by Prpo [A, 2 aj and Pr. [Ai 2 a] respectively. Following simple arguments (see App. and App. A.2 for generalization to classification.), we obtain the following bound on the log posterior
log (Prs [A, 2 a]) < Pk/ (2k) + log (Przo [A, 2 a)),
-0 E [(f(z)) - 21 is the only training-ser set dependent quantity and is generally of order one. The Bayesian interpretation of successful learning is having Pr. (A, 2 al R O(1). P Since
where k- = P-P MP
See also Lavie and Ringel (2025), for a similar data-agnostic bound in the context of lazy learning. More concretely, can assume some tolerance and then (A, 1) (e) So dem(Ar (A,
c)
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a random network is unlikely to achieve strong alignment, log Prya [A, 2 a] would typically be highly negative for large a. Therefore, sufficiently large data term is required to cancel this effect. Explicitly,
P 2 -2n log Prpo [A, a) /k:
(5)
Thus, up to the ridge parameter and the ((1) factor k. depending on the training set, the log probability of prior aligmment with the target lower bounds the sample complexity. Here, it is worth noting that the bound becomes tight when overfitting effects are small, which is typically the case for k/1 ~ O(1). Taking 6 + encourages overfitting (though often benignly BBrrlett et al. (2020)) and trivializes this bound. We conjecture that, in this case, should be kept (1)) based on the effective ridge treatment Canatar et al. (202Ib); Cohen et al. (2021); Bartlett et al. (2020). Establishing this conjecture is outside the scope of this work. From PAC-Bayesian perspective, an analogous bound would require to be much larger than the KL-divergence between the prior and posterio (e.g.. McAllester (1999).. More recently. prior-posterior relations have been studied in the context of complex Boolean functions Mingard et al. (2025).
Following the Chernoff inequality, we can find an upper bound for the probability (and a lower bound for P) via
P2 2x/k Prpo [A, al E(a), E(a)
log inf " Eps [ctAr]. 1>0
(6)
Where we refer to E(a) as the energy. We can thus express the minimal sample size necessary for learning, P.. through the energy as P. x E(a). In App. B we provide an asymptotically exact solution for E(a). and compute it explicitly for two layer network. We also argue and demonstrate that our bound inherently tied to FI.. Indeed, a network sampled from the prior that achieves such alignment is statistical outlier, driven by the emergence of an internal structure which mimics FL (see also Fig. 2 Nevertheless, such a direct LDT approach is computationally prohibitive in most cases of interest. We therefore introduce a heuristic LDT-based method for evaluating P.. This method not only enables predicting the scaling of P. but also the FL effects that lead to successful learning.
In this section, we adopt a variational approach to estimating P, by comparing different modes of feature learning under a certain loss (see (11) below). While many approaches predict different FL mechanisms Pacelli et al. ) ; Fischer et al. (2024); Meegen and Sompolinsky (2024); Buzaglo et al. (2025): Li and Sompolinsky (2021); Seroussi et al. (2023c); Rubin al. (2025) 2024); Barbier et al. (2025). they are often case-dependent, highly detailed, and complex. Thus, we propose a method that abstracts key FL mechanisms from these frameworks into distinct, comparable patterns.
4 VARIATIONAL. ANALYSIS
Our next objective is to make the sample complexity bound tractable. This requires estimating the prior probability term, Prpo [A, 2 a), for alignments = 1. As a first step, we simplify this by relating the cumulative distribution function to the probability density denoted by (a). As shown in App. A.3] for large aligmments, we have E(a) z - log (a). This allows us to re-express P. in terms of the density: P. = - 2x log.l,, (a) (a)/k. However, computing Pa,(a) (a) directly remains intractable. We therefore turn to a variational approach estimate it. As explained in the next section, we wish to express the variational probability density in terms of pre-activations f (I. Accordingly, in App. C.I mechanics techniques express this density as
we follow standard statistical
L-1 N
PAy (a) =
f Dh. (a 0, (v. kL-1.8) v)
II [=1 1=1
(1) 1.
'Following the data- data-proceee peneesssinnniqqualiiyy one can lower-bound the KL-divergence between the full prior and poste- rior probabilitics by the KL-divergence of a coarser probability of an A, 2 a event in the prior and posterior. The latter KL. divergence is given by - log Prpo [A, 2 a)
Viewed here formally emergent weighh/pre-activationn structures enabling the outlier.
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Where for each the kernels K themselves depend on the preactivations of layer 1, and similarly the fluctuations in depend on the preactivations of the penultimate layer, given by
Riso(o,
0
Ni
(b((()) a (k((Z')), Ko(x,x')
σi
(8)
N
d
Using statistical mechanics based notation, we define the Hamiltonian Hp,a and fluctuating "partition func- tion" Za,
a²
Ni
Hp.a (h)
h1. K kr M!)
2 2 KL V, kL.111
i=1
log ZA, (h) log (y. + Σ Tr log K1-1.
Consequently, PA, (a) takes the form PA,, (a) = S exp (- (h) - log Za, where the integrand defines an effective measure on the preactivation space for a given a. We next approximate this measure per a by an analytically tractable variational estimate, 9. We follow a similar convention here: for any q, a, we define (h) := Lg (h),((),, requiring that the minimum value of Hq.0 (h) be zero. For a Gaussian
q. this reduces exactly to Eq. with kernels Kj-1 that are independent of h. The variational computation follows by looking for (f) that minimizes the KL divergence between the measure on h which defines PAy, and q. The KL divergence can also be used in the estimation of E(a) = ( following the provide a brief description for the full derivation see App. C.2 By applying the Haaammann-oooluboo inequality, we obtain upper bound on E(a)
inequality Kuzemsky (2015); (Bogolubov and Jr (2009); Huber (9688). Here we
E(a:) = min(Enogga [log( (A (h)/ + Eg(a)),
Eg(a) = Enugg [Hp (h) Hg.a(h) (h)] (10)
9a
We argue in App. C.4 that for 2 the log terms are subleading w.r.t Ey(a). Defining q.,a to be the (a). Next, we turn to estimating the variational
measure that minimizes (a). we obtain E(a) energy (a) Eq. (10) for a ~ 1, omitting all a indices for brevity. In App. C.I we simplify PAy, and show that the distribution in each layer depends only on the previous through fluctuating non-linear operator. Next, we assume that this kernel is weakly fluctuating, and replace it with its expectation w.r.t the variational distribution. This choice of approximation aligns with various works deep non-linear networks, where layer-wise kernels are identified as the relevant and sufficient set order parameters Rubin et al.) (2025); Fischer et al. (2024); Seroussi et al. (2023c); Ringel et al. (2025). We further take decoupled Gaussian variational ansatz so that q()) I
(h.(h)) where Qt is Gaussian with mean μ4, and variance Q1, As shown in App. C.3 the variational energy estimate is then given by
Ni
Eg x
(n, K,, (T..hh) + (pss Q7, (T...1..))) + (y. KL-1,8)-1
(11)
=:A
where we define K2 = [K]. Here, the A.s terms arise from the difference between the approximated kernel and the actual one, and the ay term results from enforcing an alignment
1. Requiring that q minimize Eg and a xe 1. we estimate E, x E(a se 1) x P.. Another interpretation of Â discussed in App. In D. the excess weight due to FL. This viewpoint is useful for FL patterns involving circuits, the latter have a sharp imprint in weight-space. The above kemnel viewpoint is, however, more general and can be used both for circuits and for more distributed learning patterns.
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5 HEURISTICS FOR MANUAL COMPUTATION OF VARIATIONAL APPROXIMATION
5.1 FEATURE LEARNING PATTERNS
While the above variational approach allows a variety of candidate q's, we focus on the previously mentioned set of feature-leaming scenarios that have been extensively studied in the literature. Although this subset may appear restrictive, by varying behaviors among layers and between different neurons of the same layer, it already captures a wide range of phenomena. We then need to compare the variational energy (Eg), as detailed in Sec. 4 for such combinations and select the minimizer. The optimal pattern is an indication of the FL that emerges in the network to enable strong alignment, as motivated in App. A.3 Concretely, per layer and neuron pre-activation (z), we allow one of the following choices:
(1) Gaussian Process (GP). Here, Qı is a Gaussian process (GP) so that h4, ~ (O, with K;-1 defined as the expectation of the kernel defined in (8) This choice defines the "base model" of FL. For FCNs 51 it implies that the network propagates feature structure forward without altering latent features (see Sec. 5.2 When all layers and neurons follow this distribution, the network reduces to the neural network GP (NNGP) Neal (1996), where no FL occurs. Introducing any of the patterns below in subset of neurons enables FL to emerge.
(2) Gaussian Feature Learning (GFL). In this scenario, pre-activations remain Gaussian with zero mean, but the covariance is modified relative to the GP scenario (I): the kernel of the previous layer is amplified by a factor D in the direction of specific feature (e.g., eigenfunction of K2-1) & one may also consider generalizations to several features). Thus, here too, the distribution is a GP but with a different covariance Q1,i given by
Q1a(z,x)) = KI-1(1, + Ki-1, k.)) &' (x)
(12)
(3) Specialization. In this scenario, a given neuron specializes to a particular feature d with proportionality constant Mr,i. This pattern corresponds to Gaussian distribution which is sharply peaked around a non-zero mean n1, db 6 Explicitly, we define the distribution of the specialized neuron
(.h 1 (hd, の - me,l. q,,((hh, Φ4 )) N0 (d) KI-1, 1)
(13)
5.2 LAYER-WISE FEATURE PROPAGATION
Since the variational energy of each layer depends the kernel of the previous layer, important element in our heuristic is understanding how the choice of pattern in a given layer affects the kernel and its spectrum the subsequent layer. To this end, we define feature learning as any deviation from the baseline GP pattern (see Sec. }. such as introducing non-zero mean to the distribution (i.e., specialization) or altering its covariance structure (i.e., GFL). In our framework, a "feature" refers either to the mean 14, of qt. or an eigenfunction of its covariance operator Q1, x').
We now outline several key claims concerning how features typically propagate between layers in FCNs. In this context, we consider a data measure that is i.i.d. Gaussian with zero mean and variance 1, not because it approximates the data well, but rather because it provides an unbiased baseline (see also|Lavie and Ringel (2025) for measuring function overlaps. Depending on the input, other choices can also be considered (e.g., permutation-symmetric measures over discrete tokens Lavie et al. (2024)). The following claims with their justifications should be understood as heuristic principles rationalizations of empirically observed
For CNNs, even in the lazy regime, deeper kernels have different input scope and hence generate a new structure. 'Taking equilibrated networks and increasing the amount of data, specialization was shown in Rubin et al. (2024) to emerge first-order phase transition where the average of preactivations suddenly shifts to Pr... This behavior further associated with Grokking, suggesting potential ppccialiaationgrookiin link.
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phenomena. Proving them in general or augmenting for different architectures left for future work. For further details and empirical results, see App. C.5
Claim (i): Neuron specialization creates spectral spike. Assume that M neurons in layer spe- cialize on a single feature (z)), the subsequent kemel K2 develops new, dominant spectral fea- ture corresponding to a a (z:). The corresponding RKHS norm of this feature amplified, scaling (M). Justification: When M neurons specialize, the next layer's kernel is approximately M
Ki(x,,=) = A(x,x²) + N. ( ( (z')), where A is the contribution from the non-specialized neu- rons. Treating the specialization term as a rank-1 update, the Sherman-Morrionn formula shows that its RKHS norm becomes (RT² M/N)-1 where R is the RKHS norm of A, which satisfies R M/N in typical high-dimensional settings.
Claim (ii): Amplified features in the pre-activation kernel create amplified higher-order features in the post-activation kernel. If a feature & (I) in kernel K, has its eigenvalue enhanced by factor D (i.e., A_ + X, D), then the corresponding m-th order power of this feature (02))(( (m())) (z) will have the bulk of its spectral decomposition, under the downstream kernel, shifted up by Dm, with similar effect on the inverse RKHS norm. Justification: A Taylor expansion of K1+1 in terms of the eigenfunctions of K2 shows that the term corresponding to (Ø') (zz)) will have a coefficient scaling with (). We argue that this term is difficult to span using other terms in this expansion, allowing us to treat it as a spectral spike and analyze it similarly to Claim (i). A numerical demonstration of this effect is shown in Fig. 5
Claim (iii): Lazy layers preserve the relative scale of features from the previous layer. In the absence of FL., a properly normalized lazy layer approximately preserves the eigenspectrum of the previous kernel. If a feature & (x) has an eigenvalue A. with respect to the pre-activation kernel given by K;-1, its effective eigenvalue with respect to the post-activation kernel K2 will also be proportional A.. Justification: Follows from Claim (ii) taking D = 1.
Propagation rules for FCNs
(1) Specialization: Layer specialized M neurons on Φ!. For any feature & satisfying (a( +) *
0, we approximate (0, 1 b)
x
Σ
sp. NJ
where we sum over all specializing neurons.
(2) GFL: Layer amplified fluctuation along t by D so that (+) KT- Φ = (D>,) where da is the GP value of the inner product. Then for any we have ((0)) )m KT (Φ.)) (DX.)
6 CONCRETE EXAMPLES
We now apply the heuristic principles of Sec. 4 5.1 5.2 to derive sample complexity bounds in few exam- ples. In App. E.II we benchmark method wo-layer FCNs and simple CNNs with non-overlapping patches. There we reproduce both the sample complexity exponent P. = d3/1 identified for CNNs in Ringel jtt al. (202)) and further predict that P. for two-layer FCNs studying a Hermite-3 target as well as the scaling of the number of specializing neurons with width (see Fig. 2 The latter is also setting for which the prior's 's upper bound can be computed directly from E(a) using Large Deviation Theory, leading to a good match with experiment (Fig. 2 panel (a)).
Going to what we believe is beyond the current analytical state of the art, in App. E.3 we predict P. VL,
where L is the context length, of softmax attention layer learning a cubic target (see Fig. 3 Another such instance, discussed in detail below, is that of 3-layer non-linear network learning non-linear target. App. A.2 we show that this approach can be extended to classification tasks as well, on Ganssis data. In App. E.2 we apply our heuristics to concrete setting, of two-layer ReLU network trained
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(4)
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(c)
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Figure 2: Numerical and experimental results for two-layer erf network trained the normalized third Hermite polynomial (m = 3). In panel (a) we compare the experimental results and exact theoretical predictions (computed utilizing LDT, see App. B for the distribution of the alignment of the hidden layer pre-activation with the linear feature. Here we follow the same notation as in (9). so that H₂ is the negative log posterior of the preactivations up to an additive constant that enforces zero minimum. We also find the pre-activation distribution corresponds to q(h) for q M-Sp, as predicted by our heuristic approach. Panel (b) compares theoretical and experimental predictions for P., defined as alignment alignment as a function of sample size). Both theoretical and experimental results agree on P. x d. In (c), we increase N and keep P and d fixed, and plot the number of specialized neurons in the hidden layer. In agreement with our heuristic predictions, the number of neurons increases linearly with v NNd..
0.1 (inset shows
parity task. We find there as well are able to predict the emergent feature pattern, which qualitatively differs from erf networks. Rather than identifying the scaling number of specializing neurons, we find that there a finite number of neurons and we are able to predict their scale.
6.1 THE THREE-LAYER NETWORK
Here consider thyee-layer FCNs given by f(z) = M a. (E wij ow z)). where Rd is drawn from (0, Ta). We train these networks polynomial target of degree given by y(x) Hem(w, z) where Hem is the m-th probabilist Hermite polynomial, which is the standard polynomial choice under our choice of data measure, and w. S some normalized vector. The networks are trained via Langevin dynamics Welling and Tch (2011b), with ridge parameter quadratic weight decay, and standard scaling. For an extension mean mean-field scaling. see App. E.I.I
(2)
As starting point for our analysis, consider the simplest pattern where taking an l'th layer be lazy means Qt, K and H4, 0. Following the choice of pattern, our goal is to estimate the scaling of Ey. Examining Eq. 11p. we find that the A contributions all cancel by above choice of Q's ss and 2 The only non-zero contribution thus comes from the final layer and is given by the inverse of (y. y) Because of lazy learning, K2(2, x') is a standard dot product FCN kernel which can be expanded Σm=1 0n(x xx /d)", with an = =O(1) w.r.t. Ni,2, d, P. It can then be shown that ky. K2.9) v) O(d-m) m) Leading to ÉGP-CP = a, x d
having two GP/lazy layers
Next, we consider FL pattern wherein the first layer is GP distributed but the second has FL GgPS=" Specifically, for the I = layer, we take Q1,i = M1,
thereby nullifying again A Eq. [T for Fgg. For the second layer, we assume M2 specializing neurons (e.g., = 1..M2) which fluctuate around the linear feature (w, .2), while others lazy, namely Q2,i>M K K1.12,0M = and Q Q2
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K1, 12, (z) = (w.+ Examining Σx 2, i in Eq. 11 we get zero contributions from 22,i> = and M2((w, x), KJ (uc,- x)) from == 1.. As Ki is again simple FCN dot product kernel with no FL effects, normalized linear functions such as w, x have an ((d) RKHS norm. We thus obtain an overall contribution to Eg from the 2 layer equal to M2d. Finally, we need to estimate dy (gl)) Note that K2 is not a standard FCN kernel anymore, since Q2, which contains target information, is used in its definition as the expectation of the kernel appearing in (Eq. Bp. According to feature propagation rule (i), with 4 = (w, I), we have ay = M2/M2. Given M2, we thus obtain a variational energy of M2d + /2 We next need to minimize over free parameters, namely M2 leading to M2 v /N2/d and
finally ECP-Sh, Sp. V22d. Provided N₂ scales less than d2m-1 (N2 = o(d2m-1)) this pattern is favorable to GP GP.
Finally, we consider what turns out be the favorable pattern consisting of M1 neurons specializing on (w, - x) in the first layer and all neurons in the second layer specializing Hem(w, z), with a small propor- tionality constant μız,4 + /B/N22 (q"=
We refer to the second-layer pattern as magnetization. Following straightforward adaptation previous argument to this pattern, the variational energy for this pattern well as others, for m 3, can be found in Table 1
Feature Pattern
A
A
ay
Minimizing Parameters
E
GP-GP
ds
d
GP-Sp.
0
M2d
Na/MM2
M2 VNaa
v N2d
Sp.-Mag.
Mid
Nr.9/M,
Na/B
(N3/Nga)
1/3
M1 (NaN1/88)1/8
(N₂ N2d) 1/3
Table Variational energy E for different choices of feature-leariing patterns a three-layer FCN trained on g(x) = Hes( (w. .z). The patterns shown here are (first/second layer): GP-GP, GP aP-Specialization, and Specialization-Mett -aaattiinn-annnniaatiinnnn. For each pattern, the components of the variational energy (Ai, A2, ag) together with the corresponding minimizing parameters are shown. We comment that the GP-GP pattern is favorable only for d > N,, and otherwise FL will emerge.
In the non-GP g patterns, namely, P./K x d. This observation is validated experimentally in Fig. where the transition to non-zero alignment becomes sharper in the thermodynamic limit (d oc). However, the mechanism by which this scaling is realized changes. In the N//33 therefore, it increases with Ni. However, under the GP-specialization pattern, sample complexity does not scale with N1, making this pattern preferable. This prediction is in line with experimental results (see Figs. aand panel (c)) where increasing N1 causes the described change FL patterns. Our prediction also accurately determines the scaling of the number of specializing neurons with N1.
obtain the same scaling of Fyg in the proportional limit (Ni N₂ d).
pattern the sample complexity scales with
6.2 SOFTMAX ATTENTION
Here, we consider an attention block of the form
H
f(x)
Σ
@at.h (X):n(()(e (X)(wn z²), @a)i = cla""] [on]JTALT Anz" clur Anoeg-1, (14)
VL h= a
where X E IR Lxd, A E Rdxd, and xa E Rdd is the a-th row of X, and wh E RRd, Our prior on network weights is Ih N(O, La/d L/P;; An,, (dH)) wx). The only dependence on the context length L arises from the fre-fa I/VI, which ensures that for X -N(0,,), 1), we have f(X) O(1). The target
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function is given by y(X) Ea.b VL(L-1] 22ag.. also normalized be (1). Following our approach, we propose two patterns for this architecture: GP (or lazy learning) and specialization (where we take A, NNuo @ (a--)), I for og [1, 0; 0, 1] and optimize
μ)) As detailed in E.3 the variational energy scales as Ld for the GP pattern and as VHLd3 for the specialization pattern, the latter thus being favorable for H scaling less than Ldd. As shown in Fig. this scaling of P with L and d indeed matches the dependence of the sample complexity on L.
10
(k)
(4)
No. of outlers
Theory Hounstic
M-Sp GP
P.ad
LE
(N
Figure 3: Sample complexity: Heuristic predictions accurately capture sample complexity in both three- layer erf FCNs and softmax attention heads, as well as feature learning scaling. Panels (a) and (b) track network alignment against the ratio between the sample size, P, and the predicted sample complexity- P/d for FCNs in panel (a), and P/v Ld3 for attention heads in panel (b). In both cases, alignment collapses onto a single curve, confirming the sample complexity predictions. See Fig. for MSE comparison and Apps. for further details. FL patterns: Panel (c) tracks linearly specialized neurons in the first (blue) and second (purple) layers against N1 (with fixed P. d and N2). The number of first-layer specializing neurons initially follows the predicted (N1 /d)(1/33 scaling until the transition where second- neurons specialize on the linear feature, and the first layer approaches the GP distribution.
E.3
7 DISCUSSION
This paper presents a novel methodology for analyzing the scaling behavior of sample complexity, through which we can also understand how distinct learning mechanisms emerge. Its strength lies in abstracting away from fine-grained details to isolate the core principles at play. We hope such a strategy would remove barriers and expedite connections between mechanistic interpretability and first-principles scientific approaches.
Limitations. Notwithstandingg several avenues for improvement remain. In particular, quantifying feature propagation in more general CNNs and transformers, and addressing multi-feature interactions in the context of superposittor/llhaee et al. (2022). It would also be desirable to extend our heuristic to dynamics of learning, potentially drawing insights from previous work relating equilibrium and dynamical phenomena Power et al. (002): y Rubin et al. (2024); Bahri et al. (2024); [aa et al. (2024). Since Bayesian convergence times be slow, correctly predicting the emergence of FL. in early stages of training may also be highly advantageous. Finally, in some cases, such as under mean-field scaling or vanishing ridge, overfitting effects can emerge, leaving our lower bound vacuous. Extending our approach to patterns that align only on the training set and incorporating effective ridge ideas Canatar et al. (202 la) is thus desirable.
ACKNOWLEDGMENTS
Funded by the the Israeli Science Foundation - 374/23.
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Reviews:
[{"review_id": "KQJV1hqRkh", "reviewer": "Reviewer_eTgY", "summary": "The paper uses variational energy bound and proposes a set of heuristics to characterize the scaling property of sample complexity ($P^*$ such that $logPr_{p_0}(A_f \\geq \\alpha)$ . The heuristics (which is an approximation of the variational energy) are developed for the feature learning patterns of Gaussin Process, Gaussain Feature Learning, and Feature Specialization. The paper then uses these heuristics to estimate the variational energy of these feature learning patterns of Fully-Connected 2-layer networks, CNN with Non-overlapping Patches, and Three-layer Network", "questions": "1. In Eq. 7 and Eq. 8, how does $\\tilde{E}_q$ depends on $\\alpha$?\n2. (Fig 2B) How do we compute the number of specialist neurons in each layer?\n3. How does the three feature learning patterns (GP, GFL, and Specialization) get selected? Would these patterns cover most feature learning space?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper addresses a well-defined problem, which is how to estimate the scaling property of sample complexity, in deep network, with is usually analytically intractable. To address this problem, the author uses lower-bound variational energy and a set of heuristics.\nThe paper supports their claim by both theoretical results and simulations, and show that these heuristics can estimate the lower bound variational energy and derive the sample complexity for fully-connected network and CNN with non-overlapping patches, which match with previous analytical results in the literature.\nThe paper also shows that their method can be applied to a more complicated network such as three-layer neural networks, which is difficult to get analytical results", "weaknesses": "As the paper employs heuristics to estimate the the lower variational bound for a specific set of feature learning patterns (Gaussian Process, Gaussian Feature Learning, Feature Specialization), under a specific task setup (polynomial of degree m), it's difficult to evaluate how these heuristics of specific feature learning patterns can be applicable for different tasks (classification, regression with a more general basis functions). I would appreciate if the authors could discuss the expected applicability or limitations of their framework beyond the polynomial setting.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "lLxlJd1QhT", "reviewer": "Reviewer_tHzt", "summary": "This paper proposes a theoretical framework for understanding feature learning through an alignment-based measure that connects learned representations with test mean squared error (MSE). The authors derive inequalities suggesting that when alignment is close to one, the model generalizes well, and they further develop probabilistic bounds on alignment exceeding certain thresholds. The motivation is strong: linking alignment to generalization performance could offer a compact diagnostic tool for feature learning across architectures.\n\nHowever, I found the mathematical exposition—particularly in Sections 2, 3, and 4—very confusing and difficult to follow. The direction of key inequalities, the interpretation of constants, and the definitions of several introduced symbols are unclear. As a result, I cannot confidently assess the soundness of the theoretical claims, even though the overall goal of the paper is promising.\n\nThe paper presents an interesting and well-motivated idea, but the core theoretical exposition is too opaque to evaluate. The logical direction of key inequalities and the meaning of several symbols are unclear, which makes the central claims about alignment as a generalization proxy hard to verify. With clearer derivations, well-defined notation, and a complementary upper bound relating alignment to test MSE, this work could become significantly more compelling.", "questions": "1. In line 98, how does the lower bound on test MSE justify using alignment as a reliable proxy? Is there also an upper bound, or an argument showing that alignment close to 1 implies low error in practice?\n2. In Section 3, what is the intended range for α, and what does O(1) mean precisely in this context?\n3. In Section 4, should PAF(α) serve as an upper or lower bound on the probability of high alignment? Please clarify the direction and logic of this inequality.\n4. Please define all symbols that appear in main derivations (e.g., H_{p,α}, q, etc.) in the text rather than referring only to the appendix.\n5. Could you include a clear schematic or conceptual figure illustrating how alignment links to generalization, and summarize the assumptions under which the inequalities hold?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 1, "presentation": 1, "contribution": 2, "strengths": "* Strong motivation and relevance: The paper tackles an important and timely question—how to quantify representation quality via alignment—and aims to connect it to generalization performance.\n* Potential for theoretical contribution: If clarified, the proposed framework could bridge geometric and probabilistic views of feature learning, an area of broad interest to both machine learning and neuroscience communities.\n* Clear high-level narrative: The intuition that alignment captures “how well learned features match task structure” is appealing and consistent with empirical trends observed in deep learning.", "weaknesses": "* Unclear and inconsistent mathematical exposition: The logic behind the central inequality (around line 98) is confusing. It provides only a lower bound on test MSE in terms of alignment, so high alignment is at best a necessary—but not sufficient—condition for good generalization. The paper does not establish or even discuss whether an upper bound exists, which weakens the claim that alignment close to 1 can reliably indicate low test error.\n* Ambiguity in probabilistic statements: In Section 3, the analysis of “probability of alignment > α” is difficult to interpret because the relevant range of α is not specified and the meaning of O(1)  is unclear—does it denote “around one” or simply “a constant independent of dimension”? This ambiguity propagates through subsequent results.\n* Notation and definition issues: Several key quantities (e.g., H_{p,α}, q, and related expressions around lines 146–152) are either undefined or only referenced indirectly via the appendix. The paper should clearly define each symbol in the main text, especially when these terms appear in central inequalities.\n* Direction of inequalities unclear: In Section 4, it is unclear whether P(A)F(α)) is meant to upper bound or lower bound the probability of alignment exceeding α. The sign and logical direction of several inequalities seem inconsistent, making it hard to reconstruct the intended argument.\n* Difficult to evaluate validity: Because of these ambiguities, it is impossible to judge whether the mathematical derivations are correct or merely stated heuristically. The empirical illustrations in later sections cannot compensate for this lack of clarity in the core theory.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "EN90UjVHxr", "reviewer": "Reviewer_9jc9", "summary": "The paper proposes a scaling-based recipe for predicting when and where specific feature learning attributes (covariance spikes, neuron specialization) appear in deep nets. It uses a Bayesian description of neural networks to analyze the sample-size threshold needed to have $O(1)$ alignment  between a function (a kernel eigenmode) and the network. Heuristic propagation rules connect layers (e.g., specialization induces a downstream spectral spike; spikes amplify higher-order features), letting predictions compose across depth. Worked cases for data scaling exponents are given for 2/3-HL FCNs, as well as CNNs.", "questions": "Why is the error function being used for the activation function? If any can be used, why not just the standard ReLU?\nCan you clarify the point of the claims in 4.4?\nCan plots of the ConvNet case be provided? In other words, are the details you’re trying to abstract away the same ones that make ConvNets and MLPs appear differently? If so, why perform this analysis? Those differences seem key to understanding ConvNets.\nThis is framed as a new framework that is easier to apply than other approaches, so why isn’t it applied to make new predictions? Show me this goes beyond specific cases like 3 layer MLPs on isotropic data with the (seemingly specific) erf activation function", "limitations": "", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "There appears to be quality work done on the mathematics behind this paper’s analysis, as well as an original approach to mech interp. It’s clear not enough similar work is done in that community, so the authors should be applauded for their instinct to pursue this direction. Another major strength is connecting the results obtained back to previous findings, this time under a new light. The main math-heavy sections are well presented. This work appears to be quite significant to those in the mech interp field willing to read through the paper to understand its ideas and implications.", "weaknesses": "This paper would greatly benefit from plots showing how the theory matches results, not just empirical results which are quite difficult to connect back to specifically this work and not some other broad notions of the field. Another weakness is in the downplay of grokking: since this theory is entirely about near-convergence networks, it seems like there should be a much greater focus on things like weight decay or grokking; outside of the intro, I don’t see those mentioned again. Finally, some things need to be defined instead of assumed (take Figure 1(a)’s y-axis, what is H_pi?)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "HQzJGM9P3y", "reviewer": "Reviewer_d7wZ", "summary": "This paper proposes a new method to estimate sample complexity, i.e., the amount of data needed in order to learn a target function (measured in terms of alignment). The authors propose a bound in function space depending on how well a randomly drawn NN would align with the target. As this bound is uncomputable, they derive a variational approach based on a mean-field decoupling. They hypothesize two feature learning mechanisms that go beyond the NNGP baseline and derive certain signatures of feature learning, like eigenvalue spikes in the conjugate kernel and a description of feature learning in a three-layer model. This is verified on one empirical target function.", "questions": "- How does a bound on the prior correctly imply sample complexity for a feature learning network? Even after reading the appendix, I do not fully understand how this is possible.\n\n- How does the structure of the target function influence the sample complexity in this framework? I.e., if the target function only depends on $k\\ll d$ input coordinates, can it be shown that the sample complexity of the NN in that framework scales with $k$ not $d$ more generally?\n\n- How is $P_{*}$ depending on the input distribution of the data?\n\n- How do we know that the proposed minimizers of the variational densities NNGP, GFL and specialization are the true minimizers?\n\n- How does this connect to PAC-Bayes bounds? (https://arxiv.org/pdf/2110.11216)", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "- Clear quantitative predictions for general architectures which go way beyond the usual toy models or qualitative results.\n\n- Very general framework that can be applied independently of architecture, data distribution, etc.\n\n- New analysis of a three-layer non-linear network.", "weaknesses": "- Hard to follow at some places, especially the three-layer NN section. Lines 282-301 are very dense and should be made clearer in the camera-ready version.\n\n- The theory needs much more empirical validation. I.e., what happens with other target functions, other input distributions, etc.? Are the predictions still correct?\n\n- How do we know the proposed distributions actually minimize the variational energy? The paper would greatly benefit from a comparison of an SGLD-trained network and how close the neuron distribution of this network is to the ones proposed as the minimizers of the variational energy.\n\n- Some typos: line 70 misses a \".\" after the eq., lines 70 and 81 miss a \",\" after the eq., line 84 $x_{\\nu}$ is not the dataset but a point, line 219 has a \",\" at the end instead of a \".\", $\\Theta$ is not defined.\n\n- The central idea of the paper is related to [1] and especially to [2] where analytical bounds on the prior are derived in a simplified setting, but this work is not cited.\n\n[1]: https://arxiv.org/abs/2006.15191\n[2]: https://arxiv.org/abs/2505.24060", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Poster)