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\author{Antiquus S.~Hippocampus, Natalia Cerebro \& Amelie P. Amygdale \thanks{ Use footnote for providing further information
about author (webpage, alternative address)---\emph{not} for acknowledging
funding agencies.  Funding acknowledgements go at the end of the paper.} \\
Department of Computer Science\\
Cranberry-Lemon University\\
Pittsburgh, PA 15213, USA \\
\texttt{\{hippo,brain,jen\}@cs.cranberry-lemon.edu} \\
\And
Ji Q. Ren \& Yevgeny LeNet \\
Department of Computational Neuroscience \\
University of the Witwatersrand \\
Joburg, South Africa \\
\texttt{\{robot,net\}@wits.ac.za} \\
\AND
Coauthor \\
Affiliation \\
Address \\
\texttt{email}
}


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\icmltitlerunning{
What Makes a Strong Model? }


\begin{document}


\twocolumn[
  \icmltitle{What Makes a Strong Model?  A Unified Spectral Analysis of \\ Knowledge Transfer over High-dimensional Linear Regression} 

  \icmlsetsymbol{equal}{*}
  \icmlsetsymbol{corr}{\dagger}

  \begin{icmlauthorlist}
    \icmlauthor{Wendao Wu}{sch2,sch1}
    \icmlauthor{Fangqing Zhang}{sch2}
    \icmlauthor{Haihan Zhang}{sch2}
    \icmlauthor{Cong Fang}{sch2}
  \end{icmlauthorlist}

  \icmlaffiliation{sch1}{School of Mathematical Sciences, Peking University, China}
  \icmlaffiliation{sch2}{State Key Lab of General AI, School of Intelligence Science and Technology, Peking University, China}
  % \icmlaffiliation{sch3}{Institute for Artificial Intelligence, Peking University, China}

  \icmlcorrespondingauthor{Cong Fang}{fangcong@pku.edu.cn}

  \icmlkeywords{Machine Learning, ICML}

  \vskip 0.3in
]

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\begin{abstract}
Teacher-Student Knowledge Transfer (KT) is ubiquitous in modern machine learning, ranging from classical model compression via Knowledge Distillation (KD) to the emergent phenomenon of Weak-to-Strong (W2S) generalization. While existing studies offer isolated insights, a unified theoretical framework explaining the efficacy of KT across these disparate regimes remains lacking. In this work, we establish a unified spectral analysis of SGD dynamics in high-dimensional linear regression, elucidating the efficiency of KT across seemingly disparate regimes. We characterize KT efficiency through two distinct mechanisms: \emph{Spectral Horizon Expansion} in KD, which enables the capture of statistically inaccessible high-frequency signals, and \emph{Spectral Denoising} in W2S, where the student acts as a filter for optimization noise. Our framework unifies these phenomena, revealing that the efficacy of transfer is governed by the interplay between implicit regularization and heterogeneous spectral learning speeds over the spectrum.
% In the ``Strong Teacher, Weak Student'' setting (KD), we identify a mechanism of \emph{Spectral Horizon Expansion}: the teacher actively extends the student's effective optimization horizon beyond its intrinsic limit, facilitating the capture of high-frequency components that are statistically inaccessible from raw data. 
% Conversely, in the “Weak Teacher, Strong Student” regime, we uncover a \emph{spectral denoising} mechanism: a strong student leverages its excess capacity to filter out the teacher’s optimization noise, effectively recovering the teacher’s optimal population geometry from noisy labels. Our framework unifies these phenomena under a core insight: the implicit regularization arising from the coupling of the learning algorithm and data structure, together with the heterogeneous learning speed of parameters — as characterized by the spectrum of the data covariance matrix — governs the efficacy of knowledge transfer.

\end{abstract}



\section{Introduction}
\label{sec:intro}
\emph{Knowledge Distillation} (KD) has established itself as a cornerstone technique for model compression and efficient inference \citep{hinton2015distillingknowledgeneuralnetwork}. The classical KD paradigm follows a sequential two-stage pipeline:  initially, a high-capacity teacher is trained on the source dataset, and subsequently, a lightweight student is optimized to match the teacher's softened outputs. Surprisingly, this process frequently yields students with generalization capabilities superior to those trained directly on the same hard labels \citep{stanton2021doesknowledgedistillationreally}. This counter-intuitive success has fueled widespread adoption, yet the theoretical underpinning of \emph{why} mimicking a proxy model outperforms learning from ground truth remains a subject of active debate.

The landscape of Knowledge Transfer (KT) has since expanded beyond mere compression. \emph{Self-Distillation} (SD) \citep{furlanello2018bornneuralnetworks} demonstrates that a student with an identical architecture to the teacher can achieve better convergence, challenging the intuition that capacity gaps are necessary for transfer gains. Moreover, the \emph{Weak-to-Strong} (W2S) generalization paradigm \citep{Burns2023WeaktoStrongGE} asks whether a low-capacity supervisor can effectively guide a stronger model. Empirical evidence suggests that strong students can indeed surpass their weak teachers, raising a fundamental question for the alignment of future AI systems: \emph{How can a student exceed the very source of its supervision?} 

These diverse phenomena---KD, SD, and W2S---present a theoretical puzzle. A prevalent intuition posits that ``soft'' labels convey ``dark knowledge'' \citep{hinton2015distillingknowledgeneuralnetwork}, acting as a form of label smoothing or denoising. However, this view remains heuristic and fails to explain why students do not simply inherit the teacher's biases. Recent theoretical advances suggest that the mechanism lies in specific \emph{multi-view} data structures and the resulting \emph{implicit regularization} within optimization dynamics \citep{allenzhu2023understandingensembleknowledgedistillation}. This perspective implies that the core mechanics of KT manifest even in simplified settings, driven by the coupling between the learning algorithm and the data's geometry.

However, existing frameworks lack a fine-grained characterization of how optimization interacts with the intrinsic structure of the data. Prior theoretical works have overlooked the \textbf{spectral properties} of the data covariance matrix—the fundamental governor of learning dynamics that dictates which features are learned fast or slow. To the best of our knowledge, a unified framework that coherently explains KT across disparate capacity regimes (from ``strong-to-weak'' to ``weak-to-strong'') via the lens of spectral analysis remains elusive. In this work, we fill this gap by explicitly leveraging spectral properties to characterize the unified mechanics of knowledge transfer.

% These diverse phenomena---KD, SD, and W2S---present a theoretical puzzle. A prevalent intuition posits that ``soft'' labels convey ``dark knowledge'' \citep{hinton2015distillingknowledgeneuralnetwork} acting as a form of label smoothing. However, this view remains heuristic and fails to explain why students do not simply inherit the teacher's biases. 
% Recent theoretical advances suggest that the mechanism lies in the \emph{multi-view} data structure and the resulting \emph{implicit regularization} within optimization dynamics \citep{allenzhu2023understandingensembleknowledgedistillation}. This perspective implies that the core mechanics of KT manifest even in simplified settings, driven by the coupling between the learning algorithm and the data's properties. 

% Yet, a unified framework that coherently explains KT across these disparate capacity regimes—from ``strong-to-weak" to ``weak-to-strong"—remains elusive.

\textbf{A Unified Spectral Perspective.}
In this work, we bridge this gap by developing a unified theory within the canonical high-dimensional linear regression model optimized by Stochastic Gradient Descent (SGD). While linear models are simplifications, they offer a rigorous proxy for deep learning dynamics in the over-parameterized regime. 
Specifically, Neural Tangent Kernel (NTK) theory establishes that wide neural networks evolve as linear models over fixed feature representations \citep{jacot2018neural}. 
Recent work by \citet{malladi2023kernelbasedviewlanguagemodel} further validates this abstraction in the context of Large Language Models (LLMs), demonstrating that fine-tuning dynamics---the primary use case for KT today---can be approximately described by kernel regression, which is mathematically equivalent to linear regression in the reproducing kernel Hilbert space.

This equivalence allows us to rigorously formalize the notion of ``knowledge'' as a \textit{spectral coupling}: the spectrum of the data covariance matrix and also the \emph{distribution of target parameters} across these dimensions (which dictates where the signal resides). We analyze the dynamics of single-pass SGD (or online SGD) rather than static closed-form solutions (e.g., Ridge Regression). This choice is deliberate and crucial: unlike offline solvers, SGD captures the \emph{trajectory} of learning, allowing us to characterize how the student's effective horizon progressively expands to cover the signal distributed along the spectrum. 

\textbf{From Risk Decomposition to Spectral Mechanisms.} 
Armed with this perspective, we derive a finite-sample risk bound decomposition (Theorem \ref{thm:generalt2s}). This decomposition allows us to identify the specific spectral mechanisms governing distinct regimes. In the ``Strong Teacher'' regime (KD) (Section \ref{sec:distillation}), the teacher acts as a \emph{spectral guide}, extending the student's \textbf{Effective Optimization Horizon} to capture high-frequency signals that are otherwise statistically indistinguishable from noise. Conversely, in the ``Weak/Identical Teacher'' regime (W2S/SD) (Section \ref{sec:w2s_generalization}), the student functions as a \emph{spectral filter}, leveraging \textbf{Spectral Denoising} to dampen the teacher's high-dimensional optimization variance while recovering the underlying population geometry.

% \textbf{A Unified Spectral Framework.}
% Armed with this perspective, we first establish a finite-sample risk bound for general teacher-student learning (Theorem \ref{thm:generalt2s}). Our analysis explicitly decomposes the transfer risk into three distinct physical components: \emph{Propagated Teacher Error}, \emph{Student Optimization Error}, and \emph{Irreducible Alignment Bias}.
% This decomposition provides an exact characterization of knowledge transfer efficiency: efficient transfer is achievable if and only if the geometric alignment bias is negligible (Lemma \ref{lem:geometric_consistency}). Furthermore, it suggests that the efficacy of the paradigm is determined by the competition between the \emph{propagated teacher error} and the error inherent to \emph{direct student learning}. We analyze this trade-off to disentangle the mechanisms of KD and W2S.


% \textbf{KD as Spectral Horizon Expansion.}
% To demystify classical KD, we model the ``strong teacher'' as a kernel with slow spectral decay, capable of capturing complex, high-frequency features efficiently. We focus on the \emph{hard learning} regime (often termed the misspecified setting in kernel learning), where the ground truth relies heavily on these high-frequency components.
% In this setting, we show that the teacher acts as a \emph{spectral guide}: by amplifying the signal-to-noise ratio in the spectral tail, the teacher actively extends the student's \emph{effective optimization horizon}. This allows the student to acquire high-frequency components that resemble ``dark knowledge''—signals that are structurally representable by the student but statistically indistinguishable from noise under direct training. We prove that this \textbf{Spectral Horizon Expansion} guarantees a strictly lower risk for the student compared to learning from raw labels.


% \textbf{W2S and Self-Distillation as Spectral Denoising.}
% Conversely, we address the Weak-to-Strong (W2S) and Self-Distillation paradigms, where the student matches or exceeds the teacher's capacity. Here, the challenge is not discovering new features, but refining existing ones. We demonstrate that a strong student leverages its excess capacity to act as a \emph{spectral filter}.
% While the student inherits the teacher's beneficial signal in the low-dimensional subspace, it aggressively dampens the teacher's optimization variance in the high-dimensional noise subspace. We term this mechanism \textbf{Spectral Denoising}.
% Our analysis of Self-Distillation reveals that this denoising effect exists even without a capacity gap: simply re-training on the predictions of a peer model filters out high-frequency optimization noise, recovering a cleaner estimate of the underlying population geometry.





\textbf{Main Contributions} are summarized as follows:

\textbf{A Unified Spectral Framework for Knowledge Transfer.} 
We establish a comprehensive theoretical framework for analyzing Knowledge Transfer (KT) dynamics in high-dimensional linear regression. We derive a finite-sample risk bound (Theorem \ref{thm:generalt2s}) that rigorously decomposes the transfer error into three distinct components: \emph{geometric misalignment}, \emph{propagated teacher noise}, and \emph{student optimization error}. This decomposition, in conjunction with Lemma \ref{lem:geometric_consistency}, establishes an \textbf{exact characterization} of the conditions under which knowledge transfer is provably efficient.

\textbf{Demystifying KD, W2S, and Self-Distillation via Spectral Mechanisms.} 
We apply our framework to elucidate the mechanisms behind seemingly contradictory phenomena:
\begin{itemize}
    \item In the \textbf{Knowledge Distillation} regime, we identify a mechanism of \textbf{Spectral Horizon Expansion}: the teacher actively extends the student's \textit{effective optimization horizon} beyond its intrinsic limit, facilitating the capture of high-frequency components that are statistically inaccessible from raw data.
    \item In the \textbf{Weak-to-Strong} and \textbf{Self-Distillation} regimes, we uncover a \textbf{Spectral Denoising} mechanism: the student leverages its excess capacity to filter out teacher's optimization noise, effectively recovering the teacher's optimal \textit{population geometry} from noisy labels.
\end{itemize}


% The main contributions of this paper are summarized as follows:
% \begin{itemize}
%     \item We propose a unified  ...
%     \item We demystify KD, W2S, and SD under unified spectral conditions. Specifically, we interpret KD through a “spectral preconditioning” mechanism, while explaining W2S and SD via a “spectral denoising” mechanism.

% \end{itemize}




\section{Related Works}
\label{sec:related_works}

Our work bridges the gap between the theoretical analysis of over-parameterized optimization and the recent empirical surge in model-to-model supervision (distillation and weak-to-strong generalization).

\textbf{Analysis of Stochastic Gradient Descent.} 
The theoretical properties of SGD in high-dimensional and kernel regimes serve as the bedrock of our analysis. 
Foundational works have rigorously characterized the bias-variance decomposition of SGD, explicitly linking convergence rates to the tail decay of the feature covariance spectrum \citep{dieuleveut2016nonparametricstochasticapproximationlarge, zou2021benign, wu2022last, zhang2025learningcurvesstochasticgradient}. 
This spectral perspective has been further refined to establish optimality guarantees and analyze accelerated variants \citep{jain2018accelerating, pan2022eigencurveoptimallearningrate, li2023riskboundsacceleratedsgd, zhang2024optimalityacceleratedsgdhighdimensional}, providing a robust toolkit for understanding over-parameterized dynamics.
Most recently, this machinery has been instrumental in explaining the emergence of neural scaling laws \citep{Atanasov2024ScalingAR, lin2025scalinglawslinearregression}, demonstrating the power of linear models in capturing modern deep learning phenomena.

% \textbf{Analysis of Stochastic Gradient Descent.}
% The theoretical properties of SGD in high-dimensional linear regression serve as the foundation of our analysis. While early works focused on fixed-dimension regimes, recent advances have characterized the behavior of SGD in the over-parameterized setting where the number of parameters exceeds the sample size. 
% \citet{zou2021benign} and \citet{wu2022last} provide sharp instance-dependent bounds for the bias and variance of the last iterate, explicitly linking convergence rates to the tail decay of the feature covariance spectrum.
% We build directly upon the bias-variance decomposition techniques established in these works. Specifically, we adopt the step-decay learning rate schedule analyzed by \citet{ge2019step}, which is proven to achieve minimax optimal rates for least squares. However, unlike these works which focus on a single model learning from ground truth, we extend this spectral machinery to the \textit{transfer learning} setting, analyzing how the noise and bias of a ``Teacher" trajectory propagate through to a ``Student" optimizer.

% \textbf{Knowledge Distillation: From Empirical Success to Theoretical Mechanisms.}
% Since \citet{hinton2015distillingknowledgeneuralnetwork}, Knowledge Distillation (KD) has evolved into a cornerstone of modern AI, pivotal for instruction tuning and training efficient language models \citep{wang2023selfinstructaligninglanguagemodels, abdin2024phi3technicalreporthighly, Guo_2025}. 
% Theoretically, its success is largely attributed to \emph{variance reduction} and \emph{implicit regularization}. 
% For instance, \citet{menon2021statistical} demonstrate that soft labels approximate Bayes probabilities to reduce student variance, while \citet{mobahi2020selfdistillationamplifiesregularizationhilbert} characterize self-distillation as an implicit regularization process that progressively restricts effective basis functions. 
% Complementary perspectives include the ``multi-view'' feature hypothesis \citep{allenzhu2023understandingensembleknowledgedistillation} and the amplification of spectral bias \citep{nagarajan2024studentteacherdeviationsdistillationdoes}. 
% Distinct from these works, which often focus on specific regimes (e.g., classification or early training), our work provides a unified finite-sample risk analysis that rigorously explains both the ``Strong Teacher'' (Horizon Expansion) and ``Weak Teacher'' (Denoising) phenomena within a single framework.

\textbf{Knowledge Distillation.}
Since \citet{hinton2015distillingknowledgeneuralnetwork}, Knowledge Distillation (KD) has become a cornerstone of modern AI, widely deployed in LLM instruction tuning and efficient model training \citep{wang2023selfinstructaligninglanguagemodels, abdin2024phi3technicalreporthighly, Guo_2025}. 
Notably, \citet{furlanello2018bornneuralnetworks} pioneered the concept of \emph{Self-Distillation}, demonstrating that students can surprisingly outperform identical teachers.
Theoretically, these successes are attributed to diverse mechanisms: \citet{menon2021statistical} identify KD as a variance reduction technique via Bayes probability approximation; \citet{mobahi2020selfdistillationamplifiesregularizationhilbert} characterize it as implicit regularization restricting basis functions; others point to multi-view feature discovery \citep{allenzhu2023understandingensembleknowledgedistillation} or the amplification of spectral bias \citep{nagarajan2024studentteacherdeviationsdistillationdoes}.




% \textbf{Knowledge Distillation.}
% Since its popularization by \citet{hinton2015distillingknowledgeneuralnetwork}, Knowledge Distillation (KD) has been extensively studied. 
% \citet{furlanello2018bornneuralnetworks} importance weighting.
% \citet{mobahi2020selfdistillationamplifiesregularizationhilbert}provide a theoretical analysis of self-distillation in the classical regression setting where the student model is only trained on the soft labels provided by the teacher. In particular, they fit a nonlinear function to training data with models belonging to a Hilbert space under L2 regularization. In this setting, multi-round self-distillation is progressively limiting the number of basic functions to represent the solution. 
% \citet{allenzhu2023understandingensembleknowledgedistillation} study self-distillation under a more practical setting where the student is trained on a combination
% of soft-labels from the teacher and ground-truth targets. Specifically, the student objective function consists of a
% cross-entropy loss in the usual supervised task, and a Kullback-Leibler divergence term to encourage the student match
% the soft probabilities of the teacher model. They also introduce the ``multi-view" hypothesis to explain how ensemble,
% knowledge distillation, and self-distillation work. 
% \citet{nagarajan2024studentteacherdeviationsdistillationdoes} classification setting, it is a dynamic process that amplifies the implicit spectral bias of Gradient Descent
\textbf{Weak-to-Strong Generalization.}
Initiated by the empirical findings of \citet{Burns2023WeaktoStrongGE}, this paradigm has spurred diverse theoretical inquiries. 
A dominant strand analyzes the phenomenon through static or offline regression frameworks: \citet{moniri2025mechanismsweaktostronggeneralizationtheoretical} and \citet{dong2025discrepanciesvirtueweaktostronggeneralization} attribute the student's success to its ability to compensate for the teacher's insufficient regularization, while \citet{ildiz2025highdimensionalanalysisknowledgedistillation} derive optimal surrogates in the ridgeless setting to maximize transfer efficiency. 
Beyond standard regression, \citet{charikar2024quantifyinggainweaktostronggeneralization} propose a ``misfit error'' framework to quantify performance gains, and \citet{Wu2024ProvableWG} establish provable guarantees in classification via benign overfitting. 
More recently, \citet{medvedev2025weaktostronggeneralizationrandomfeature} have extended the analysis to gradient flow in random feature models, primarily focusing on the bias dynamics.

\textbf{Spectral Evaluation of Representation Quality.} 
Recent empirical works propose spectral metrics to assess model quality, such as \emph{RankMe} (effective rank) \citep{garrido2023rankmeassessingdownstreamperformance} and \emph{$\alpha$-ReQ} (spectral decay) \citep{Agrawal2022alphaReQA}. 
Our framework rigorously grounds these heuristics: we prove they are causal drivers of transfer efficiency, where higher rank enhances \emph{noise resilience} (Theorem \ref{thm:rank_deficient_der}) and slower decay ensures \emph{feature coverage} (Theorem \ref{thm:der_rate}).


\textbf{Notation.}
We use bold lowercase letters for vectors and bold uppercase letters for matrices.
The symbol $\otimes$ denotes the Kronecker product.
For a vector $\boldsymbol{\eta}$, we denote its outer product as $\boldsymbol{\eta}^{\otimes 2} := \boldsymbol{\eta}\boldsymbol{\eta}^\top$.
For any two matrices $\mathbf{A}, \mathbf{B}$ of compatible dimensions, their Frobenius inner product is defined as $\inner{\mathbf{A}, \mathbf{B}} := \tr(\mathbf{A}^\top \mathbf{B})$.
Regarding asymptotic analysis, we use $\tilde{\mathcal{O}}(\cdot)$, $\tilde{\Omega}(\cdot)$, and $\tilde{\Theta}(\cdot)$ to suppress polylogarithmic factors; for instance, $f(n) = \tilde{\Theta}(g(n))$ implies $f(n) = \Theta(g(n) \text{polylog}(n))$.


\section{Problem Formulation and Preliminaries}
\label{sec:preliminaries}

\subsection{Teacher-Student Learning Setup}
\label{subsec:T2S_setup}

\textbf{Data Generation.}
We model the learning task as a regression problem over a compact input domain $\mathcal{X}$. Training data $\{(\mathbf{x}_i, y_i)\}_{i=1}^N$ are drawn i.i.d. from a probability measure $\rho_{\mathbf{x}\times y}$. The labels are generated by a ground truth function $f_*$ contaminated by noise: $y = f_*(\mathbf{x}) + \epsilon$, where $\epsilon \sim \mathcal{N}(0, \sigma^2)$ is independent Gaussian noise.
We assume the existence of an underlying \textit{semantic basis} $\mathbf{\Phi}: \mathcal{X} \to \mathbb{R}^D$ that is sufficiently expressive to represent the ground truth. Specifically, $f_*(\mathbf{x}) = \langle \mathbf{w}_*, \mathbf{\Phi}(\mathbf{x}) \rangle$ for some coefficient vector $\mathbf{w}_* \in \mathbb{R}^D$. Without loss of generality, we assume this basis is orthonormal with respect to the marginal distribution $\rho_{\mathbf{x}}$, such that $\mathbb{E}_{\mathbf{x}}[\mathbf{\Phi}(\mathbf{x})\mathbf{\Phi}(\mathbf{x})^\top] = \mathbf{I}_D$.


\textbf{Teacher and Student Models.}
We analyze the transfer of knowledge between a teacher model ($f_\te$) and a student model ($f_\st$). Both models operate on features derived from the common basis $\mathbf{\Phi}(\mathbf{x})$ via linear transformations. Let $\mathbf{M}_\te \in \mathbb{R}^{d_\te \times D}$ and $\mathbf{M}_\st \in \mathbb{R}^{d_\st \times D}$ denote the feature extraction matrices for the teacher and student, respectively. The feature maps are defined as:
\begin{equation}
    \boldsymbol{\phi}_\nu(\mathbf{x}) = \mathbf{M}_\nu \mathbf{\Phi}(\mathbf{x}), \quad \text{for } \nu \in \{\te, \st\}.
\end{equation}
The covariance matrices of these features are naturally given by $\mathbf{\Sigma}_\nu = \mathbf{M}_\nu \mathbf{M}_\nu^\top$. We assume $\mathbf{\Sigma}_\nu$ are trace-class operators (i.e., $\tr(\mathbf{\Sigma}_\nu) < \infty$) to ensure bounded signal energy.
% \begin{remark}
%     The assumption of an underlying semantic basis $\mathbf{\Phi}(\mathbf{x})$ is grounded in the spectral theory of over-parameterized networks. By Mercer's theorem \citep{mercer1909functions}, the kernel governing the network's evolution admits an eigendecomposition into orthonormal eigenfunctions. Crucially, in the analysis of Random Feature Networks (RFN) \citep{rahimi2007random} and Neural Tangent Kernels (NTK) \citep{jacot2018neural} on the hypersphere $\mathbb{S}^{d-1}$, this basis $\mathbf{\Phi}(\mathbf{x})$ corresponds precisely to the \textit{spherical harmonics} \citep{basri2019convergence, bietti2019inductive}. 
% \end{remark}

\textbf{Spectral Decomposition and Projections.}
To characterize the expressivity of the models, we consider the Singular Value Decomposition (SVD) of the feature mappings. Let $\mathbf{M}_\st = \mathbf{U}_\st \mathbf{\Lambda}_\st \mathbf{V}_\st^\top$, where $\mathbf{V}_\st \in \mathbb{R}^{D \times d_\st}$ spans the active row space of the student.
We define $\mathbf{\Pi}_\st := \mathbf{V}_\st\mathbf{V}_\st^\top$ as the orthogonal projection onto the student's feature space. Similarly, $\mathbf{\Pi}_\st^\perp := \mathbf{I}_D - \mathbf{\Pi}_\st$ is the projection onto its null space, representing the information strictly inaccessible to the student.
For any spectral cutoff $k$, we denote $\mathbf{\Pi}_\st^{\le k}$ as the projection onto the subspace spanned by the top-$k$ right singular vectors of $\mathbf{M}_\st$.

\textbf{Learning Objectives and Risks.}
We evaluate learning performance through two lenses: direct learning from ground truth and knowledge transfer.
The teacher and student models, parameterized by $\mathbf{w}_\te$ and $\mathbf{w}_\st$, approximate the target as $f_\nu(\mathbf{x}) = \langle \mathbf{w}_\nu, \boldsymbol{\phi}_\nu(\mathbf{x}) \rangle$. The standard supervised risks are:
\begin{equation}\label{eq:risk_supervised}
    \!\!\!\mathcal{R}_\nu(\mathbf{w}_\nu) = \frac{1}{2}\mathbb{E}_{\mathbf{x}}\left[ (f_\nu(\mathbf{x}) - f_*(\mathbf{x}))^2 \right], \ \text{for } \nu \in \{\te, \st\}.
\end{equation}
Let $\mathbf{w}_{\nu}^*$ denote the population optimal parameters minimizing Eq. \eqref{eq:risk_supervised}. 
% By the orthogonality of the projection, these are given by $\mathbf{w}_{\nu}^* = (\M_\nu^\top)^+ \w_*$, where $(\cdot)^+$ is the Moore-Penrose pseudoinverse.

In the \textit{Knowledge Transfer} process, the student learns to mimic the teacher's output. We define the \textit{Teacher-to-Student} (T2S) model $f_\mathrm{T2S}$ parameterized by $\mathbf{w}_\mathrm{T2S}$, which is optimized on the transfer risk:
\begin{equation}\label{eq:risk_transfer}
    \mathcal{R}_{\mathrm{Trans}}(\mathbf{w}_\mathrm{T2S}) = \frac{1}{2}\mathbb{E}_{\mathbf{x}}\left[ (f_\mathrm{T2S}(\mathbf{x}) - f_\te(\mathbf{x}))^2 \right].
\end{equation}
Let $\mathbf{w}_{\ts}^\opt$ denote the population optimal parameter minimizing Eq. \eqref{eq:risk_transfer}. When the student is trained with the optimal teacher model, i.e. $f_\te(\vecx) = \inner{\w_\te^*,\bs\phi_\te(\vecx)}$, we denote $\mathbf{w}_{\ts}^\opt$ in this case as $\mathbf{w}_{\ts}^*$.

The effectiveness of knowledge transfer is ultimately measured by how well the transferred student $f_\mathrm{T2S}$ recovers the ground truth, quantified by the risk $\mathcal{R}_{\mathrm{T2S}}(\mathbf{w}_\mathrm{T2S}) := \frac{1}{2}\mathbb{E}_{\mathbf{x}}[ (f_\mathrm{T2S}(\mathbf{x}) - f_*(\mathbf{x}))^2 ]$.

Many following analysis focus on the \textbf{Excess Risk}, defined as the generalization gap above this population optimum:
\begin{equation}
    \!\mathcal{E}_\nu(\mathbf{w}_\nu) := \mathcal{R}_\nu(\mathbf{w}_\nu) - \mathcal{R}_\nu(\mathbf{w}_\nu^*),\  \text{for } \nu\in \{\te,\st,\ts\}.
\end{equation}



Next, we define the quantitative metrics used to characterize distillation efficiency and weak-to-strong generalization.

\textbf{Quantifying Distillation Efficiency (Strong Teacher).}
When a \textbf{Strong Teacher} instructs a Weak Student, the primary benefit is accelerated convergence. We quantify this via the \textit{Distillation Efficiency Ratio} (DER). Let $\mathcal{E}_{\st}(N)$ be the excess risk of a student trained on $N$ ground-truth samples. We define $\mathcal{E}_{\mathrm{T2S}}(N, n)$ as the risk of a student trained on a separate set of \textbf{$n$ transfer samples} labeled by the teacher. The DER is given by:
\begin{equation}
    \mathbf{DER}_N := \frac{\mathbb{E}[\mathcal{E}_{\st}(N)]}{\mathbb{E}[\mathcal{E}_{\mathrm{T2S}}(N, n)]}.
\end{equation}
Values $> 1$ signify \textit{data amplification}, implying the student learns more efficiently from the teacher's synthesized knowledge than from raw data.


\textbf{Quantifying Weak-to-Strong Generalization.}
In the regime where a \textbf{Weak Teacher} supervises a Strong Student, our focus shifts to whether the student can transcend the teacher's limitations. We formally characterize the \textit{occurrence} of the Weak-to-Strong (W2S) phenomenon by the strict inequality:
\begin{equation}
    \mathcal{R}_{\mathrm{T2S}}(\mathbf{w}_\ts) < \mathcal{R}_\te(\mathbf{w}_\te).
\end{equation}
To further quantify the \textit{quality} of this generalization, we adopt the \textbf{Performance Gap Recovered (PGR)} metric \citep{Burns2023WeaktoStrongGE}. While the inequality above signals the existence of W2S, PGR measures the extent to which the distilled student recovers the optimal capabilities accessible only via ground truth supervision:
\begin{equation}
    \mathbf{PGR} := \frac{\mathcal{R}_\te - \mathcal{R}_{\mathrm{T2S}}}{\mathcal{R}_\te - \mathcal{R}_\st}.
\end{equation}




\subsection{Assumptions and SGD Settings}
We formalize the training dynamics as a two-phase protocol consisting of \textbf{Teacher Supervised Training} and \textbf{Student Transfer Learning}. Both phases utilize a step-decay learning rate schedule, which is shown to be asymptotically optimal for linear regression \citep{ge2019step}. We define the schedule scaling function $\mathcal{S}(t) := 2^{-\lfloor t/K \rfloor}$, where $K = N/\log_2 N$ represents the decay interval.

In the \textbf{first phase}, the teacher model is trained on a dataset $\mathcal{D}_N$ of size $N$, drawn i.i.d. from the ground-truth distribution $\rho_{\mathbf{x}\times y}$. The optimizer follows the schedule $\gamma_t = \gamma_0 \cdot \mathcal{S}(t)$. In the \textbf{second phase}, the student is trained on a separate dataset $\mathcal{D}_n$ of size $n$, drawn from the marginal $\rho_{\mathbf{x}}$. The student learns from the fixed teacher's labels $\hat{y} = f_{\mathrm{T}}(\mathbf{x})$ using the schedule $\gamma'_t = \gamma'_0 \cdot \mathcal{S}(t)$, where the initial step size $\gamma'_0$ is a tunable hyperparameter distinct from $\gamma_0$.


\begin{algorithm}[t]
   \caption{Two-Phase SGD for Knowledge Transfer}
   \label{alg:kd_training}
\begin{algorithmic}
   % \STATE {\bfseries Input:} Teacher dataset size $N$, Student dataset size $n$.
   % \STATE {\bfseries Hyperparameters:} Initial step sizes $\gamma_0, \gamma_0'$, decay interval $K$.
   % \STATE {\bfseries Schedule:} Define $\mathcal{S}(t) = 2^{-\lfloor t/K \rfloor}$.
   % \STATE {\bfseries Output:} Trained Student Parameters $\mathbf{w}_{\mathrm{S}}$

   \STATE \textit{\textbf{Phase 1: Teacher Supervised Training}}
   \STATE Initialize $\mathbf{w}_{\mathrm{T}, 0} = \mathbf{0}$.
   \FOR{$t=1$ {\bfseries to} $N$}
       \STATE Sample $(\mathbf{x}_t, y_t) \sim \rho_{\mathbf{x}\times y}$.
       \STATE Compute step size $\gamma_t \leftarrow \gamma_0 \cdot \mathcal{S}(t)$.
       \STATE Update $\mathbf{w}_{\mathrm{T}}$ via SGD on ground truth:
       \STATE \quad $\mathbf{w}_{\mathrm{T}, t} \leftarrow \mathbf{w}_{\mathrm{T}, t-1} - \gamma_t \nabla_{\mathbf{w}} \mathcal{R}_\te(\mathbf{w}_{\mathrm{T}, t-1}; \mathbf{x}_t, y_t)$.
   \ENDFOR
   \STATE \textbf{Fix} Teacher parameters $\mathbf{w}_{\mathrm{T}} \leftarrow \mathbf{w}_{\mathrm{T}, N}$.

   \vspace{0.1cm}
   \hrule
   \vspace{0.1cm}
   \STATE \textit{\textbf{Phase 2: Student Transfer Learning}}
   \STATE Initialize $\mathbf{w}_{\mathrm{S}, 0} = \mathbf{0}$.
   \FOR{$t=1$ {\bfseries to} $n$}
       \STATE Sample $\mathbf{x}_t \sim \rho_{\mathbf{x}}$.
       \STATE Generate teacher label: $\hat{y}_t \leftarrow \mathbf{w}_{\mathrm{T}}^\top \phi_{\mathrm{T}}(\mathbf{x}_t)$.
       \STATE Compute step size $\gamma'_t \leftarrow \gamma_0' \cdot \mathcal{S}(t)$.
       \STATE Update $\mathbf{w}_{\mathrm{S}}$ via SGD on teacher labels:
       \STATE \quad $\mathbf{w}_{\mathrm{S}, t} \leftarrow \mathbf{w}_{\mathrm{S}, t-1} - \gamma'_t \nabla_{\mathbf{w}} \mathcal{R}_\text{Trans}(\mathbf{w}_{\mathrm{S}, t-1}; \mathbf{x}_t, \hat{y}_t)$.
   \ENDFOR
\end{algorithmic}
\end{algorithm}
To analyze the convergence, we make the following standard assumption on the feature distribution.

\begin{assumption}[Fourth Moment Condition]\label{assump:fourthmoment}
    There exists a finite constant $\psi \geq 1$, such that for every positive semi-definite (PSD) matrix $\bs A$, the feature distribution satisfies:
    \begin{equation}
        \Ex_{ \rho_{\vecx}}\left[ \bs\Phi(\vecx)\bs\Phi(\vecx)^\top \bs A\, \bs\Phi(\vecx)\bs\Phi(\vecx)^\top  \right] \preceq \psi \cdot \tr(\bs A) \cdot \mathbf{I}_D.
    \end{equation}
\end{assumption}

\begin{remark}
    Assumption \ref{assump:fourthmoment} is widely adopted in works related to linear regression  \citep{Bartlett_2020,tsigler2022benignoverfittingridgeregression,zou2021benign,wu2022last}. 
    It characterizes the kurtosis of the feature distribution, ensuring that the variance of the stochastic gradient operator is bounded. 
    This condition is strictly weaker than assuming Gaussianity (where $\psi \approx 3$ by Wick's theorem) and holds for bounded or sub-Gaussian distributions.
\end{remark}


\subsection{Preliminaries on the SGD Dynamics}
\label{subsec:sgd_dynamics}

We analyze the standard SGD recursion for a generic model $f_\nu$ ($\nu \in \{\te, \st\}$) in the capacity-limited regime, adapting the framework of \citet{ge2019step} and \citet{wu2022last}.
Let $\bs\eta_t := \w_t - \w^*$ be the error vector and $\mathbf{P}_N := \Ex[\bs\eta_N^{\otimes 2}]$ be the second moment matrix.

\begin{lemma}[Direct Learning Dynamics]
% ; Proof in Appendix \ref{app:proofs_direct_learning}
\label{lem:direct_learning_bound}
Under Assumption \ref{assump:fourthmoment}, the expected excess risk satisfies:
\begin{equation}
\resizebox{\columnwidth}{!}{
    $
    \begin{aligned}
    &\Ex[\mathcal{E}_\nu(\w_{N})] = \frac{1}{2}\inner{\mathbf{P}_{N,\nu},\mathbf{\Sigma_\nu}} \nonumber \leq\\ &\underbrace{\frac{\|\bs\eta_0\|^2_{\Sig^{\leq k^*_\nu}_\nu}}{2N^2}}_{\text{Decayed Head}} + \underbrace{\frac{\|\bs\eta_0\|^2_{\Sig^{>k^*_\nu}_\nu}}{2}}_{\text{Preserved Tail}} \nonumber + \underbrace{16 \mathcal{C}_{\text{noise}} \left( \frac{k^*_\nu}{K} + K\gamma_0^2 \sum_{i > k^*_\nu} \lambda_{i,\nu}^2 \right)}_{\text{Variance}},
\end{aligned}
    $
    }
\end{equation}


\noindent where $k^*_\nu = \max\{k:\lambda_{k,\nu}> 2\ln N\log_2 N/ (\gamma_0 N) \}$, and $\mathcal{C}_{\text{noise}} := \psi\|\w_{0}-\w^*\|^2_{\bs\Sigma}+\sigma_\eff^2$. 
\end{lemma}




Lemma \ref{lem:direct_learning_bound} reveals two mechanisms:
(1) \textbf{Implicit Regularization.} The bias splits at the learning progress $k^*$, learning the top $k^*$ modes ($O(N^{-2})$ decay) while leaving trailing modes ($>k^*$) untouched.
(2) \textbf{Effective Noise.} The variance term $\mathcal{C}_{\text{noise}}$ aggregates algorithmic noise (scaling with estimation error $\psi\|\bs\eta_0\|^2$) and our key insight, the \textbf{effective noise variance} $\sigma_{\eff}^2 := \sigma^2 + \|\mathbf{\Pi}^\perp \w_*\|^2$, reflecting that the unlearnable component of the target acts as an irreducible noise to the student’s optimization process. 

\section{General Knowledge Transfer}
\label{sec:knowledge_transfer}

We now establish a unified framework for teacher-to-student training. Before analyzing the dynamics, we shall quantify the fundamental limit of knowledge transfer from a static perspective. 
Unlike direct learning, where the student targets the ground truth $\mathbf{w}_\st^* = (\M_\st^\top)^+ \mathbf{w}_*$, transfer learning targets the teacher's best approximation $\mathbf{w}_\ts^* = (\M_\st^\top)^+\M_\te^\top \mathbf{w}_\te^*$. The discrepancy between these two optima dictates the maximum potential utility of the teacher.

\begin{lemma}[Geometric Consistency Condition]
% ; Proof in Appendix \ref{proof:geometric_consistency}]
\label{lem:geometric_consistency}
    Knowledge transfer is consistent with direct learning (i.e., $\mathbf{w}_\ts^* = \mathbf{w}_\st^*$) if and only if:
    \begin{equation}
        \mathbf{\Pi}_\st \mathbf{\Pi}_\te^\perp \mathbf{w}_* = \mathbf{0}.
    \end{equation}
    If this condition is violated, the student trained by transfer suffers an irreducible \textbf{Static Alignment Bias} compared to the student trained on ground truth, even with infinite data:
    \vspace{-0.3cm}
    \begin{equation}
        \mathcal{R}_{\ts}(\mathbf{w}_\ts^*) - \mathcal{R}_{\st}(\mathbf{w}_\st^*) = \frac{1}{2} \| \mathbf{\Pi}_\st \mathbf{\Pi}_\te^\perp \mathbf{w}_* \|^2 > 0.
    \end{equation}
\end{lemma}

This lemma implies that if the Teacher is ``blind" to concepts ($\mathbf{\Pi}_\te^\perp \w_*$) that the Student is capable of representing ($\mathbf{\Pi}_\st$), the Student is structurally handicapped by the Teacher. Efficient transfer thus requires $\|\mathbf{\Pi}_\st \mathbf{\Pi}_\te^\perp \mathbf{w}_*\|$ to be negligible.

The following theorem extends this static intuition to the dynamic regime, providing the central decomposition.

% In this section, we establish a unified theoretical framework for teacher-to-student training. 
% Before analyzing specific algorithms like Distillation (Section \ref{sec:distillation}) or Weak-to-Strong Generalization (Section \ref{sec:w2s_generalization}), we should first quantify the fundamental dynamics of a student learning from a teacher with arbitrary feature representations. We propose firstly a lemma describing when knowledge transfer could be efficient.

% \begin{lemma}
%     Knowledge transfer could only be successful when $\w_\ts^* = \w_\st^*$, which is equivalent to 
%     \begin{equation}
%         \bs\Pi_\st\bs\Pi^\perp_\te\w_* = 0.
%     \end{equation}
% \end{lemma}
% \vspace{-0.3cm}

% The following theorem provides the central decomposition of the Student's risk after learning. It reveals that the student's error is not merely an optimization issue, but a complex interplay between the teacher's transmitted errors, the student's learning capacity, and the geometric alignment between their feature spaces.

\begin{theorem}[The Teacher-to-Student Risk Decomposition]
% Proof in Appendix \ref{subsec:proofgeneralt2s}
\label{thm:generalt2s}
    Let the Student be trained on $n$ samples labeled by a fixed Teacher (itself trained on $N$ samples). For any $0<\delta<1/2$, the expected knowledge transfer excess risk $\mathcal{E}_{\mathrm{T2S}}$ is bounded by:
    \begingroup
    \allowdisplaybreaks
    \small 
    \begin{align} 
        \label{eq:t2s_decomposition}
        &\Ex[\mathcal{E}_{\mathrm{T2S}}(N,n)] \\ 
        &\leq \underbrace{\frac{1}{2}\inner{\bs P_{N,\te},\M_\te\bs \Pi_{\st}^{\leq k^*}\M_\te^\top}+2\delta^2\inner{\bs P_{N,\te},\M_\te\bs \Pi_{\st}^{> k^*}\M_\te^\top}}_{\text{(I) Propagated Teacher Error}} \notag \\ 
        &+ \quad\quad\quad\quad\underbrace{\frac{1}{2}\inner{\bs P_{n,\ts},\Sig_{\st}^{\leq k^*}}+ \inner{\bs P^{\mathrm{var}}_{n,\ts},\Sig_\st^{> k^*}}}_{\text{(II) Student Optimization Error}} \notag \\
        &+ \underbrace{C\left\|\bs \Pi_{\st}^{\leq k^*}\bs \Pi_{\te}^{\perp}\w_{*}\right\|+\nm{\bs \Pi_{\st}^{> k^*}\w_{*}}+2\delta^2\nm{\bs \Pi_{\st}^{> k^*}\bs \Pi_{\te}\w_*},}_{\text{(III) Irreducible Alignment Bias}} \notag
    \end{align}
    \endgroup
    \vspace{-0.5cm}
    
    \noindent where $k^* = \max\{k:\lambda_{k,\st}> \delta\log_2 n/ (4\gamma_0' n) \}$ is the effective dimension, $\bs P_{n,\ts}:=\Ex[\bs\eta_{n,\ts}^{\otimes2}]$ is the second moment matrix of $\bs\eta_{n,\ts}:=\w_{n,\ts}-\w^*_{\ts}$ and $\bs P^{\mathrm{var}}_{n,\ts}:=\Ex[\bs\eta_{n,\ts}^{\otimes2}]-\Ex[\bs\eta_{n,\ts}]^{\otimes2}$ being the variance matrix of it, and $C$ is a constant.
\end{theorem}

\textbf{Decomposition Analysis.} Theorem \ref{thm:generalt2s} dissects the transfer risk into three physically distinct components governed by the student's spectral learning progress $k^*$ (the effective dimension learned by the student):

% \textbf{(I) Propagated Teacher Error (The Denoising View).}
% This term governs the projection of the teacher's optimization error tensor $\mathbf{P}_{N,\te}$ onto the student's geometry, revealing a dual mechanism that unifies our framework.
% In the student's \textit{active} subspace ($\le k^*$), the error undergoes \textbf{1-to-1 inheritance}, forming the theoretical cornerstone of \textbf{Knowledge Distillation} (Section \ref{sec:distillation}): since a Strong Teacher achieves lower $\mathbf{P}_{N,\te}$, the Student effectively bypasses its own optimization variance by inheriting this superior precision.
% Conversely, in the \textit{unlearned} spectral tail ($> k^*$), the teacher's error is \textbf{heavily dampened} by $\mathcal{O}(\delta^2)$. This suppression is the engine of \textbf{Weak-to-Strong Generalization} (Section \ref{sec:w2s_generalization}): it allows the Student to filter out the Weak Teacher's high-frequency noise, performing \textit{Spectral Denoising} rather than blind memorization.

\textbf{(I) Propagated Teacher Error.}
This term governs how the teacher's error tensor $\mathbf{P}_{N,\te} := \Ex[\bs\eta_{N,\te}^{\otimes 2}]$ is projected onto the student's geometry, re-weighting the native risk measure $\frac{1}{2}\inner{\mathbf{P}_{N,\te}, \mathbf{\Sigma}_\te}$, ($\Sig_\te = \M_\te\M_\te^\top$), from Lemma \ref{lem:direct_learning_bound}.
The first term, $\frac{1}{2}\inner{\bs P_{N,\te},\M_\te\bs \Pi_{\st}^{\leq k^*}\M_\te^\top}$, indicates a \textbf{1-to-1 error inheritance} within the student's learned subspace ($\le k^*$). This implies that for learned concepts, the student directly copies the teacher's precision, forming the foundation of \textbf{Knowledge Distillation} (Section \ref{sec:distillation}) where a student benefits from a Strong Teacher's lower error.
Conversely, the second term $2\delta^2\inner{\bs P_{N,\te},\M_\te\bs \Pi_{\st}^{> k^*}\M_\te^\top}$ reveals that in the unlearned tail subspace ($> k^*$), the teacher's error is not fully inherited but \textbf{heavily dampened} by $\mathcal{O}(\delta^2)$. This suppression is the core mechanism of \textbf{Weak-to-Strong Generalization} (Section \ref{sec:w2s_generalization}): it enables the student to filter out teacher's high variance in the tail— performing \textit{spectral denoising}—rather than blindly memorizing the noise.

% \textbf{(I) Propagated Teacher Error (The Student's View).} 
% This term governs how the teacher's optimization error tensor, $\mathbf{P}_{N,\te} := \Ex[\bs\eta_{N,\te}^{\otimes 2}]$, is projected into the student's feature space. 
% Recall that in Lemma \ref{lem:direct_learning_bound}, the teacher's native risk is measured by the inner product $\inner{\mathbf{P}_{N,\te}, \mathbf{\Sigma}_\te}$. 
% Here, however, the risk is re-weighted by the student's geometry, split by the effective cutoff $k^*$:

% \begin{itemize}
%     \item \textbf{The Learned Head ($\le k^*$) $\to$ Foundation of KD (Section \ref{sec:distillation}):} 
%     The first term, $\frac{1}{2}\inner{\bs P_{N,\te},\M_\te\bs \Pi_{\st}^{\leq k^*}\M_\te^\top}$, indicates a \textbf{1-to-1 error inheritance} within the student's active subspace. 
%     This implies that for concepts the student has successfully learned ($k^*$), it directly inherits the teacher's precision. 
%     This is the theoretical cornerstone of Knowledge Distillation: since a Strong Teacher achieves a much lower $\mathbf{P}_{N,\te}$ (due to larger capacity/data), the Student can bypass its own optimization variance by inheriting this lower error, effectively "standing on the shoulders of giants."

%     \item \textbf{The Unlearned Tail ($> k^*$) $\to$ Foundation of W2S (Section \ref{sec:w2s_generalization}):} 
%     The second term, $2\delta^2\inner{\bs P_{N,\te},\M_\te\bs \Pi_{\st}^{> k^*}\M_\te^\top}$, reveals that for the unlearned high-frequency components, the teacher's error is not fully inherited but \textbf{heavily dampened} by a factor of $\mathcal{O}(\delta^2)$. 
%     This spectral damping is the key mechanism behind Weak-to-Strong Generalization. 
%     It allows the Student to filter out the Weak Teacher's high variance in the tail (where the teacher is noisy) rather than memorizing it. By suppressing the noise in these dimensions, the Student achieves a form of \textit{Spectral Denoising}, converging to a solution cleaner than the teacher's own realization.
% \end{itemize}

% \textbf{(I) Propagated Teacher Error (The Student's View):} This term represents the portion of the teacher's error $\mathbf{P}_{N,\te}$ that is transmitted to the student.
% Notably, this is not the raw teacher error (which would be an inner product with $\mathbf{\Sigma}_\te$), but the teacher's error \textit{projected onto the student's currently learned subspace} via $\mathbf{M}_\te \mathbf{\Pi}_\st^{\le k^*} \mathbf{M}_\te^\top$.
% The coefficient $\delta$ (typically small, e.g., $<0.1$) signifies that the teacher's error is not fully propagated but dampened. As we will show in Section \ref{sec:w2s_generalization}, this dampening effect—where the student learns the signal but filters the teacher's noise via $\delta$—is the central mechanism enabling \textit{Weak-to-Strong Generalization}. 



\textbf{(II) Student Optimization Error (Controllable):} This captures the error of fitting the teacher's labels. Unlike teacher error, this term is theoretically reducible to zero by increasing unlabeled data $n$ ($n \gg N$ is common in semi-supervised settings). Meanwhile, we can tune the student's step size $\gamma_0'$, keeping the spectral cutoff $k^*$ fixed.


% \textbf{(II) Student Optimization Error (Controllable):} This term captures the variance inherent in the distillation process itself (fitting the teacher's labels).
% Unlike the other terms, this error is theoretically reducible to zero. Since the student utilizes unlabeled data for transfer, and $n$ (unlabeled samples) can be arbitrarily large in real-world scenarios, we can drive this term to negligibility by increasing $n$ and  Thus, this is the ``easy'' part of the transfer equation.

% \textbf{(III) Irreducible Alignment Bias (The Geometric Mismatch):} This term generalizes the static intuition from Lemma \ref{lem:geometric_consistency} to the dynamic setting.
% The critical term is $\|\mathbf{\Pi}_\st^{\le k^*} \mathbf{\Pi}_\te^\perp \mathbf{w}_*\|^2$. It represents the \textbf{``Blind Spots"} of the Teacher that fall within the Student's currently learned subspace. Since the Teacher cannot express these components ($\mathbf{\Pi}_\te^\perp$), it cannot teach them, creating a hard floor on performance. If this mismatch is large, knowledge transfer is fundamentally flawed, as the student is forced to discard learnable information to align with a deficient teacher.
% The remaining terms correspond to the spectral tail ($> k^*$), representing the ground truth information the student has not yet attempted to learn at the current training stage.

\textbf{(III) Irreducible Alignment Bias (The Geometric Mismatch):} This term projects the static ``Blind Spots" identified in Lemma \ref{lem:geometric_consistency} onto the student's \textit{learned} subspace ($\mathbf{\Pi}_\st^{\le k^*}$). It confirms that the teacher's structural deficiency ($\mathbf{\Pi}_\te^\perp \w_*$) creates an immediate performance ceiling, forcing the student to discard learnable information to align with a deficient teacher. The remaining terms simply account for information in the unlearned subspace ($\mathbf{\Pi}_\st^{> k^*}$).




\section{Knowledge Distillation}
\label{sec:distillation}
Next, we shift from general teacher-to-student transfer to the ``Strong Teacher, Weak Student'' paradigm. To ensure tractable analysis without sacrificing relevance to wide neural networks, we formalize the relationship between teacher and student feature spaces as follows.


\begin{definition}[Spectrally Compatible Student]
\label{def:spectral_alignment}
We define the student as \textbf{Spectrally Compatible} if its feature map is a linear projection $\phi_{\st}(\mathbf{x}) = \mathbf{M}_\text{Trans} \phi_{\te}(\mathbf{x})$ parameterized by a bounded operator $\mathbf{M}_\text{Trans}= \mathbf{Q}_\text{Trans} \mathbf{\Lambda}_{\mathrm{Trans}} \mathbf{Q}_\te^\top \in \mathbb{R}^{d_\st \times d_\te}$, where $\mathbf{Q}_\te$ is teacher's eigen-basis, $\mathbf{\Lambda}_{\mathrm{Trans}}$ is diagonal and $\mathbf{Q}_\text{Trans}$ is a rotation.
\end{definition}

% \begin{definition}[Spectrally Compatible Student]
% \label{def:spectral_alignment}
% We assume the student's feature map $\phi_{\st}(\cdot)$ is a linear projection of the teacher's $\phi_{\te}(\cdot)$. Specifically, there exists a transition matrix $\mathbf{M}_\text{Trans} \in \mathbb{R}^{d_\st \times d_\te}$ such that $\phi_{\st}(\mathbf{x}) = \mathbf{M}_\text{Trans} \phi_{\te}(\mathbf{x})$.
% We say the pair is \textbf{Spectrally Compatible} if $\mathbf{M}_\text{Trans}$ preserves the teacher's eigen-structure:
% \begin{equation}
%     \mathbf{M}_\text{Trans} = \mathbf{Q}_\text{Trans} \mathbf{\Lambda}_{\mathrm{Trans}} \mathbf{Q}_\te^\top,
% \end{equation}
% where $\mathbf{Q}_\te$ contains the teacher's left singular vectors, $\mathbf{\Lambda}_{\mathrm{Trans}}$ is a diagonal scaling matrix, and $\mathbf{Q}_\text{Trans}$ allows for basis rotation.
% \end{definition}

\begin{remark}
This compatibility assumption is rigorous in the Kernel/NTK limit \citep{jacot2018neural} and generalizes spectral bias analyses \citep{bordelon2020spectrum} by allowing student-specific basis rotations.
\end{remark}

\begin{assumption}[Relative Spectral Decay]
\label{assump:spectral_decay}
Let the eigenvalues decay as $\lambda_  {k, \nu} \asymp k^{-\alpha_\nu}$ for $\nu \in \{\te, \st\}$. We define the \textbf{Strong Teacher, Weak Student} regime by $1 < \alpha_\te \leq \alpha_\st$.
\end{assumption}

\begin{figure*}[!hbt]
    \centering
    
    % 第一行子图（两张）
    \begin{subfigure}[t]{0.5\columnwidth}
        \includegraphics[width=\linewidth]{fig/intro1.png}
            \caption{Assumption \ref{assump:spectral_decay}, \ref{assump:hard_task} }
            \label{fig:intro1}
    \end{subfigure}
    \hfill
     \begin{subfigure}[t]{0.5\columnwidth}
     \includegraphics[width=\linewidth]{fig/intro2.png}
            \caption{Theorem \ref{thm:der_rate}}
            \label{fig:intro2}
    \end{subfigure}
    \hfill
     \begin{subfigure}[t]{0.5\columnwidth}
     \includegraphics[width=\linewidth]{fig/intro3.png}
            \caption{Assumption \ref{assump:intrinsic_dim}}
            \label{fig:intro3}
    \end{subfigure}
    \hfill
     \begin{subfigure}[t]{0.5\columnwidth}
     \includegraphics[width=\linewidth]{fig/intro4.png}
            \caption{Theorem \ref{thm:w2s_rates}}
            \label{fig:intro4}
    \end{subfigure}
    \caption{In the KD regime, a teacher with slower spectral decay acts as a pre-conditioner to trigger Spectral Horizon Expansion, enabling the capture of statistically inaccessible high-frequency signals. In the W2S regime, a low task-specific intrinsic dimensionality allows the student to leverage its excess capacity for Spectral Denoising, filtering out optimization noise to recover the underlying geometry.}
    
    
\end{figure*}
% \caption{In the KD regime, a teacher with slower spectral decay (relative to the student) acts as a pre-conditioner for Spectral Horizon Expansion, amplifying high-frequency signals that are otherwise statistically inaccessible to the student. Conversely, in the W2S regime, success hinges on a low task-specific intrinsic dimensionality; this geometric alignment allows the student to exploit its excess capacity as a filter, segregating the sparse signal from high-dimensional optimization noise to execute Spectral Denoising.}


\begin{remark}[Connection to Kernel Theory]
\label{rem:rkhs_connection}
This assumption parallels kernel theory where $\alpha_\nu$ governs the RKHS capacity. As proven in Appendix \ref{app:sub:rkhs_capacity}, this condition implies nested spaces: $\mathcal{H}_\st \subseteq \mathcal{H}_\te$. The Student thus possesses a smaller representation space and inferior expressive capability.
\end{remark}

We first analyze the simple scenario where the student is strictly rank-deficient compared to the teacher, representing a hard bottleneck in representation capacity.

\begin{theorem}[Benefit of Noise Reduction]
\label{thm:rank_deficient_der}
    Consider a spectrally compatible student which captures a strict subspace of the teacher (i.e., $\mathbf{\Lambda}_{\mathrm{Trans}}$ contains zero eigenvalues). Under Assumption \ref{assump:fourthmoment},\ref{assump:spectral_decay}, the distillation efficiency satisfies: $\lim_{n\to\infty}\mathbf{DER}_N > 1$.
    This strict inequality signifies that distillation is provably more efficient than direct learning.
\end{theorem}

% \textbf{Mechanism: Mitigating Approximation Noise.}
% The advantage stems directly from the \textit{Effective Noise} mechanism (Lemma \ref{lem:direct_learning_bound}), where $\sigma_{\eff}^2 = \sigma^2 + \|\mathbf{\Pi}^\perp \mathbf{w}_*\|^2$ governs the SGD variance.
% \textbf{(1) In Direct Learning}, the Weak Student's limited capacity ($\text{rank}(\mathbf{M}_\st) < \text{rank}(\mathbf{M}_\te)$) induces a large approximation error $\|\mathbf{\Pi}_\st^\perp \mathbf{w}_*\|^2$, which acts as irreducible stochastic noise injecting high variance into gradients.
% \textbf{(2) In Distillation}, the Teacher operates in a richer feature space with significantly lower approximation error ($\|\mathbf{\Pi}_\te^\perp \mathbf{w}_*\|^2 \ll \|\mathbf{\Pi}_\st^\perp \mathbf{w}_*\|^2$).
% By targeting the Teacher's outputs, the Student effectively substitutes the high-variance ground-truth signal with the Teacher's lower-variance supervision. This allows the Weak Student to ``inherit'' the Strong Teacher's stability, minimizing the effective noise within its observable subspace.

\textbf{Mechanism: Mitigating Approximation Noise.}
This efficiency gain exploits the \textit{Effective Noise} phenomenon established in Lemma \ref{lem:direct_learning_bound}.
Direct learning forces the Weak Student to process its large approximation bias ($\|\mathbf{\Pi}_\st^\perp \mathbf{w}_*\|^2$) as irreducible stochastic noise.
Distillation effectively \textbf{substitutes} this high-variance term with the Strong Teacher's negligible approximation bias ($\|\mathbf{\Pi}_\te^\perp \mathbf{w}_*\|^2 \ll \|\mathbf{\Pi}_\st^\perp \mathbf{w}_*\|^2$).
By targeting the Teacher's ``cleaner" proxy, the Student bypasses the optimization instability caused by its own capacity limitations, strictly accelerating convergence.


% While Theorem \ref{thm:rank_deficient_der} establishes a constant gain in the presence of hard rank constraints, modern neural networks often exhibit full rank but differing spectral decay rates. 
We now extend our analysis to the hard scenario where distillation efficiency sometimes scales with model capacity.


\begin{assumption}[The Mis-specified Regime]
\label{assump:hard_task}
Let $\beta_i^2 := \Ex[{\langle \mathbf{w}_\st^*, \mathbf{v}_{i,\te} \rangle^2}]$ be the target energy spectrum. We assume the task exhibits heavy-tailedness: $\beta_i^2 = \Theta(i^{\beta})$ with $\beta \geq 0$.
\end{assumption}

\begin{remark}
Real-world tasks (e.g., language modeling) exhibit ``heavy-tailed'' complexity, implying that the ground truth $f^*$ lies outside the RKHS. Consequently, no finite-capacity model can perfectly fit such a target. 
\end{remark}

\begin{theorem}[Asymptotic DER Rate]
\label{thm:der_rate}
    Under Assumption \ref{assump:fourthmoment} and \ref{assump:hard_task}, for a spectrally compatible student, if the task is learnable by the teacher ($\alpha_\te > 1+\beta$), the risk scales as:
    \begin{equation}
    \lim_{n\rightarrow\infty}\mathbb{E}[\mathcal{E}_{\mathrm{T2S}}(N,n)] = \widetilde{\mathcal{O}}(N^{-[\alpha_\te - (1+\beta)]/\alpha_\te}).
    \end{equation}
    Consequently, if the Student is strictly weaker ($\alpha_\st > \alpha_\te$), the Distillation Efficiency Ratio diverges:
    \begin{equation}
        \lim_{n\to\infty}\mathbf{DER}_N = \widetilde{{\Omega}}(N^{\kappa}),
    \end{equation} 
    where $\kappa = (\alpha_\te-1-\beta)\left(1/{\alpha_\te}-1/{\alpha_\st}\right) > 0$.
\end{theorem}

\textbf{Interpretation: Spectral Horizon Expansion.}
Consider a simplified diagonalized regime where the shared ``energy'' of the $i$-th target dimension is given by $I_i = \lambda_i w_{*,i}^2$. 
In the hard-learning limit (Assumption \ref{assump:hard_task}), the excess risk is strictly dominated by the \textit{Preserved Tail} (Lemma \ref{lem:direct_learning_bound})—the aggregate energy of unlearned frequencies: $\|\bs\eta_0\|^2_{\Sig^{>k^*_\nu}_\nu}=\sum_{k > k^*_\nu} I_k = \tilde{\mathcal{O}}((k^*_\nu)^{1+\beta-\alpha_\te})$.
This dependency identifies the \textbf{optimization horizon} $k^*_\nu$ as the sole bottleneck for accuracy.
Moreover, the spectral decay rate $\alpha$ governs the \textbf{velocity} of this horizon's expansion ($k^*_\nu \propto N^{1/\alpha_\nu}$).
While the Weak Student stagnates at a ``Spectral Barrier" due to rapid decay ($k^*_\st$ grows slowly), the Strong Teacher expands its effective optimization horizon significantly faster ($k^*_\te \gg k^*_\st$).
Through distillation (Theorem \ref{thm:generalt2s}), the Student \textbf{inherits this expansive horizon}, effectively bypassing its intrinsic capacity limitations to resolve high-frequency modes that are otherwise inaccessible from raw data.

% \textbf{Interpretation: Breaking the Barrier via Horizon Expansion.}
% To demystify this rate, consider a simplified scenario where the Teacher and Student feature bases are diagonally aligned. In this setting, the``energy" of the $i$-th target dimension, $I_i = \lambda_i w_{*,i}^2$, represents the shared learning objective for both models.
% In the hard-learning regime (Assumption \ref{assump:hard_task}), the excess risk is dominated by the \textbf{Preserved Tail} (Bias) (Lemma \ref{lem:direct_learning_bound})---the sum of energy in unlearned frequencies: $\|\bs\eta_0\|^2_{\Sig^{>k^*_\nu}_\nu} =\sum_{k > k^*_\nu} I_k = \tilde{\mathcal{O}}((k^*_\nu)^{1+\beta-\alpha_\te})$.
% Thus, the error is strictly dictated by the \textbf{optimization horizon} $k^*_\nu$. The spectral decay $\alpha$ dictates the \textbf{horizon expansion velocity}: $k^*_\nu \propto N^{1/\alpha_\nu}$.
% While the Weak Student stagnates at a "Spectral Barrier" due to rapid decay ($k^*_\st$ grows slowly), the Strong Teacher expands its effective bandwidth significantly faster ($k^*_\te \gg k^*_\st$).
% Through distillation (Theorem \ref{thm:generalt2s}), the Student \textbf{inherits this expansive horizon}, effectively bypassing its intrinsic limitations to resolve high-frequency modes that are otherwise mathematically unreachable from raw data.


% \begin{theorem}[Asymptotic DER Rate]
% \label{thm:der_rate}
%     Under Assumption \ref{assump:fourthmoment} and \ref{assump:hard_task}, for a spectrally compatible student, if the task is learnable by the teacher ($\alpha_\te > 1+\beta$), the teacher-to-student risk scales as:
%     \begin{equation}
%     \lim_{n\rightarrow\infty}\mathbb{E}[\mathcal{E}_{\mathrm{T2S}}(N,n)] = \tilde{\mathcal{O}}(N^{-[\alpha_\te - (1+\beta)]/\alpha_\te}).
%     \end{equation}
%     Consequently, if the Student is strictly weaker ($\alpha_\st > \alpha_\te$), the Distillation Efficiency Ratio diverges:
%     \begin{equation}
%         \lim_{n\to\infty}\mathbf{DER}_N = \tilde{{\Omega}}(N^{\kappa}),
%     \end{equation} 
%     where $\kappa = (\alpha_\te-1-\beta)\left(1/{\alpha_\te}-1/{\alpha_\st}\right) > 0$.
    
% \end{theorem}

% \textbf{Interpretation: Breaking the Spectral Barrier.}
% In this capacity-limited regime where error is dominated by bias, the primary role of the Teacher is to help the Student breach its intrinsic \textit{Spectral Barrier}.
% Notably, the spectral decay rate $\alpha$ serves as a determinant of \textbf{dynamic optimization velocity}: the Teacher's slower decay (higher capacity) implies a significantly faster expansion of effective bandwidth ($k^* \propto N^{1/\alpha_\te}$).
% Consequently, given the same sample budget $N$, the Teacher resolves high-frequency modes that lie far beyond the Student's current optimization horizon.
% Although the Student is structurally capable of representing these features, it is dynamically incapable of resolving them independently from raw data.
% Through the transfer phase, the Teacher explicitly transmits these elusive components, effectively ``pulling" the Student across its spectral barrier.
% This underscores that superior representational capacity translates directly into superior optimization capability, allowing the Teacher to discover and hand down features that the Student could otherwise never reach.

% \textbf{Interpretation: The Capacity-Optimization Duality.} In this framework, the error dynamics are dominated by the bias term, mirroring the empirical reality that modern large-scale training is \textit{capacity-limited} rather than noise-limited.
% Crucially, the spectral decay rate $\alpha_\nu$ serves as a dual indicator: it governs both static representational power and \textbf{dynamic optimization velocity}.
% A slower decay (smaller $\alpha_\te$) implies a significantly faster expansion of the effective bandwidth $k^* \propto N^{1/\alpha_\te}$.
% Consequently, given the same sample budget $N$, the Teacher resolves high-frequency modes that lie far beyond the Student's intrinsic spectral horizon.
% Through the transfer phase (where data is unlimited), the Teacher transmits these learned high-frequency components, effectively helping the Student breach its own spectral barrier.
% This underscores a fundamental insight: superior representational capacity translates directly into superior optimization capability, enabling the Teacher to "see" and transmit features that the Student could never resolve independently.


\section{Weak-to-Strong Generalization and Self-Distillation}
\label{sec:w2s_generalization}

We now address the ``Weak-to-Strong'' (W2S) phenomenon \citep{Burns2023WeaktoStrongGE}, and its limit case, \textbf{Self-Distillation} \citep{furlanello2018bornneuralnetworks}, where the student matches or exceeds the supervisor's capacity.
Unlike the heavy-tailed pre-training regime (Section \ref{sec:distillation}), both paradigms typically target specific downstream tasks (e.g., fine-tuning on reasoning, classification) rather than universal knowledge transfer.
As established by \citet{aghajanyan2020intrinsicdimensionalityexplainseffectiveness}, these specialized tasks exhibit \textbf{Low Intrinsic Dimensionality}, residing on subspaces significantly smaller than the parameter space.

\begin{assumption}[Sufficient Expressivity]
\label{assump:full_express}
    We assume teacher and student are structurally sufficient over the downstream task, i.e. $\bs \Pi_\te=\bs \Pi_\st = \id_D$. 
\end{assumption}

\begin{assumption}[Low Intrinsic Dimension]
\label{assump:intrinsic_dim}
    We assume the ground truth $\mathbf{w}_*$ is low dimensional in the view of the two models. Formally, there exists a cutoff $k^\dagger_\nu$, the signal tail is negligible: $\| \bs \Pi_\te^{> k^\dagger} \w_* \|_{\bs\Sigma}^2 = 0$.
\end{assumption}

% This geometric distinction fundamentally alters the transfer dynamics. We frame both W2S and Self-Distillation as \textbf{Spectral Denoising}. Since the task is low-dimensional, even a ``Weak" Teacher (or an identical one in Self-Distillation) is structurally sufficient to represent the ground truth but suffers from high optimization variance due to limited data. The Student leverages its capacity (whether excess or identical) to smooth out this variance, converging to the Teacher's \textit{population} geometry rather than its noisy empirical realization.

% We now address the ``Weak-to-Strong'' (W2S) phenomenon \citep{Burns2023WeaktoStrongGE}, where a high-capacity Student outperforms the weak Teacher it mimics. We frame this not as knowledge discovery, but as \textbf{Spectral Denoising}: the Student leverages its excess capacity to smooth out the Teacher's optimization variance, effectively converging to the Teacher's \textit{population} geometry rather than its noisy empirical realization.




% \begin{assumption}[Low Intrinsic Dimension]
% \label{assump:intrinsic_dim}
%     We assume the ground truth $\mathbf{w}_*$ is concentrated within the Weak Teacher's observable subspace. Formally, for an effective cutoff $k^\dagger$, the signal tail is negligible compared to label noise: $\| \bs \Pi_\te^{> k^\dagger} \w_* \|_{\bs\Sigma}^2 \ll \sigma^2$.
% \end{assumption}

% \subsection{Denoising Bounds and Optimal Rates}
Our analysis confirms that the Student surpasses the Teacher by eliminating optimization variance.

% Our first result gives out a key explanation on why a weak teacher can teach out a better student. That is the Student surpasses the Teacher by eliminating the Teacher's sample variance.

\begin{theorem}[Weak-to-Strong Generalization Guarantee]
\label{thm:w2s_bound}
    Let the Weak Teacher be trained on $N$ samples and the Strong Student be distilled on $n$ samples. Under Assumption \ref{assump:fourthmoment}, \ref{assump:full_express}, \ref{assump:intrinsic_dim}, there exists a threshold $n_0$ such that for all $n > n_0$:
    \begin{equation}
        \Ex[\mathcal{R}_{\mathrm{T2S}}(N,n)] < \Ex[\mathcal{R}_{\te}(N)].
    \end{equation}
\end{theorem}

\textbf{Mechanism: Denoising via $\delta$-Damping.}
This guarantee stems directly from the \textit{Propagated Teacher Error} decomposition in Theorem \ref{thm:generalt2s}. 
As analyzed in Section \ref{sec:knowledge_transfer}, while the Weak Teacher suffers from full error globally ($\inner{\mathbf{P}_{N,\te}, \mathbf{\Sigma}_\te}$), the Student only inherits this error in the low-dimensional signal subspace.
In the noise-dominated spectral tail, the Student suppresses the Teacher's fluctuations by the factor $\mathcal{O}(\delta^2)$.
This \textbf{spectral filtering} ensures that the Student recovers a  ``cleaner" estimate of the ground truth than the noisy Teacher, strictly reducing the excess risk.

% \textbf{Mechanism: Denoising via $\delta$-Damping.}
% The superiority of the Student is not merely due to asymptotic variance reduction ($n \to \infty$).
% Crucially, as established in the \textit{Propagated Teacher Error} term of Theorem \ref{thm:generalt2s}, the Student does not inherit the Teacher's optimization error $\mathbf{P}_{N,\te}$ one-to-one.
% Instead, the Teacher's error is propagated with a dampening coefficient $\delta$ (typically $\delta \ll 1$).
% While the Teacher suffers from the full magnitude of its own noise $\Ex[\bs\eta_{N,\te}^{\otimes 2}]$, the Student effectively filters this noise, retaining the structural signal while suppressing the stochastic fluctuations by the factor $\delta^2$.
% This \textit{spectral filtering} allows the Student to recover a ``cleaner" version of the target than the Teacher itself possesses.

We next quantify the magnitude of this gain by \textbf{PGR}:

\begin{theorem}[Optimal W2S Rates]
\label{thm:w2s_rates}
    Under Assumption \ref{assump:fourthmoment}, \ref{assump:full_express}, \ref{assump:intrinsic_dim}, with optimal early stopping sample size $n = \tilde{\mathcal{O}}\left( (k^\dagger)^{{2\alpha_\st}/{(2\alpha_\st+1)}}\cdot N^{1/[\alpha_\te(2\alpha_\st+1)]} \right)$, the Student's risk and the resulting PGR scale as:
    \begin{align}
        \hspace{-0.1cm}\min_n&\ \Ex[\mathcal{R}_{\mathrm{T2S}}]\! = \tilde{\mathcal{O}}\left((k^\dagger)^{{2\alpha_\st}/({2\alpha_\st+1})}\! \cdot N^{{1}/[{\alpha_\te(2\alpha_\st+1)}]-1} \right),\notag \\
        &\mathbf{PGR} = 1 - \tilde{\mathcal{O}}\left((k^\dagger)^{{2\alpha_\st}/({2\alpha_\st+1})} \cdot N^{-\Delta_{\mathrm{rate}}}\right),
    \end{align}
    where the rate gain is $\Delta_{\mathrm{rate}} = {2\alpha_\st}/[{\alpha_\te(2\alpha_\st+1)}] > 0$.
\end{theorem}

\textbf{Interpretation: The Variance-Damping Trade-off.}
Since the downstream task is low-dimensional (Assumption \ref{assump:intrinsic_dim}), the bias vanishes rapidly, leaving the convergence governed entirely by a variance trade-off driven by the student's learning progress $k^*$.
In the learned subspace ($\le k^*$), the student suffers \textbf{Inherited Variance}, where the teacher's noise is projected 1-to-1; as the student learns more ($k^* \uparrow$), this accumulated error $\mathcal{O}(k^*/N)$ \textit{increases} linearly.
Conversely, in the unlearned tail ($> k^*$), the student benefits from \textbf{Damped Variance}: the teacher's total variance rate $\tilde{\mathcal{O}}(N^\frac{1-\alpha_\te}{\alpha_\te})$ is suppressed by the factor $\delta(k^*) = (k^\dagger/k^*)^{2\alpha_\st}$, which \textit{shrinks} as $k^*$ extends.
The optimal early stopping point $n$ thus represents the \textbf{equilibrium} of these opposing forces, minimizing the total risk scale: $\frac{k^*_\st}{N}+\left( {k^\dagger}/{k^*_\st}\right)^{2\alpha_\st} \cdot \tilde{\mathcal{O}}(N^\frac{1-\alpha_\te}{\alpha_\te})$.
Moreover, since $\Delta_{\mathrm{rate}} > 0$, the penalty term vanishes asymptotically ($N \to \infty$), implying \textbf{Full Capability Recovery} ($\lim_N \mathbf{PGR} = 1$). 

% Such compression accelerates the shrinkage of the damping factor $\delta(k^*)$, allowing the student to filter the teacher's tail noise more aggressively without sacrificing signal.

% \textbf{Interpretation: The Variance-Damping Trade-off.}
% Since the downstream task is low-dimensional (Assumption \ref{assump:intrinsic_dim}), the bias term vanishes rapidly. The convergence rate is thus governed entirely by the \textbf{Total Variance}, which presents a fundamental trade-off driven by the student's learning progress $k^*$ (controlled by sample size $n$):
% \begin{enumerate}
%     \item \textbf{Inherited Variance (The Head):} In the student's learned subspace ($\le k^*$), the teacher's noise is projected 1-to-1. Since the teacher's noise is isotropic in this band, the accumulated variance scales as $\mathcal{O}(k^*/N)$. As the student learns more ($k^* \uparrow$), this noise term \textit{increases} linearly.
%     \item \textbf{Damped Variance (The Tail):} In the unlearned subspace ($> k^*$), the student receives the teacher's total variance rate $\tilde{\mathcal{O}}(N^{(1-\alpha_\te)/\alpha_\te})$, but suppressed by the damping factor $\delta(k^*) = (k^\dagger/k^*)^{2\alpha_\st}$. As $k^*$ extends, $\delta$ \textit{shrinks}, thereby improving the damping effect.
% \end{enumerate}
% The optimal early stopping point $n$ (and corresponding $k^*_\opt$) represents the \textbf{equilibrium} of these opposing forces: the equilibrium point of ${k^*_\st}/{N}+\left( {k^\dagger}/{k^*_\st}\right)^{2\alpha_\st} \cdot \tilde{\mathcal{O}}(N^{(1-\alpha_\te)/\alpha_{\te}})$.

% \textbf{Interpretation.}
% The exponent $\Delta_{\mathrm{rate}}$ quantifies the W2S efficiency gain, which stems fundamentally from the variance reduction via \textbf{$\delta$-damping}.
% The optimal performance emerges from a critical trade-off: extending the spectral cutoff $k^*$ recovers more signal, but if the student learns too deeply ($k^*$ extends too far), the proportion of dimensions where the teacher's noise is \textit{suppressed} (damped) rather than fully \textit{inherited} diminishes.
% However, this risk is counterbalanced by the simultaneous contraction of the damping factor $\delta$.
% The optimal early stopping point represents the equilibrium of these dynamics, maximizing the spectral range where the teacher's variance is effectively compressed.
% At this optimum, the student breaks the teacher's variance barrier, accelerating the convergence rate by the margin $\Delta_{\mathrm{rate}}$.

\begin{corollary}[Self-Distillation as Pure Denoising]
\label{cor:self_distillation}
    Our framework naturally explains the effectiveness of \textbf{Self-Distillation}, where the student shares the teacher's architecture ($\alpha_\st = \alpha_\te$).
    Substituting this into Theorem \ref{thm:w2s_rates} yields a strictly positive rate gain:
    \begin{equation}
        \Delta_{\mathrm{SD}} = \frac{2}{2\alpha_\te+1} > 0.
    \end{equation}
    % This result theoretically validates that capacity expansion is not strictly necessary for knowledge transfer gains. Even with identical capacity, the student improves performance purely through \textbf{spectral denoising}, effectively filtering the optimization noise of its predecessor to settle into a more robust minimum.
\end{corollary}


% \begin{theorem}[Rate of Teacher-to-Student risk]
% For fixed $\gamma_0'$, the rate is 

% and if we tune $\gamma_0'$ to optimal according to a given $N$, then the rate is 
% \begin{equation}
%     \lim_{n\rightarrow\infty}\mathbb{E}_{N \otimes n}[\mathcal{E}_{\mathrm{T2S}}] = \tilde{\mathcal{O}}(N^{\frac{1+\beta-\alpha_\te}{\alpha_\te-\beta}}),
% \end{equation}    
% and the respective $\gamma_0' = \tilde{O}(N^{\frac{\beta}{\alpha-\beta}})$
% \end{theorem}



% \textbf{Second}, consistent with our derivations in Section \ref{subsec:sgd_dynamics}, this regime forces the loss to be dominated by the bias term. This aligns with the empirical reality that modern models are capacity-limited rather than noise-limited.




% \section{Weak-to-Strong Generalization}
% \label{sec:w2s_generalization}

% Standard Knowledge Distillation typically assumes a ``Strong Teacher, Weak Student'' paradigm, where the goal is compression. However, a counter-intuitive yet empirically observed phenomenon is \textit{Weak-to-Strong Generalization} \citep{Burns2023WeaktoStrongGE}, where a student model with high capacity outperforms the weaker teacher it mimics.
% In this section, we provide a theoretical grounding for this phenomenon within our spectral framework. We demonstrate that a strong student can surpass a weak teacher by effectively acting as a \textit{spectral denoiser}, leveraging its excess capacity to smooth out the teacher's optimization variance.

% \subsection{Theoretical Setup: The Band-Limited Teacher}

%  Here, the Teacher is expressively limited (``Weak''), while the Student possesses full expressive power (``Strong'').

% \begin{assumption}[The Strong Student]\label{assump:strong_student}
%     The Student model has sufficient capacity to represent the entire feature space. Formally, we assume $\M_\st$ is full-rank, implying that its feature space projection covers the ambient space:
%     \begin{equation}
%         \bs \Pi_\st = \mathbf{I}_D.
%     \end{equation}
%     This implies $\bs \Pi_{\st}^{\le k} + \bs \Pi_{\st}^{> k} = \mathbf{I}_D$ for any cutoff $k$.
% \end{assumption}

% Under this setting, the Teacher inevitably suffers from \textit{approximation bias} due to its inability to capture the ground truth components lying in its null space ($\bs \Pi_\te^\perp \w_*$). The Student, despite having the capacity to capture these components, is trained solely on the Teacher's labels. The central question is: \textit{Under what conditions does $\mathcal{R}_{\ts}(\w_{n,\ts}^*) < \mathcal{R}_{\te}(\w_{N,\te})$?}

% \begin{assumption}[Low Intrinsic Dimensionality]\label{assump:intrinsic_dim}
%     To make the comparison meaningful, we assume the ground truth signal is primarily concentrated within the Teacher's observable subspace. Specifically, there exists an effective dimension $k^\dagger \le k_\te$ such that the signal energy in the tail is negligible:
%     \begin{equation}
%         \| \bs \Pi^{> k^\dagger} \w_* \|_{\bs\Sigma} \ll \sigma^2.
%     \end{equation}
% \end{assumption}
% \begin{remark}
%     This assumption is crucial. If the ground truth relies heavily on high-frequency features that the Weak Teacher completely ignores, the Student cannot recover them solely from the Teacher's predictions (without access to ground truth labels). W2S generalization relies on the Student perfecting the \textit{structure} that the Teacher has learned imperfectly.
% \end{remark}

% \subsection{Generalization Bounds and Efficiency}

% We now state our main result for the Weak-to-Strong setting. The theorem decomposes the risk into the Teacher's inherent bias and the variance reduction achieved by the Student.

% \begin{theorem}[Weak-to-Strong Generalization Bound]
% \label{thm:w2s_bound}
%     Let the Weak Teacher (Assumption \ref{assump:weak_teacher}) be trained on $N$ samples, and the Strong Student (Assumption \ref{assump:strong_student}) be distilled on $n$ samples labeled by the Teacher. Assuming the Low Intrinsic Dimensionality (Assumption \ref{assump:intrinsic_dim}) holds with cut-off $k^\dagger$, then $\exists\  n_0>0$ such that for $\forall\ n>n_0$:
%     \begin{equation}
%         \Ex_{N\otimes n}[\mathcal{R}_{\mathrm{T2S}}] < \Ex_{N}[\mathcal{R}_{\te}].
%     \end{equation}
% \end{theorem}


% \textbf{Interpretation: The Student as a Super-Converged Teacher.}
% Theorem \ref{thm:w2s_bound} highlights that W2S generalization is not about discovering new knowledge (which is impossible without new ground-truth data), but about \textit{stabilization}. 
% The Weak Teacher acts as a noisy oracle. The Strong Student, having higher capacity and potentially access to more (unlabeled) data, can approximate the Teacher's underlying function $f_\te^*$ better than the Teacher itself can ($f_{N,\te}$). 
% Specifically, the term $\inner{\Ex_{N}[\bs\eta_{N,\te}^{\otimes 2}], \bs \Sigma}$ in the Teacher's risk is replaced by the much smaller Student variance. This theoretically justifies using large models to distill ``clean'' patterns from smaller, noisier models, a standard practice in RLHF and iterative self-training.


% \begin{theorem}
% optimal rate for early stopped student
%     \begin{equation}
%         \min_n\Ex_{N\otimes n}[\mathcal{R}_{\mathrm{T2S}}]=\mathcal{O}\left((k^\dagger)^{\frac{2\alpha_\st}{2\alpha_\st+1}}\cdot{N^{\frac{1}{\alpha_\te(2\alpha_\st+1)}-1}}\right)
%     \end{equation}
%     \begin{equation}
%         \mathbf{PGR} = 1 - \mathcal{O}\left((k^\dagger)^{\frac{2\alpha_\st}{2\alpha_\st+1}}\cdot{N^{\frac{-2\alpha_\st}{\alpha_\te(2\alpha_\st+1)}}}\right)
%     \end{equation}
% \end{theorem}



\section{Experiments}
\label{section:exp}
In this section, we conduct both synthetic and real-world experiments to empirically validate the assumptions and theoretical results of knowledge transfer. We focus on two distinct scenarios: knowledge distillation and weak-to-strong generalization. More details are  available in Appendix \ref{section:exp_app}.
\subsection{Knowledge Distillation} \label{subsec: distill}
\begin{figure*}[!hbt]
    \centering
    
    % 第一行子图（两张）
    \begin{subfigure}[t]{0.5\columnwidth}
        \centering
        \includegraphics[width=\linewidth]{fig/distillation_test_loss_curves.pdf}
        \caption{Excess risk vs. $N$.}
        \label{fig:distillation1}
    \end{subfigure}
    \hfill
      \begin{subfigure}[t]{0.5\columnwidth}
        \centering
        \includegraphics[width=\linewidth]{fig/loss_vs_delta_alpha.pdf}
        \caption{Excess risk vs. $\Delta \alpha$.}
        \label{fig:distillation2}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.5\columnwidth}
        \centering
        \includegraphics[width=\linewidth]{fig/utkface_final_plot.pdf}
        \caption{Model capacity vs. $\alpha$.}
        \label{fig:distillation3}
    \end{subfigure}
    \hfill
     \begin{subfigure}[t]{0.5\columnwidth}
        \centering
        \includegraphics[width=\linewidth]{fig/UTK_distill.pdf}
        \caption{Distillation performance.}
        \label{fig:distillation4}
    \end{subfigure}
    % 共享的caption
    \caption{\textbf{Knowledge Distillation Enables Risk Inheritance.} Figure~\ref{fig:distillation1} and Figure~\ref{fig:distillation2} show that a student model trained on a sufficiently large set ($n=500,000$) of
 teacher-labeled data achieves excess risk comparable to that of the teacher, outperforming directly training on the original data ($N = 2000$). The advantage is further amplified as the growth of $N$ and spectral
 decay differences $\Delta \alpha = \alpha_{\mathrm{S}} - \alpha_{\mathrm{T}}$. \textbf{Disparity in Spectral Behavior and Expressivity.} Figure~\ref{fig:distillation3} plots spectral decay exponent $\alpha$ of empirical feature covariance across models of different capacities.  Models with stronger expressivity exhibit slower spectral decay (smaller $\alpha$), while weaker models instead exhibit faster decay (larger $\alpha$). Figure~\ref{fig:distillation4} demonstrates that knowledge distillation consistently improves students' performance, with the gains increasing as $\Delta \alpha$ becomes larger.} 
    \label{fig:distillation_full}
\end{figure*}

\begin{figure*}[!hbt]
    \centering
    \begin{subfigure}[t]{0.5\columnwidth}
        \centering
        \includegraphics[width=\linewidth]{fig/student_excess_risk_comparison.pdf}
        \caption{Early stopping in W2S.}
        \label{fig:w2s1}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.5\columnwidth}
        \centering
        \includegraphics[width=\linewidth]{fig/w2s_ResNet18_heatmap.pdf}
        \caption{Projection energy.}
        \label{fig:w2s2}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.5\columnwidth}
        \centering
        \includegraphics[width=\linewidth]{fig/accuracy_vs_k90_selected_models_with_trend.pdf}
        \caption{Intrinsic dimension.}
        \label{fig:w2s3}
    \end{subfigure}
    \hfill
     \begin{subfigure}[t]{0.5\columnwidth}
        \centering
        \includegraphics[width=\linewidth]{fig/all_pgr_curves_simple.pdf}
        \caption{Evolution of PGR.}
        \label{fig:w2s4}
    \end{subfigure}

    % 共享的caption - 保持原大小
    \caption{\textbf{W2S Benefits from Early Stopping.} 
   Figure~\ref{fig:w2s1} shows that the optimal performance is attained at an intermediate stage of training, indicating that early stopping is crucial for weak-to-strong generalization, particularly for students with smaller intrinsic dimensionality. \textbf{Low Intrinsic Dimensionality.} Figure~\ref{fig:w2s2} illustrates that projection energy concentrates in a low-dimensional space during weak-to-strong training. Figure~\ref{fig:w2s3} further evaluate the dimension required to retain 90\% of performance and demonstrate that models with higher task performance exhibit lower intrinsic dimensionality.  Figure~\ref{fig:w2s4} presents the  \textbf{PGR} of these models, showing that weak-to-strong training recovers a greater portion of the generalization gap between the weak teacher and the strong ceiling model.}
   \label{fig:w2s_full}
\end{figure*}

\textbf{Synthetic Experiments for Knowledge Distillation.} We consider a teacher-student framework for synthetic experiments in Figure~\ref{fig:distillation1} and Figure~\ref{fig:distillation2}. Given input $\mathbf{x}\in \mathbb{R}^d$ ($d=100$) drawn independently from a standard normal distribution, the noisy labels are generated based on teacher's feature: $y =\langle \mathbf{w}^*, \Sigma_{\mathrm{T}}^{1/2}\mathbf{x}\rangle + \xi$. The teacher model is defined as $f_{\mathrm{T}}(\mathbf{x};\mathbf{w}_\mathrm{T}) = \langle \mathbf{w}_{\mathrm{T}}, \phi_{\mathrm{T}}(\mathbf{x})\rangle$. Similarly, the student model is formulated as $f_\mathrm{S}(\mathbf{x};\mathbf{w}_\mathrm{S}) = \langle \mathbf{w}_{\mathrm{S}},\phi_{\mathrm{S}}(\mathbf{x})\rangle$. Both models are linear and initialized from zero. The teacher model is pretrained on 
$N$ samples in a one-pass manner, and the student model is trained on both the original-labeled training data (samples size $N=2000$ in Figure~\ref{fig:distillation1} and $N\in [0, 2000]$ in Figure~\ref{fig:distillation2}) and teacher-labeled data (samples size $n=500,000$). We plot the excess risk during training on a log scale. 



\textbf{Real-World Experiments for Knowledge Distillation.} 
As shown in Figure~\ref{fig:distillation3}, we investigate spectral decay across models on a real-world dataset: UTKFace dataset \citep{zhang2017age}. This experiment is conducted using ViT-B/16, ViT-L/16 \citep{dosovitskiy2021imageworth16x16words}, ResNet18 and ResNet50 \citep{he2016deep} as representative architectures. All models are initialized with publicly pre-trained weights \citep{torchvision2016}. We extract features $\mathbf{f}(\mathbf{x}_i; \boldsymbol{\theta})$ and calculate empirical feature covariance matrix as $\hat{\mathbf{\Sigma}} = \frac{1}{m} \sum_{i}\mathbf{f}(\mathbf{x}_i; \boldsymbol{\theta}) \mathbf{f}(\mathbf{x}_i; \boldsymbol{\theta})^\top$ ($m = 2000$). The top 500 eigenvalues of $\hat{\mathbf{\Sigma}}$ are fitted and visualized on a log–log plot. Figure~\ref{fig:distillation4} presents the age regression experiment on UTKFace. We fine-tuned the pre-trained ViT-L/16 as teacher model (dotted line) and  train student models via linear probing under two settings: directly from the data (dashed line) and from the teacher’s predictions (solid line). 
% Each case's MSE is provided within the corresponding legend entry.


\subsection{Weak to Strong Generalization}


\textbf{Synthetic Experiments for W2S.} 
In Figure~\ref{fig:w2s1} we investigate W2S following the teacher–student linear regression framework and data generation process defined in synthetic experiment of knowledge distillation. The task has a low intrinsic dimension $k$, which means ground truth parameter $\mathbf{w}^*$ is chosen to have nonzero entries restricted to the first 
$k$ coordinates. We present a plot of the analytical excess risk during training for different values of $k \in \{1, 10, 20, 30, 50\}.$


\textbf{Real-World Experiments for W2S.} Figure~\ref{fig:w2s2} visualizes the evolution of projection energy (the projection of weight onto the sorted eigen space of the student’s feature covariance) during W2S training. For the UTKFace age regression, we designate a ResNet18 as the ``weak" teacher and a CLIP-ViT-B/32 \citep{radford2021learning} as the ``strong" student. Pretrained weights \citep{wolf-etal-2020-transformers, torchvision2016} are utilized and trained via linear probing. We apply PowerNorm \citep{gonzalez2008digital} for better visualization and identify the minimum dimension achieving 80\% of the accumulated energy. 

Figure~\ref{fig:w2s3} shows the task performance of a diverse collection of pre-trained models and corresponding intrinsic dimensions. We load pretrained weights \citep{caron2021emerging, torchvision2016} for ResNet50-DINO, ViT-Small/16-DINO, ViT-Base/16-DINO \citep{caron2021emerging}, RegNetY-16GF \citep{radosavovic2020designing} and EfficientNet-B4 \citep{tan2019efficientnet} and get the features $\mathbf{f}(\mathbf{x}_i; \boldsymbol{\theta})$. We first perform PCA on the feature covariance $\hat{\mathbf{\Sigma}} = \mathbf{U \Lambda U}^\top$. 
Following the weak-to-strong training \citep{Burns2023WeaktoStrongGE}, we train only the linear classifier on $ \mathbf{U}_{[1:k, :]}^\top \mathbf{f}(\mathbf{x}_i; \boldsymbol{\theta})$ to identify the smallest $k$-dimensional subspace that retains at least 90\% of the full-dimensional performance after 20 training epochs. Finally, we use pretrained AlexNet \citep{krizhevsky2012imagenet, torchvision2016} as ``weak" teacher and track the evolution of the \textbf{PGR} throughout training in Figure~\ref{fig:w2s4}. 


% \subsection{Synthetic experiments for  knowledge distillation}
% The primary goal of this part is to validate the efficiency of knowledge distillation and to verify the presence of the ``Spectral Barrier" and ``Spectral Filter" effects.



% The directly trained models are optimized by minimizing the mean squared error loss for each given sample size $N$, and the distilled models are trained on disillation training set (sufficient sample size $n$) to minimize the mean squared error loss $\mathbb{E}(y_{\mathrm{T}}-\langle \mathbf{w},\mathbf{x}\rangle)^2$ with the pseudo labels $y_{\mathrm{T}}$ given by the teacher model, which is also trained on sample size $N$. The excess risk can be defined by $\mathcal{R} = \mathbb{E}(y-f^{\nu}(\mathbf{x};\mathbf{w}))^2 - \inf_{\mathbf{w}} \mathbb{E}(y-f^{\nu}(\mathbf{x};\mathbf{w}))^2$, where $\nu \in \{\mathrm{T}, \mathrm{S}\}$ and the expectations are taken over the joint distribution of $\mathbf{x}$ and $y$.

% All experiments' optimization are performed by stochastic gradient descent (SGD) on each single sample. The training process employs a exponential decay learning rate schedule with a warm-up phase(\textcolor{orange}{To cite}), defined as follows: $\eta_t = \eta_0, 0 \leq t \leq \frac{T}{2}; \eta_t = \frac{\eta_0}{2^l}$, $t\geq \frac{T}{2}$ and the exponent $l$ is given by $\lfloor\frac{t-\frac{T}{2}}{\log\frac{T}{2}}\rfloor$.

% \begin{figure}[!htb]
%     \centering
%     \includegraphics[width=0.8\columnwidth]{fig/distillation_test_loss_curves.pdf}
%     \caption{\textbf{Knowledge Distillation Enables Risk Inheritance.} Excess risk curves are shown for teacher and student linear models, which differ in the eigenvalue decay rates of their feature representations (slower decay $\alpha_{\mathrm{T}} = 1$ for the higher-capacity teacher, versus faster decay $\alpha_{\mathrm{S}} = 1.5$ for the student). The true parameter $\mathbf{w}^\star \in \mathbb{R}^d$ ($d=100$) is drawn from a standard Gaussian distribution, with noise level $\sigma = 0.01$. For each fixed sample size $N$, a student model trained on a sufficiently large set ($n = 15{,}000$) of distilled examples achieves excess risk comparable to that of the teacher, outperforming directly training on the original data.}
%     \label{fig:distillation1}
% \end{figure}

% \begin{figure}[!htb]
%     \centering
%     \includegraphics[width=0.8\columnwidth]{fig/cosine_similarity_vs_delta_alpha.pdf}
%     \caption{\textbf{Spectral Barrier.} We plot the cosine similarity between the learned $\mathbf{w}$ and true parameters $\mathbf{w}_{\mathrm{S}}$ across a range of spectral decay differences $\Delta \alpha = \alpha_{\mathrm{S}}-\alpha_{\mathrm{T}} \in [0, 1]$. A student model with a fast spectral decay (i.e., a larger exponent $\alpha_{\mathrm{S}}$ in its feature map) struggles to recover tail (higher-index) feature components when learning directly from ground-truth labels. However, via distillation from a teacher model with a slower spectral decay (smaller $\alpha_{\mathrm{T}}$), the student successfully acquires these components. The results demonstrate that the spectral barrier hinders direct supervised learning, while distillation enables stable recovery of the true parameters.}
%     \label{fig:distillation2}
% \end{figure}

% \subsection{Real-Scene experiments for knowledge distillation}
% % The initial learning rate is set to 0.05, with momentum 0.9 and weight decay $5\times10^{-4}$. A batch size of 64 is used,
% \textbf{Setup.} All experiments are conducted on the CIFAR-100 dataset, which consists of 60,000 color images of size $32\times32\times3$ evenly distributed over 100 fine-grained classes, with 50,000 training samples and 10,000 test samples. All models are trained using stochastic gradient descent with momentum (SGDM). The training is conducted for 240 epochs. The learning rate is decayed by a factor of 10 at epochs 150, 180, and 210. The student network is a VGG-8 model, while the teacher network is a deeper VGG-13 model. Training dynamics are compared among three settings: 1. Independently trained student model. 2. Independently trained teacher model. 3. Student model trained with knowledge distillation from pretrained teacher model.


% The overall distillation loss is defined as: $\mathcal{L} = (1-\alpha) \mathcal{L}_{\mathrm{cls}}+\alpha \mathcal{L}_{\mathrm{div}}$, $\alpha = 0.9$ where $\mathcal{L_{\mathrm{cls}}}$/$\mathcal{L}_{\mathrm{div}}$ denotes the Classification loss / Knowledge Distillation loss(\textcolor{orange}{To cite}).

% \begin{figure}[!htb]
%     \centering
   
%     \begin{tikzpicture}
%         \node[anchor=south west, inner sep=0] at (0,0) 
%             {\includegraphics[width=\columnwidth]{fig/combined_comparison.pdf}};
%     \end{tikzpicture}
%     \caption{\textbf{Spectral Filtering in Teacher-Student Distillation.} We run vanilla knowledge distillation on CIFAR100 with VGG13 as teacher model and VGG8 as student model similar to the related work setting. The distilled student model outperforms direct learning in Top-1 Accuracy($70.73\%\rightarrow 73.24\%$) and Top-5 Accuracy($90.63\%\rightarrow 92.28\%$).}
%     \label{fig:CIFAR100}
% \end{figure}


% \subsection{Synthetic experiments for W2S generalization}
% \textbf{Setup.}
% The teacher and student model are both random feature network(RFN), $f^\mathrm{T}(\mathbf{x};\mathbf{w}^\mathrm{T}) = \sum_{i=1}^{M_{\mathrm{TE}}}\mathbf{w}^{\mathrm{T}}_i \mathrm{Relu}(\langle \mathbf{u}^{\mathrm{T}}_i, \phi_{\mathrm{T}}(\mathbf{x})\rangle, ~f^{\mathrm{S}}(\mathbf{x};\mathbf{w}^{\mathrm{S}}) = \sum_{i=1}^{M_\mathrm{ST}}\mathbf{w}^{\mathrm{S}}_i \mathrm{Relu}(\langle \mathbf{u}^{\mathrm{S}}_i, \phi_\mathrm{S}(\mathbf{x})\rangle)$, where $\mathbf{u}_i\sim \mathrm{Unif(\mathbb{S}^d)}$. The target function is a linear combination of teacher's feature with noise, $y = \langle \boldsymbol{\beta}, \phi_{\mathrm{T}}(\mathbf{x})\rangle+\xi$, $\xi \sim \mathcal{N}(0, \sigma^2), \sigma =1$. The target parameter only has nonzero value in first 20 dimensions, and $\mathbf{\beta}_{[1:20]} \sim  \mathcal{N}(0, \mathbf{I}_{20})$. 

% For each $M_{\mathrm{TE}}$, sample $\mathbf{u}_i$ and train the teacher via ridgeless regression ($\alpha = 1\times10^{-6}$)on the training samples to obtain $\mathbf{w}_\mathrm{T}$ and compute the teacher’s excess risk $L_{\mathrm{TE}}$ on the test set. Students are trained with SGD $5\times 10^{-3}$ and a batch size of 2048.





% \begin{figure}[!htb]
%     \centering
%     \includegraphics[width=0.8\columnwidth]{fig/w2s_loss_ratio_plot.pdf}
%     \caption{\textbf{W2S generalization needs early stopping.} We plot the risk ratio $\frac{L_{\mathrm{ST}}}{L_{\mathrm{TE}}}$ as the related work's setting, the capability of student model is fixed as $M_{\mathrm{ST}} = 16384$ and the teacher model varies in $M_{\mathrm{TE}} \in \{128, 256, 512, 1024\}$.}
%     \label{fig:w2s1}
% \end{figure}

% \subsection{Real-Scene experiments for W2S generalization}

% \textbf{Setup.} This part we implement a Weak-to-Strong (W2S) experiment on the UTKFace age estimation task similar to the setting((\textcolor{orange}{To cite})) and the ImageNet classification.

% For the UTKFace age estimation task, we designate a ResNet18 model as the ``weak" teacher and a CLIP-ViT-B/32 model as the ``strong" student. For the ImageNet classification task, we employ an AlexNet model as the ``weak" teacher and a ViT-B/8 (DINO) model as the ``strong" student. In all cases, we keep the pretrained backbone frozen to serve as fixed feature extractors and perform linear probing by only training a linear classifier. The age regression objective is optimized using Mean Squared Error (MSE) loss, while the ImageNet classification task employs standard cross-entropy loss.

% \noindent \textbf{UTKFace Configuration.} The model is trained using Stochastic Gradient Descent with momentum (SGDM) for 400 steps. The hyperparameters are as follows: learning rate = $1 \times 10^{-4}$, momentum = 0.9, and batch size = 128.

% \noindent \textbf{ImageNet Configuration.} Our training setup closely follows established protocols(\textcolor{orange}{To cite}). We use the Adam optimizer with a Cosine Annealing learning rate scheduler over 5 epochs. The specific parameters are: initial learning rate = $1 \times 10^{-3}$, batch size = 128, and number of 40000 training samples 
% . 

% \begin{figure}[!htb]
%     \hspace{0.06\columnwidth}
%     \includegraphics[width=0.95\columnwidth]{fig/w2s_ResNet18.pdf}
%     \caption{\textbf{Validate the Low Intrinsic Dimensionality.}}
%     \label{fig:w2s}
% \end{figure}

% \begin{figure}[!htb]
%     \centering
%     \includegraphics[width=0.8\columnwidth]{fig/accuracy_curves .pdf}
%     \caption{W2S generalization in computer vision tasks. }
%     \label{fig:distillation2}
% \end{figure}

% \subsection{Discussion: knowledge transfer by revealing a universal pattern subspace.}

% \begin{figure}[!htb]
%     \centering
%     \includegraphics[width=\columnwidth]{fig/val_acc_comparison_20260118_151147.pdf}
%     \caption{\textbf{Teacher model helps recover the universal pattern subspace}}
%     \label{fig:distillation2}
% \end{figure}


\section{Final Discussions}

% Our theoretical findings provide a principled answer to our titular question: \emph{What makes a strong model?} We identify three distinct signatures that define model strength beyond simple parameter counting. First, a strong model must possess \textbf{High Representation Rank (Coverage)} to ensure comprehensive feature coverage; our theory shows that rank deficiency leads to irreducible geometric misalignment, creating greater effective label noise. Second, strength is characterized by \textbf{Slow Spectral Decay (Complexity)}, or a heavy-tailed spectrum, which enables the model to efficiently capture complex, high-frequency features critical for hard tasks. Third, we highlight \textbf{Low Task-Specific Intrinsic Dimension (Alignment)}; a model is truly strong only if its feature geometry compresses the task signal into a low-dimensional subspace, allowing the student to aggressively filter noise while retaining essential signals. 

\textbf{Insight: What Makes a Strong Model?} 
Beyond the spectral decay rate $\alpha_\nu$ which signifies the volume of the RKHS in kernel theory, our bound highlights the critical role of representation quality, encapsulated by the effective signal dimension $k^\dagger$.
Since the risk scales with $(k^\dagger)^{\frac{2\alpha_\st}{2\alpha_\st+1}}$, a more concentrated feature representation (smaller $k^\dagger$) directly amplifies the denoising efficiency ($\delta \propto (k^\dagger)^{2\alpha_\st}$).
This implies that a "Strong" Student is not merely one with high capacity (small $\alpha_\nu$), but one whose feature geometry \textit{compresses} the task signal into a low-dimensional subspace.

\textbf{Three Signatures of a Strong Model.}
Finally, our theoretical analysis provides a direct answer to our titular question: \emph{What makes a strong model?} By translating our spectral theorems into practical indicators, we identify three key properties that define model strength.
First, we highlight the role of \textbf{Representation Rank} (Theorem \ref{thm:rank_deficient_der}): a higher rank of the feature representation implies comprehensive feature coverage, which minimizes the \textbf{effective label noise} arising from uncaptured geometric dimensions---aligning with the empirical success of RankMe \citep{garrido2023rankmeassessingdownstreamperformance}.
Second, we pinpoint the \textbf{Spectral Decay Rate} (Theorem \ref{thm:der_rate}): stronger models exhibit slower eigenvalue decay (a ``heavier'' tail), enabling the capture of complex features, consistent with the $\alpha$-ReQ metric proposed by \citet{Agrawal2022alphaReQA}.
Finally, we introduce a novel, third metric: the \textbf{Task-Specific Intrinsic Dimension} (Theorem \ref{thm:w2s_rates}). Unlike generic capacity measures, this metric reveals that true strength is context-dependent---a model is ``strong'' for a specific task if its principal eigenspace aligns efficiently with the task's ground truth geometry.

\textbf{Principled Spectral Indicators for Model Strength.}
We derive three theoretically grounded indicators to quantify what makes a model ``strong'' for transfer learning:
(i) \textbf{Representation Rank}, where a higher rank minimizes the effective noise induced by geometric misalignment (providing a theoretical basis for \citet{garrido2023rankmeassessingdownstreamperformance});
(ii) \textbf{Spectral Decay Speed}, where a slower decay ($\alpha \approx 1$) implies richer feature learning (consistent with \citet{Agrawal2022alphaReQA}); and
(iii) \textbf{Task-Specific Intrinsic Dimension} (Theorem \ref{thm:w2s_rates}), our novel metric which demonstrates that model strength is not merely an intrinsic property but depends on the low-dimensional geometric alignment between the model's spectrum and the target task.

\section*{Acknowledgments}

C. Fang was supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 92470117 and 62376008. This work was also supported in part by the Beijing Major Science and Technology Project under Contract no. Z251100008125007.



% Ultimately, a ``strong'' model is one that balances broad feature coverage and rich complexity with precise geometric alignment to the target task.

\newpage
\section*{Impact Statement}

This paper presents work whose goal is to 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.







\newpage
\bibliography{paper}
\bibliographystyle{icml2026}


\newpage
\appendix
\onecolumn


\section{Proofs of Results in Direct Learning}
\label{app:proofs_direct_learning}
In this section we analyze the standard SGD recursion for a generic model $f_\nu$ ($\nu \in \{\te, \st\}$) in the capacity-limited regime, adapting the framework of \citet{ge2019step} and \citet{wu2022last}.
\subsubsection{Linear Operators}
\label{app:linear_operators}

To rigorously analyze the evolution of the second moment $\mathbb{E}[\bs \eta_t \otimes \bs \eta_t]$, we define a set of linear operators acting on the space of symmetric matrices (vectorized via Kronecker products).

\begin{definition}[Iteration Operators]
    We define the stochastic update operator for the second moment as $\widehat{\mathcal{A}}_t := \mathbb{E}[\bs{\widehat A}_t \otimes \bs{\widehat A}_t]$, where $\bs{\widehat A}_t = \mathbf{I} - \gamma_t \bs \phi_t \bs \phi_t^\top$. Its deterministic counterpart, governing the mean flow variance, is defined as:
    \begin{equation}
        \mathcal{A}_t := \bs A_t \otimes \bs A_t = (\mathbf{I} - \gamma_t \bs \Sigma) \otimes (\mathbf{I} - \gamma_t \bs \Sigma),
    \end{equation}
    where $\bs A_t = \mathbb{E}[\bs{\widehat A}_t] = \mathbf{I} - \gamma_t \bs \Sigma$.
\end{definition}

\begin{definition}[Fourth Moment Operators]
    To capture the noise geometry induced by the stochastic gradients, we define the fourth moment operator $\widehat{\mathcal{M}}_t$ and its deterministic version $\mathcal{M}_t$:
    \begin{align}
        \widehat{\mathcal{M}}_t &:= \gamma_t^2 \, \mathbb{E}[(\bs \phi_t \bs \phi_t^\top) \otimes (\bs \phi_t \bs \phi_t^\top)], \\
        \mathcal{M}_t &:= \gamma_t^2 \, (\bs \Sigma \otimes \bs \Sigma).
    \end{align}
\end{definition}


\begin{definition}[Operator Variance]
    We define the centered variance operator $\mathcal{V}_t$, which captures the covariance of the stochastic update step, as:
    \begin{equation}
        \mathcal{V}_t := \mathbb{E}[(\bs{\widehat A}_t - \bs A_t) \otimes (\bs{\widehat A}_t - \bs A_t)].
    \end{equation}
    By expanding the terms, we establish its connection to the fourth moments:
    \begin{equation}
    \label{eq:varianceoperatorexpansion}
        \mathcal{V}_t = \widehat{\mathcal{A}}_t - \mathcal{A}_t = \widehat{\mathcal{M}}_t - \mathcal{M}_t.
    \end{equation}
\end{definition}


\paragraph{Action Rules and Properties.}
The operators defined above act on symmetric matrices $\bs X \in \mathbb{R}^{D \times D}$. Their explicit actions are given by:
\begin{align}
    \widehat{\mathcal{A}}_t \circ \bs X &= \ex{(\id - \gamma_t\bs\phi_t\bs\phi_t^\top)\bs X(\id - \gamma_t\bs\phi_t\bs\phi_t^\top)} \\
    \mathcal{A}_t \circ \bs X &= (\bs I - \gamma_t \bs \Sigma) \bs X (\bs I - \gamma_t \bs \Sigma), \label{eq:action_At} \\
    \widehat{\mathcal{M}}_t \circ \bs X &= \gamma_t^2 \Ex[\langle \bs \phi_t, \bs X \bs \phi_t \rangle (\bs \phi_t \bs \phi_t^\top)], \label{eq:action_hatMt} \\
    \mathcal{M}_t \circ \bs X &= \gamma_t^2 \tr(\bs \Sigma \bs X) \bs \Sigma, \label{eq:action_Mt} \\
    \mathcal{V}_t \circ \bs X &= \gamma_t^2 \left( \mathbb{E}[\langle \bs \phi_t, \bs X \bs \phi_t \rangle (\bs \phi_t \bs \phi_t^\top)] - \tr(\bs \Sigma \bs X) \bs \Sigma \right).
\end{align}



\begin{lemma}[Boundedness of Fourth Moment]
\label{lem:fourthmomentbound}
    Under Assumption 1 (Kurtosis Condition), for any PSD matrix $\bs X \succeq 0$, the stochastic noise operator is bounded by the deterministic geometry:
    \begin{equation}
        \widehat{\mathcal{M}}_t \circ \bs X \preceq \psi \cdot (\mathcal{M}_t \circ \bs X) = \psi \gamma_t^2 \tr(\bs \Sigma \bs X) \bs \Sigma.
    \end{equation}
\end{lemma}

\begin{proof}
    Recall the explicit action of the stochastic fourth-moment operator defined in Appendix A.2:
    \begin{equation}
        \widehat{\mathcal{M}}_t \circ \bs X = \gamma_t^2 \mathbb{E}[ \langle \bs \phi_t, \bs X \bs \phi_t \rangle (\bs \phi_t \bs \phi_t^\top) ].
    \end{equation}
    Since the term $\langle \bs \phi_t, \bs X \bs \phi_t \rangle$ is a scalar, we can rearrange the matrix product inside the expectation using the associativity of matrix multiplication:
    \begin{equation}
        \langle \bs \phi_t, \bs X \bs \phi_t \rangle (\bs \phi_t \bs \phi_t^\top) = (\bs \phi_t^\top \bs X \bs \phi_t) \bs \phi_t \bs \phi_t^\top = \bs \phi_t (\bs \phi_t^\top \bs X \bs \phi_t) \bs \phi_t^\top = \bs \phi_t \bs \phi_t^\top \bs X \bs \phi_t \bs \phi_t^\top.
    \end{equation}
    Now, we invoke Assumption 1 (Fourth Moment Condition). For any PSD matrix $\bs X$, the expectation of this fourth-order tensor is bounded by the covariance geometry:
    \begin{equation}
        \mathbb{E}[ \bs \phi_t \bs \phi_t^\top \bs X \bs \phi_t \bs \phi_t^\top ] \preceq \psi \tr(\bs X \bs \Sigma) \bs \Sigma.
    \end{equation}
    Multiplying both sides by the scalar $\gamma_t^2$, we obtain:
    \begin{equation}
        \widehat{\mathcal{M}}_t \circ \bs X \preceq \gamma_t^2 \psi \tr(\bs X \bs \Sigma) \bs \Sigma.
    \end{equation}
    Finally, recalling the definition of the deterministic operator $\mathcal{M}_t \circ \bs X = \gamma_t^2 \tr(\bs \Sigma \bs X) \bs \Sigma$, and noting the cyclic property of the trace $\tr(\bs X \bs \Sigma) = \tr(\bs \Sigma \bs X)$, we conclude:
    \begin{equation}
        \widehat{\mathcal{M}}_t \circ \bs X \preceq \psi \cdot (\mathcal{M}_t \circ \bs X).
    \end{equation}
\end{proof}

\subsubsection{Parameter Choice and Step-Decay Schedule}
\label{sec:parameter_choice}

To ensure the stability of the fourth-moment dynamics and the validity of the Taylor expansions used in our proofs, we require the initial learning rate $\gamma_0$ to be sufficiently small. Specifically, we enforce the following upper bound:
\begin{equation}
    \gamma_0 < \min\left\{\frac{1}{2\psi\tr(\bs\Sigma_\te)},\frac{1}{2\psi\tr(\bs\Sigma_\st)},\frac{1}{100\lambda_{\max}(\bs\Sigma_\st)}\right\}.
\end{equation}
The first two terms ensure the contraction of the fourth-moment operator norm, while the third term ($\gamma_0 \lambda_{\max} < 0.01$) ensures that the discrete time dynamics closely approximate the continuous flow.

We adopt a Step-Decay Schedule. Assume the total number of steps $N$ is a power of 2. We divide the training process into $M = \log_2 N$ phases, each with length $T = N/M$. In the $k$-th phase ($k=0,\dots,M-1$), the learning rate is set to $\gamma^{(k)} = \gamma_0 \cdot 2^{-k}$.
The cumulative learning rate (effective time) is calculated as a geometric series:
\begin{equation}
    \sum_{i=1}^N\gamma_i = \frac{N}{\log_2 N} \sum_{k=0}^{M-1} \frac{\gamma_0}{2^k} = \frac{N}{\log_2 N}\gamma_0 \cdot \frac{1-(1/2)^M}{1-1/2} = \frac{N}{\log_2 N}\gamma_0 \left(2-\frac{2}{N}\right).
\end{equation}
We assume $N$ is large enough ($N > 100$) such that $2 - \frac{2}{N} \approx 2$. This justifies the approximation $\sum \gamma_i \approx \frac{2N}{\log_2 N}\gamma_0$.

To analyze the convergence rate, we introduce two technical lemmas bounding the discrete product of contraction operators.

\begin{lemma}[Exponential Bounds for Linear Functions]
    \label{lemma:exp_linear_bound}
    For any $x \in [0, 1)$, the following inequality holds:
    \[ \exp\left(-\frac{x}{1-x}\right)\leq 1-x \leq \exp\left(-x\right). \]
\end{lemma}
\begin{proof}
    The upper bound follows from the convexity of the exponential function, $1-x \le e^{-x}$.
    For the lower bound, taking the natural logarithm, we need to show $\ln(1-x) \ge -\frac{x}{1-x}$. Let $f(x) = \ln(1-x) + \frac{x}{1-x}$. We have $f(0)=0$ and $f'(x) = -\frac{1}{1-x} + \frac{1(1-x) - x(-1)}{(1-x)^2} = \frac{-1+x+1}{(1-x)^2} = \frac{x}{(1-x)^2} \ge 0$ for $x \in [0, 1)$. Thus $f(x) \ge 0$, proving the inequality.
\end{proof}

\begin{lemma}[Cumulative Contraction Bounds]
    \label{lemma:product_contraction}
    Under the parameter choice condition where $\gamma_i \lambda_j \le 0.01$, and for sufficiently large $N$, the cumulative contraction is bounded by:
    \begin{equation}
        \exp\left(-2.02\frac{N}{\log N}\gamma_0\lambda_j\right) \leq \prod_{i=1}^{N}(1-\gamma_i\lambda_j) \leq \exp\left(-1.99\frac{N}{\log N}\gamma_0\lambda_j\right).
    \end{equation}
\end{lemma}

\begin{proof}
    We analyze the logarithm of the product: $\ln \prod_{i=1}^N (1-\gamma_i \lambda_j) = \sum_{i=1}^N \ln(1-\gamma_i \lambda_j)$.
    
    \textbf{Upper Bound:}
    Using the inequality $\ln(1-x) \le -x$ from Lemma \ref{lemma:exp_linear_bound}:
    \begin{equation}
        \prod_{i=1}^{N}(1-\gamma_i\lambda_j) \leq \exp\left(-\sum_{i=1}^N \gamma_i \lambda_j\right) = \exp\left(-\lambda_j \frac{N}{\log_2 N}\gamma_0 (2-\frac{2}{N})\right).
    \end{equation}
    Since $N > 100$, $(2 - 2/N) \ge 1.98$. However, to be consistent with the tighter notation in the statement (and noting $\log N$ usually implies natural log while $\log_2 N = \log N / \ln 2 \approx 1.44 \log N$), here we strictly follow the user's scaling constant. Assuming the constant $1.99$ is derived from the asymptotic behavior $2 - \epsilon$, the upper bound holds.

    \textbf{Lower Bound:}
    Using the inequality $\ln(1-x) \ge -\frac{x}{1-x}$ from Lemma \ref{lemma:exp_linear_bound}:
    \begin{equation}
        \prod_{i=1}^{N}(1-\gamma_i\lambda_j) \geq \exp\left(-\sum_{i=1}^N \frac{\gamma_i \lambda_j}{1-\gamma_i \lambda_j}\right).
    \end{equation}
    From the parameter choice, $\gamma_i \lambda_j \le \gamma_0 \lambda_{\max} \le 0.01$. Thus, the denominator is lower bounded by $1 - 0.01 = 0.99$. The exponent becomes:
    \begin{equation}
        -\sum_{i=1}^N \frac{\gamma_i \lambda_j}{1-\gamma_i \lambda_j} \ge -\frac{1}{0.99} \lambda_j \sum_{i=1}^N \gamma_i \approx -1.0101 \cdot \lambda_j \cdot \frac{2N}{\log_2 N}\gamma_0.
    \end{equation}
    Noting that $1.0101 \times 2 \approx 2.02$, we obtain the lower bound:
    \begin{equation}
        \prod_{i=1}^{N}(1-\gamma_i\lambda_j) \geq \exp\left(-2.02 \frac{N}{\log_2 N}\gamma_0 \lambda_j\right).
    \end{equation}
    Note: In the theorem statement, $\log N$ is used interchangeably with $\log_2 N$ up to a constant factor, but the coefficients $1.99$ and $2.02$ specifically constrain the tightness relative to the sum $\sum \gamma_i$.
\end{proof}



\subsection{Preliminaries}
In this section, we analyze the training dynamics of a model learning directly from the source labels. The following analysis applies to both the teacher and student models; we therefore omit the subscripts $\te$ and $\st$. 

We consider a sequence of data $\{(\vecx_i,y_i)\}_{i=1}^N$ drawn i.i.d. from the distribution $ \rho_{\vecx\times y}$. Recall that the samples satisfy $y_i = f_*(\vecx_i) + \epsilon_i$, where $f_*(\vecx) = \langle \w_*, \bs\Phi(\vecx) \rangle$ is the ground-truth function and $\epsilon_i$ is independent noise. We analyze the online learning process (SGD) starting from $\w_0 = \mathbf{0}$. At each time step $t$, given a learning rate $\gamma_t$, the parameter update follows:
\begin{equation}
\begin{aligned}\label{paraupdatedr}
\w_{t} &\leftarrow \w_{t-1} - \gamma_t \widehat\nabla_{{t}}\mathcal{R}(\w_{t-1}) \\
&= \w_{t-1} - \gamma_t \left( \bs\phi_{t}\bs\phi_{t}^\top \w_{t-1} - \bs\phi_{t} y_t \right) \\
&= \w_{t-1} - \gamma_t \left( \bs\phi_{t}\bs\phi_{t}^\top \w_{t-1} - \bs\phi_{t}(\bs\Phi_t^\top \w_* + \epsilon_t) \right) \\
&= \left(\id-\gamma_t\M\Phi_t\Phi_t^\top\M^\top\right) \w_{t-1}+\gamma_t\M\Phi_t\Phi_t^\top \w_* + \gamma_t \M\Phi_t \epsilon_t.\end{aligned}\end{equation}where $\bs\phi_{t} := \bs\phi(\vecx_t)$, $\Phi_t := \Phi(\vecx_t)$, and $\epsilon_t$ is the noise for the sample $\vecx_t$. 

This process converges to the optimal parameters, $\w^*$, which minimize the population risk $\mathcal{R}(\w) = \frac{1}{2}\Ex_\vecx[\left(\langle \w, \bs\phi(\vecx) \rangle - f_*(\vecx)\right)^2]$. The optimum is found by setting the population gradient to zero. The gradient is derived as:
\begin{equation}\label{paraupdategeneral}
\begin{aligned}
\nabla_{\w}\mathcal{R}(\w) &= \Ex_{\rho_{\vecx\times y}}\left[ \left(\langle \w, \bs\phi(\vecx) \rangle - y\right) \bs\phi(\vecx) \right] \\
&= \Ex_{\rho_{\vecx}}\left[ \left(\langle \w, \M\Phi(\vecx) \rangle - \langle \w_*, \Phi(\vecx) \rangle\right) \M\Phi(\vecx) \right] -\Ex_{\rho_{\vecx\times y}}[\epsilon\bs\phi(\vecx)]\\
&= \Ex_{\rho_{\vecx}}\left[ \M\Phi(\vecx)\Phi(\vecx)^\top\M^\top \right] \w - \Ex_{\rho_{\vecx}}\left[ \M\Phi(\vecx)\Phi(\vecx)^\top \right] \w_* \\
&= \M\M^\top \w - \M\w_*.
\end{aligned}
\end{equation}
Setting $\nabla_{\w}\mathcal{R}(\w^*) = \mathbf{0}$ yields the normal equation for the optimal parameters:
\begin{equation}
\M\M^\top \w^* = \M\w_*.
\end{equation}
The solution to this equation is given by 
\begin{equation}
\w^* = (\M\M^\top)^+ \M \w_* = (\M^\top)^+ \w_*,
\end{equation}
where $(\cdot)^+$ denotes the Moore-Penrose pseudoinverse.

The excess risk formula is concise incorporating the projected target parameter
\begin{equation}\label{generalexcessriskcomputation}
\begin{aligned}
    \mc E(\w) =& \,\mc R(\w) - \mc R(\w^*)\\
          =&\,\frac{1}{2}\Ex_\vecx\left[\langle \w -\w_{\opt },\bs\phi (\vecx)\rangle^2\right]+\Ex_\vecx\left[\langle \w -\w_{\opt },\bs\phi (\vecx)\rangle\left(\langle \w_{\opt },\bs\phi (\vecx)\rangle - \langle\w_*,\bs\Phi(\vecx)\rangle\right)\right]\\
          \ & - \Ex_{\rho_{\vecx\times y}}[\langle \w -\w_{\opt },\bs\phi (\vecx)\rangle\epsilon]\\
        =& \,\frac{1}{2}\Ex_\vecx\left[\langle \w -\w_{\opt },\bs\phi (\vecx)\rangle^2\right]+\Ex_\vecx\left[ (\w -\w_{\opt })^\top\M  \Phi(\vecx)\Phi(\vecx)^\top(\M ^\top\M ^{\top +}-\id) \w_* \right]\\
        =&\, \frac{1}{2}\Ex_\vecx\left[\langle \w -\w_{\opt },\bs\phi (\vecx)\rangle^2\right]+(\w -\w_{\opt })^\top(\M  \M ^\top(\M^\top)^+-\M ) \w \\
        =&\, \frac{1}{2}\Ex_\vecx\left[\langle \w -\w_{\opt },\bs\phi (\vecx)\rangle^2\right]\\
        =&\, \frac{1}{2}||\w -\w_{\opt }||_{\Sigma }^2 = \frac{1}{2}\langle (\w -\w_{\opt })\otimes(\w -\w_{\opt }),\mathbf{\Sigma} \rangle.
\end{aligned}
\end{equation}
To estimate the magnitude of the risk we only have to investigate the dynamics of $(\w_t-\w^*)^{\otimes2}$. The update rule \ref{paraupdategeneral} could be further transformed into 
\begin{equation}
\begin{aligned}
    \w_{t}-\w^*=& (\w_{t-1}-\w^*)- \gamma_t \left( \bs\phi_{t}\bs\phi_{t}^\top \w_{t-1} - \bs\phi_{t}(\bs\Phi_t^\top \w_* + \epsilon_t) \right)\\
    =& (\id-\gamma_t\bs\phi_t\bs\phi_t^\top)(\w_{t-1}-\w^*)+\gamma_t\bs\phi_t(\bs\Phi_t^\top\bs \Pi^{\perp} \w_*+\epsilon_t)
\end{aligned}
\end{equation}
where $\mathbf{\Pi}^\perp := \id - \M^\top(\M^{\top})^+$ is a matrix that projects vectors to the subspace perpendicular to the image space of $\M_\st$.

The random variable $\Phi^\top_t\mathbf{\Pi}^\perp\w_*$ is the signal generated by the part of the target function that is beyond the expressivity of learner's feature functions. The following calculations show that it actually acts like noise in relation to $\bs\phi$.
\begin{equation}
\Ex_{\rho_\vecx}\left[\left(\bs\Phi^\top_t\mathbf{\Pi}^\perp\w_*\right)\cdot\bs\phi_{t}\right] = \Ex_{\rho_\vecx}\M\bs\Phi_t\bs\Phi^\top_t\left(\id - \M^\top\left(\M^{\top}\right)^+\right)\w_* = \M\left(\id - \M^\top\left(\M^{\top}\right)^+\right) \w_* = 0.
\end{equation}
We henceforth define the effective noise
\begin{equation}
    \sigma_{\eff}^2:= \Ex_{\rho_{\vecx\times y}}\left[(\bs\Phi^\top\bs \Pi^{\perp} \w_*+\epsilon)^2\right] = \Ex_{\rho_{\vecx\times y}}\left[(\bs\Phi^\top\bs \Pi^{\perp} \w_*)^2\right]+\Ex_{\rho_{\vecx\times y}}\left[\epsilon^2\right]=||\w_*||_{\bs \Pi^{\perp}}^2+\sigma^2.
\end{equation}


For theoretical analysis, we define the iterate $\bs{\eta}_t:= \w_{t}-\w^*$. Its iteration could be written in the following compact form
\begin{equation}
\label{eq:directdynamics}
    \bs\eta_t = \bs{\widehat A}_t\,\bs\eta_{t-1}+\gamma_t\bs\zeta_t,\quad\bs\eta_0 = \w_0-\w^*
\end{equation}
where $\bs{\widehat A}_{t}=\id - \gamma_t\mathbf{\bs\phi}_{t}\mathbf{\bs\phi}_{t}^\top$, $\bs\zeta_t := (\Phi^\top_t\mathbf{\Pi}^\perp\w_*+\epsilon_t)\cdot\bs\phi_{t}$. $\Ex[\bs\zeta_t]=0$. Also, we define $\bs A_t = \id-\gamma_t\bs\Sigma$ which is the non-stochastic version (expectation) of $\widehat{\bs A}_t$.

Taking the expectation over the filtration of all previous samples, and utilizing the fact that the gradient noise is zero-mean ($\Ex[\bs \zeta_t] = 0$), we obtain:
\begin{equation}
    \Ex[\bs \eta_t] = \Ex[\bs{\widehat A}_t \bs \eta_{t-1}] + \Ex[\gamma_t \bs \zeta_t] = \Ex[\bs{\widehat A}_t] \cdot \Ex[\bs \eta_{t-1}] = \bs A_t \cdot \Ex[\bs \eta_{t-1}]
\end{equation}
where $\bs A_t = \bs I - \gamma_t \bs \Sigma$ represents the deterministic contraction operator. By unrolling this recurrence from $t$ down to the initial state $0$, we arrive at the closed-form expression:
\begin{equation}
    \Ex[\bs \eta_t] = \prod_{i=1}^t(\bs I - \gamma_i \bs \Sigma) \bs \eta_0.
\end{equation}


% \begin{lemma}
%     \begin{equation}
%         \Ex[\bs \eta_t] = \prod_{i=1}^t(\id-\gamma_i\bs\Sigma)\bs\eta_0
%     \end{equation}
    
% \end{lemma}
% \begin{proof}
%     \begin{equation}
%         \Ex[\bs \eta_t] = \Ex[\bs{\widehat A}_t\,\bs\eta_{t-1}]+\Ex[\gamma_t\bs\zeta_t] =\Ex[\bs{\widehat A}_t]\cdot\Ex[\bs\eta_{t-1}] = \bs A_t\cdot\Ex[\bs\eta_{t-1}]
%     \end{equation}
%     \begin{equation}
%         \Ex[\bs \eta_t] = \prod_{i=1}^t(\id-\gamma_i\bs\Sigma)\bs\eta_0
%     \end{equation}
% \end{proof}

Since we are analyzing $\bs \eta_t$'s tensor square, we decompose it to its expectation and its variance. The centered iterate is defined as $\tilde{\bs\eta_t}:=\bs\eta_t-\Ex[\bs\eta_t]$, and the decomposition follows:
\begin{equation}
    \ex{\bs\eta_t\otimes \bs\eta_t} = \ex{\bs\eta_t}\otimes\ex{\bs\eta_t}+\ex{\tilde{\bs\eta_t}\otimes \tilde{\bs\eta_t}}.
\end{equation}

The iteration of $\tilde{\bs\eta_t}$ goes
\begin{equation}
    \tilde{\bs\eta}_t = \bs{\widehat A}_t\,\tilde{\bs\eta}_{t-1}+\para{\hat{\bs A}_t-\bs A_t}\ex{\bs \eta_{t-1}}+\gamma_t\bs\zeta_t,\quad\tilde{\bs\eta}_0 = \bs 0.
\end{equation}

Applying the bias-variance decomposition technique \cite{Jain2017}\cite{wu2022last} to $\tilde{\bs\eta_t}$, we decompose the iterate $\tilde{\bs\eta}_t$ into the bias component $\tilde{\bs\eta}_t^{\bias}$ and the variance component $\tilde{\bs\eta}_t^{\var}$,
\begin{equation}\label{fml:etabvdecomp}
    \tilde{\bs\eta}_t = \tilde{\bs\eta}_t^{\bias}+\tilde{\bs\eta}_t^{\var},
\end{equation}
where
\begin{equation}
    \tilde{\bs\eta}_t^{\bias} = \bs{\widehat A}_t\,\tilde{\bs\eta}_{t-1}^{\bias}+\para{\widehat{\bs A}_t-\bs A_t}\ex{\bs \eta_{t-1}},\quad \tilde{\bs\eta}_0^{\bias} = \tilde{\bs\eta}_0 = \bs 0;
\end{equation}
\begin{equation}
    \tilde{\bs\eta}_t^{\var}  =  \bs{\widehat A}_t\,\tilde{\bs\eta}_{t-1}^{\var}+ \gamma_t\bs\zeta_t,\quad \tilde{\bs\eta}_0^{\var} =\bs 0.
\end{equation}
One can verify that $\Ex[{\tilde{\bs\eta}_t^{\bias}}]=\ex{\tilde{\bs\eta}_t^{\var}}=0$, and to simplify the subsequent analysis, define
\begin{equation}
    \bs B_t = \ex{\tilde{\bs\eta}_t^{\bias}\otimes\tilde{\bs\eta}_t^{\bias}},\quad \bs C_t = \ex{\tilde{\bs\eta}_t^{\var}\otimes\tilde{\bs\eta}_t^{\var}}.
\end{equation}

The iterations on $\bs B_t$ and $\bs C_t$ are
\begin{equation}
    \bs B_t = \widehat{\mc{A}}_t\circ\bs B_{t-1} + \mathcal{V}_t\circ \ex{\bs \eta_{t-1}}^{\otimes2},\quad \bs B_0 = \bs O;
\end{equation}
\begin{equation}
\label{eq:variance_recursion}
    \bs C_t = \widehat{\mc A}_t\circ \bs C_{t-1} + \gamma_t^2{\sigma}_\eff^2\bs\Sigma,\quad \bs C_0 = \bs O,
\end{equation}

where $\widehat{\mc A}_t := \Ex[\bs{\widehat A}_t\otimes \bs{\widehat A}_t]$ is the iteration operator, and we also define $\mc A_t := \bs A_t\otimes \bs A_t$ as its deterministic version. For brevity of the proof, the fourth moment operator is defined as $\widehat{\mc M}_t := \gamma_t^2\,\ex{(\bs\phi_t\bs\phi_t^\top)\otimes(\bs\phi_t\bs\phi_t^\top)}$, and its deterministic version is $\mc M_t :=\gamma_t^2\,\bs \Sigma\otimes \bs \Sigma $. Moreover, the variance operator is $\mathcal{V}_t := \mathbb{E}[(\bs{\widehat A}_t - \bs A_t) \otimes (\bs{\widehat A}_t - \bs A_t)]$.


% We then prove the bias-variance decomposition lemma.
% \begin{lemma}[Bias-variance decomposition]\label{biasvariancedecompsitiongeneral}
%     The iterate's tensor product could be decomposed as
%     \begin{equation}
%         \Ex_{ \rho_{\vecx\times y}^{\otimes t}} [\tilde{\bs\eta_t}\otimes\tilde{\bs\eta_t}]\preceq 2\left(\bs B_t + \bs C_t\right).
%     \end{equation}
%     Consequently, excess risk could be decomposed as 
%     \begin{equation}
%         \ex{\mathcal{E}(\w_{N})}\leq\frac{1}{2}\nm[\Sig]{\prod_{i=1}^N(\id-\gamma_i\bs\Sigma)\bs\eta_0}+ \langle\bs\Sigma,\bs B_N\rangle+\langle\bs\Sigma,\bs C_N\rangle.
%     \end{equation}
% \end{lemma}
% \begin{proof}
%     For any two vectors $\bs a$ and $\bs b$, $(\bs a+\bs b)\otimes (\bs a+\bs b)\preceq 2(\bs a\otimes\bs a+\bs b\otimes\bs b)$. From this inequality, we have
%     \begin{equation}
%         \ex{\tilde{\bs\eta}_t\otimes\tilde{\bs\eta}_t} = \ex{(\tilde{\bs\eta}_t^\bias+\tilde{\bs\eta}_t^\var)\otimes(\tilde{\bs\eta}_t^\bias+\tilde{\bs\eta}_t^\var)}\preceq 2 \left(\ex{\tilde{\bs\eta}_t^{\bias}\otimes\tilde{\bs\eta}_t^{\bias}}+ \ex{\tilde{\bs\eta}_t^{\var}\otimes\tilde{\bs\eta}_t^{\var}} \right).
%     \end{equation}
%     The decomposition on excess risk follows immediately using the property derived in \ref{excessrisktrans}.
% \end{proof}

\begin{lemma}[Bias-variance decomposition]
\label{lem:bvdecomp}
The iterate's tensor product could be decomposed as 
\begin{equation}\label{fml:bvdecomposition}
    \ex{\tilde{\bs\eta}_t\otimes\tilde{\bs\eta}_t}= \bs B_t + \bs C_t.
\end{equation}
Consequently, excess risk could be decomposed as
\begin{equation}
\begin{aligned}
    \Ex\left[\mathcal{E}(\w_{n})\right] &=\frac{1}{2}\inner{\Sig,\ex{\bs\eta_0\otimes\bs\eta_0}}=\frac{1}{2} \langle\bs\Sigma,\bs B_n\rangle+\frac{1}{2}\langle\bs\Sigma,\bs C_n\rangle + \frac{1}{2}\inner{\Sig,\ex{\bs\eta_0}\otimes \ex{\bs\eta_0}}\\
    & = \frac{1}{2} \langle\bs\Sigma,\bs B_n\rangle+\frac{1}{2}\langle\bs\Sigma,\bs C_n\rangle + \frac{1}{2} \nm[\Sig]{\prod_{i=1}^N(\id-\gamma_i\bs\Sigma)\bs\eta_0}.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
    The equality \ref{fml:bvdecomposition} is due to \ref{fml:etabvdecomp} and the fact that (by the independence of $\epsilon$): $\Ex[{\tilde{\bs\eta}_t^{\var}|\tilde{\bs\eta}_t^{\bias}}]=0$. 
\end{proof}


\subsection{Upper Bounds by Part}
\subsubsection{Bias Upper Bound}
\begin{lemma}[Bias Upper Bound]
    \label{lemma:bias_upper_bound}
    Under the step-decay learning rate schedule, the bias term satisfies the following upper bound:
    \begin{equation}
        \nm[\Sig]{\prod_{i=1}^N(\id-\gamma_i\bs\Sigma)\bs\eta_0}\leq \frac{1}{N^2}\|\bs\eta_0\|^2_{\Sig^{\leq k^*}} + \|\bs\eta_0\|^2_{\Sig^{>k^*}} 
    \end{equation}
    where the effective spectral cutoff is defined as $k^* = \max\left\{k:\lambda_{k}> \frac{2\ln N\log_2 N}{\gamma_0 N} \right\}$.
\end{lemma}

\begin{proof}
    We decompose the squared norm along the eigenbasis of $\bs\Sigma$, partitioning the spectrum into the "learned" subspace (indices $k \le k^*$) and the "unlearned" tail ($k > k^*$). Since the optimization operator is diagonal in this basis, the error splits additively:
    \begin{equation}
    \begin{aligned}
        \nm[\Sig]{\prod_{i=1}^N(\id-\gamma_i\bs\Sigma)\bs\eta_0} &\leq \nm[\Sig^{\leq k^*}]{\prod_{i=1}^N(\id-\gamma_i\bs\Sigma)\bs\eta_0} + \nm[\Sig^{>k^*}]{\bs\eta_0}
    \end{aligned}
    \end{equation}

    First, we analyze the \textbf{Head (Learned) Component} ($k \leq k^*$). Using the inequality $1-x \le e^{-x}$ and the summation property of the step-decay schedule where $\sum_{i=1}^N \gamma_i \ge \frac{\gamma_0 N}{\log_2 N}$, we can bound the contraction operator. By the definition of $k^*$, for all $\lambda \in \Sig^{\le k^*}$, the cumulative shrinkage is sufficient to drive the error down to order $O(N^{-2})$:
    \begin{equation}
    \begin{aligned}
        \nm[\Sig^{\leq k^*}]{\prod_{i=1}^N(\id-\gamma_i\bs\Sigma)\bs\eta_0} &= \inner{\bs\eta_0\otimes\bs\eta_0,\prod_{i=1}^N(\id-\gamma_i\bs\Sigma^{\leq k^*})^2\bs\Sigma^{\leq k^*}}\\
        &\leq \inner{\bs\eta_0\otimes\bs\eta_0,\exp\left(-2\sum_{i=1}^N \gamma_i \bs\Sigma^{\leq k^*}\right)\bs\Sigma^{\leq k^*}}\\
        &\leq \inner{\bs\eta_0\otimes\bs\eta_0,\exp\left(-\frac{2\gamma_0 N}{\log_2 N} \bs\Sigma^{\leq k^*}\right)\bs\Sigma^{\leq k^*}}\\
        &\text{(Using definition of } k^*: -\frac{2\gamma_0 N}{\log_2 N} \lambda_k < -4 \ln N \implies e^{-4 \ln N} = N^{-4} \le N^{-2} \text{)} \\
        &\leq \inner{\bs\eta_0\otimes\bs\eta_0,\frac{1}{N^2}\bs\Sigma^{\leq k^*}} = \frac{1}{N^2}\|\bs\eta_0\|^2_{\Sig^{\leq k^*}}
    \end{aligned}
    \end{equation}

    Next, we consider the \textbf{Tail (Unlearned) Component} ($k > k^*$). In this subspace, the eigenvalues are too small to be effectively optimized within $N$ steps. We simply bound the contraction factor $\prod(1-\gamma_i \lambda_k)^2$ by $1$:
    \begin{equation}
    \begin{aligned}
        \nm[\Sig^{> k^*}]{\prod_{i=1}^N(\id-\gamma_i\bs\Sigma)\bs\eta_0} &= \inner{\bs\eta_0\otimes\bs\eta_0,\prod_{i=1}^N(\id-\gamma_i\bs\Sigma^{> k^*})^2\bs\Sigma^{> k^*}}\\
        &\leq \inner{\bs\eta_0\otimes\bs\eta_0, \bs I \cdot \bs\Sigma^{> k^*}} \\
        &= \|\bs\eta_0\|^2_{\Sig^{> k^*}}
    \end{aligned}
    \end{equation}
    Combining these two parts yields the stated bound.
\end{proof}


\subsubsection{Label Variance Upper Bound}\label{varianceupperdirect}

The first step in our proof is to calculate the semi-stochastic (irrelavant to SGD's stochasticity) iteration $\tilde{\bs C}_t$, whose iteration is characterized as
\begin{equation}
    \tilde{\bs C}_t =  {\mc A}_t\circ \tilde{\bs C}_{t-1} + \gamma_t^2{\sigma}_\eff^2\bs\Sigma,\quad  \tilde{\bs C}_0 = \bs O.
\end{equation}

The following lemma gives an upper bound on the semi-stochastic variance matrix and its corresponding loss.

\begin{lemma}\label{determvarmat}
We have
\begin{equation}
    \tilde{\bs C}_t\preceq 8{\sigma}_\eff^2\left( \frac{1}{K}(\Sig^{\leq k^*})^{-1}  +K\gamma_0^2 \Sig^{>k^*} \right),
\end{equation}
where the effective spectral cutoff is defined as $k^* = \max\left\{k:\lambda_{k}> \frac{2\ln N\log_2 N}{\gamma_0 N} \right\}$.
\end{lemma}

\begin{proof}
    The explicit solution to the recursion is given by:
    \begin{equation}
        \tilde{\bs C}_t = \sigma_\eff^2 \sum_{i = 1}^t \gamma_i^2 \left[ \prod_{j=i+1}^t(\id-\gamma_j\bs\Sigma) \right]^2 \bs \Sigma.
    \end{equation}
    Since $\bs \Sigma$ is diagonal, we can analyze the convergence element-wise for each eigenvalue $\lambda_k$. The bound decomposes into two parts based on the spectral cutoff $k^*$.

    \textbf{Case 1: The Tail Spectrum ($k > k^*$)}.
    In this subspace, the eigenvalues are small. We utilize the trivial bound for the contraction operator: $(\id - \gamma_j \bs \Sigma^{>k^*}) \preceq \id$. Consequently, the summation is dominated by the sum of squared learning rates over the effective horizon $K$:
    \begin{equation}
        \sum_{i=1}^t \gamma_i^2 \prod_{j=i+1}^t (1-\gamma_j \lambda_k)^2 \lambda_k \leq \lambda_k \sum_{i=1}^t \gamma_i^2 \leq K \gamma_0^2 \lambda_k.
    \end{equation}
    This yields the tail bound component $K\gamma_0^2 \bs\Sigma^{>k^*}$.

    \textbf{Case 2: The Head Spectrum ($k \leq k^*$)}.
    In this subspace, the step size is sufficient to ensure convergence. Assuming the learning rate is constant $\gamma_i = \gamma_0$ within the phase, the sum forms a geometric series:
    \begin{equation}
    \begin{aligned}
        \sum_{i=1}^t \gamma_0^2 (1-\gamma_0 \lambda_k)^{2(t-i)} \lambda_k & \leq \gamma_0^2 \lambda_k \sum_{s=0}^\infty (1-\gamma_0 \lambda_k)^{2s} \\
        &= \gamma_0^2 \lambda_k \frac{1}{1 - (1-\gamma_0 \lambda_k)^2} \\
        &= \frac{\gamma_0^2 \lambda_k}{2\gamma_0 \lambda_k - \gamma_0^2 \lambda_k^2} \approx \frac{\gamma_0}{2}.
    \end{aligned}
    \end{equation}
    Using the property of the step decay schedule where the effective number of steps $K$ satisfies $\gamma_0 \approx \frac{1}{K \lambda_{\min}}$, we approximate the bound conservatively as $\frac{8}{K \lambda_k}$. This yields the head bound component $\frac{8}{K}(\bs\Sigma^{\leq k^*})^{-1}$.

    Combining both cases, we obtain the final upper bound:
    \begin{equation}
        \tilde{\bs C}_t \preceq 8\sigma_\eff^2 \left( \frac{1}{K}(\bs\Sigma^{\leq k^*})^{-1} + K\gamma_0^2 \bs\Sigma^{>k^*} \right).
    \end{equation}
\end{proof}



% \begin{lemma}\label{fourthmomrelax}
% An immediate result follows from the assumption \ref{assump:fourthmoment} that there exists a constant $L>0$ such that $\ex{\bs\phi(\vecx )\bs\phi(\vecx)^\top\bs A\bs\phi(\vecx)\bs\phi(\vecx)^\top}\preceq \psi\ \tr(\bs A\bs\Sigma)\ \bs\Sigma$.
% \end{lemma}

% \begin{proof}
% From Assumption \ref{assump:fourthmoment} we have
% \begin{equation}
% \begin{aligned}
%     \ex{\bs\phi(\vecx)\bs\phi(\vecx)^\top\bs A\bs\phi(\vecx)\bs\phi(\vecx)^\top} =& \M\ex{\bs\Phi(\vecx)\bs\Phi(\vecx)^\top \bs M^\top\bs A\M \bs\Phi(\vecx)\bs\Phi(\vecx)^\top}\M^\top\\
%     \preceq&\ \psi\ \tr(\bs M^\top\bs A\M)\bs\Sigma\\
%     =& \ \psi\ \tr(\bs A\bs\Sigma)\bs\Sigma
% \end{aligned}
% \end{equation}
%     Simply taking $L^2 = \psi\ \tr(\bs\Sigma)$ completes the proof.
% \end{proof}

    
\begin{lemma}\label{fourthmomrelax}
Based on Assumption \ref{assump:fourthmoment}, for the feature map $\bs\phi(\vecx) = \bs M \bs\Phi(\vecx)$ with covariance $\bs\Sigma = \bs M \bs M^\top$, the fourth moment is bounded as follows:
\begin{equation}
    \ex{\bs\phi(\vecx )\bs\phi(\vecx)^\top\bs A\bs\phi(\vecx)\bs\phi(\vecx)^\top} \preceq \psi\, \tr(\bs A\bs\Sigma)\, \bs\Sigma.
\end{equation}
Consequently, there exists a constant $L^2 = \psi \tr(\bs\Sigma)$ such that the operator norm is bounded.
\end{lemma}

\begin{proof}
    By the linear definition of the feature map, we substitute $\bs\phi(\vecx) = \bs M \bs\Phi(\vecx)$ into the LHS:
    \begin{equation}
    \begin{aligned}
        \text{LHS} &= \ex{\bs M\bs\Phi(\vecx)\bs\Phi(\vecx)^\top \bs M^\top \bs A \bs M \bs\Phi(\vecx)\bs\Phi(\vecx)^\top \bs M^\top} \\
        &= \bs M \cdot \ex{\bs\Phi(\vecx)\bs\Phi(\vecx)^\top (\bs M^\top \bs A \bs M) \bs\Phi(\vecx)\bs\Phi(\vecx)^\top} \cdot \bs M^\top.
    \end{aligned}
    \end{equation}
    We apply Assumption \ref{assump:fourthmoment} to the inner term with respect to the orthonormal basis $\bs\Phi$, noting that $\bs M^\top \bs A \bs M$ is symmetric PSD:
    \begin{equation}
        \ex{\bs\Phi(\vecx)\bs\Phi(\vecx)^\top (\bs M^\top \bs A \bs M) \bs\Phi(\vecx)\bs\Phi(\vecx)^\top} \preceq \psi \, \tr(\bs M^\top \bs A \bs M) \, \bs I_D.
    \end{equation}
    Substituting this back into the expression, we utilize the linearity of matrix multiplication and the cyclic property of the trace operator ($\tr(\bs M^\top \bs A \bs M) = \tr(\bs A \bs M \bs M^\top) = \tr(\bs A \bs\Sigma)$):
    \begin{equation}
    \begin{aligned}
        \text{LHS} &\preceq \bs M \left( \psi \, \tr(\bs M^\top \bs A \bs M) \, \bs I_D \right) \bs M^\top \\
        &= \psi \, \tr(\bs A \bs\Sigma) \, (\bs M \bs M^\top)\\
        &= \psi \, \tr(\bs A \bs\Sigma) \, \bs\Sigma.
    \end{aligned}
    \end{equation}
    This completes the proof. The constant $L$ is effectively bounded by $\sqrt{\psi \tr(\bs\Sigma)}$.
\end{proof}


\begin{lemma}\label{varmatupbound}
    Under Assumption \ref{assump:fourthmoment}, and assuming the step size satisfies $\gamma_0 \psi \tr(\bs \Sigma) < 1$. For every $t$, the variance matrix is bounded by:
    \begin{equation}
        \bs C_t \preceq \frac{\gamma_0\sigma_\eff^2}{1-\gamma_0 \psi\,\tr(\bs \Sigma)}\cdot \id
    \end{equation}
\end{lemma}

\begin{proof}
    We prove this by induction. For $t=0$, $\bs C_0 = \bs O$, the bound holds trivially.
    Assume the bound holds for $t-1$. Let $U := \frac{\gamma_0\sigma_\eff^2}{1-\gamma_0 \psi\,\tr(\bs \Sigma)}$ denote the upper bound constant.
    
    From the recursion of $\bs C_t$ defined in Eq.~\eqref{eq:variance_recursion}, and noting that $\ex{\bs\zeta_t\bs\zeta_t^\top} = \sigma_\eff^2 \bs\Sigma$, we have:
    \begin{equation}
        \begin{aligned}
            \bs C_t =&\ \mc{\widehat A}_t\circ\bs C_{t-1}+\gamma_t^2 \sigma_\eff^2 \bs\Sigma\\
            \preceq&\ \mc{\widehat A}_t\circ (U \cdot \id) +\gamma_t^2 \sigma_\eff^2 \bs\Sigma\\
            =&\ U \cdot \ex{(\id - \gamma_t \bs\phi_t\bs\phi_t^\top)(\id)(\id - \gamma_t \bs\phi_t\bs\phi_t^\top)} + \gamma_t^2\sigma_\eff^2\bs\Sigma\\
            =&\ U \cdot (\id-2\gamma_t\bs\Sigma + \gamma_t^2 \ex{\bs\phi(\vecx)\bs\phi(\vecx)^\top\bs\phi(\vecx)\bs\phi(\vecx)^\top}) +\gamma_t^2\sigma_\eff^2\bs\Sigma
        \end{aligned}
    \end{equation}
    Applying Assumption \ref{assump:fourthmoment} (Kurtosis condition) to bound the fourth moment term:
    \begin{equation}
        \begin{aligned}
            \bs C_t \preceq&\ U \cdot (\id-2\gamma_t\bs\Sigma) + \gamma_t^2 U \cdot \psi\,\tr(\bs\Sigma)\bs\Sigma+\gamma_t^2\sigma_\eff^2\bs\Sigma\\
            =&\ U \cdot \id - \gamma_t \bs\Sigma \left[ 2U - \gamma_t (U \psi \tr(\bs\Sigma) + \sigma_\eff^2) \right]
        \end{aligned}
    \end{equation}
    Rearranging the definition of $U$, we have $U(1-\gamma_0\psi\tr(\bs\Sigma)) = \gamma_0\sigma_\eff^2$, which implies $U \psi\tr(\bs\Sigma) + \sigma_\eff^2 = \frac{U}{\gamma_0}$. Substituting this back:
    \begin{equation}
        \begin{aligned}
            \bs C_t \preceq&\ U \cdot \id - \gamma_t \bs\Sigma \left[ 2U - \gamma_t \frac{U}{\gamma_0} \right]\\
            =&\ U \cdot \id - \frac{\gamma_t U}{\gamma_0} (2\gamma_0 - \gamma_t) \bs\Sigma
        \end{aligned}
    \end{equation}
    Since the step size decays, $\gamma_t \le \gamma_0$, it follows that $(2\gamma_0 - \gamma_t) > 0$. Therefore, the subtraction term is negative definite (or zero), yielding:
    \begin{equation}
        \bs C_t \preceq U \cdot \id = \frac{\gamma_0\sigma_\eff^2}{1-\gamma_0 \psi\,\tr(\bs \Sigma)}\cdot \id
    \end{equation}
    This completes the inductive step.
\end{proof}


\begin{lemma}
We have
\begin{equation}
    {\bs C}_t\preceq 16{\sigma}_\eff^2\left( \frac{1}{K}(\Sig^{\leq k^*})^{-1}  +K\gamma_0^2 \Sig^{>k^*} \right),
\end{equation}
 where the effective spectral cutoff is defined as $k^* = \max\left\{k:\lambda_{k}> \frac{2\ln N\log_2 N}{\gamma_0 N} \right\}$.
\end{lemma}

\begin{proof}
    We decompose the total variance matrix into the semi-stochastic component calculated previously and a residual term: $\bs C_t = \tilde{\bs C}_t + \bs \Delta_t$, where $\bs \Delta_t$ captures the additional variance induced by the stochasticity of the operator $\widehat{\mc A}_t$.

    Subtracting the recursion for $\tilde{\bs C}_t$ from that of $\bs C_t$, we obtain the recursion for the residual:
    \begin{equation}
        \bs \Delta_t = \mc A_t \circ \bs \Delta_{t-1} + (\widehat{\mc A}_t - \mc A_t) \circ \bs C_{t-1}.
    \end{equation}
    Recall from the operator definitions that $\widehat{\mc A}_t - \mc A_t = \widehat{\mc M}_t - \mc M_t$. We bound the source term using Assumption \ref{assump:fourthmoment} and the coarse bound derived in Lemma \ref{varmatupbound}. Let $\rho := \gamma_0 \psi \tr(\bs\Sigma)$. Assuming the step size is small enough such that $\rho \le 1/2$, we have:
    \begin{equation}
    \begin{aligned}
        (\widehat{\mc M}_t - \mc M_t) \circ \bs C_{t-1} &\preceq \widehat{\mc M}_t \circ \bs C_{t-1} \\
        &\preceq \gamma_t^2 \psi \tr(\bs \Sigma \bs C_{t-1}) \bs \Sigma \\
        &\text{(Using Lemma \ref{varmatupbound}: } \bs C_{t-1} \preceq \frac{\gamma_0 \sigma_\eff^2}{1-\rho} \id \text{)} \\
        &\preceq \gamma_t^2 \psi \tr(\bs \Sigma) \frac{\gamma_0 \sigma_\eff^2}{1-\rho} \bs \Sigma \\
        &= \frac{\rho}{1-\rho} (\gamma_t^2 \sigma_\eff^2 \bs \Sigma) \preceq \gamma_t^2 \sigma_\eff^2 \bs \Sigma.
    \end{aligned}
    \end{equation}

    Notice that the driving term for $\bs \Delta_t$ (which is $\preceq \gamma_t^2 \sigma_\eff^2 \bs \Sigma$) is bounded by the driving term of $\tilde{\bs C}_t$. Since both evolve under the same contraction operator $\mc A_t$ and start from zero, by the linearity of the recurrence, we strictly have:
    \begin{equation}
        \bs \Delta_t \preceq \tilde{\bs C}_t.
    \end{equation}

    Consequently, the total variance is bounded by:
    \begin{equation}
        \bs C_t = \tilde{\bs C}_t + \bs \Delta_t \preceq 2 \tilde{\bs C}_t.
    \end{equation}
    Substituting the bound for $\tilde{\bs C}_t$ from Lemma \ref{determvarmat} (which has a coefficient of 8), we obtain the final bound with a coefficient of 16:
    \begin{equation}
        \bs C_t \preceq 16{\sigma}_\eff^2\left( \frac{1}{K}(\Sig^{\leq k^*})^{-1} +K\gamma_0^2 \Sig^{>k^*} \right).
    \end{equation}
\end{proof}

\subsubsection{SGD Variance Upper Bound}
We follow a similar approach and calculate the iteration on the part that is irrelevant to SGD's stochasticity. The iteration on $\tilde{\bs B}_t$ is defined as
\begin{equation}
    \tilde{\bs B}_t = \mc A_t\circ \tilde{\bs B}_{t-1} + \mathcal{V}_t\circ \ex{\bs \eta_{t-1}}^{\otimes2},\quad {\bs B}_0 = \bs O.
\end{equation}
% It is straightforward to see that $\tilde{\bs B}_t$ can be expressed in the closed-form
% \begin{equation}
%     \tilde{\bs B}_t = \mc A_t\circ\mc A_{t-1}\circ\cdots\circ\mc A_1 \circ\tilde{\bs B}_0 = \left(\prod_{i=1}^t(\id-\gamma_i\bs\Sigma )\bs\eta_0\right)^{\otimes2}.
% \end{equation}

% Next, we construct an auxiliary sequence $\bs B_t^\dagger$ to estimate the stochasticity of SGD iteration. Its recursion is as follows
% \begin{equation}
%     \bs B_t^\dagger = \widehat{\mc A}_t \circ \bs B_{t-1}^\dagger + \widehat{\mc M}_t\circ \tilde{\bs B}_{t-1},\quad \bs B^\dagger_0 = \bs O.
% \end{equation}

Notice that $\prod_{i=1}^t(\id-\gamma_i\bs\Sigma )$ is a shrinking operator, and therefore $\ex{\bs \eta_{t-1}}^{\otimes2}\preceq \bs\eta_0^{\otimes 2}$. Thus,
\begin{equation}
\begin{aligned}
    \tilde{\bs B}_t &= {\mc A}_t \circ \tilde{\bs B}_{t-1} + \mathcal{V}_t\circ \ex{\bs \eta_{t-1}}^{\otimes2}\\
    &\preceq {\mc A}_t \circ \tilde{\bs B}_{t-1} + \mathcal{V}_t\circ {\bs \eta_0}^{\otimes2}\\
    &\preceq {\mc A}_t \circ \tilde{\bs B}_{t-1} + \widehat{\mc M}_t\circ {\bs \eta_0}^{\otimes2}\quad(\text{using decomposition \ref{eq:varianceoperatorexpansion}})\\
    &\preceq {\mc A}_t \circ \tilde{\bs B}_{t-1} +\gamma_t^2\psi \,\tr(\bs\eta_0^{\otimes 2}\bs\Sigma)\bs\Sigma \quad(\text{using lemma \ref{lem:fourthmomentbound}})\\
    &= {\mc A}_t \circ \tilde{\bs B}_{t-1} +\gamma_t^2\psi ||\w_0-\w^*||^2_{\bs\Sigma}\,\bs\Sigma.
\end{aligned}
\end{equation}

The following steps parallel \ref{varianceupperdirect}, since we can replace $\bs B_t$ with $\bs C_t$ and $\sigma^2_\eff$ with $\psi ||\w_0-\w^*||^2_{\bs\Sigma}$. Then, we can obtain the following lemma.

\begin{lemma}
    \begin{equation}
    \bs B_t\preceq 16\psi ||\w_0-\w^*||^2_{\bs\Sigma}\left( \frac{1}{K}(\Sig^{\leq k^*})^{-1}  +K\gamma_0^2 \Sig^{>k^*} \right),
\end{equation}
and consequently
\begin{equation}
    \langle\bs\Sigma,\bs B_t\rangle\leq 16\psi ||\w_0-\w^*||^2_{\bs\Sigma}\left( \frac{k^*}{K}  +K\gamma_0^2 \sum_{i = k^*+1}^{d} \lambda_{i}^2 \right).
\end{equation}
\end{lemma}

\subsection{Direct Learning Upper Bound}
\label{subsec:directupper}
We now utilize the previous results to provide a comprehensive bound on the parameter error and excess risk for the general SGD dynamics. This theorem characterizes the trade-off between the optimization error (bias), the sampling noise (variance), and the approximation error.

\begin{theorem}\label{instanceuppergeneral}
    Let $\bs \eta_t = \bs w_t - \bs w^*$. Under the step-decay schedule and Assumption \ref{assump:fourthmoment}, the second moment matrix satisfies:
    \begin{equation}
        \Ex[\bs\eta_N\otimes\bs \eta_N]\preceq 32 \left(\psi\|\bs\eta_0\|^2_{\bs\Sigma}+\sigma_\eff^2\right)\left( \frac{1}{K}(\bs\Sigma^{\leq k^*})^{-1} +K\gamma_0^2 \bs\Sigma^{>k^*} \right) + \bs B_N,
    \end{equation}
    where $\bs B_N = \frac{1}{N^2}\id^{\leq k^*}(\bs\eta_0\otimes\bs \eta_0)\id^{\leq k^*}+\id_{k^*:d}(\bs\eta_0\otimes\bs \eta_0)\id_{k^*:d}$ represents the bias decay and $k^* = \max\{k:\lambda_{k}> 2\ln N\log_2 N/ (\gamma_0 N) \}$.
    
    Consequently, the excess risk $\mathcal{E}(\bs w_N)$ is bounded by:
    \begin{equation}
        \ex{\mathcal{E}(\bs w_{N})}\leq\frac{1}{2N^2}\|\bs\eta_0\|^2_{\Sig^{\leq k^*}} + \frac{1}{2}\|\bs\eta_0\|^2_{\Sig^{>k^*}} + 16\left(\psi\|\bs\eta_0\|^2_{\bs\Sigma}+\sigma_\eff^2\right)\left( \frac{k^*}{K} + K\gamma_0^2 \sum_{i = k^*+1}^{d} \lambda_{i}^2 \right).
    \end{equation}
    
    Finally, the total risk (generalization error) includes the approximation residual:
    \begin{equation}
        \ex{\mathcal{R}(\bs w_{N})} = \ex{\mathcal{E}(\bs w_{N})} +\frac{1}{2}\|\bs w_*\|_{\bs \Pi^{\perp}}^2.
    \end{equation}
\end{theorem}

\begin{proof}
    \textbf{1. Decomposition of Second Moment.}
    By the bias-variance decomposition lemma (Lemma \ref{lem:bvdecomp}), the total second moment is the sum of the squared bias and the variance:
    \begin{equation}
        \Ex[\bs\eta_N \otimes \bs\eta_N] = \bs B_N + \bs C_N.
    \end{equation}
    For the bias term $\bs B_N$, we apply the result from Lemma \ref{lemma:bias_upper_bound}. The step-decay schedule ensures that the error in the head subspace ($k \le k^*$) decays at a rate of $O(1/N^2)$, while the tail error ($k > k^*$) remains proportional to the initial condition. This yields the term:
    \begin{equation}
        \bs B_N \preceq \frac{1}{N^2}\id^{\leq k^*}(\bs\eta_0\otimes\bs \eta_0)\id^{\leq k^*} + \id_{k^*:d}(\bs\eta_0\otimes\bs \eta_0)\id_{k^*:d}.
    \end{equation}
    
    For the variance term $\bs C_N$, we recall the Total Variance Bound derived in Appendix \ref{varianceupperdirect}. However, in the general case, the stochastic gradient noise is driven not just by the label noise $\sigma_\eff^2$, but also by the fluctuations arising from the current estimation error. Under Assumption \ref{assump:fourthmoment}, the magnitude of this gradient noise is bounded by $\sigma_{noise}^2 \approx \sigma_\eff^2 + \psi \|\bs\eta_0\|^2_{\bs\Sigma}$ (taking the conservative bound using the initial error energy). Substituting this combined noise level into the variance bound (and applying the coefficient 32 to account for the loose relaxation of the fourth moments):
    \begin{equation}
        \bs C_N \preceq 32 (\psi\|\bs\eta_0\|^2_{\bs\Sigma}+\sigma_\eff^2)\left( \frac{1}{K}(\bs\Sigma^{\leq k^*})^{-1} + K\gamma_0^2 \bs\Sigma^{>k^*} \right).
    \end{equation}
    Summing $\bs B_N$ and $\bs C_N$ yields the first inequality of the theorem.

    \textbf{2. Derivation of Excess Risk.}
    The expected excess risk is given by $\ex{\mathcal{E}(\bs w_N)} = \frac{1}{2} \tr(\bs\Sigma \, \Ex[\bs\eta_N \otimes \bs\eta_N])$. We take the trace of the RHS of Eq. (1) against $\frac{1}{2}\bs\Sigma$.
    
    \textit{Variance Part:}
    \begin{align}
        \frac{1}{2} \tr\left( \bs\Sigma \cdot (\bs\Sigma^{\leq k^*})^{-1} \right) &= \frac{1}{2} \sum_{i=1}^{k^*} \lambda_i \cdot \frac{1}{\lambda_i} = \frac{k^*}{2}, \\
        \frac{1}{2} \tr\left( \bs\Sigma \cdot \bs\Sigma^{> k^*} \right) &= \frac{1}{2} \sum_{i=k^*+1}^d \lambda_i^2.
    \end{align}
    Applying the pre-factor $32 (\dots)$, the variance contribution becomes $16 (\psi\|\bs\eta_0\|^2_{\bs\Sigma}+\sigma_\eff^2) (\frac{k^*}{K} + K\gamma_0^2 \sum \lambda_i^2)$.
    
    \textit{Bias Part:}
    Similarly, $\frac{1}{2} \tr(\bs\Sigma \bs B_N)$ directly translates to the weighted norms $\frac{1}{2N^2}\|\bs\eta_0\|^2_{\Sig^{\leq k^*}}$ and $\frac{1}{2}\|\bs\eta_0\|^2_{\Sig^{>k^*}}$.
    
    \textbf{3. Total Risk.}
    The total risk $\mathcal{R}(\bs w_N)$ is defined over the entire input space. If the target function $\bs w^*$ is not fully realizable within the feature space (i.e., $\bs w^*$ has a component in the null space of $\bs \Sigma$), an irreducible approximation error exists. By the Pythagorean theorem in the Hilbert space, this error is orthogonal to the estimation error, yielding the additive term $\frac{1}{2}\|\bs w_*\|_{\bs \Pi^{\perp}}^2$.
\end{proof}

\subsection{Lower Bounds by Part}
\label{subsec:lowerboundsbypart}

\subsubsection{Bias Lower Bound}
\label{subsubsec:biaslowerbound}

We now show that the bias in the tail subspace ($k > k^*$) cannot be reduced arbitrarily. Due to the limited capacity of the step-decay schedule, the components of the initial error aligned with small eigenvalues are effectively preserved.

\begin{lemma}[Tail Bias Lower Bound]
    \label{lemma:bias_lower_bound}
    The bias energy in the tail subspace satisfies the following lower bound:
    \begin{equation}
        \nm[\Sig^{> k^*}]{\prod_{i=1}^N(\id-\gamma_i\bs\Sigma)\bs\eta_0} \geq \frac{1}{100} \|\bs\eta_0\|^2_{\Sig^{> k^*}}
    \end{equation}
    where the effective spectral cutoff is defined as $k^* = \max\left\{k:\lambda_{k}> \frac{\log_2 N}{\gamma_0 N} \right\}$.
\end{lemma}

\begin{proof}
    We analyze the squared weighted norm in the tail subspace.
    First, recall the lower bound for the cumulative contraction from Lemma \ref{lemma:product_contraction}:
    \[ \prod_{i=1}^N (1-\gamma_i \lambda_k) \geq \exp\left( -2.02 \frac{N}{\log_2 N} \gamma_0 \lambda_k \right). \]
    Since the bias term involves the square of the contraction operator, the exponent coefficient doubles to $4.04$.
    
    By the definition of the tail indices $k > k^*$, the eigenvalues satisfy $\lambda_k \le \frac{\log_2 N}{\gamma_0 N}$. Consequently, for any direction $u$ in the tail subspace $\bs\Sigma^{>k^*}$, the cumulative contraction is bounded away from zero:
    \begin{equation}
        \prod_{i=1}^N (1-\gamma_i \lambda_k)^2 \geq \exp\left( -4.04 \frac{N}{\log_2 N} \gamma_0 \lambda_k \right) \geq \exp\left( -4.04 \frac{N}{\log_2 N} \gamma_0 \cdot \frac{\log_2 N}{\gamma_0 N} \right) = e^{-4.04}.
    \end{equation}
    
    Calculating the constant, we have $e^{-4.04} \approx 0.0176 > 0.01$. Applying this to the matrix norm:
    \begin{equation}
    \begin{aligned}
        \nm[\Sig^{> k^*}]{\prod_{i=1}^N(\id-\gamma_i\bs\Sigma)\bs\eta_0} &= \inner{\bs\eta_0\otimes\bs\eta_0,\prod_{i=1}^N(\id-\gamma_i\bs\Sigma^{> k^*})^2\bs\Sigma^{> k^*}}\\
        &\geq \inner{\bs\eta_0\otimes\bs\eta_0, \exp\left(-4.04\frac{N}{\log_2 N}\gamma_0\bs\Sigma^{>k^*}\right) \bs\Sigma^{> k^*}} \\
        &\geq \inner{\bs\eta_0\otimes\bs\eta_0, e^{-4.04}\cdot \id \cdot \bs\Sigma^{> k^*}}\\
        &\geq \frac{1}{100} \inner{\bs\eta_0\otimes\bs\eta_0, \bs\Sigma^{> k^*}}\\
        &= \frac{1}{100} \|\bs\eta_0\|^2_{\Sig^{> k^*}}.
    \end{aligned}
    \end{equation}
    This confirms that at least 1\% of the initial error energy in the tail subspace remains uncorrected after $N$ steps.
\end{proof}


\subsubsection{Variance Lower Bound}
\label{subsubsec:variance_lower_bound}

To establish the tightness of our analysis, we explicitly derive the lower bound of the variance term. This confirms that the error scaling with respect to the learning rate and spectrum is unavoidable under the SGD dynamics.

\begin{lemma}[Variance Lower Bound]
\label{lem:variance_lower_bound}
    Under Assumption \ref{assump:fourthmoment}, and assuming the step size is sufficiently small such that $\gamma_0 \lambda_{\max} \leq 1$, the variance matrix has the following lower bound:
    \begin{equation}
        \bs C_t \succeq \frac{\sigma_\eff^2}{4} \left( \gamma_0 \id^{\le k^*} +  K \gamma_0^2 \bs\Sigma^{>k^*} \right),
    \end{equation}
    where $K$ is the effective number of steps in the final constant learning rate phase.
\end{lemma}

\begin{proof}
    We start with the variance recursion $\bs C_t = \mc{\widehat A}_t \circ \bs C_{t-1} + \gamma_t^2 \sigma_\eff^2 \bs \Sigma$.
    Recall the operator decomposition $\widehat{\mc A}_t = \mc A_t + \mc B_t$, where $\mc B_t$ is the covariance of the operator itself (as defined in Appendix A.2). Since $\mc B_t$ is a covariance operator, it is positive semi-definite, meaning $\mc B_t \circ \bs X \succeq \bs O$ for any PSD matrix $\bs X$. Consequently, $\mc{\widehat A}_t \circ \bs X \succeq \mc A_t \circ \bs X$.
    
    This allows us to lower bound the total variance $\bs C_t$ by the semi-stochastic variance $\tilde{\bs C}_t$ (which ignores operator noise):
    \begin{equation}
        \bs C_t \succeq \tilde{\bs C}_t = \sigma_\eff^2 \sum_{i = 1}^t \gamma_i^2 \left[ \prod_{j=i+1}^t(\id-\gamma_j\bs\Sigma) \right]^2 \bs \Sigma.
    \end{equation}
    We analyze the diagonal elements $[\tilde{\bs C}_t]_k$ for the Head and Tail subspaces separately.
    
    \textbf{Case 1: The Head Spectrum ($k \leq k^*$)}.
    For the learned directions, the eigenvalues are large enough that the memory of the initial state decays. We focus on the last phase of length $K$ with constant learning rate $\gamma_0$. The sum becomes a geometric series:
    \begin{equation}
        [\tilde{\bs C}_t]_k \ge \sigma_\eff^2 \lambda_k \gamma_0^2 \sum_{s=0}^{K-1} (1-\gamma_0 \lambda_k)^{2s}.
    \end{equation}
    Using the geometric series sum formula $\sum_{s=0}^{K-1} r^s = \frac{1-r^K}{1-r}$ with $r = (1-\gamma_0 \lambda_k)^2$:
    \begin{equation}
        \sum_{s=0}^{K-1} (1-\gamma_0 \lambda_k)^{2s} = \frac{1 - (1-\gamma_0 \lambda_k)^{2K}}{1 - (1-\gamma_0 \lambda_k)^2}.
    \end{equation}
    By the definition of the spectral cutoff $k^*$, for $k \le k^*$, the term $(1-\gamma_0 \lambda_k)^{2K}$ is negligible (specifically, we can bound $(1-\gamma_0 \lambda_k)^{2K} \le 1/2$ for sufficiently large $K$). For the denominator, we use the identity $1-(1-x)^2 = 2x - x^2$. Since $\gamma_0 \lambda_k \le 1$, we have $2\gamma_0 \lambda_k - (\gamma_0 \lambda_k)^2 \le 2\gamma_0 \lambda_k$. Thus:
    \begin{equation}
        [\tilde{\bs C}_t]_k \ge \sigma_\eff^2 \lambda_k \gamma_0^2 \cdot \frac{1/2}{2\gamma_0 \lambda_k} = \frac{\gamma_0 \sigma_\eff^2}{4}.
    \end{equation}
    This establishes that the head variance is isotropic and bounded below by $O(\gamma_0)$.

    \textbf{Case 2: The Tail Spectrum ($k > k^*$)}.
    In the tail, the contraction is minimal. We use the inequality $(1-x)^2 \ge 1-2x$. The contraction factor over the last $K$ steps satisfies:
    \begin{equation}
        \prod_{j=t-K+1}^t (1-\gamma_0 \lambda_k)^2 = (1 - \gamma_0 \lambda_k)^{2K} \ge 1 - 2K\gamma_0 \lambda_k.
    \end{equation}
    Since $k > k^*$, we have $K \gamma_0 \lambda_k \ll 1$, so the product is lower bounded by a constant (conservatively $\ge 1/2$). The summation over $K$ steps simply accumulates:
    \begin{equation}
        [\tilde{\bs C}_t]_k \ge \sigma_\eff^2 \lambda_k \sum_{i=t-K}^t \gamma_0^2 \cdot \frac{1}{2} = \frac{1}{2} K \gamma_0^2 \sigma_\eff^2 \lambda_k.
    \end{equation}
    
    Combining both cases yields the stated matrix lower bound.
\end{proof}

\subsection{Direct Learning Lower Bound}
\label{subsec:directlowerbound}

We combine the results from the bias and variance lower bounds to establish the fundamental limit of direct learning.

\begin{theorem}
    The excess risk $\mathcal{E}(\bs w_N)$ is bounded by:
    \begin{equation}
        \ex{\mathcal{E}(\bs w_{N})}\geq \frac{1}{200}\|\bs\eta_0\|^2_{\Sig^{>k^*}} + \frac{\sigma_\eff^2}{8}\left( \frac{k^*}{K} + K\gamma_0^2 \sum_{i = k^*+1}^{d} \lambda_{i}^2 \right)
    \end{equation}
    where the effective spectral cutoff is defined as $k^* = \max\left\{k:\lambda_{k}> \frac{\log_2 N}{\gamma_0 N} \right\}$.
\end{theorem}

\begin{proof}
    The expected excess risk is defined as $\Ex[\mathcal{E}(\bs w_N)] = \frac{1}{2} \tr(\bs \Sigma \Ex[\bs \eta_N \otimes \bs \eta_N])$. Using the bias-variance decomposition $\Ex[\bs \eta_N \otimes \bs \eta_N] = \bs B_N + \bs C_N$, we lower bound each component separately.

    \textbf{1. Bias Component:}
    From Lemma \ref{lemma:bias_lower_bound}, we established the lower bound for the bias energy in the tail subspace:
    \begin{equation}
        \tr(\bs \Sigma \bs B_N) \ge \nm[\Sig^{> k^*}]{\prod_{i=1}^N(\id-\gamma_i\bs\Sigma)\bs\eta_0} \geq \frac{1}{100} \|\bs\eta_0\|^2_{\Sig^{> k^*}}.
    \end{equation}
    Multiplying by the factor $\frac{1}{2}$ from the risk definition yields the first term $\frac{1}{200}\|\bs\eta_0\|^2_{\Sig^{>k^*}}$.

    \textbf{2. Variance Component:}
    From Lemma \ref{lem:variance_lower_bound}, the variance matrix satisfies $\bs C_N \succeq \frac{\sigma_\eff^2}{4} ( \gamma_0 \id^{\le k^*} + K \gamma_0^2 \bs\Sigma^{>k^*} )$. Applying the trace operator $\frac{1}{2}\tr(\bs\Sigma \cdot)$:
    
    For the \textbf{Tail part} ($k > k^*$):
    \begin{equation}
        \frac{1}{2} \tr\left( \bs \Sigma \cdot \frac{\sigma_\eff^2}{4} K \gamma_0^2 \bs\Sigma^{>k^*} \right) = \frac{\sigma_\eff^2}{8} K \gamma_0^2 \sum_{i=k^*+1}^d \lambda_i^2.
    \end{equation}
    
    For the \textbf{Head part} ($k \le k^*$):
    \begin{equation}
        \frac{1}{2} \tr\left( \bs \Sigma \cdot \frac{\sigma_\eff^2}{4} \gamma_0 \id^{\le k^*} \right) = \frac{\sigma_\eff^2}{8} \sum_{i=1}^{k^*} \gamma_0 \lambda_i.
    \end{equation}
    We now connect the term $\sum \gamma_0 \lambda_i$ to $k^*/K$. By the definition of the effective cutoff $k^* = \max\{k:\lambda_{k}> \frac{\log_2 N}{\gamma_0 N} \}$ and recalling that the effective phase length is $K \approx \frac{N}{\log_2 N}$, we have the inequality $\gamma_0 \lambda_i \ge \frac{1}{K}$ for all $i \le k^*$.
    Consequently, the summation is bounded by:
    \begin{equation}
        \sum_{i=1}^{k^*} \gamma_0 \lambda_i \geq \sum_{i=1}^{k^*} \frac{1}{K} = \frac{k^*}{K}.
    \end{equation}
    Substituting this back yields the head variance term $\frac{\sigma_\eff^2}{8} \frac{k^*}{K}$.

    Summing the Bias, Head Variance, and Tail Variance terms completes the proof.
\end{proof}

\subsection{Rate in Polynomial-Decay Scenario}
\label{subsec:polydecaybound}

We explicitly calculate the convergence rate under the polynomial decay assumption, which is a standard setting for high-dimensional analysis.

\begin{theorem}
    Assume the eigenvalues of $\bs\Sigma$ follow a polynomial decay law $\lambda_k \asymp k^{-\alpha}$, and the target parameter coefficients satisfy $(w_k^*)^2 \asymp k^{\beta}$. Then the excess risk $\mathcal{E}(\bs w_N)$ has the matching upper and lower rate:
    \begin{equation}
        \tilde{{\Theta}}\left(N^{\frac{1+\beta-\alpha}{\alpha}}\cdot\gamma_0^{\frac{1+\beta-\alpha}{\alpha}}\right) + \tilde{{\Theta}}\left(N^{\frac{1-\alpha}{\alpha}}\cdot\gamma_0^{\frac{1}{\alpha}}\right).
    \end{equation}
\end{theorem}

\begin{proof}
    The proof relies on substituting the spectral decay rates into the general bounds derived in Theorem \ref{instanceuppergeneral} and Section \ref{subsec:directlowerbound}.

    \textbf{Step 1: Effective Spectral Cutoff $k^*$.}
    The cutoff index $k^*$ is determined by the condition $\gamma_0 N \lambda_{k^*} \asymp 1$ (ignoring logarithmic factors). Substituting $\lambda_k \asymp k^{-\alpha}$:
    \begin{equation}
        \gamma_0 N (k^*)^{-\alpha} \asymp 1 \implies k^* \asymp (\gamma_0 N)^{\frac{1}{\alpha}}.
    \end{equation}

    \textbf{Step 2: Bias Decay Rate.}
    The bias is dominated by the tail error in the unlearned subspace ($k > k^*$). Using the integral approximation for the sum:
    \begin{equation}
        \text{Bias} \asymp \sum_{k=k^*}^\infty \lambda_k (w_k^*)^2 \asymp \int_{k^*}^\infty x^{-\alpha} x^{\beta} \, dx = \int_{k^*}^\infty x^{\beta-\alpha} \, dx.
    \end{equation}
    Assuming $\beta - \alpha < -1$ for convergence, the integral evaluates to:
    \begin{equation}
        \left[ \frac{x^{\beta-\alpha+1}}{\beta-\alpha+1} \right]_{k^*}^\infty \asymp (k^*)^{1+\beta-\alpha}.
    \end{equation}
    Substituting $k^* \asymp (\gamma_0 N)^{\frac{1}{\alpha}}$:
    \begin{equation}
        \text{Bias} \asymp \left( (\gamma_0 N)^{\frac{1}{\alpha}} \right)^{1+\beta-\alpha} = N^{\frac{1+\beta-\alpha}{\alpha}} \gamma_0^{\frac{1+\beta-\alpha}{\alpha}}.
    \end{equation}

    \textbf{Step 3: Variance Decay Rate.}
    The variance is dominated by the head dimension term $\frac{k^*}{N}$ (assuming the effective noise $\sigma_\eff^2$ is constant).
    \begin{equation}
        \text{Variance} \asymp \frac{k^*}{N} \asymp \frac{(\gamma_0 N)^{\frac{1}{\alpha}}}{N} = N^{\frac{1}{\alpha}-1} \gamma_0^{\frac{1}{\alpha}} = N^{\frac{1-\alpha}{\alpha}} \gamma_0^{\frac{1}{\alpha}}.
    \end{equation}
    (Note: The tail variance term $N \gamma_0^2 \sum_{k>k^*} \lambda_k^2$ is typically of a lower order or comparable magnitude depending on $\alpha$, so the head term determines the main rate).

    \textbf{Step 4: Conclusion.}
    Combining the Bias and Variance rates yields the final expression:
    \begin{equation}
        \mathcal{E}(\bs w_N) \asymp N^{\frac{1+\beta-\alpha}{\alpha}}\gamma_0^{\frac{1+\beta-\alpha}{\alpha}} + N^{\frac{1-\alpha}{\alpha}}\gamma_0^{\frac{1}{\alpha}}.
    \end{equation}
\end{proof}









\section{Proofs of Results in Knowledge Transfer}
\subsection{SGD Dynamics for Knowledge Transfer}
We model the knowledge transfer as an online training process. The student model, $f_\ts(\vecx) = \langle \w_\ts, \bs\phi_\st(\vecx) \rangle$, is trained over $n$ samples from $\w_{0,\ts} = \bs 0$ to mimic a fixed, pre-trained teacher, $f_\te(\vecx) = \langle \w_\te, \bs\phi_\te(\vecx) \rangle$, using Stochastic Gradient Descent (SGD) on the transfer risk $\mathcal{R}_{\mathrm{Trans}}$ (defined in Eq. \ref{eq:risk_transfer}).


At each time step $t$, given a learning rate $\gamma_t$ and a new sample $\vecx_t$, the student's parameters $\w_{\ts}$ are updated as follows:
\begin{equation}
\begin{aligned}
\label{paraupdatets}
\w_{t,\ts} &\leftarrow \w_{t-1,\ts} - \gamma_t \widehat\nabla_{t}\mathcal{R}_\mathrm{Trans}(\w_{t-1,\ts}) \\
&= \w_{t-1,\ts} - \gamma_t \left( \bs\phi_{t,\st}\bs\phi_{t,\st}^\top \w_{t-1,\ts} - \bs\phi_{t,\st}f_\te(\vecx_t) \right) \\
&= \w_{t-1,\ts} - \gamma_t \left( \bs\phi_{t,\st}\bs\phi_{t,\st}^\top \w_{t-1,\ts} - \bs\phi_{t,\st}\bs\phi_{t,\te}^\top \w_\te \right) \\
&= \left(\id-\gamma_t\M_\st\Phi_t\Phi_t^\top\M_\st^\top \right)\w_{t-1,\ts}+\gamma_t \M_\st\Phi_t\Phi_t^\top\M_\te^\top \w_\te.
\end{aligned}
\end{equation}
Here, we use the shorthand $\bs\phi_{t, \st} := \bs\phi_\st(\vecx_t)$, $\bs\phi_{t, \te} := \bs\phi_\te(\vecx_t)$, and $\Phi_t := \Phi(\vecx_t)$ for the feature vectors evaluated at $\vecx_t$. The training is assumed to start from $\w_{0,\ts} = \mathbf{0}$. The final line substitutes the feature basis definitions from our setup.

This iterative process converges to the optimal parameters $\w^*_\ts$ that minimize the population transfer risk. We find this optimum by setting the gradient of the population risk to zero:
\begin{equation}
\begin{aligned}
\nabla_{\w_\ts}\mathcal{R}_{\mathrm{Trans}} &= \nabla_{\w_\ts}\left[\frac{1}{2}\Ex_\vecx\left(\langle \w_\ts,\bs\phi_\st(\vecx)\rangle-\langle \w_\te,\bs\phi_\te(\vecx)\rangle\right)^2 \right] \\
&= \Ex_\vecx\left[ \bs\phi_\st(\vecx)\bs\phi_\st(\vecx)^\top \w_\ts - \bs\phi_\st(\vecx)\bs\phi_\te(\vecx)^\top \w_\te \right] \\
&= \Ex_\vecx\left[ \M_\st\Phi(\vecx)\Phi(\vecx)^\top\M_\st^\top \right] \w_\ts - \Ex_\vecx\left[ \M_\st\Phi(\vecx)\Phi(\vecx)^\top\M_\te^\top \right] \w_\te \\
&= \M_\st\M_\st^\top \w_\ts - \M_\st\M_\te^\top \w_\te = \mathbf{0}.
\end{aligned}
\end{equation}
This is a standard normal equation, where we used the assumption $\Ex[\Phi(\vecx)\Phi(\vecx)^\top]=\id$. The solution $\w^*_\ts$, which represents the optimal parameters the student can learn from the teacher, is given by:
\begin{equation}
\w^*_\ts = (\M_\st\M_\st^\top)^+ \M_\st\M_\te^\top \w_\te = (\M_\st^\top)^+ \M_\te^\top \w_\te,
\end{equation}
where $(\cdot)^+$ denotes the Moore-Penrose pseudoinverse.

The excess risk formula is in a concise form similar to \ref{generalexcessriskcomputation}:
\begin{equation}\label{excessrisktrans}
\begin{aligned}
    \mathcal{E}_{\mathrm{Trans}} =& \mathcal{R}_\mathrm{Trans}(\w_\ts)-\mathcal{R}_\mathrm{Trans}(\w^*_\ts)\\
    =&\frac{1}{2}\Ex_\vecx\left[\langle \w_\ts-\w^*_\ts,\bs\phi_\st(\vecx)\rangle\right]^2+\Ex_\vecx\left[(\langle \w_\ts-\w^*_\ts,\bs\phi_\st(\vecx)\rangle)(\langle \w^*_\ts,\bs\phi_\st(\vecx)\rangle-\langle \w_\te,\bs\phi_\te(\vecx)\rangle)\right]\\
    =& \frac{1}{2}\Ex_\vecx\left[\langle \w_\ts-\w^*_\ts,\bs\phi_\st(\vecx)\rangle\right]^2+\Ex_\vecx\left[ (\w_\ts-\w^*_\ts)^\top\M_\st \Phi(\vecx)\Phi(\vecx)^\top(\M_\st^\top\M_\st^{\top +}-\id)\M_\te^\top \w_\te\right]\\
    =& \frac{1}{2}\Ex_\vecx\left[\langle \w_\ts-\w^*_\ts,\bs\phi_\st(\vecx)\rangle\right]^2+(\w_\ts-\w^*_\ts)^\top(\M_\st \M_\st^\top\M_\st^{\top +}-\M_\st)\M_\te^\top \w_\te\\
    =& \frac{1}{2}\Ex_\vecx\left[\langle \w_\ts-\w^*_\ts,\bs\phi_\st(\vecx)\rangle\right]^2\\
    =& \frac{1}{2}||\w_\ts-\w^*_\ts||_{\Sigma_\st}^2 = \frac{1}{2}\langle (\w_\ts-\w^*_\ts)\otimes(\w_\ts-\w^*_\ts),\mathbf{\Sigma}_\st\rangle.
\end{aligned}
\end{equation}
To estimate the magnitude of the risk we only have to investigate the dynamics of $(\w_\ts-\w^*_\ts)^{\otimes2}$. The update rule \ref{paraupdatets} could be further transformed into 
\begin{equation}
\begin{aligned}
    \w_{t,\ts}-\w^*_\ts=& (\w_{t-1,\ts}-\w^*_\ts)-\gamma_t\left(\M_\st\Phi_t\Phi_t^\top\M_\st^\top \w_{t-1,\ts}- \M_\st\Phi_t\Phi_t^\top\M_\te^\top \w_\te\right)\\
    =& (\id-\gamma_t\mathbf{\bs\phi}_{t,\st}\mathbf{\bs\phi}_{t,\st}^\top)(\w_{t-1,\ts}-\w^*_\ts)+\gamma_t\M_\st\Phi_t\Phi^\top_t\mathbf{\Pi}^\perp_\st\M_\te^\top \w_\te\\
    =& (\id-\gamma_t\mathbf{\bs\phi}_{t,\st}\mathbf{\bs\phi}_{t,\st}^\top)(\w_{t-1,\ts}-\w^*_\ts)+\gamma_t(\Phi^\top_t\mathbf{\Pi}^\perp_\st\M_\te^\top \w_\te)\cdot\bs\phi_{t,\st}\\
    =&  (\id-\gamma_t\mathbf{\Sigma}_\st)(\w_{t-1,\ts}-\w^*_\ts)+\gamma_t(\mathbf{\Sigma}_\st-\mathbf{\bs\phi}_{t,\st}\mathbf{\bs\phi}_{t,\st}^\top)(\w_{t-1,\ts}-\w^*_\ts)\\
    &+\gamma_t(\Phi^\top_t\mathbf{\Pi}^\perp_\st\M_\te^\top \w_\te)\cdot\bs\phi_{t,\st},
\end{aligned}
\end{equation}
where $\mathbf{\Pi}^\perp_\st := \id - \M_\st^\top\M_\st^{\top+}$ is a matrix that projects vectors to the subspace perpendicular to the image space of $\M_\st$.


The random variable $\Phi^\top_t\mathbf{\Pi}^\perp\M_\te^\top \w_\te$ could be understood as the noise generated by the parts of the teacher model that the student model could not understand, i.e., beyond its expressivity. The following calculations show that it actually acts like noise relative to $\bs\phi_\st$.
\begin{equation}
\Ex_\vecx(\Phi^\top_t\mathbf{\Pi}^\perp\M_\te^\top \w_\te)\cdot\bs\phi_{t,\st} = \Ex_\vecx\M_\st\Phi_t\Phi^\top_t(\id - \M_\st^\top\M_\st^{\top+})\M_\te^\top \w_\te = \M_\st(\id - \M_\st^\top\M_\st^{\top+})\M_\te^\top \w_\te = 0.
\end{equation}

We henceforth define the transfer noise
\begin{equation}
    \hat\sigma^2:= \Ex_{\rho_{\vecx}}\left[\nm{\bs\Phi^\top\bs \Pi^{\perp}_\st \w_*}\right] =||\w_*||_{\bs \Pi^{\perp}_\st}^2.
\end{equation}

For theoretical analysis, we define the iterate $\bs{\eta}_t:= \w_{t,\ts}-\w^*_\ts$. Its iteration could be written in the following compact form
\begin{equation}
\label{eq:transferiterateevolution}
    \bs\eta_t = \bs{\widehat A}_t\,\bs\eta_{t-1}+\gamma_t\bs\zeta_t,
\end{equation}
where $\bs{\widehat A}_{t}=\id - \gamma_t\mathbf{\bs\phi}_{t,\st}\mathbf{\bs\phi}_{t,\st}^\top$, $\bs\zeta_t := (\Phi^\top_t\mathbf{\Pi}^\perp\M_\te^\top \w_\te)\cdot\bs\phi_{t,\st}$. $\Ex[\bs\zeta_t]=0$. Also, we define $\bs A_t = \id-\gamma_t\bs\Sigma_\st$ which is the non-stochastic version (expectation) of $\widehat{\bs A}_t$.

We notice the evolution of the transfer iterate (equation \ref{eq:transferiterateevolution}) follows exactly same dynamics as the direct learning iterate (equation \ref{eq:directdynamics}). Apply similar procedures we shall derive a bound on the transfer risk.


\begin{theorem}\label{thm:transferriskupperbound}
    The excess risk $\mathcal{E}(\bs w_N)$ is bounded by:
    \begin{equation}
        \ex{\mathcal{E}(\bs w_{n,\ts})}\leq\frac{1}{2n^2}\ex{\|\w_{N,\te}\|^2_{\Sig^{\leq k^*}}} + \frac{1}{2}\ex{\|\w_{N,\te}\|^2_{\Sig^{>k^*}}} + 16\left(\psi\ex{\|\w_{N,\te}\|^2_{\bs\Sigma}}+\hat\sigma^2\right)\left( \frac{k^*}{K'} + K'(\gamma_0')^2 \sum_{i = k^*+1}^{d} \lambda_{i}^2 \right).
    \end{equation}
    where $k^* = \max\{k:\lambda_{k,\st}> 2\ln n\log_2 n/ (\gamma_0' n) \}$.
\end{theorem}

\begin{proof}
    To prove Theorem \ref{thm:transferriskupperbound}, we map the specific dynamics of the knowledge transfer process derived in Eq. \eqref{paraupdatets} to the general SGD dynamics analyzed in Theorem \ref{instanceuppergeneral}.

    Theorem \ref{instanceuppergeneral} provides the upper bound for excess risk under step-decay SGD:
    \begin{equation}
        \ex{\mathcal{E}(\bs w_n)} \leq \underbrace{\frac{1}{2n^2}\|\bs\eta_0\|^2_{\Sig^{\leq k^*}} + \frac{1}{2}\|\bs\eta_0\|^2_{\Sig^{>k^*}}}_{\text{Bias Term}} + \underbrace{16(\psi\|\bs\eta_0\|^2_{\bs\Sigma}+\sigma_\eff^2)\left( \frac{k^*}{K} + K\gamma_0^2 \sum_{i > k^*} \lambda_{i}^2 \right)}_{\text{Variance Term}}.
    \end{equation}
    
    Substituting the mapped variables into this equation:
    \begin{enumerate}
        \item Replace the sample size $N$ with the student's training steps $n$.
        \item Replace the generic covariance $\bs\Sigma$ with $\bs\Sigma_\st$ and its eigenvalues $\lambda_{k,\st}$.
        \item Replace the initial error norms $\|\bs\eta_0\|^2_{(\cdot)}$ with the teacher's norms $\|\w_{N,\te}\|^2_{(\cdot)}$.
        \item Replace the noise variance $\sigma_\eff^2$ with the transfer residual $\hat\sigma^2$.
        \item Replace the stepsize interval $K$ with $K' = n/\log_2n$.
    \end{enumerate}
    Then we shall have the desired upper bound.
\end{proof}

\subsection{Proof of Geometric Consistency Condition (Lemma \ref{lem:geometric_consistency})}
\label{proof:geometric_consistency}

We derive the consistency condition by explicitly comparing the optimal parameters obtained via Direct Learning versus Knowledge Transfer using the Moore-Penrose pseudoinverse properties.

\begin{proof}
    Let the ground truth function be represented by $\w_*$ in the feature space $\Phi$. The student and teacher parameterize their functions as $f(\vecx) = \w^\top \M \Phi(\vecx) = (\M^\top \w)^\top \Phi(\vecx)$.
    Define the orthogonal projection operators onto the image spaces of the student and teacher feature maps as $\bs\Pi_\st = \M_\st^\top (\M_\st^\top)^+$ and $\bs\Pi_\te = \M_\te^\top (\M_\te^\top)^+$, respectively.

    \textbf{1. Optimal Parameters Derivation}
    
    \textit{Direct Learning:} The student directly approximates $\w_*$. The minimal-norm solution to $\min_\w \| \M_\st^\top \w - \w_* \|^2$ is:
    \begin{equation}
        \w^*_\st = (\M_\st^\top)^+ \w_*.
    \end{equation}

    \textit{Knowledge Transfer:} The teacher first learns the ground truth, yielding $\w^*_\te = (\M_\te^\top)^+ \w_*$. The student then mimics the teacher's effective parameter $\w_{\te,\text{eff}} = \M_\te^\top \w^*_\te = \bs\Pi_\te \w_*$. The student's optimal solution $\w^\opt_\ts$ for this target is:
    \begin{equation}
        \w_{\ts}^\opt = (\M_\st^\top)^+ (\M_\te^\top \w^*_\te) = (\M_\st^\top)^+ \bs\Pi_\te \w_*.
    \end{equation}

    \textbf{2. Consistency Condition}
    
    The gap between the direct solution and the transfer solution is:
    \begin{equation}
    \begin{aligned}
        \w^*_\st - \w_{\ts}^\opt &= (\M_\st^\top)^+ \w_* - (\M_\st^\top)^+ \bs\Pi_\te \w_* \\
        &= (\M_\st^\top)^+ (\id - \bs\Pi_\te) \w_* \\
        &= (\M_\st^\top)^+ \bs\Pi_\te^\perp \w_*.
    \end{aligned}
    \end{equation}
    Using the property that $(\M_\st^\top)^+ = (\M_\st^\top)^+ \bs\Pi_\st$ (since the pseudoinverse acts on the range space), we can rewrite this as:
    \begin{equation}
        \w^*_\st - \w_{\ts}^\opt = (\M_\st^\top)^+ \bs\Pi_\st \bs\Pi_\te^\perp \w_*.
    \end{equation}
    Thus, consistency ($\w^*_\st = \w_{\ts}^\opt$) holds if and only if $\bs\Pi_\st \bs\Pi_\te^\perp \w_* = \bs 0$ ($\M_\st$ has the same null space as $\bs\Pi_\st$).

    \textbf{3. Static Alignment Bias (Risk Gap)}
    
    The generalization risk is measured in the feature space: $\mathcal{R}(\w) = \frac{1}{2} \| \M_\st^\top \w - \w_* \|^2$.
    
    For the transfer student, the prediction is $\M_\st^\top \w_{\ts}^\opt = \bs\Pi_\st \bs\Pi_\te \w_*$. The residual vector can be decomposed orthogonally:
    \begin{equation}
    \begin{aligned}
        \M_\st^\top \w_{\ts}^\opt - \w_* &= (\bs\Pi_\st \bs\Pi_\te \w_* - \bs\Pi_\st \w_*) + (\bs\Pi_\st \w_* - \w_*) \\
        &= -\bs\Pi_\st (\id - \bs\Pi_\te) \w_* - (\id - \bs\Pi_\st) \w_* \\
        &= \underbrace{-\bs\Pi_\st \bs\Pi_\te^\perp \w_*}_{\in \text{Range}(\M_\st^\top)} \quad \underbrace{-\bs\Pi_\st^\perp \w_*}_{\in \text{Range}(\M_\st^\top)^\perp}.
    \end{aligned}
    \end{equation}
    Since the two terms are orthogonal, the squared norm splits:
    \begin{equation}
        2\mathcal{R}_\ts(\w^\opt_\ts) = \| \bs\Pi_\st \bs\Pi_\te^\perp \w_* \|^2 + \| \bs\Pi_\st^\perp \w_* \|^2.
    \end{equation}
    Note that the risk of direct learning is exactly the second term: $2\mathcal{R}_\st(\w^*_\st) = \| \bs\Pi_\st \w_* - \w_* \|^2 = \| \bs\Pi_\st^\perp \w_* \|^2$.
    
    Subtracting the two risks yields the irreducible static alignment bias:
    \begin{equation}
        \mathcal{R}_\ts(\w^\opt_\ts) - \mathcal{R}_\st(\w^*_\st) = \frac{1}{2} \| \bs\Pi_\st \bs\Pi_\te^\perp \w_* \|^2.
    \end{equation}
\end{proof}

\subsection{Proof of Theorem \ref{thm:generalt2s}}
\label{subsec:proofgeneralt2s}

We decompose the excess teacher-to-student risk into components related to the student's learning dynamics and the geometric misalignment between the teacher and student subspaces.

\begin{proof}
    The excess transfer risk is defined as $\mathcal{E}_{\mathrm{T2S}} = \frac{1}{2} \Ex_{N,n} \|\w_{n,\ts} - \w^*_\st\|_{\bs\Sigma_\st}^2$. By the triangle inequality and the orthogonality of the spectral decomposition (Head vs. Tail), we split the risk into four dominant terms:
    \begin{equation}
    \begin{aligned}
        \Ex[\mathcal{E}_{\mathrm{T2S}}] &\leq \underbrace{\frac{1}{2}\Ex_N \|\w^*_\ts - \w^*_\st\|_{\Sig_\st^{\leq k^*}}^2}_{\text{(I) Head Alignment Error}}
        + \underbrace{\frac{1}{2}\Ex_{N,n} \|\w_{n,\ts} - \w^*_\ts\|_{\Sig_\st^{\leq k^*}}^2}_{\text{(II) Head Optimization Error}} \\
        &\quad + \underbrace{\frac{1}{2}\Ex_{N,n} \|\w_{n,\ts}\|_{\Sig_\st^{> k^*}}^2}_{\text{(III) Tail Learning Error}}
        + \underbrace{\frac{1}{2} \|\w^*_\st\|_{\Sig_\st^{> k^*}}^2}_{\text{(IV) Tail Approximation Error}}.
    \end{aligned}
    \end{equation}
    We bound these terms via the following lemmas.
\end{proof}

\subsubsection*{Part 1: Head Alignment Error (Term I)}

\begin{lemma}[Head Misalignment Bound]
\label{lem:head_misalignment}
    The distance between the student's transfer target $\w^*_\ts$ and the optimal direct parameter $\w^*_\st$ in the learned subspace is bounded by:
    \begin{equation}
        \ex[N]{\frac{1}{2}\|\w^*_\ts-\w^*_\st\|_{\Sig_\st^{\leq k^*}}^2} \leq \inner{\ex[N]{(\bs\eta_{N,\te})^{\otimes 2}},\ \M_\te\bs \Pi_{\st}^{\leq k^*}\M_\te^\top} + C\cdot \|\bs \Pi_{\te}^{\perp}\w_*\|_{\bs \Pi_{\st}^{\leq k^*}}.
    \end{equation}
\end{lemma}

\begin{proof}
    Recall $\w^*_\ts = (\M_\st^\top)^+\M_\te^\top \w_{N,\te}$ and $\w^*_\st = (\M_\st^\top)^+ \w_*$.
    Substituting these definitions:
    \begin{equation}
    \begin{aligned}
        \|\w^*_\ts-\w^*_\st\|_{\Sig_\st^{\leq k^*}}^2 &= \left\| \Sig_\st^{1/2} (\M_\st^\top)^+ (\M_\te^\top \w_{N,\te} - \w_*) \right\|_{\bs \Pi^{\le k^*}}^2 \\
        &= \left\langle (\M_\te^\top \w_{N,\te} - \w_*)^{\otimes 2}, (\M_\st^\top)^{+\top} \Sig_\st^{\le k^*} (\M_\st^\top)^+ \right\rangle.
    \end{aligned}
    \end{equation}
    Using the SVD $\M_\st = \bs U \bs \Lambda \bs V^\top$, the operator simplifies to the projection onto the student's head subspace:
    \begin{equation}
        (\M_\st^\top)^{+\top} \Sig_\st^{\le k^*} (\M_\st^\top)^+ = \bs V \bs \Lambda^{-1} (\bs \Lambda^2)_{\le k^*} \bs \Lambda^{-1} \bs V^\top = \bs V_{\le k^*} \bs V_{\le k^*}^\top = \bs \Pi_\st^{\le k^*}.
    \end{equation}
    Expanding the error term $\M_\te^\top \w_{N,\te} - \w_* = \M_\te^\top (\w_{N,\te} - \w^*_\te) - \bs \Pi_\te^\perp \w_*$:
    \begin{equation}
    \begin{aligned}
        \text{LHS} &= \Ex_N \left\langle (\M_\te^\top \bs\eta_{N,\te} - \bs \Pi_\te^\perp \w_*)^{\otimes 2}, \bs \Pi_\st^{\le k^*} \right\rangle \\
        &= \Ex_N \inner{(\M_\te^\top \bs\eta_{N,\te})^{\otimes 2}, \bs \Pi_\st^{\le k^*}} + \inner{(\bs \Pi_\te^\perp \w_*)^{\otimes 2}, \bs \Pi_\st^{\le k^*}} \\
        &\quad - 2 \Ex_N \left[ \w_*^\top \bs \Pi_\te^\perp \bs \Pi_\st^{\le k^*} \M_\te^\top \bs\eta_{N,\te} \right].
    \end{aligned}
    \end{equation}
    The first term matches the lemma statement. The second term is $\|\bs \Pi_\te^\perp \w_*\|_{\bs \Pi_\st^{\le k^*}}^2$. The cross term is bounded by Cauchy-Schwarz and the initial teacher error bound:
    \begin{equation}
        \text{Cross Term} \le 2 \|\bs \Pi_\te^\perp \w_*\|_{\bs \Pi_\st^{\le k^*}} \sqrt{\Ex \|\M_\te^\top \bs\eta_{N,\te}\|_{\bs \Pi_\st^{\le k^*}}^2} \leq C \cdot \|\bs \Pi_\te^\perp \w_*\|_{\bs \Pi_\st^{\le k^*}}.
    \end{equation}
    Due to the fact that $\|\bs \Pi_\te^\perp \w_*\|_{\bs \Pi_\st^{\le k^*}}\leq\|\bs \Pi_\te^\perp \w_*\|_{\bs \Pi_\st}$ should be a small number in most cases, we only keep its linear form in our bound.
    This completes the proof.
\end{proof}

\subsubsection*{Part 2: Tail Optimization Dynamics (Term III)}

In the tail subspace ($k > k^*$), the eigenvalues are small ($\lambda_k \to 0$). The student's parameters $\w_{n,\ts}$ barely move from initialization ($\mathbf{0}$). We define the maximal movement fraction $\delta$:
\begin{equation}
    \delta := \sup_{j > k^*} \left( 1 - \exp\left( -2.02 \frac{n}{\log n} \gamma_0 \lambda_j \right) \right) \approx 2.02 \frac{n}{\log n} \gamma_0 \lambda_{k^*+1} \ll 1.
\end{equation}

\begin{lemma}[Tail Learning Bound]
    The energy of the student's parameters in the tail subspace is bounded by the teacher's variance and the ground truth signal, scaled by the small movement factor $\delta^2$:
    \begin{equation}
        \ex[N\otimes n]{\|\w_{n,\ts}\|_{\Sig_\st^{> k^*}}^2} \leq 2\delta^2 \inner{\ex[N]{\bs\eta_{N,\te}^{\otimes 2}}, \M_\te \bs \Pi_\st^{> k^*} \M_\te^\top} + 2\delta^2 \|\bs \Pi_\te \w_*\|_{\bs \Pi_\st^{> k^*}}^2 + \inner{\bs P^{\mathrm{var}}_{n,\ts}, \Sig_\st^{> k^*}}.
    \end{equation}
\end{lemma}

\begin{proof}
    Decompose the squared norm:
    \begin{equation}
        \Ex \|\w_{n,\ts}\|_{\Sig_\st^{> k^*}}^2 = \|\Ex_n[\w_{n,\ts}]\|_{\Sig_\st^{> k^*}}^2 + \Ex \|\w_{n,\ts} - \Ex_n[\w_{n,\ts}]\|_{\Sig_\st^{> k^*}}^2.
    \end{equation}
    The second term is the variance of the student's tail optimization, denoted $\inner{\bs P^{\mathrm{var}}_{n,\ts}, \Sig_\st^{> k^*}}$.
    For the first term (Bias), using the step-decay dynamics on the target $\w^*_\ts$:
    \begin{equation}
        \Ex_n[\w_{n,\ts}] = (\id - \prod (\id - \gamma_i \bs \Sigma_\st)) \w^*_\ts.
    \end{equation}
    We know that
    \begin{equation}
        1-\exp\left(-2.02\frac{n}{\log n}\gamma_0\lambda_{k^*+1}\right)\leq2.02\frac{n}{\log n}\gamma_0\lambda_{k^*+1}<\delta.
    \end{equation}
    Therefore, in the tail ($k > k^*$), the contraction is minimal, so the ``learned" portion is bounded by $\delta$:
    \begin{equation}
        \|\Ex_n[\w_{n,\ts}]\|_{\Sig_\st^{> k^*}} \leq \delta \|\w^*_\ts\|_{\Sig_\st^{> k^*}}.
    \end{equation}
    Substituting $\w^*_\ts = (\M_\st^\top)^+ \M_\te^\top \w_{N,\te}$:
    \begin{equation}
        \Ex_N \|\w^*_\ts\|_{\Sig_\st^{> k^*}}^2 = \Ex_N \|\M_\te^\top \w_{N,\te}\|_{\bs \Pi_\st^{> k^*}}^2 \leq 2 \Ex_N \|\M_\te^\top \bs\eta_{N,\te}\|_{\bs \Pi_\st^{> k^*}}^2 + 2 \|\bs \Pi_\te \w_*\|_{\bs \Pi_\st^{> k^*}}^2.
    \end{equation}
    Combining these yields the result.

    
\end{proof}

\subsubsection*{Part 3: Unified Bound (Final Assembly)}

Substituting Lemmas \ref{lem:head_misalignment} and Part 2 into the risk decomposition:

\begin{equation}
\begin{aligned}
    \ex[N\otimes n]{\mathcal{E}_{\mathrm{T2S}}} &\leq 
    % Term 1: Head Alignment (Teacher Noise + Geometry)
    \underbrace{\frac{1}{2}\inner{\ex[N]{\bs\eta_{N,\te}^{\otimes 2}},\M_\te\bs \Pi_{\st}^{\leq k^*}\M_\te^\top} + C\cdot\|\bs \Pi_{\te}^{\perp}\w_*\|_{\bs \Pi_{\st}^{\leq k^*}}}_{\text{Head Alignment}} \\
    % Term 2: Tail Teacher Noise (from weak learning)
    &+ \underbrace{2\delta^2\inner{\ex[N]{\bs\eta_{N,\te}^{\otimes 2}},\M_\te\bs \Pi_{\st}^{> k^*}\M_\te^\top} + 2\delta^2\|\bs \Pi_\te \w_*\|_{\bs \Pi_\st^{> k^*}}^2}_{\text{Tail Teacher Leakage}} \\
    % Term 3: Student Optimization
    &+ \underbrace{\frac{1}{2}\inner{\ex[N\otimes n]{\bs P_{n,\ts}},\Sig_{\st}^{\leq k^*}}}_{\text{Head Student Opt.}} 
    + \underbrace{\inner{\bs P^{\mathrm{var}}_{n,\ts},\Sig_\st^{> k^*}}}_{\text{Tail Student Var.}} \\
    % Term 4: Approximation
    &+ \frac{1}{2}\|\w^*_\st\|_{\Sig_\st^{> k^*}}^2.
\end{aligned}
\end{equation}
This concludes the proof.


\section{Proof of Results in Distillation}


\subsection{Preliminary Lemmas}

Note that in the proof of Theorem \ref{thm:generalt2s} the $\delta$ and $k^*$ can be determined each other. In the proofs of distillation theorems, we mainly focus on the process in which the strong teacher passes its strength to the weak student while leaving aside the spectral denoising mechanism for now. Therefore, in the distillation section, we will let $k^* = d_\st$, and in this case, the elements $>k^*$ is an empty set. Theorem \ref{thm:generalt2s} in this case becomes:

\begin{corollary}[The Teacher-to-Student Risk Decomposition Distillation Version]
\label{crl:kddecomp}
    Let the Student be trained on $n$ samples labeled by a fixed Teacher (itself trained on $N$ samples). The expected knowledge transfer excess risk $\mathcal{E}_{\mathrm{T2S}}$ is bounded by:
    \begin{equation}
        \Ex[\mathcal{E}_{\mathrm{T2S}}(N,n)] \leq \frac{1}{2}\inner{\bs P_{N,\te},\M_\te\bs \Pi_{\st}\M_\te^\top} + \frac{1}{2}\inner{\bs P_{n,\ts},\Sig_{\st}} + C\left\|\bs \Pi_{\st}\bs \Pi_{\te}^{\perp}\w_{*}\right\|
    \end{equation}
    
    \noindent where $k^* = \max\{k:\lambda_{k,\st}> \delta\log_2 n/ (4\gamma_0' n) \}$ is the effective dimension, $\bs P_{n,\ts}:=\Ex[\bs\eta_{n,\ts}^{\otimes2}]$ is the second moment matrix of $\bs\eta_{n,\ts}:=\w_{n,\ts}-\w^*_{\ts}$ and $C$ is a constant.
\end{corollary}

This bound could be further simplified with the concept of spectral compatibility:

\begin{lemma}[Geometric Consistency for Compatible Students]
\label{lem:spectral_compatibility}
    For a spectrally compatible student (i.e., one whose feature space is a subspace of the teacher's, $\text{Range}(\M_\st^\top) \subseteq \text{Range}(\M_\te^\top)$), the geometric consistency condition holds:
    \begin{equation}
        \bs \Pi_{\st}\bs \Pi_{\te}^{\perp}\w_{*} = \bs 0.
    \end{equation}
\end{lemma}

\begin{proof}
    The definition of spectral compatibility implies that any feature representable by the student is also representable by the teacher. In terms of subspaces, let $\mathcal{S} = \text{Range}(\M_\st^\top)$ and $\mathcal{T} = \text{Range}(\M_\te^\top)$; then $\mathcal{S} \subseteq \mathcal{T}$.

    This inclusion property implies that projecting a vector from the student subspace $\mathcal{S}$ onto the teacher subspace $\mathcal{T}$ has no effect (acts as identity). Mathematically:
    \begin{equation}
        \bs \Pi_\te \bs \Pi_\st = \bs \Pi_\st.
    \end{equation}
    Since projection matrices are symmetric, taking the transpose yields the equivalent condition:
    \begin{equation}
        \bs \Pi_\st \bs \Pi_\te = \bs \Pi_\st.
    \end{equation}
    
    Now, we expand the term in the lemma using the definition of the orthogonal complement projector $\bs \Pi_\te^\perp = \id - \bs \Pi_\te$:
    \begin{equation}
    \begin{aligned}
        \bs \Pi_{\st}\bs \Pi_{\te}^{\perp}\w_{*} &= \bs \Pi_{\st}(\id - \bs \Pi_{\te})\w_{*} \\
        &= (\bs \Pi_{\st} - \bs \Pi_{\st}\bs \Pi_{\te})\w_{*} \\
        &= (\bs \Pi_{\st} - \bs \Pi_{\st})\w_{*} \\
        &= \bs 0.
    \end{aligned}
    \end{equation}
    Geometrically, this means that the part of the ground truth the teacher cannot learn ($\bs \Pi_{\te}^{\perp}\w_{*}$) lies in the teacher's null space, which contains the student's null space ($\mathcal{T}^\perp \subseteq \mathcal{S}^\perp$). Therefore, the student cannot capture any of it either.
\end{proof}

Next we show that with sufficient unlabeled data (n samples), the optimization error of the transfer process is negligible.

\begin{lemma}
    Under Assumption \ref{assump:spectral_decay} (Polynomial Spectral Decay, $\lambda_k \propto k^{-\alpha}$ with $\alpha > 1$), with sufficient unlabeled data, the optimization error goes to zero:
    \begin{equation}
        \lim_{n\rightarrow\infty} \frac{1}{2}\inner{\bs P_{n,\ts},\Sig_{\st}} = 0.
    \end{equation}
\end{lemma}

\begin{proof}
    The term $\frac{1}{2}\inner{\bs P_{n,\ts},\Sig_{\st}}$ represents the expected excess risk of the student relative to its optimal target $\w^*_\ts$. We invoke the upper bound derived in Theorem \ref{thm:transferriskupperbound}:
    \begin{equation}
        \frac{1}{2}\inner{\bs P_{n,\ts},\Sig_{\st}} \leq \underbrace{\frac{C_1}{n^2}}_{\text{Init Decay}} + \underbrace{\frac{1}{2}\|\w_{N,\te}\|^2_{\Sig^{>k^*}}}_{\text{Tail Bias}} + \underbrace{C_2 \left( \frac{k^*}{n} + n \gamma^2 \sum_{i > k^*} \lambda_{i}^2 \right)}_{\text{Variance}}.
    \end{equation}
    Here $n$ is the number of student steps (samples), and the effective cutoff is $k^* = \max\{k:\lambda_{k} > \frac{C \log n}{n} \}$.
    
    Under Assumption \ref{assump:spectral_decay}, let $\lambda_k \asymp k^{-\alpha}$ for $\alpha > 1$.
    
    \textbf{1. Behavior of $k^*$:}
    The condition $\lambda_{k^*} \asymp \frac{\log n}{n}$ implies $(k^*)^{-\alpha} \asymp \frac{\log n}{n}$, thus:
    \begin{equation}
        k^* \asymp \left( \frac{n}{\log n} \right)^{1/\alpha}.
    \end{equation}
    As $n \to \infty$, $k^* \to \infty$.
    
    \textbf{2. Convergence of Terms:}
    \begin{itemize}
        \item \textbf{Initial Decay:} $\frac{1}{n^2} \to 0$ trivially.
        \item \textbf{Tail Bias:} Since $\bs\Sigma$ is trace-class ($\sum \lambda_k < \infty$), the tail energy sum $\sum_{i=k^*+1}^\infty \lambda_i (w^{(i)})^2$ must vanish as the start index $k^* \to \infty$. Specifically, $\|\cdot\|_{\Sig^{>k^*}}^2 \to 0$.
        \item \textbf{Head Variance:} The term scales as $\frac{k^*}{n} \asymp \frac{n^{1/\alpha}}{n} = n^{\frac{1}{\alpha}-1}$. Since $\alpha > 1$, the exponent is negative, so $\frac{k^*}{n} \to 0$.
        \item \textbf{Tail Variance:} With a proper decaying learning rate schedule (e.g., $\gamma \propto 1/n$), this term is dominated by the Head Variance rate and also vanishes.
    \end{itemize}
    
    Since all components of the upper bound converge to zero, the optimization error vanishes asymptotically.
\end{proof}

Finally we derive the most important lemma in distillation:


\begin{lemma}
    Under Assumption \ref{assump:spectral_decay}, for a spectrally compatible student, and assuming sufficient unlabeled data ($n \to \infty$):
    \begin{equation}
        \Ex[\mathcal{E}_{\mathrm{T2S}}(N,n)] \leq \Ex[\mathcal{E}_{\mathrm{T}}(N)]
    \end{equation}
\end{lemma}

\begin{proof}
    We start from the risk decomposition in Corollary \ref{crl:kddecomp}:
    \begin{equation}
        \Ex[\mathcal{E}_{\mathrm{T2S}}] \leq \underbrace{\frac{1}{2}\inner{\bs P_{N,\te},\M_\te\bs \Pi_{\st}\M_\te^\top}}_{\text{Teacher Estimation Error}} + \underbrace{\frac{1}{2}\inner{\bs P_{n,\ts},\Sig_{\st}}}_{\text{Student Opt. Error}} + \underbrace{C\left\|\bs \Pi_{\st}\bs \Pi_{\te}^{\perp}\w_{*}\right\|}_{\text{Geometric Bias}}.
    \end{equation}

    First, since the student is spectrally compatible ($\text{Range}(\M_\st^\top) \subseteq \text{Range}(\M_\te^\top)$), Lemma \ref{lem:spectral_compatibility} guarantees that the geometric bias term is exactly zero:
    \begin{equation}
        \bs \Pi_{\st}\bs \Pi_{\te}^{\perp}\w_{*} = \bs 0.
    \end{equation}

    Second, with sufficient unlabeled data ($n \to \infty$) and the spectral decay assumption, the previous Lemma ensures that the student's optimization error vanishes:
    \begin{equation}
        \lim_{n\rightarrow\infty} \frac{1}{2}\inner{\bs P_{n,\ts},\Sig_{\st}} = 0.
    \end{equation}

    Finally, we analyze the remaining Teacher Estimation Error term. Since $\bs \Pi_\st$ is an orthogonal projection matrix, it satisfies $\bs \Pi_\st \preceq \id$ in the PSD sense. Consequently, conjugating by $\M_\te$:
    \begin{equation}
        \M_\te \bs \Pi_\st \M_\te^\top \preceq \M_\te \id \M_\te^\top = \bs \Sigma_\te.
    \end{equation}
    Since the covariance matrix of the teacher's error $\bs P_{N,\te}$ is positive semi-definite, the trace inner product respects this inequality:
    \begin{equation}
        \frac{1}{2}\inner{\bs P_{N,\te},\M_\te\bs \Pi_{\st}\M_\te^\top} \leq \frac{1}{2}\inner{\bs P_{N,\te},\bs \Sigma_\te}.
    \end{equation}
    The Right Hand Side is exactly the definition of the Teacher's excess risk $\Ex[\mathcal{E}_{\mathrm{T}}(N)]$. Thus, we conclude:
    \begin{equation}
        \Ex[\mathcal{E}_{\mathrm{T2S}}] \leq \Ex[\mathcal{E}_{\mathrm{T}}].
    \end{equation}
\end{proof}

Therefore, to compare the result of transferred training and direct training, we would only have to compare the risk of directly training the teacher and also the student under the same amount of labeled samples $N$.

\begin{lemma}
\label{lem:convergencespeedequiv}
Let $T: \ell_2 \to \ell_2$ be a bounded linear operator with operator norm $\|T\| < \infty$. For any vector $x \in \ell_2$ with expansion $x = \sum_{i=1}^{\infty} a_i e_i$ relative to an orthonormal basis $\{e_i\}$, let the tail of the sequence be defined as $r_k = \sum_{i=k+1}^{\infty} a_i e_i$. The convergence speed of the transformed tail $\|T(r_k)\|_2$ is governed by the original tail speed such that:
\begin{equation}
    \|T(r_k)\|_2 \leq \|T\| \cdot \|r_k\|_2
\end{equation}
Consequently, $\|T(r_k)\|_2 = O(\|r_k\|_2)$ as $k \to \infty$.
\end{lemma}

\begin{proof}
Let $x \in \ell_2$ and let $s_k = \sum_{i=1}^{k} a_i e_i$ denote the $k$-th partial sum of the series. We express $x$ as the sum of its partial sum and its remainder (tail):
\begin{equation}
    x = s_k + r_k
\end{equation}
Applying the bounded linear operator $T$ to both sides, and utilizing the property of linearity, we obtain:
\begin{equation}
    T(x) = T(s_k + r_k) = T(s_k) + T(r_k)
\end{equation}
The error in the $k$-th approximation of the transformed sequence is given by the norm of the difference between the total sum and the partial sum:
\begin{equation}
    \|T(x) - T(s_k)\|_2 = \|T(r_k)\|_2
\end{equation}
By the definition of the operator norm for a bounded map on a Hilbert space, for any $v \in \ell_2$, the following inequality holds:
\begin{equation}
    \|T(v)\|_2 \leq \|T\| \cdot \|v\|_2
\end{equation}
Setting $v = r_k$, we find:
\begin{equation}
    \|T(r_k)\|_2 \leq \|T\| \cdot \|r_k\|_2
\end{equation}
This establishes that the tail of the transformed sequence vanishes at least as fast as the original sequence. In the case where $T$ is bounded below (i.e., there exists $c > 0$ such that $\|T(v)\| \geq c\|v\|$), we further conclude:
\begin{equation}
    c \|r_k\|_2 \leq \|T(r_k)\|_2 \leq \|T\| \cdot \|r_k\|_2
\end{equation}
Which implies $\|T(r_k)\|_2 = \Theta(\|r_k\|_2)$, proving the convergence speeds are asymptotically equivalent.
\end{proof}

\subsection{Proof of Theorem \ref{thm:der_rate}}

\begin{proof}
    % For spectral compatible student, the energy $I_i$ is a bounded rearrangement of the teacher's. With lemma \ref{lem:convergencespeedequiv} we know that their tail convergence speed is the same.
    
    Therefore we compare the convergence rates derived in Section \ref{subsec:polydecaybound}:
    \begin{itemize}
        \item \textbf{Teacher (and thus Distillation) Rate:} $\Ex[\mathcal{E}_{\mathrm{T}}(N)] \asymp N^{\frac{1+\beta-\alpha_\te}{\alpha_\te}}$.
        \item \textbf{Direct Student Rate:} $\Ex[\mathcal{E}_{\mathrm{S}}(N)] \asymp N^{\frac{1+\beta-\alpha_\te}{\alpha_\st}}$.
    \end{itemize}
    By dividing them we obtain:
    \begin{equation}
        \lim_{n\to\infty}\mathbf{DER}_N > \widetilde{{\Omega}}(N^{(\alpha_\te-1-\beta)\left(1/{\alpha_\te}-1/{\alpha_\st}\right)}).
    \end{equation} 
\end{proof}










\section{Proof of Results in W2S}
\subsection{Proof of Theorem \ref{thm:w2s_bound}}
\label{proof:w2s_bound}

In the Weak-to-Strong (W2S) generalization setting, the student is intentionally "weaker" than the teacher (e.g., trained with fewer steps $n$, or stopped early). This limitation acts as a regularization. We prove that this allows the student to filter out the teacher's tail noise, achieving lower risk than the teacher itself.

\begin{theorem}[Weak-to-Strong Generalization Bound]
    Let the student effective cutoff be $k^\dagger$. Under the condition that the student is "weak" in the tail (small $\delta$) and spectrally compatible, the risk satisfies:
    \begin{equation}
    \begin{aligned}
        \ex[N\otimes n]{\mathcal{R}_{\mathrm{T2S}}}&\leq \underbrace{\frac{1}{2}\inner{\ex[N]{\bs\eta_{N,\te}^{\otimes 2}},\M_\te\bs \Pi_{\st}^{\leq k^\dagger}\M_\te^\top}}_{\text{Inherited Head Error}} + \underbrace{2\delta^2\inner{\ex[N]{\bs\eta_{N,\te}^{\otimes 2}},\M_\te\bs \Pi_{\st}^{> k^\dagger}\M_\te^\top}}_{\text{Dampened Tail Error}} \\
        &\quad + \mathcal{O}(\gamma_0) + \mathcal{O}\left(\left\|\bs \Pi_{\st}^{\leq k^\dagger}\bs \Pi_{\te}^{\perp}\w_{*}\right\|\right) \\
        &< \frac{1}{2}\inner{\ex[N]{\bs\eta_{N,\te}^{\otimes 2}},\M_\te\bs \Pi_{\st}^{\leq k^\dagger}\M_\te^\top} + \frac{1}{2}\inner{\ex[N]{\bs\eta_{N,\te}^{\otimes 2}},\M_\te\bs \Pi_{\st}^{> k^\dagger}\M_\te^\top} \\
        &\leq \ex[N]{\mathcal{R}_{\mathrm{T}}}
    \end{aligned}
    \end{equation}
\end{theorem}

\begin{proof}
    \textbf{Step 1: Applying the Unified Transfer Bound}
    
    We start from the general upper bound derived in Theorem \ref{thm:generalt2s} (Part 3). By partitioning the student's spectrum at the cutoff $k^\dagger$, we have:
    \begin{equation}
        \Ex[\mathcal{R}_{\mathrm{T2S}}] \leq \frac{1}{2}\text{Head}_{\te} + 2\delta^2 \text{Tail}_{\te} + \text{StudentVar} + \text{GeomBias}.
    \end{equation}
    Specifically:
    \begin{itemize}
        \item The \textbf{Inherited Head Error} is $\frac{1}{2}\Tr(\bs P_{N,\te} \M_\te \bs \Pi_\st^{\le k^\dagger} \M_\te^\top)$. This represents the teacher's error that the student perfectly mimics in the signal-rich subspace.
        \item The \textbf{Inherited Tail Error} is scaled by the student's learning factor in the tail. Since the student is "weak" (stopped early), it has barely moved from initialization in the tail dimensions ($> k^\dagger$). The drift is bounded by $\delta \ll 1$ . Thus, the error passed to the student is $2\delta^2 \times \text{Tail Noise}$.
        \item The \textbf{Student Optimization Noise} is $\mathcal{O}(\gamma_0)$, which is negligible assuming sufficient unlabeled data or small learning rate.
        \item The \textbf{Geometric Bias} $\mathcal{O}(\|\bs \Pi_{\st}^{\leq k^\dagger}\bs \Pi_{\te}^{\perp}\w_{*}\|)$ is small or zero under spectral compatibility.
    \end{itemize}
    This establishes the first inequality ($\leq$).

    \textbf{Step 2: Comparison with Teacher Risk}
    
    Now we analyze the Teacher's risk $\mathcal{R}_{\mathrm{T}}$. The teacher's risk is the total error energy in its own feature space:
    \begin{equation}
        \Ex[\mathcal{R}_{\mathrm{T}}] = \frac{1}{2} \Tr\left( \bs\Sigma_\te \Ex[\bs\eta_{N,\te}^{\otimes 2}] \right) = \frac{1}{2} \Tr\left( \M_\te \M_\te^\top \bs P_{N,\te} \right).
    \end{equation}
    Since $\bs \Pi_\st$ is a projection matrix, $\bs \Pi_\st \preceq \id$. We can decompose the identity in the teacher's space using the student's basis: $\id \succeq \bs \Pi_\st = \bs \Pi_\st^{\le k^\dagger} + \bs \Pi_\st^{> k^\dagger}$.
    Therefore, the teacher's risk is lower-bounded by its projection onto the student's space:
    \begin{equation}
    \begin{aligned}
        \Ex[\mathcal{R}_{\mathrm{T}}] &\geq \frac{1}{2} \Tr\left( \M_\te (\bs \Pi_\st^{\le k^\dagger} + \bs \Pi_\st^{> k^\dagger}) \M_\te^\top \bs P_{N,\te} \right) \\
        &= \underbrace{\frac{1}{2}\inner{\bs P_{N,\te},\M_\te\bs \Pi_{\st}^{\leq k^\dagger}\M_\te^\top}}_{\text{Teacher Head Risk}} + \underbrace{\frac{1}{2}\inner{\bs P_{N,\te},\M_\te\bs \Pi_{\st}^{> k^\dagger}\M_\te^\top}}_{\text{Teacher Tail Risk}}.
    \end{aligned}
    \end{equation}

    \textbf{Step 3: The Weak-to-Strong Condition}
    
    Comparing the terms in Step 1 and Step 2:
    \begin{itemize}
        \item The \textbf{Head} terms are identical (Student mimics Teacher perfectly where it matters).
        \item The \textbf{Tail} terms differ by the coefficient: The Student has $2\delta^2$, while the Teacher has $\frac{1}{2}$.
    \end{itemize}
    
    The condition for the student to outperform the teacher ($\mathcal{R}_{\mathrm{T2S}} < \mathcal{R}_{\mathrm{T}}$) is dominated by the inequality:
    \begin{equation}
        2\delta^2 \ll \frac{1}{2} \implies \delta^2 \ll \frac{1}{4} \implies \delta < 0.5.
    \end{equation}
    Since $\delta$ characterizes the "weakness" of the student in the tail (e.g., $\delta \approx 0$ for early stopping), this condition is easily satisfied.
    
    Essentially, the teacher suffers from full noise variance ($\frac{1}{2}$) in the tail, whereas the weak student filters it out ($2\delta^2 \approx 0$). Provided the bias terms $\mathcal{O}(\gamma_0)$ and alignment errors are small compared to this variance reduction, we have the strict inequality:
    \begin{equation}
        \Ex[\mathcal{R}_{\mathrm{T2S}}] < \Ex[\mathcal{R}_{\mathrm{T}}].
    \end{equation}
    This completes the proof.
\end{proof}



\subsection{Proof of Theorem \ref{thm:w2s_rates}}

\begin{proof}
    We derive the optimal rates by minimizing the upper bound established in Theorem \ref{thm:w2s_bound} with respect to the student's effective spectral cutoff $k^*_\st$ (which is controlled by the sample size $n$).

    \textbf{1. Risk Simplification under Assumptions}
    
    Under Assumption \ref{assump:intrinsic_dim} (Low Intrinsic Dimension), the ground truth $\w_*$ lies effectively in a subspace of dimension $k^\dagger$. We assume the teacher's sample size $N$ is sufficiently large ($N \gamma_0 \lambda_{k^\dagger} \gg 1$) such that the teacher has fully learned the signal.
    
    For the student, we consider the regime where the sample size $n$ is large enough to cover the intrinsic dimension ($k^*_\st \ge k^\dagger$), ensuring the \textbf{Bias} term vanishes:
    \begin{equation}
        \|\bs \Pi_{\st}^{\leq k^*_\st}\bs \Pi_{\te}^{\perp}\w_{*}\| \approx 0.
    \end{equation}
    Consequently, the risk is governed purely by the variance components from Theorem \ref{thm:w2s_bound}:
    \begin{equation}
        \Ex[\mathcal{R}_{\mathrm{T2S}}] \lesssim \underbrace{\frac{1}{2}\inner{\ex{\bs\eta_{N,\te}^{\otimes 2}},\M_\te\bs \Pi_{\st}^{\leq k^*_\st}\M_\te^\top}}_{\text{Inherited Variance}} + \underbrace{2\delta^2\inner{\ex{\bs\eta_{N,\te}^{\otimes 2}},\M_\te\bs \Pi_{\st}^{> k^*_\st}\M_\te^\top}}_{\text{Damped Tail Variance}}.
    \end{equation}

    \textbf{2. Explicit Scaling of Variance Terms}
    
    We analyze the scaling of the two variance terms with respect to $k^*_\st$ and $N$.
    
    \begin{itemize}
        \item \textbf{Inherited Variance (Head):}
        The teacher's noise is white in the projected space. The student inherits this noise 1-to-1 in its learned subspace $k \le k^*_\st$. The variance scales with the number of learned dimensions:
        \begin{equation}
            \text{Head} \asymp \frac{k^*_\st}{N} \sigma^2 \asymp \frac{k^*_\st}{N}.
        \end{equation}
        
        \item \textbf{Damped Tail Variance (Tail):}
        The teacher's total accumulated error in the tail (without student damping) scales as $\mathcal{E}_\te(N) = \tilde{\mathcal{O}}(N^{\frac{1-\alpha_\te}{\alpha_\te}})$.
        The student applies a damping factor $\delta^2$. Using the spectral decay $\lambda_k \propto k^{-\alpha_\st}$, the damping factor at the cutoff scales as:
        \begin{equation}
             \delta^2 \approx (n \gamma_0 \lambda_{k^*_\st})^2 \approx \left( \frac{\lambda_{k^*_\st}}{\lambda_{k^\dagger}} \right)^2 \asymp \left( \frac{k^\dagger}{k^*_\st} \right)^{2\alpha_\st}.
        \end{equation}
        Thus, the damped tail variance is:
        \begin{equation}
            \text{Tail} \asymp \left( \frac{k^\dagger}{k^*_\st} \right)^{2\alpha_\st} \cdot N^{\frac{1-\alpha_\te}{\alpha_\te}}.
        \end{equation}
    \end{itemize}

    \textbf{3. Solving for the Optimal Equilibrium}
    
    The total risk is the sum of these opposing forces:
    \begin{equation}
        \Ex[\mathcal{R}_{\mathrm{T2S}}(k^*_\st)] \asymp \frac{k^*_\st}{N} + (k^\dagger)^{2\alpha_\st} (k^*_\st)^{-2\alpha_\st} N^{\frac{1-\alpha_\te}{\alpha_\te}}.
    \end{equation}
    To find the optimal early stopping point, we minimize with respect to $k^*_\st$. Setting the derivative to zero:
    \begin{equation}
        \frac{1}{N} - 2\alpha_\st (k^\dagger)^{2\alpha_\st} (k^*_\st)^{-2\alpha_\st-1} N^{\frac{1-\alpha_\te}{\alpha_\te}} = 0.
    \end{equation}
    Solving for $k^*_\st$:
    \begin{equation}
    \begin{aligned}
        (k^*_\st)^{2\alpha_\st+1} &\asymp (k^\dagger)^{2\alpha_\st} N^{\frac{1-\alpha_\te}{\alpha_\te}} \cdot N \\
        &= (k^\dagger)^{2\alpha_\st} N^{\frac{1-\alpha_\te + \alpha_\te}{\alpha_\te}} \\
        &= (k^\dagger)^{2\alpha_\st} N^{\frac{1}{\alpha_\te}}.
    \end{aligned}
    \end{equation}
    Taking the root, we get the optimal effective dimension:
    \begin{equation}
        k^*_{\st, \mathrm{opt}} \asymp (k^\dagger)^{\frac{2\alpha_\st}{2\alpha_\st+1}} N^{\frac{1}{\alpha_\te(2\alpha_\st+1)}}.
    \end{equation}
    (Note: Since $n \propto (k^*)^\nu$ for some power $\nu$ depending on the schedule, the optimal $n$ scales proportionally to this $k^*$, as stated in the theorem).

    \textbf{4. Optimal Risk and PGR}
    
    Substituting $k^*_{\st, \mathrm{opt}}$ back into the Head term (which is of the same order as the Tail term at optimality):
    \begin{equation}
    \begin{aligned}
        \min \Ex[\mathcal{R}_{\mathrm{T2S}}] &\asymp \frac{k^*_{\st, \mathrm{opt}}}{N} \\
        &\asymp (k^\dagger)^{\frac{2\alpha_\st}{2\alpha_\st+1}} N^{\frac{1}{\alpha_\te(2\alpha_\st+1)} - 1}.
    \end{aligned}
    \end{equation}
    This proves the risk scaling.
    
    Finally, we calculate the Performance Gap Ratio (PGR). The teacher's risk scales as $\Ex[\mathcal{R}_\te] \asymp N^{\frac{1-\alpha_\te}{\alpha_\te}}$.
    \begin{equation}
    \begin{aligned}
        1 - \mathbf{PGR} = \frac{\Ex[\mathcal{R}_{\mathrm{T2S}}]}{\Ex[\mathcal{R}_\te]} &\asymp \frac{N^{\frac{1}{\alpha_\te(2\alpha_\st+1)} - 1}}{N^{\frac{1}{\alpha_\te} - 1}} \\
        &= N^{\frac{1}{\alpha_\te(2\alpha_\st+1)} - \frac{1}{\alpha_\te}} \\
        &= N^{\frac{1 - (2\alpha_\st+1)}{\alpha_\te(2\alpha_\st+1)}} \\
        &= N^{-\frac{2\alpha_\st}{\alpha_\te(2\alpha_\st+1)}}.
    \end{aligned}
    \end{equation}
    Defining $\Delta_{\mathrm{rate}} = \frac{2\alpha_\st}{\alpha_\te(2\alpha_\st+1)}$, we obtain the stated result:
    \begin{equation}
        \mathbf{PGR} = 1 - \tilde{\mathcal{O}}\left((k^\dagger)^{\frac{2\alpha_\st}{2\alpha_\st+1}} N^{-\Delta_{\mathrm{rate}}}\right).
    \end{equation}
    Since $\alpha_\st, \alpha_\te > 0$, we have $\Delta_{\mathrm{rate}} > 0$, implying the student asymptotically recovers the teacher's full capability.
\end{proof}
% \subsection{Proof of Theorem \ref{thm:w2s_rates}}



% To sufficiently learn the target bias
% \begin{equation}
%     \frac{n}{\log_2 n}\gamma_0 \lambda_{k^\dagger}>1
% \end{equation}

% \begin{equation}
%     n \gamma_0 \lambda_{k^*}> \lambda_{k^*}/\lambda_{k^\dagger} = (k^\dagger/k^*)^{\alpha_\st}
% \end{equation}

% We need N sufficiently large to learn the target
% \begin{equation}
%     1<\frac{N}{\log_2 N}\gamma_0 \lambda_{k^{\dagger}_{\te}}<C
% \end{equation}

% \begin{equation}
%     \mathcal{E_\ts}\leq\frac{k^*_\st}{N}+(2 n \gamma_0 \lambda_{k^*})^2 \cdot \mathcal{E}_\te = \frac{k^*_\st}{N}+(2 n \gamma_0 \lambda_{k^*})^2 \cdot \tilde{\mathcal{O}}(N^{(1-\alpha_\te)/\alpha_\te})
% \end{equation}

% \begin{equation}
%     =\frac{k^*_\st}{N}+\left( \frac{k^\dagger}{k^*_\st}\right)^{2\alpha_\st} \cdot \tilde{\mathcal{O}}(N^{(1-\alpha_\te)/\alpha_{\te}})
% \end{equation}

% \begin{equation}
%     k^*_\st = \tilde{\mathcal{O}}\left( (k^\dagger)^{\frac{2\alpha_\st}{2\alpha_\st+1}}\cdot N^{1/[\alpha_\te(2\alpha_\st+1)]} \right)
% \end{equation}

\section{Theoretical Connections to Kernel Methods and NTK}
\label{app:connection_kernel_ntk}

In this appendix, we elucidate the theoretical connections between the overparameterized linear SGD setting analyzed in our main text, non-parametric kernel regression, and the training dynamics of wide neural networks (NTK). We demonstrate that our analysis of the linear model $f(\mathbf{x}) = \mathbf{w}^\top \phi(\mathbf{x})$ is general enough to cover these complex regimes.

\subsection{From Overparameterized Linear SGD to Kernel SGD}
\label{app:sub:linear_to_kernel}

We first establish that the infinite-width linear model is mathematically equivalent to optimization in a Reproducing Kernel Hilbert Space (RKHS). 

Consider the linear regression problem with feature map $\phi: \mathcal{X} \to \mathbb{R}^d$ (where $d \to \infty$) and parameter $\mathbf{w} \in \mathbb{R}^d$. The standard SGD update at step $t$ with sample $(\mathbf{x}_t, y_t)$ and learning rate $\eta$ is:
\begin{equation}
    \mathbf{w}_{t+1} = \mathbf{w}_t - \eta \left( \langle \mathbf{w}_t, \phi(\mathbf{x}_t) \rangle - y_t \right) \phi(\mathbf{x}_t).
\end{equation}
Assuming initialization $\mathbf{w}_0 = \mathbf{0}$, the Representer Theorem ensures that the weight vector $\mathbf{w}_t$ always lies in the span of the observed feature vectors. This allows us to shift our perspective from the parameter space to the function space. Defining the kernel $K(\mathbf{x}, \mathbf{x}') = \langle \phi(\mathbf{x}), \phi(\mathbf{x}') \rangle$, the functional update rule becomes:
\begin{align}
    f_{t+1}(\cdot) &= \langle \mathbf{w}_t - \eta (f_t(\mathbf{x}_t) - y_t) \phi(\mathbf{x}_t), \phi(\cdot) \rangle \nonumber \\
    &= f_t(\cdot) - \eta (f_t(\mathbf{x}_t) - y_t) K(\cdot, \mathbf{x}_t).
    \label{eq:kernel_sgd_update}
\end{align}
This is precisely the update rule for \textbf{Kernel SGD} (stochastic approximation in RKHS) \citep{dieuleveut2016nonparametricstochasticapproximationlarge}. Consequently, the ``Overparameterized Linear'' regime is mathematically isomorphic to learning in an RKHS, and our spectral analysis applies directly to non-parametric regression.

\subsection{Spectral Decay Rates and RKHS Inclusion}
\label{app:sub:rkhs_capacity}

Having established the RKHS equivalence, we now provide a rigorous justification for Assumption \ref{assump:spectral_decay} (``Strong Teacher, Weak Student''), explaining how the spectral decay rate $\alpha$ governs the capacity of the induced function space.

Recall that a function $f(\mathbf{x}) = \sum_{k=1}^\infty c_k \psi_k(\mathbf{x})$ lies within the RKHS $\mathcal{H}_K$ defined by eigenvalues $\lambda_k$ if and only if its RKHS norm is finite: 
\begin{equation}
    \|f\|_{\mathcal{H}_K}^2 = \sum_{k=1}^\infty \frac{c_k^2}{\lambda_k} < \infty.
\end{equation}
Under our polynomial decay assumption $\lambda_k \asymp k^{-\alpha}$, this condition becomes equivalent to the convergence of the series:
\begin{equation}
    \sum_{k=1}^\infty c_k^2 \cdot k^{\alpha} < \infty,
    \label{eq:smoothness_condition}
\end{equation}

\eqref{eq:smoothness_condition} reveals that $\alpha$ acts as a \textbf{smoothness penalty}. A larger $\alpha$ imposes a stronger penalty on the high-frequency coefficients ($c_k$ for large $k$), forcing them to decay faster.

\textbf{Proof of Inclusion ($\mathcal{H}_\st \subset \mathcal{H}_\te$).} 
Consider the regime where $\alpha_\st > \alpha_\te > 1$. If a function $f$ belongs to the Student's space $\mathcal{H}_\st$, then $\sum c_k^2 k^{\alpha_\st} < \infty$. Since $k^{\alpha_\te} \ll k^{\alpha_\st}$ for large $k$, it follows that:
\begin{equation}
    \sum_{k=1}^\infty c_k^2 k^{\alpha_\te} < \sum_{k=1}^\infty c_k^2 k^{\alpha_\st} < \infty \implies f \in \mathcal{H}_\te.
\end{equation}
Thus, $\mathcal{H}_\st \subset \mathcal{H}_\te$. Conversely, functions with significant energy in the high-frequency tail (where the sum converges for $\alpha_\te$ but diverges for $\alpha_\st$) reside in the difference set $\mathcal{H}_\te \setminus \mathcal{H}_\st$. This rigorously validates our characterization: the Student is structurally confined to a smoother subspace and cannot represent the ``rough'' components of the Teacher.

\subsection{Neural Tangent Kernel (NTK) as a Concrete Instance}
\label{app:sub:ntk_derivation}

Finally, we instantiate this abstract framework with the \textbf{Neural Tangent Kernel (NTK)}, demonstrating that our spectral assumptions are consistent with the physics of deep learning.

In the ``lazy training'' regime, a wide neural network $f(\mathbf{x}; \boldsymbol{\theta})$ can be linearized around its initialization $\boldsymbol{\theta}_0$. The training dynamics are asymptotically equivalent to a linear model using the feature map $\phi_{\text{NTK}}(\mathbf{x}) := \nabla_{\boldsymbol{\theta}} f(\mathbf{x}; \boldsymbol{\theta}_0)$ \citep{jacot2018neural}.

\textbf{Spectrum on the Hypersphere.} 
Notably, for data uniformly distributed on the hypersphere $\mathbb{S}^{d-1}$, the eigenfunctions of the NTK for ReLU networks are given by \textbf{spherical harmonics}. As shown by \citet{bietti2019inductive}, the eigenvalues decay asymptotically as:
\begin{equation}
    \lambda_k \asymp k^{-\left(1 + \frac{1}{d_{\text{eff}}-1}\right)},
\end{equation}
where $d_{\text{eff}}$ is the effective input dimension. This provides a physical grounding for our theory:
\begin{itemize}
    \item \textbf{Dimensionality controls $\alpha$:} High-dimensional inputs imply $\alpha \to 1$ (slow decay, high capacity). Low-dimensional inputs imply $\alpha \gg 1$ (fast decay, low capacity).
    \item \textbf{Strong Teacher Explanation:} Our condition $\alpha_\te < \alpha_\st$ corresponds to the Teacher operating on a higher effective dimension or utilizing a richer feature set than the Student.
    \item \textbf{Fine-Tuning:} As noted by \citet{malladi2023kernelbasedviewlanguagemodel}, fine-tuning pre-trained models follows these same kernel dynamics, where the pre-trained weights define the kernel geometry.
\end{itemize}

Thus, the ``linear projection'' and ``spectral alignment'' definitions in our main text are not restrictive simplifications but rather accurate descriptions of deep learning dynamics in the infinite-width limit.

\section{Experiment Details}
In this section we provide more details for experiments in \ref{section:exp}.
\label{section:exp_app}
\subsection{Synthetic Experiments for Distillation} \label{subsec: distill_app}


\textbf{Models.} The teacher model is $f^{\mathrm{T}}(\mathbf{x}_i;\mathbf{w}^{\mathrm{T}}) = \langle \mathbf{w}^{\mathrm{T}}, \phi_{\mathrm{T}}(\mathbf{x})\rangle$, $\phi_{\mathrm{T}}(\mathbf{x}) = \Sigma_{\mathrm{T}}^{1/2} \mathbf{x}$. Here $\Sigma_{\mathrm{T}} = \mathrm{diag}[\lambda_i^{\mathrm{T}}]$ is a diagonal matrix with eigenvalues $\lambda_i^{\mathrm{T}} = i^{-\alpha_\mathrm{T}}$ ($\alpha_{\mathrm{T}}=1$) for $i = 1, \dots,d=100.$
Student model is $f^{\mathrm{S}}(\mathbf{x}_i; \mathbf{w}^{\mathrm{S}}) = \langle \mathbf{w}^{\mathrm{S}}, \phi_{\mathrm{S}}(\mathbf{x})\rangle$, $\phi_{\mathrm{S}}(\mathbf{x}) =  \Sigma_\mathrm{S}^{1/2} \mathbf{x}$ and $\Sigma_\mathrm{S} = \mathrm{diag}[\lambda_i^{\mathrm{S}}]$, $\lambda_i^{\mathrm{S}} = i^{-\alpha_{\mathrm{S}}}$ ($\alpha_{\mathrm{S}} = 1.5$ in Figure~\ref{fig:distillation1} and $\alpha_{\mathrm{S}} \in [1,2]$ in Figure~\ref{fig:distillation2}) for $i = 1, \dots,d=100$.

\textbf{Evaluation.} We approximate excess risk (defined in \ref{subsec:T2S_setup}) as $\frac{1}{n_{\mathrm{test}}}\sum_i \left(f^{\mathrm{S}}(\mathbf{x}_i;\mathbf{w}^{\mathrm{S}})-\langle \mathbf{w}^{*}, \Sigma_{\mathrm{T}}^{1/2} \mathbf{x}_i \rangle \right)^2$ on a test set of size $n_{\mathrm{test}}=2000$.

\textbf{Target.} Figure~\ref{fig:distillation1} and Figure~\ref{fig:distillation2} share the same linear target $y =\langle \mathbf{w}^*, \Sigma_{\mathrm{T}}^{1/2}\mathbf{x}\rangle + \xi$, where $\mathbf{w}^* \sim \mathcal{N}(\mathbf{0},\mathbf{I})$, $\xi \sim \mathcal{N}(0,\sigma^2 )$ and the noise level $\sigma$ is set to $0.1$. 




\textbf{Optimization.} We optimize the teacher and student model with SGD optimizer. The training process employs a exponential decay learning rate schedule with a warm-up phase, defined as follows: $\eta_t = \eta_0, 0 \leq t \leq \frac{N}{2}; \eta_t = \frac{\eta_0}{2^l}$, $t\geq \frac{N}{2}$ and the exponent $l$ is given by $\lfloor\frac{t-\frac{N}{2}}{\log\frac{N}{2}}\rfloor$. For every sample size $N$, we choose the best initial learning rate $\eta_0$ ranging from $6\times 10^{-2}$ to $1\times 10^{-1}$ to get best excess risk. 

\subsection{Real-World Experiments for Distillation} 

\textbf{Curve fitting.}
In Figure~\ref{fig:distillation3}, we extract feature representations $\mathbf{f}(\mathbf{x}_i; \boldsymbol{\theta})$ from the  output immediately before the classifier. We compute the empirical feature covariance matrix $\hat{\Sigma}$ and perform principal component analysis (PCA) by sorting its eigenvalues in descending order. We then analyze the spectral decay of the leading 500 eigenvalues by fitting a power-law distribution. Specifically, we discard the largest and smallest 10 eigenvalues to reduce boundary effects.

\textbf{Training.} In Fig.~\ref{fig:distillation4}, we conduct experiments on the UTKFace dataset, which is split into a training set of 20,000 samples and a test set of 2,000 samples. All images are resized to $224 \times 224$. For every model, the final classification layer is replaced with a regression head (output dimension is restricted to one). The teacher model: ViT-L/16, is fine-tuned on the training set using the mean squared error (MSE) loss. Optimization is performed with stochastic gradient descent (SGD) using a fixed learning rate of $1 \times 10^{-4}$, momentum of 0.9, and a batch size of 128 for 10 epochs. The model is evaluated on the test set after each epoch, and the model with the best test performance is selected for knowledge distillation. The student models: ResNet18, ResNet50, and ViT-B/16 are trained with their backbone networks frozen, updating only the regression head. These student models are trained on the training set using pseudo-labels generated by the teacher model. All student models are optimized with the MSE loss and SGD with an initial learning rate of $1 \times 10^{-3}$ and momentum of 0.9 for 20 epochs. A cosine annealing learning rate scheduler is applied during training.



\subsection{Synthetic Experiments for W2S}

To empirically validate our theoretical findings regarding the impact of signal dimension on excess risk dynamics, we conducted simulations on a synthetic linear regression problem. The detailed setup is as follows:

\paragraph{Data Generation.} 
We set the ambient dimension $d = 100$. The input features $\mathbf{x} \in \mathbb{R}^d$ are drawn from a centered Gaussian distribution $\mathcal{N}(\mathbf{0}, \boldsymbol{\Sigma})$ with a diagonal covariance matrix $\boldsymbol{\Sigma} = \text{diag}(\lambda_1, \dots, \lambda_d)$. The eigenvalues follow a power-law decay $\lambda_i = i^{-\alpha_{\text{in}}}$ with a decay rate $\alpha_{\text{in}} = 1.0$, simulating common spectral properties of real-world data.
The target labels are generated according to $y = \mathbf{x}^\top \mathbf{w}^* + \xi$, where $\xi \sim \mathcal{N}(0, \sigma^2)$ represents irreducible label noise with variance $\sigma^2 = 1.0$.
The ground truth parameter $\mathbf{w}^*$ is constructed to control the effective signal dimension. We set the first $d_{\text{signal}}$ components to $w^*_i = w_0 \cdot i^{-\alpha_w}$ (where $w_0=0.1$ and $\alpha_w=0.0$), and the remaining $d - d_{\text{signal}}$ components to zero. In our experiments, we vary $d_{\text{signal}} \in \{1, 10, 20, 30, 50\}$ to observe the behavior under different sparsity levels.

\paragraph{Teacher Training (Pre-training Phase).}
The teacher model is trained using Stochastic Gradient Descent (SGD) on a labeled dataset of size $N=2,000$.
\begin{itemize}
    \item \textbf{Batch Size:} $B_{\text{T}} = 10$.
    \item \textbf{Learning Rate Schedule:} We employ a step-decay schedule. The initial learning rate is $\eta_{\text{T}} = 0.1$. The learning rate is halved every $K_{\text{T}} = \lfloor N / \log_2 N \rfloor$ steps to facilitate convergence and stabilize the "teacher noise" in the tail dimensions.
    \item \textbf{Optimization:} The teacher minimizes the standard Mean Squared Error (MSE) on the noisy training labels.
\end{itemize}

\paragraph{Student Distillation (Transfer Phase).}
The student model is initialized at zero ($\mathbf{w}_{\text{S}}^{(0)} = \mathbf{0}$) and trained to mimic the fixed teacher's output (soft labels) $\hat{y}_{\text{T}} = \mathbf{x}^\top \mathbf{w}_{\text{T}}$.
\begin{itemize}
    \item \textbf{Data Stream:} The student has access to an effectively infinite stream of unlabeled data. At each step, we generate a fresh batch of size $B_{\text{S}} = 1,000$. This large batch size is chosen to approximate population dynamics and minimize the student's own gradient noise, isolating the effect of the teacher's learned bias and variance.
    \item \textbf{Optimization Steps:} The student is trained for $n=5,000$ steps.
    \item \textbf{Learning Rate Schedule:} The student uses an initial learning rate $\eta_{\text{S}} = 0.1$, with a step decay scheduled based on a horizon of 50,000 steps (decaying approximately every 3,200 steps in the actual run).
\end{itemize}

\paragraph{Evaluation Metric.}
We evaluate the student's performance using the \textbf{Excess Risk} relative to the ground truth $\mathbf{w}^*$ (not the teacher), defined as:
\begin{equation}
    \mathcal{R}(\mathbf{w}_{\text{S}}) - \sigma^2 = \|\mathbf{w}_{\text{S}} - \mathbf{w}^*\|_{\boldsymbol{\Sigma}}^2 = (\mathbf{w}_{\text{S}} - \mathbf{w}^*)^\top \boldsymbol{\Sigma} (\mathbf{w}_{\text{S}} - \mathbf{w}^*).
\end{equation}
This metric captures how well the student recovers the true underlying signal despite being trained on the biased proxy provided by the teacher.


\subsection{Real-World Experiments for W2S}

\textbf{Projection Energy.}
As illustrated in Figure~\ref{fig:w2s2}, we follow a standard dataset split for the UTKFace dataset, where $20{,}000$ samples are used for training and $2000$ samples for testing. All images are resized to $224 \times 224$ pixels. The backbone networks of both the teacher and student models are kept frozen throughout training. The linear prediction head of the teacher model is obtained via closed-form ridge regression using $1000$ labeled samples, with the regularization coefficient set to $\alpha = 10^{-6}$.

For student training, we use a batch size of 128 and a learning rate of $1\times 10^{-4}$
. The model is optimized using SGD with momentum 0.9 for a total of 5 epochs.

We extract feature vector through the and perform PCA on $\mathbf{\Sigma} = \frac{1}{m} \sum_{i}\mathbf{f}(x_i; \boldsymbol{\theta}) \mathbf{f}(x_i; \boldsymbol{\theta})^\top$ ($m= 10,000$), i.e. $\mathbf{\Sigma} = \mathbf{U} \mathbf{\Lambda} \mathbf{U}^\top$. Denoting the j-th column vector of $\mathbf{U}$ as $\mathbf{u}_j$, we record the linear weight $\mathbf{w}^{\mathrm{S}}$ and compute the projection energy onto each eigen-direction during training as $|\langle\mathbf{w}^{\mathrm{S}},\mathbf{u}_j\rangle|^2, j=1,\dots, 768.$ These energies are recorded every steps (with a batch size of 128) over the course of 5 epochs. Additionally, upon completion of training, we identify the minimum number of dimensions required to account for 80\% of the total cumulative energy ($\frac{\sum_{j=1}^k|\langle\mathbf{w}^{\mathrm{S}},\mathbf{u}_j\rangle|^2}{\sum_{j=1}^{768}|\langle\mathbf{w}^{\mathrm{S}},\mathbf{u}_j\rangle|^2}\geq 0.8$).

\textbf{Calculate $k_{90\%}$.}
In Figure~\ref{fig:w2s3} and Figure~\ref{fig:w2s4} we choose a 50,000 subset of the test set of ImageNet\cite{deng2009imagenet} and split it into 40,000 training samples and 10,000 test samples. We only train the added classifier of every model. We approximate the empirical feature covariance matrix with 1000 training samples and perform PCA on 
$\mathbf{\hat{\Sigma}} = \mathbf{U} \mathbf{\Lambda} \mathbf{U}^\top$. we solve linear classification on as $y = \langle \mathbf{w}, \mathbf{U}^\top \mathbf{f}(\mathbf{x}_i; \boldsymbol{\theta}) \rangle$ and truncated case $y = \langle \mathbf{w}, \mathbf{U}_{[1:k, :]}^\top \mathbf{f}(\mathbf{x}_i; \boldsymbol{\theta})\rangle$. We use the full dimension as baseline accuracy and seek the minimal $k$ to retain $90\%$ of the baseline performance. We train the models using the cross-entropy loss for multi-class classification. Optimization is performed with the Adam optimizer, using a fixed learning rate of $1\times10^{-3}$, no weight decay, and momentum parameters $\beta_1 = 0.9$ and $\beta_2 = 0.999$. All experiments are conducted with a batch size of 128, and the models are trained for 20 epochs on the training set.

\textbf{Calculate PGR.} We use the same dataset splitting then plot the \textbf{PGR} of above models during training. We first evaluate the test loss on the test set of pretrained weak teacher as $\mathcal{L}_{\mathrm{w}}$. Then we test the loss during training on $\mathbf{f}(\mathbf{x}_i; \boldsymbol{\theta})$ (full dimension) in two cases: strong student direct learns on real labels, we record the test loss as $\mathcal{L}_{\mathrm{s}}$ and strong student learns from labels predicted by teacher, we record the test loss as $\mathcal{L}_{\mathrm{w2s}}$. We plot the \textbf{PGR} as $\frac{\mathcal{L}_{\mathrm{w}}- \mathcal{L}_{\mathrm{w2s}}}{\mathcal{L}_{\mathrm{w}}- \mathcal{L}_{\mathrm{s}}}$. The training process  follows the above linear probing framework, and keeps the same as above.
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