arXiv subjects

Theodor Misiakiewicz

Publications and source records attributed to Theodor Misiakiewicz.

At least 19 recordsLinked to original sources

arXiv2026

The anisotropic local law for sample covariance matrices under quadratic-form concentration

We study sample covariance matrices $K = \frac{1}{N} \sum_{i=1}^N \x_i \x_i^* \in \R^{n \times n}$ in the proportional regime $n \asymp N$. The columns $\x_1, \ldots, \x_N \in \R^n$ are independent and centered, with common covariance $\E \x_i \x_i^* = \Sigma$, but may otherwise have strongly and nonlinearly dependent coordinates. Assuming only that quadratic forms of the columns concentrate uniformly at the optimal rate $| \x_i^* A \x_i - \Tr \Sigma A | \prec \| A \|_F$, together with polynomial norm moments and a standard nondegeneracy condition on $\Sigma$, we prove the optimal anisotropic local law: on regular spectral domains, uniformly down to spectral scales $\eta:= \Im z \geq N^{-1 + \tau}$, \[ \big| \< \u , \big( (K-z)^{-1} - (-zI_n-z\widetilde m_0(z)\Sigma \big)^{-1} \bv \> \big| \prec \sqrt{\frac{\Im \widetilde m_0 (z)}{N\eta}} + \frac{1}{N\eta} \] for all deterministic unit vectors $\u,\bv \in \C^n$, where $\widetilde m_0(z)$ is the Stieltjes transform of the deformed Marchenko-Pastur law. This removes the higher-cumulant tensor assumption of Fan, Ma, Paquette, and Wang (2026), thereby answering the question raised in their work. The result applies, among other examples, to every centered log-concave column distribution with bounded, nondegenerate covariance, nonlinear tilts of Gaussian vectors, deep random features, and a high-temperature spherical 4-spin model for which the cumulant assumption is known to fail.

math.PR↗

arXiv2026

Generalization, memorization, and overfitting for diffusion models trained in the lazy high-dimensional regime

Modern score-based generative models have achieved remarkable empirical success in high-dimensional tasks such as image, audio, and video synthesis. These models reduce distribution learning to a sequence of regression problems that, if solved exactly on finite data, would ultimately reproduce the training samples. Their ability to generalize must therefore arise from the implicit or explicit regularization during training. In this work, we develop a generative counterpart to the theory of benign overfitting and algorithmic regularization for overparameterized neural networks in the supervised lazy-training regime. We study denoising score matching in a vector-valued reproducing kernel Hilbert space with an inner-product kernel. In the proportional high-dimensional regime $n\asymp d$, we derive exact risk trajectories under gradient flow training. These trajectories exhibit three phases governed by qualitatively distinct estimators: a spectral estimator that generalizes, a pure-noise score with localized peaks that interpolate the training objective, and an empirical Bayes estimator that memorizes the data. We then analyze how these estimators combine along the reverse-time SDE and characterize the distribution of the resulting samples. The analysis reveals familiar mechanisms from supervised learning, including kernel linearization and self-induced regularization from the nonlinear part of the kernel, but also reveals a distinct phenomenology specific to generative modeling.

stat.ML↗

arXiv2026

The sharp SAT/UNSAT phase transition in random ellipsoid fitting

Let $x_1,\ldots,x_n$ be independent standard Gaussian vectors in $\mathbb{R}^d$. An \emph{ellipsoid fit} is a matrix $S \succeq 0$ such that $x_i^\top S x_i =d$ for every $i$, so that all the points lie on the boundary of the centered ellipsoid $\{ x : x^\top S x = d\}$. Saunderson, Parrilo and Willsky conjectured that, as $n,d \to \infty$, this semidefinite feasibility problem undergoes a sharp transition at $n \sim d^2/4$. We prove this conjecture. If $\lim \sup n/d^2 = \alpha^* <1/4$, then, with probability tending to one, an ellipsoid fit exists; moreover, one can choose $S$ with all eigenvalues in a fixed interval $[\lambda_- , \lambda_+] \subset (0,\infty)$ depending only on $\alpha^*$. Conversely, if $\lim \inf n/d^2 > 1/4$, then, with probability tending to one, no ellipsoid fit exists, without any spectral restriction. Our proof builds on the Gaussian-equivalence framework developed by Bandeira and Maillard (2025) and closes the two gaps left open in their work: establishing exact fitting and removing the operator-norm constraint. On the satisfiable side, the new ingredients are a head-tail decomposition of the dual vector, exact correction of the sparse head constraints, and a Gaussian comparison principle for the low-influence tail. On the unsatisfiable side, we split a candidate into a low-rank spectral head and a Schatten-3 diffuse bulk, Gaussianize the bulk conditionally on the head, and apply a projected Gordon escape argument. The threshold is governed by the statistical dimension $d(d+1)/4$ of the positive semidefinite cone.

math.PR↗

arXiv2026

Dynamical mean-field limit and replica-symmetric free energy for the orthogonally-invariant SK model

We study a class of diffusion processes on $\mathbb{R}^n$ interacting through a symmetric matrix $X\in\mathbb{R}^{n\times n}$. When eigenvectors of $X$ are Haar-uniform on the orthogonal group, we derive a dynamical mean-field limit for the empirical law of sample paths, extending the classical Sompolinsky--Zippelius characterization for $X\sim\mathrm{GOE}$. The limit takes the form of a generalized Langevin equation with correlated Gaussian noise and memory, whose correlation and response kernels relate to those of the original dynamics through convolution equations involving the free cumulants of the eigenvalue distribution of $X$. For the overdamped Langevin diffusion associated with $\mu(\boldsymbol{\theta})\propto \exp\!\big(\frac12\boldsymbol{\theta}^{\top}X\boldsymbol{\theta}\big)\prod_{i=1}^n\nu(\mathrm{d}\theta_i)$, we analyze the mean-field limit under a rapid-mixing assumption. The correlation and response kernels admit time-translation-invariant approximants satisfying a fluctuation-dissipation relation. The generalized Langevin equation admits a Markovian approximation coupled to an auxiliary multivariate OU process and converges to a replica-symmetric prediction for the empirical coordinate law under $\mu$. This auxiliary correlation structure is characterized through the infinitesimal generator of a Markov semigroup for the lifted path-history process. Consequently, the free energy converges to a replica-symmetric limit under an explicit high-temperature condition, which for an Ising model is $\|X\|_{\mathrm{op}}<1/2$. By recent dynamical universality results, the same free-energy characterization holds for deterministic models without random disorder when $X$ satisfies a set of deterministic delocalization conditions.

math.PR↗

arXiv2026

Improved Scaling Laws via Weak-to-Strong Generalization in Random Feature Ridge Regression

It is increasingly common in machine learning to use learned models to label data and then employ such data to train more capable models. The phenomenon of weak-to-strong generalization exemplifies the advantage of this two-stage procedure: a strong student is trained on imperfect labels obtained from a weak teacher, and yet the strong student outperforms the weak teacher. In this paper, we show that the potential improvement is substantial, in the sense that it affects the scaling law followed by the test error. Specifically, we consider students and teachers trained via random feature ridge regression (RFRR). Our main technical contribution is to derive a deterministic equivalent for the excess test error of the student trained on labels obtained via the teacher. Via this deterministic equivalent, we then identify regimes in which the scaling law of the student improves upon that of the teacher, unveiling that the improvement can be achieved both in bias-dominated and variance-dominated settings. Strikingly, the student may attain the minimax optimal rate regardless of the scaling law of the teacher -- in fact, when the test error of the teacher does not even decay with the sample size.

cs.LG↗

arXiv2026

Statistical-Computational Trade-offs in Learning Multi-Index Models via Harmonic Analysis

We study the problem of learning multi-index models (MIMs), where the label depends on the input $\boldsymbol{x} \in \mathbb{R}^d$ only through an unknown $\mathsf{s}$-dimensional projection $\boldsymbol{W}_*^\mathsf{T} \boldsymbol{x} \in \mathbb{R}^\mathsf{s}$. Exploiting the equivariance of this problem under the orthogonal group $\mathcal{O}_d$, we obtain a sharp harmonic-analytic characterization of the learning complexity for MIMs with spherically symmetric inputs -- which refines and generalizes previous Gaussian-specific analyses. Specifically, we derive statistical and computational complexity lower bounds within the Statistical Query (SQ) and Low-Degree Polynomial (LDP) frameworks. These bounds decompose naturally across spherical harmonic subspaces. Guided by this decomposition, we construct a family of spectral algorithms based on harmonic tensor unfolding that sequentially recover the latent directions and (nearly) achieve these SQ and LDP lower bounds. Depending on the choice of harmonic degree sequence, these estimators can realize a broad range of trade-offs between sample and runtime complexity. From a technical standpoint, our results build on the semisimple decomposition of the $\mathcal{O}_d$-action on $L^2 (\mathbb{S}^{d-1})$ and the intertwining isomorphism between spherical harmonics and traceless symmetric tensors.

math.ST↗

arXiv2026

Positive Distribution Shift as a Framework for Understanding Tractable Learning

We study a setting where the goal is to learn a target function f(x) with respect to a target distribution D(x), but training is done on i.i.d. samples from a different training distribution D'(x), labeled by the true target f(x). Such a distribution shift (here in the form of covariate shift) is usually viewed negatively, as hurting or making learning harder, and the traditional distribution shift literature is mostly concerned with limiting or avoiding this negative effect. In contrast, we argue that with a well-chosen D'(x), the shift can be positive and make learning easier -- a perspective called Positive Distribution Shift (PDS). Such a perspective is central to contemporary machine learning, where much of the innovation is in finding good training distributions D'(x), rather than changing the training algorithm. We further argue that the benefit is often computational rather than statistical, and that PDS allows computationally hard problems to become tractable even using standard gradient-based training. We formalize different variants of PDS, show how certain hard classes are easily learnable under PDS, and make connections with membership query learning.

cs.LG↗

arXiv2025

When does Gaussian equivalence fail and how to fix it: Non-universal behavior of random features with quadratic scaling

A major effort in modern high-dimensional statistics has been devoted to the analysis of linear predictors trained on nonlinear feature embeddings via empirical risk minimization (ERM). Gaussian equivalence theory (GET) has emerged as a powerful universality principle in this context: it states that the behavior of high-dimensional, complex features can be captured by Gaussian surrogates, which are more amenable to analysis. Despite its remarkable successes, numerical experiments show that this equivalence can fail even for simple embeddings -- such as polynomial maps -- under general scaling regimes. We investigate this breakdown in the setting of random feature (RF) models in the quadratic scaling regime, where both the number of features and the sample size grow quadratically with the data dimension. We show that when the target function depends on a low-dimensional projection of the data, such as generalized linear models, GET yields incorrect predictions. To capture the correct asymptotics, we introduce a Conditional Gaussian Equivalent (CGE) model, which can be viewed as appending a low-dimensional non-Gaussian component to an otherwise high-dimensional Gaussian model. This hybrid model retains the tractability of the Gaussian framework and accurately describes RF models in the quadratic scaling regime. We derive sharp asymptotics for the training and test errors in this setting, which continue to agree with numerical simulations even when GET fails. Our analysis combines general results on CLT for Wiener chaos expansions and a careful two-phase Lindeberg swapping argument. Beyond RF models and quadratic scaling, our work hints at a rich landscape of universality phenomena in high-dimensional ERM.

math.ST↗

arXiv2025

Learning single-index models via harmonic decomposition

We study the problem of learning single-index models, where the label $y \in \mathbb{R}$ depends on the input $\boldsymbol{x} \in \mathbb{R}^d$ only through an unknown one-dimensional projection $\langle \boldsymbol{w}_*,\boldsymbol{x}\rangle$. Prior work has shown that under Gaussian inputs, the statistical and computational complexity of recovering $\boldsymbol{w}_*$ is governed by the Hermite expansion of the link function. In this paper, we propose a new perspective: we argue that $spherical$ $harmonics$ -- rather than $Hermite$ $polynomials$ -- provide the natural basis for this problem, as they capture its intrinsic $rotational$ $symmetry$. Building on this insight, we characterize the complexity of learning single-index models under arbitrary spherically symmetric input distributions. We introduce two families of estimators -- based on tensor unfolding and online SGD -- that respectively achieve either optimal sample complexity or optimal runtime, and argue that estimators achieving both may not exist in general. When specialized to Gaussian inputs, our theory not only recovers and clarifies existing results but also reveals new phenomena that had previously been overlooked.

cs.LG↗

arXiv2025

An Optimized Franz-Parisi Criterion and its Equivalence with SQ Lower Bounds

Bandeira et al. (2022) introduced the Franz-Parisi (FP) criterion for characterizing the computational hard phases in statistical detection problems. The FP criterion, based on an annealed version of the celebrated Franz-Parisi potential from statistical physics, was shown to be equivalent to low-degree polynomial (LDP) lower bounds for Gaussian additive models, thereby connecting two distinct approaches to understanding the computational hardness in statistical inference. In this paper, we propose a refined FP criterion that aims to better capture the geometric ``overlap" structure of statistical models. Our main result establishes that this optimized FP criterion is equivalent to Statistical Query (SQ) lower bounds -- another foundational framework in computational complexity of statistical inference. Crucially, this equivalence holds under a mild, verifiable assumption satisfied by a broad class of statistical models, including Gaussian additive models, planted sparse models, as well as non-Gaussian component analysis (NGCA), single-index (SI) models, and convex truncation detection settings. For instance, in the case of convex truncation tasks, the assumption is equivalent with the Gaussian correlation inequality (Royen, 2014) from convex geometry. In addition to the above, our equivalence not only unifies and simplifies the derivation of several known SQ lower bounds -- such as for the NGCA model (Diakonikolas et al., 2017) and the SI model (Damian et al., 2024) -- but also yields new SQ lower bounds of independent interest, including for the computational gaps in mixed sparse linear regression (Arpino et al., 2023) and convex truncation (De et al., 2023).

math.ST↗

arXiv2025

A Theory of Learning with Autoregressive Chain of Thought

For a given base class of sequence-to-next-token generators, we consider learning prompt-to-answer mappings obtained by iterating a fixed, time-invariant generator for multiple steps, thus generating a chain-of-thought, and then taking the final token as the answer. We formalize the learning problems both when the chain-of-thought is observed and when training only on prompt-answer pairs, with the chain-of-thought latent. We analyze the sample and computational complexity both in terms of general properties of the base class (e.g. its VC dimension) and for specific base classes such as linear thresholds. We present a simple base class that allows for universal representability and computationally tractable chain-of-thought learning. Central to our development is that time invariance allows for sample complexity that is independent of the length of the chain-of-thought. Attention arises naturally in our construction.

stat.ML↗

arXiv2024

Sparse PCA: Phase Transitions in the Critical Regime

This work studies estimation of sparse principal components in high dimensions. Specifically, we consider a class of estimators based on kernel PCA, generalizing the covariance thresholding algorithm proposed by Krauthgamer et al. (2015). Focusing on Johnstone's spiked covariance model, we investigate the "critical" sparsity regime, where the sparsity level $m$, sample size $n$, and dimension $p$ each diverge and $m/\sqrt{n} \rightarrow \beta$, $p/n \rightarrow \gamma$. Within this framework, we develop a fine-grained understanding of signal detection and recovery. Our results establish a detectability phase transition, analogous to the Baik--Ben Arous--P\'ech\'e (BBP) transition: above a certain threshold -- depending on the kernel function, $\gamma$, and $\beta$ -- kernel PCA is informative. Conversely, below the threshold, kernel principal components are asymptotically orthogonal to the signal. Notably, above this detection threshold, we find that consistent support recovery is possible with high probability. Sparsity plays a key role in our analysis, and results in more nuanced phenomena than in related studies of kernel PCA with delocalized (dense) components. Finally, we identify optimal kernel functions for detection -- and consequently, support recovery -- and numerical calculations suggest that soft thresholding is nearly optimal.

math.ST↗

arXiv2024

Dimension-free deterministic equivalents and scaling laws for random feature regression

In this work we investigate the generalization performance of random feature ridge regression (RFRR). Our main contribution is a general deterministic equivalent for the test error of RFRR. Specifically, under a certain concentration property, we show that the test error is well approximated by a closed-form expression that only depends on the feature map eigenvalues. Notably, our approximation guarantee is non-asymptotic, multiplicative, and independent of the feature map dimension -- allowing for infinite-dimensional features. We expect this deterministic equivalent to hold broadly beyond our theoretical analysis, and we empirically validate its predictions on various real and synthetic datasets. As an application, we derive sharp excess error rates under standard power-law assumptions of the spectrum and target decay. In particular, we provide a tight result for the smallest number of features achieving optimal minimax error rate.

stat.ML↗

arXiv2024

The merged-staircase property: a necessary and nearly sufficient condition for SGD learning of sparse functions on two-layer neural networks

It is currently known how to characterize functions that neural networks can learn with SGD for two extremal parameterizations: neural networks in the linear regime, and neural networks with no structural constraints. However, for the main parametrization of interest (non-linear but regular networks) no tight characterization has yet been achieved, despite significant developments. We take a step in this direction by considering depth-2 neural networks trained by SGD in the mean-field regime. We consider functions on binary inputs that depend on a latent low-dimensional subspace (i.e., small number of coordinates). This regime is of interest since it is poorly understood how neural networks routinely tackle high-dimensional datasets and adapt to latent low-dimensional structure without suffering from the curse of dimensionality. Accordingly, we study SGD-learnability with $O(d)$ sample complexity in a large ambient dimension $d$. Our main results characterize a hierarchical property, the "merged-staircase property", that is both necessary and nearly sufficient for learning in this setting. We further show that non-linear training is necessary: for this class of functions, linear methods on any feature map (e.g., the NTK) are not capable of learning efficiently. The key tools are a new "dimension-free" dynamics approximation result that applies to functions defined on a latent space of low-dimension, a proof of global convergence based on polynomial identity testing, and an improvement of lower bounds against linear methods for non-almost orthogonal functions.

cs.LG↗

arXiv2024

On the Complexity of Learning Sparse Functions with Statistical and Gradient Queries

The goal of this paper is to investigate the complexity of gradient algorithms when learning sparse functions (juntas). We introduce a type of Statistical Queries ($\mathsf{SQ}$), which we call Differentiable Learning Queries ($\mathsf{DLQ}$), to model gradient queries on a specified loss with respect to an arbitrary model. We provide a tight characterization of the query complexity of $\mathsf{DLQ}$ for learning the support of a sparse function over generic product distributions. This complexity crucially depends on the loss function. For the squared loss, $\mathsf{DLQ}$ matches the complexity of Correlation Statistical Queries $(\mathsf{CSQ})$--potentially much worse than $\mathsf{SQ}$. But for other simple loss functions, including the $\ell_1$ loss, $\mathsf{DLQ}$ always achieves the same complexity as $\mathsf{SQ}$. We also provide evidence that $\mathsf{DLQ}$ can indeed capture learning with (stochastic) gradient descent by showing it correctly describes the complexity of learning with a two-layer neural network in the mean field regime and linear scaling.

cs.LG↗

arXiv2024

Asymptotics of Random Feature Regression Beyond the Linear Scaling Regime

Recent advances in machine learning have been achieved by using overparametrized models trained until near interpolation of the training data. It was shown, e.g., through the double descent phenomenon, that the number of parameters is a poor proxy for the model complexity and generalization capabilities. This leaves open the question of understanding the impact of parametrization on the performance of these models. How does model complexity and generalization depend on the number of parameters $p$? How should we choose $p$ relative to the sample size $n$ to achieve optimal test error? In this paper, we investigate the example of random feature ridge regression (RFRR). This model can be seen either as a finite-rank approximation to kernel ridge regression (KRR), or as a simplified model for neural networks trained in the so-called lazy regime. We consider covariates uniformly distributed on the $d$-dimensional sphere and compute sharp asymptotics for the RFRR test error in the high-dimensional polynomial scaling, where $p,n,d \to \infty$ while $p/ d^{κ_1}$ and $n / d^{κ_2}$ stay constant, for all $κ_1 , κ_2 \in \mathbb{R}_{>0}$. These asymptotics precisely characterize the impact of the number of random features and regularization parameter on the test performance. In particular, RFRR exhibits an intuitive trade-off between approximation and generalization power. For $n = o(p)$, the sample size $n$ is the bottleneck and RFRR achieves the same performance as KRR (which is equivalent to taking $p = \infty$). On the other hand, if $p = o(n)$, the number of random features $p$ is the limiting factor and RFRR test error matches the approximation error of the random feature model class (akin to taking $n = \infty$). Finally, a double descent appears at $n= p$, a phenomenon that was previously only characterized in the linear scaling $κ_1 = κ_2 = 1$.

stat.ML↗

arXiv2024

A non-asymptotic theory of Kernel Ridge Regression: deterministic equivalents, test error, and GCV estimator

We consider learning an unknown target function $f_*$ using kernel ridge regression (KRR) given i.i.d. data $(u_i,y_i)$, $i\leq n$, where $u_i \in U$ is a covariate vector and $y_i = f_* (u_i) +\varepsilon_i \in \mathbb{R}$. A recent string of work has empirically shown that the test error of KRR can be well approximated by a closed-form estimate derived from an `equivalent' sequence model that only depends on the spectrum of the kernel operator. However, a theoretical justification for this equivalence has so far relied either on restrictive assumptions -- such as subgaussian independent eigenfunctions -- , or asymptotic derivations for specific kernels in high dimensions. In this paper, we prove that this equivalence holds for a general class of problems satisfying some spectral and concentration properties on the kernel eigendecomposition. Specifically, we establish in this setting a non-asymptotic deterministic approximation for the test error of KRR -- with explicit non-asymptotic bounds -- that only depends on the eigenvalues and the target function alignment to the eigenvectors of the kernel. Our proofs rely on a careful derivation of deterministic equivalents for random matrix functionals in the dimension free regime pioneered by Cheng and Montanari (2022). We apply this setting to several classical examples and show an excellent agreement between theoretical predictions and numerical simulations. These results rely on having access to the eigendecomposition of the kernel operator. Alternatively, we prove that, under this same setting, the generalized cross-validation (GCV) estimator concentrates on the test error uniformly over a range of ridge regularization parameter that includes zero (the interpolating solution). As a consequence, the GCV estimator can be used to estimate from data the test error and optimal regularization parameter for KRR.

stat.ML↗

arXiv2023

SGD learning on neural networks: leap complexity and saddle-to-saddle dynamics

We investigate the time complexity of SGD learning on fully-connected neural networks with isotropic data. We put forward a complexity measure -- the leap -- which measures how "hierarchical" target functions are. For $d$-dimensional uniform Boolean or isotropic Gaussian data, our main conjecture states that the time complexity to learn a function $f$ with low-dimensional support is $\tildeΘ(d^{\max(\mathrm{Leap}(f),2)})$. We prove a version of this conjecture for a class of functions on Gaussian isotropic data and 2-layer neural networks, under additional technical assumptions on how SGD is run. We show that the training sequentially learns the function support with a saddle-to-saddle dynamic. Our result departs from [Abbe et al. 2022] by going beyond leap 1 (merged-staircase functions), and by going beyond the mean-field and gradient flow approximations that prohibit the full complexity control obtained here. Finally, we note that this gives an SGD complexity for the full training trajectory that matches that of Correlational Statistical Query (CSQ) lower-bounds.

cs.LG↗