Limitations of SGD for Multi-Index Models Beyond Statistical Queries

In this work, we tackle the following question: Can neural networks trained with gradient-based methods achieve the optimal computational-statistical tradeoff in learning Gaussian single-index models? Prior research has shown that any polynomial-time algorithm under the statistical query (SQ) framework requires $Ω(d^{s^\star/2}\lor d)$ samples, where $s^\star$ is the generative exponent representing the intrinsic difficulty of learning the underlying model. However, it remains unknown whether neural networks can achieve this sample complexity. Inspired by prior techniques such as label transformation and landscape smoothing for learning single-index models, we propose a unified gradient-based algorithm for training a two-layer neural network in polynomial time. Our method is adaptable to a variety of loss and activation functions, covering a broad class of existing approaches. We show that our algorithm learns a feature representation that strongly aligns with the unknown signal $θ^\star$, with sample complexity $\widetilde{O} (d^{s^\star/2} \lor d)$, matching the SQ lower bound up to a polylogarithmic factor for all generative exponents $s^\star\geq 1$. Furthermore, we extend our approach to the setting where $θ^\star$ is $k$-sparse for $k = o(\sqrt{d})$ by introducing a novel weight perturbation technique that leverages the sparsity structure. We derive a corresponding SQ lower bound of order $\widetildeΩ(k^{s^\star})$, matched by our method up to a polylogarithmic factor. Our framework, especially the weight perturbation technique, is of independent interest, and suggests potential gradient-based solutions to other problems such as sparse tensor PCA.

Score-based Generative Models (SGMs) have achieved impressive performance in data generation across a wide range of applications. While the statistical properties of their sampling procedures are increasingly well understood, the optimization dynamics underlying their training remain less explored. SGMs are typically trained by minimizing a weighted denoising scorematching objective, yet optimization guarantees with stochastic gradients remain limited. In this work, we study Stochastic Gradient Descent (SGD) for SGMs, contributing results in two complementary regimes. First, for general score parameterizations, we establish a non-convex convergence rate for SGD on the weighted denoising score-matching objective, with explicit dependence on the schedule-dependent weighting factors. Second, for overparameterized two-layer ReLU networks, we develop a Neural Tangent Kernel analysis tailored to diffusion training with stochastic gradients, yielding score-approximation error bounds along the SGD trajectory. Finally, our analysis quantifies the role of the reweighting factor in the score approximation error, providing theoretical guidance for weighting choices used in practice.

We consider the Multiscale Single-Index Model (MSIM), first introduced in \cite{oymak2021learning}, as a stylized model for hierarchical learning with \emph{scale separation}. Each layer extracts a shared single-index feature at one physical scale and passes it to the next, thus defining a tractable setting in which to study how deep architectures learn multiscale representations. Under non-degeneracy and delocalization assumptions on the link function and planted features respectively, for fixed depth $K$ and local scale $d$, the first Wiener chaos of the target behaves as a perturbed spiked tensor, where the perturbation of order $d^{-1/2}$ comes from the non-linearity -- revealing the MSIM as a natural non-linear analogue of the Tensor PCA model \cite{montanari2014statistical}. While this perturbative picture is sufficient to enable efficient spectral recovery based on Tensor unfolding (as already observed in \cite{oymak2021learning}), it is not precise enough for the analysis of backpropagation gradient-based methods. In this work, we address this limitation by performing a fine-grained analysis of the Wiener chaos using Edgeworth expansions. In the first chaos, this gives a finite-rank hierarchy at scales $d^{-q/2}$. In higher chaoses, balanced flattenings exhibit staircase singular-value plateaus of size $d^{-ρ/2}$ and multiplicity $d^ρ$ under a natural higher-chaos non-cancellation condition. Using this higher-chaos structure, and under an additional slow Hermite-energy tail condition, we first establish shallow-network approximation lower bounds, quantifying the benefit of depth in this model. Next, and most importantly, we prove that online SGD on the correlation objective, where all layers evolve in the same timescale, achieves $1 - o_d(1)$ recovery with $n = \widetilde{O}( d^{K-1})$ samples, recovering the same sample complexity as in the linear counterpart.