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Improved high-dimensional estimation with Langevin dynamics and stochastic weight averaging

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Learning theoryhigh-dimensional statisticsnon-convex optimization
Abstract:

Significant recent work has studied the ability of gradient descent to recover a hidden planted direction θ⋆∈Sd−1\theta^\star \in S^{d-1} in different high-dimensional settings, including tensor PCA and single-index models. The key quantity that governs the ability of gradient descent to traverse these landscapes is the information exponent k⋆k^\star (Ben Arous et al., (2021)), which corresponds to the order of the saddle at initialization in the population landscape. Ben Arous et al., (2021) showed that n≳dmax⁡(1,k⋆−1)n \gtrsim d^{\max(1, k^\star-1)} samples were necessary and sufficient for online SGD to recover θ⋆\theta^\star, and Ben Arous et al., (2020) proved a similar lower bound for Langevin dynamics. More recently, Damian et al., (2023) showed it was possible to circumvent these lower bounds by running gradient descent on a smoothed landscape, and that this algorithm succeeds with n≳dmax⁡(1,k⋆/2)n \gtrsim d^{\max(1, k^\star/2)} samples, which is optimal in the worst case. This raises the question of whether it is possible to achieve the same rate without explicit smoothing. In this paper, we show that Langevin dynamics can succeed with n≳dk⋆/2n \gtrsim d^{ k^\star/2 } samples if one considers the average iterate, rather than the last iterate. The key idea is that the combination of noise-injection and iterate averaging is able to emulate the effect of landscape smoothing. We apply this result to both the tensor PCA and single-index model settings. Finally, we conjecture that minibatch SGD can also achieve the same rate without adding any additional noise.

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Overview

Overall Novelty Assessment

The paper studies Langevin dynamics with iterate averaging for recovering planted directions in high-dimensional tensor PCA and single-index models, achieving sample complexity n ≳ d^(k*/2) where k* is the information exponent. It resides in the Tensor PCA and Multi-Spiked Recovery leaf, which contains five papers total including this one. This leaf sits within the broader Spiked Matrix and Tensor Models branch, indicating a moderately populated research direction focused on tensor-structured signal recovery. The taxonomy shows this is an active but not overcrowded area, with sibling leaves addressing matrix-only models and Langevin-specific analysis.

The taxonomy reveals neighboring research directions that contextualize this work. The Single-Index and Multi-Index Models branch addresses related projection recovery problems but without tensor structure, while the Stochastic Gradient Descent and Optimization Dynamics branch studies non-convex landscapes more broadly. Within the same parent category, the Matrix Models and Robust PCA leaf focuses on matrix-only settings, and the Langevin Dynamics for Spiked Models leaf analyzes Langevin algorithms specifically. The paper bridges these areas by applying Langevin methods to tensor problems, connecting stochastic optimization theory with high-dimensional tensor inference.

Among twelve candidates examined across three contributions, none were found to clearly refute the claimed novelty. The first contribution on improved sample complexity examined one candidate with no refutation. The second contribution on noise injection emulating smoothing examined ten candidates, again with no refutations found. The third contribution on the algorithmic framework examined one candidate without refutation. This limited search scope suggests the specific combination of Langevin dynamics, iterate averaging, and the k*/2 sample complexity rate may not have direct precedents among semantically similar recent work, though the analysis does not claim exhaustive coverage.

Based on examination of twelve semantically related candidates, the work appears to occupy a distinct position combining stochastic noise injection with averaging to achieve smoothing-like benefits. The taxonomy structure indicates this sits in a moderately active research area with clear boundaries separating tensor methods from matrix-only and single-index approaches. The limited search scope means potentially relevant work outside the top semantic matches may exist, particularly in adjacent optimization or sampling literature not captured by the field-specific taxonomy.

Taxonomy

Core-task Taxonomy Papers
27
3
Claimed Contributions
12
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: Recovering planted directions in high-dimensional statistical models using gradient-based methods. The field organizes around several complementary perspectives on extracting low-dimensional signal from high-dimensional noise. Spiked Matrix and Tensor Models focus on settings where a few principal directions are planted in random matrices or tensors, with works like Tensor PCA Thresholds[16] and SGD Tensor PCA[4] analyzing when gradient descent can recover these structures. Single-Index and Multi-Index Models study regression-like problems where the response depends on a small number of linear projections of high-dimensional covariates, as surveyed in Multi-Index Models Survey[5]. Stochastic Gradient Descent and Optimization Dynamics examine the algorithmic landscape itself, investigating how noise and geometry shape convergence, while Projection Pursuit and Clustering address unsupervised discovery of interesting directions. Signal Recovery with Structured Priors and Constraints incorporates domain knowledge such as sparsity or smoothness, and Specialized Recovery Problems tackle application-specific challenges. Theoretical Foundations and Surveys provide unifying perspectives across these branches. Recent work has intensified around understanding when and how gradient-based algorithms succeed in tensor settings, where computational-statistical gaps can emerge. Papers like Normalized SGD Tensor[22] and Tensor QAOA Estimation[14] explore algorithmic refinements and alternative optimization strategies for multi-spiked recovery, while SGD Glassy Landscape[3] investigates the complex energy surfaces that can trap iterative methods. Langevin Stochastic Averaging[0] sits within the Tensor PCA and Multi-Spiked Recovery cluster, contributing analysis of stochastic gradient schemes that blend noise injection with averaging to navigate these landscapes. Compared to neighbors like SGD Tensor PCA[4], which studies vanilla stochastic gradient descent, or Tensor PCA Thresholds[16], which characterizes information-theoretic limits, Langevin Stochastic Averaging[0] emphasizes how controlled randomness can aid exploration and escape from poor local configurations, offering a distinct algorithmic lens on the multi-spiked tensor recovery problem.

Claimed Contributions

Langevin dynamics with iterate averaging achieves improved sample complexity

The authors demonstrate that combining Langevin dynamics with stochastic weight averaging (iterate averaging) enables recovery of the planted direction in high-dimensional estimation problems with sample complexity n ≳ dk⋆/2, matching the optimal rate achieved by explicit smoothing methods but without requiring landscape smoothing.

1 retrieved paper
Noise injection and averaging emulates landscape smoothing

The authors show that the combination of noise injection (from Langevin dynamics) and iterate averaging can achieve the same effect as explicit landscape smoothing approaches, providing a new mechanism for improving sample complexity in high-dimensional estimation.

10 retrieved papers
Algorithm for tensor PCA and single-index models via time-averaged Langevin

The authors develop Algorithm 1, which runs Langevin dynamics and returns either the time-averaged iterate (for odd information exponent) or the top eigenvector of the time-averaged second moment (for even information exponent), successfully recovering the planted direction in both tensor PCA and single-index model settings.

1 retrieved paper

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Langevin dynamics with iterate averaging achieves improved sample complexity

The authors demonstrate that combining Langevin dynamics with stochastic weight averaging (iterate averaging) enables recovery of the planted direction in high-dimensional estimation problems with sample complexity n ≳ dk⋆/2, matching the optimal rate achieved by explicit smoothing methods but without requiring landscape smoothing.

Contribution

Noise injection and averaging emulates landscape smoothing

The authors show that the combination of noise injection (from Langevin dynamics) and iterate averaging can achieve the same effect as explicit landscape smoothing approaches, providing a new mechanism for improving sample complexity in high-dimensional estimation.

Contribution

Algorithm for tensor PCA and single-index models via time-averaged Langevin

The authors develop Algorithm 1, which runs Langevin dynamics and returns either the time-averaged iterate (for odd information exponent) or the top eigenvector of the time-averaged second moment (for even information exponent), successfully recovering the planted direction in both tensor PCA and single-index model settings.

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