Heavy-Tailed Distributions

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3 papers in the last four weeks, level with the four weeks before. 0.0% of all new papers.

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Latest papers 39

Oct 6, 2026cond-mat.dis-nn

Asymptotic Analysis of Empirical Risk Minimization on Entry-wise i.i.d. Heavy-Tailed Data

Many real-world datasets exhibit unusually large values far more frequently than predicted by Gaussian models. Heavy-tailed distributions capture this behavior, yet evaluating learning performance under them remains challenging because rare, large feature entries retain non-vanishing effects even in high dimensions. Even in the canonical setting of empirical risk minimization for linear regression with entry-wise i.i.d. symmetric

α

-stable data, a precise asymptotic characterization of prediction has been lacking. In this work, we introduce a functional order parameter that describes the random effective problem associated with each coefficient. Using the replica method, we fully characterize the generalization error in the proportional high-dimensional limit where the sample size and feature dimension diverge at a fixed ratio. Additionally, this analysis establishes a heavy-tail universality law, scaling laws relating typical errors to prediction reliability, and the Bayes-optimal prediction error. In addition to characterizing the effects of extreme entries on the learning process, our method applies broadly to other systems with persistent local heterogeneity.

Kaito Takanami, Takashi Takahashi, Yoshiyuki Kabashima

Graduate School of Science, The University of Tokyo · Institute for Physics of Intelligence, The University of Tokyo · Trans-Scale Quantum Science Institute, The University of Tokyo

Statistical Physics of Learning Linear Regression Heavy-Tailed Distributions Empirical Risk Minimization

Sep 30, 2026cs.LG

Attention Kernels for Learning Maps Between Heavy-Tailed Measures

Operator learning on probability measures can be accomplished with transformers. For measures with polynomial tails, the exponential weighting in softmax can make the corresponding measure-level attention integrals diverge. This motivates replacing the exponential with slower-growing functions. We construct two benchmarks for operator learning on measures with closed-form targets. We use these benchmarks to study attention kernel growth and data transformation in post-norm transformers. Without data transformation, the softmax models exhibit ensemble collapse on both heavy-tailed benchmarks, while the three slower-growing kernels avoid collapse. Symlog preprocessing allows softmax to avoid collapse on the matrix inverse task but not on the sheared swap task. On the Gaussian control, all four kernels perform similarly. We also examine how sample size affects the sensitivity of empirical energy and Wasserstein distances to tail differences. These results support slower-growing attention kernels as an effective design choice for post-norm transformers learning from heavy-tailed ensembles.

Kailen Hargenrader, Edoardo Calvello, Bohan Chen

Computing and Mathematical Sciences California Institute of Technology · Lawrence Berkeley National Laboratory University of California, Berkeley ICSI

Softmax Attention Attention Mechanisms Heavy-Tailed Distributions Operator Learning

Sep 23, 2026math.OC

Distributed Stochastic Approximation Algorithms and Heavy-Tailed Age of Information

Algorithms in multi-agent systems such as federated learning, mobile robotic swarming, and consensus control can be designed and analyzed as distributed stochastic approximation algorithms. Such algorithms involve information exchanges between agents for various computations. The freshness of the information can be quantified using the Age of Information (AoI) metric. Consider robotic teams operating in highly obstructed geographical settings, such as subterranean or dense urban environments. Because of spatial disconnections, AoI has empirically been observed to be heavy-tailed with unbounded moments. However, most analyses assume AoI with bounded moments, creating a gap between theory and practice. To the best of our knowledge, ours is the first analysis under general heavy-tailed AoI with potentially infinite mean. We study the stability (almost sure boundedness of the distributed iterates) and convergence of multi-agent systems that are strictly dissipative in the scaling limit (system at ``infinity''). Examples include most gradient-based and consensus algorithms under the Robbins-Monro step-size regime.

Adrian Redder, Arunselvan Ramaswamy, Holger Karl

Dept. of Electrical Engineering and Information Technology, Paderborn University, Germany · Dept. of Industrial Engineering and Operations Research, Indian Institute of Technology Bombay, India · Hasso-Plattner Institute (HPI), Digital Engineering Faculty, University of Potsdam, Germany

Stochastic Approximation Age of Information Heavy-Tailed Distributions

Sep 14, 2026cs.LG

Dimension-Corrected Hitting Times for Heavy-Tailed Spectral Emergence in Neural Optimizer Dynamics

Heavy-tailed empirical spectral densities of neural-network weight matrices are widely used as diagnostics of implicit self-regularization, but the step complexity of heavy-tail emergence remains poorly understood. We formulate spectral heavy-tail formation as a right-censored hitting-time problem: a run that does not reach a heavy-tail diagnostic within the observation horizon is treated as censored rather than discarded. In controlled full-batch teacher--student dynamics, we find that the first-step spike--bulk gap alone does not explain onset time. Instead, finite-onset regression supports a dimension-corrected spectral-gap law, (\tau_{\mathrm{HT}}\approx C\Delta_1^{-\gamma}d^\rho), with (R^2=0.683), (\gamma=0.626), and (\rho=0.772) across 330 completed runs. Right-censored lognormal accelerated-failure-time models further favor the dimension-corrected model over a gap-only model, improving AIC from 706.62 to 628.70. Theoretically, we prove that exact early loss dynamics in linear networks do not determine factor spectral tails, that Adam recurrences alone do not imply spectral redistribution, and that projected singular-basis spreading implies contraction of a spectral-tail potential and hence a dimension-corrected hitting-time bound. Empirically, projected-kernel profiles support the sufficient spreading mechanism, Adam and AdamW agree under tested grids, GD and signGD do not reach onset in the same regimes, and real pretrained Qwen2.5-0.5B and Pythia-70M transformer weights show non-Gaussian spectral-tail structure relative to matched Gaussian nulls. The result is a reproducible spectral hitting-time law with rigorous conditional theory, not a claim that Adam necessarily generates heavy tails from first principles.

Zongmin Liu

Stanford University

Heavy-Tailed Distributions Neural Network Training Dynamics

Sep 3, 2026cs.LG

A location-invariant estimator of extremal quantile treatment effects for heavy-tailed distributions

Quantile treatment effects (QTEs) measure the effect of a treatment on the distribution of an outcome, and their estimation at extreme quantile levels is of central interest in applications where the target quantiles lie far beyond the range of the data. For heavy-tailed potential outcomes, existing extremal QTE estimators rely on extrapolation combined with a causal extreme value index (EVI) estimator, but the resulting estimator is not invariant under a common location shift of the potential outcome distributions, even though the population QTE is. We address this issue in two steps. First, we adapt the location-invariant Fraga estimator of the EVI to the causal setting using inverse propensity score weighting. Second, we replace the original extrapolation formula with a difference-based scheme, under which the location parameter cancels when quantile differences are taken. The resulting QTE estimator is therefore location invariant. We establish the consistency and asymptotic normality of the proposed extremal QTE estimators, and provide a consistent variance estimator, leading to asymptotically valid inference. A simulation study confirms the location invariance, the stability with respect to the threshold, and the coverage of the proposed methods.

Xin Yu, Shuwei Huang, Jicheng Liu +4

Huazhong University of Science and Technology · Tencent

Causal Effect Estimation Heavy-Tailed Distributions Extreme Value Theory

Sep 3, 2026cs.LG

Restricted Eigenvalues Beyond Gaussian Width: Threshold Occupancy under Heavy Tails

Restricted eigenvalue (RE) bounds govern stable recovery by norm-regularized estimators. For isotropic sub-Gaussian measurements, the benchmark sample size is

1+w(A)^2

, where

w(A)

is the Gaussian width of the normalized descent cone. The COLT 2015 open-problem note (Banerjee et al., 2015) asked whether the same law follows for heavy-tailed designs from a uniform small-ball condition alone. We give an explicit and systematic negative answer to the general question as formulated there: the proposed law fails in its full dimension-free, arbitrary-set form, and the missing obstruction is simultaneous threshold occupancy. A constant-width polyhedral descent cone with fixed small-ball constants has zero empirical RE on every sample path up to half the ambient dimension. More generally, every finite range space admits exact threshold encoding in an arbitrarily narrow spherical cap and a lift to a full polyhedral descent-cone section. For every fixed threshold VC dimension

d

, as

β\downarrow0

, the sharp worst-case sample complexity is

Θ(β^{-1}[d\log(1/β)+\log(1/δ)])

. The separation persists under exact isotropy and all finite moments: on the same constant-width cone, Gaussian measurements succeed with

O(1+\log(1/δ))

samples, whereas an isotropic heavy-tailed design fails pathwise for

n\lesssim\sqrt{p/\log p}

. Gaussian smoothing yields an everywhere-positive

C^\infty

density while retaining arbitrarily poor RE. Under isotropy, a distribution-free fallback governed by affine dimension times squared enclosing radius is sharp on this family.

Shi Fu, Huibo Xu, Qixin Zhang +1

Generative AI Lab, College of Computing and Data Science, Nanyang Technological University, Singapore

Sparse Recovery Heavy-Tailed Noise Heavy-Tailed Distributions

Aug 12, 2026stat.ML

Fine-Tuning Generative Models for Extreme Events via CVaR-Penalized Wasserstein Gradient Flows

We propose CVaR-penalized Generative Particle Algorithm (CVaR-GPA), a robust, tail-agnostic algorithm for fine-tuning generative models to learn heavy-tailed distributions and capture extreme events, requiring no prior knowledge or estimation of the target's tail characteristics. The method is the Wasserstein gradient flow of the Lipschitz-regularized Kullback-Leibler (KL) divergence penalized by a Conditional Value-at-Risk (CVaR) discrepancy term: the Lipschitz-regularized KL divergence enables robust learning under minimal assumptions on the target distribution, while the CVaR penalty restores the velocity that otherwise vanishes prematurely in the under-sampled tails. The penalized flow admits a bounded but non-Lipschitz velocity field. This departs from the Lipschitz transport maps of standard generators, which preserve the tail behavior of a light-tailed source, and enables transport toward heavier-tailed targets. To define this flow on empirical measures, we derive the first-variation subgradients of CVaR from its Rockafellar-Uryasev representation, valid precisely where the classical density-based formula fails. The particle algorithm CVaR-GPA fine-tunes the output samples of any pre-trained model, without access to its architecture, and runs on an adaptive time horizon set by a kinetic-energy stopping criterion rather than a preset depth. On synthetic isotropic and anisotropic Student-

t

target distributions, Neal's funnel distribution, and the real-world high-dimensional Fama-French 25 portfolio dataset, CVaR-GPA dramatically improves global and tail accuracy on heavy-tailed targets over the pre-trained baseline.

Thejani Gamage, Hyemin Gu, Zhizhen Zhang +3

Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA · School of Data and Information Sciences (SDIS) and Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC

Wasserstein Gradient Flows Generative Modeling Heavy-Tailed Distributions

Jul 31, 2026cs.LG

Parameter-Free Heavy-Tailed Bandits

Heavy-tailed distributions arise naturally in sequential decision-making problems such as financial investment, online advertising, and network management, where rare but extreme outcomes can dominate performance. Heavy-tailed bandits model online decision-making in these settings by assuming only that rewards

X

satisfy

\mathbb{E}[|X|^{1+ε}]\leq u

, for some tail exponent

ε\in(0,1]

and moment bound

u<+\infty

. However, most existing regret minimization algorithms require these parameters to be known. This assumption is particularly restrictive in practice:

ε

and

u

govern the frequency and magnitude of rare events and are therefore precisely the quantities that are hardest to infer reliably from limited observations. Motivated by an open problem posed by Genalti and Metelli at COLT 2025, we resolve the assumption-free adaptation problem for heavy-tailed bandits and characterize the price in the regret of not knowing the tail parameters. We first study adaptation to the moment bound

u

for a fixed tail exponent

ε

. We prove that every algorithm unaware of

u

, or of any upper bound on it, must obey a sharp trade-off between its distribution-dependent and distribution-free regret guarantees. We then introduce a scheduled-exploration algorithm that requires no knowledge of

u

and matches the resulting adaptation frontier up to logarithmic factors. Finally, we show that the same algorithm can be instanced without knowing

ε

by calibrating its exploration schedule to the endpoint

ε=1

. It achieves sublinear regret for every fixed

ε>0

, while no algorithm can guarantee sublinear regret uniformly over all

ε\in(0,1]

. Altogether, our results resolve the COLT open problem without additional distributional assumptions and provide a sharp characterization of the statistical cost of adapting to unknown heavy tails.

Gianmarco Genalti, Alberto Maria Metelli

Politecnico di Milano

Multi-Armed Bandits Heavy-Tailed Noise Heavy-Tailed Distributions

Jul 29, 2026stat.ML

HOMER: Huber-of-Means for Efficient and Robust Estimation in Hilbert Spaces

Heavy tails weaken high-confidence control for the empirical mean. Geometric median-of-means (MOM) also lacks a threshold that moves toward mean efficiency. We propose \emph{HOMER}, or Huber-of-Means for Efficient and Robust Estimation. HOMER aggregates block means through a radial Huber center. Its canonical and pseudo-Huber forms bound each block score and interpolate between median-like robustness and the empirical mean. We establish a Hilbert-space majority theorem and a MOM-order deviation bound under a finite second moment. Canonical HOMER recovers the sample mean inside its quadratic region. Pseudo-HOMER approaches the sample mean as the threshold grows. It also admits asymptotic linearity and consistent sandwich covariance estimation around the population block-Huber target. Under a finite third moment, fixed finite-dimensional projections support mean inference at the usual parametric rate. This result requires growing block sizes and counts, with block sizes increasing faster. Heavy-tailed simulations show that HOMER remains stable when a minority of block summaries is displaced. On clean Gaussian data, both versions closely approach the empirical mean's efficiency. Finite-block sandwich intervals undercovered, especially for skewed functional data. Further studies show failure when contamination affects most blocks or compromises ordinary within-block means.

Kisung You, Boram Cho

Baruch College · Yale University

Heavy-Tailed Distributions Robust Estimation

Jul 15, 2026cs.LG

Heavy-Tailed Flow Matching via Random Clocks

Heavy-tailed data arise in many domains where rare events carry disproportionate importance, such as imbalanced image datasets, financial returns, and weather extremes. Standard diffusion and flow-matching models typically begin from Gaussian noise or Gaussian source distributions, which yield tractable training targets but provide a poor inductive match for heavy-tailed data. We propose Heavy-Tailed Flow Matching via Random Clocks (HTFM), a framework that portrays heavy-tailed sources as mixtures of clock-conditioned Gaussian sources. Conditioning on a given clock path, the source distribution and flow are Gaussian; marginalizing over the clock gives a Gaussian scale mixture covering Gaussian,

α

-stable, and Student-t families. To make the clock-conditioned vector field practical, we encode the path-valued clock using truncated logsignature features, allowing the velocity field to adapt to the realized conditional space with negligible overhead. Empirically, on 2D imbalanced

α

-stable mixtures, CIFAR10-LT, and HRRR weather fields, HTFM improves mode coverage, sample quality, and tail-statistic recovery over Gaussian flow matching and competitive heavy-tailed baselines, while retaining the low-NFE sampling advantage of flow matching. Moreover, the random-clock formulation further provides a practical tail-control interface: by varying only the clock law or tail parameter, the same architecture can calibrate the ``heaviness'' of generated tails across different distribution families.

Zhouhao Yang, Yezhen Wang, Kenji Kawaguchi +2

Department of Applied Mathematics and Statistics, Johns Hopkins University. · School of Computing, National University of Singapore. · Department of Computer Science, Johns Hopkins University.

Flow Matching Heavy-Tailed Noise Generative Modeling Flow-Based Generative Modeling Heavy-Tailed Distributions

Jul 9, 2026cs.LG

Vanilla SGD with Momentum Survives Heavy-Tailed Noise: Convergence Analysis without Gradient Clipping or Normalization

Stochastic gradient descent (SGD) is a cornerstone of modern optimization. While its performance under heavy-tailed noise is often addressed through specialized modifications such as gradient clipping or normalization, we investigate a more fundamental question: how does vanilla SGD, particularly with momentum, perform in the presence of heavy-tailed noise? In this paper, we refine existing convergence results for vanilla SGD and, more importantly, provide the first comprehensive convergence analysis of vanilla SGD with momentum for strongly convex, convex, and nonconvex objectives, without employing any gradient control mechanisms. Our results demonstrate that the obtained convergence rates are inferior to the optimal rates achieved by clipped or normalized variants of SGD, thereby revealing inherent limitations of vanilla methods under heavy-tailed noise. The theoretical findings are supported by experiments on synthetic functions.

Ryusei Yamada, Naoki Sato, Hideaki Iiduka

Meiji University, Japan

Heavy-Tailed Noise Stochastic Optimization Convergence Heavy-Tailed Distributions Heavy-Tailed Stochastic Optimization Stochastic Gradient Descent

Jul 2, 2026cs.LG

Model Merging as Probabilistic Inference in Fine-Tuning Parameter Space

Model merging aims to combine existing single-task solutions into a multi-task solution without additional data-driven fine-tuning.~Most existing approaches achieve this using geometric properties of local solution spaces. However, such geometric views provide limited guidance for scoring how statistically useful each task-specific update direction is across tasks during merging. To address this, we formulate model merging from a new perspective of probabilistic inference under a product-of-experts (PoE) scenario where each single-task solution defines an energy-based expert model (EBM) over the merged parameters. We show that several existing model merging methods arise as special cases of our framework under energy designs that impose implicit Gaussian assumptions on directional residuals between merged and task-specific models. Empirically, we find that these residuals are often heavy-tailed which exposes a mismatch with the imposed light-tailed Gaussian structures. We address this with a heavy-tailed PoE design based on Cauchy experts, which better captures the observed residual behavior while admitting a provably convergent inference procedure. Experiments across multiple tasks and architectures show significant improvements over state-of-the-arts baselines. Our code is available at https://github.com/MinhLong210/PoE-EBM-Merging.git.

Long Minh Bui, Tuan Anh Le Van, Tung Phi Duc +3

Washington State University · Hanoi University of Science and Technology

Energy-Based Models Heavy-Tailed Distributions Model Merging

Jun 29, 2026cs.LG

Heads, Not Backbones: Output Heads Dominate Architectures on Fat-Tailed Returns

In a deep forecasting pipeline for fat-tailed financial returns at short horizons, which matters more - the backbone architecture or the output head? We compare four modern backbones (TimesNet, DLinear, N-BEATS, iTransformer) under three output heads: a point head, a single-Gaussian density head, and a Gaussian mixture density head with K=4 components. On S and P 500 monthly log-returns (1871-2023) under anchored walk-forward validation, the three heads form a strict gradient: switching from point to Gaussian improves CRPS by about 1.3 percent; switching from Gaussian to mixture adds a further about 2.4 percent. Switching between backbones, in contrast, changes CRPS by less than 1.5 percent on the point-head row and on the backbone-mean axis; density-head backbone spread is larger (up to 5.1 percent on the h=1 Gaussian row, driven by N-BEATS) but the head gradient (3.7 percentage points) still dominates. The Model Confidence Set on squared errors does not exclude any of the 12 variants at the 5 percent level: the head separates them only on distributional metrics (CRPS, pinball, coverage), not on squared error. The mixture head incremental value over a single Gaussian is largest in the highest-volatility regimes (13.9 percent in 1970s stagflation at h=12), confirming the mixture captures tail risk beyond what a unimodal Gaussian can express. The picture is horizon-dependent: the head dominates at short horizons, but at long horizons (h >= 6) the backbone re-takes the lead - an h-split we document against classical baselines (section 5.1). We conclude that on fat-tailed returns at short horizons, the head dominates the backbone, and the mixture distribution adds genuine value over a single Gaussian during crisis periods when risk-management decisions actually matter.

Sichao He, Yansong Zhang

Peking University · Academy for Advanced Interdisciplinary Studies · Beijing, China +2

Quantitative Finance Financial Forecasting Heavy-Tailed Distributions Financial Time Series Finance Probabilistic Time Series Forecasting

Jun 29, 2026stat.ML

Adjusted Wasserstein distances for bridging empirical and true distributions with applications to MDS

This paper examines how metric adjustments to Multidimensional Scaling (MDS) can enhance its effectiveness as a visual tool for pattern recognition. The distance under consideration, referred to as Max-D-SW, is an adjustment of the Max-Sliced Wasserstein distance. In contrast to the original formulation, which optimizes over single unit directions, Max-D-SW aggregates contributions over orthonormal bases. This modification provides a clear numerical advantage in MDS outcomes, particularly when applied to heavy-tailed distributions. We also establish sample-complexity bounds showing that Max-D-SW remains statistically tractable, with rates comparable to those of its max-sliced counterpart. Moreover, we show that a better sample complexity for a metric does not necessarily translate into better performance when the metric is used as an input for MDS.

Flor Martinez-Sermeno, Arturo Jaramillo, Johan Van Horebeek

Dimensionality Reduction Heavy-Tailed Distributions Sliced Wasserstein Distance Wasserstein Distance

Jun 25, 2026stat.ML

The Decision Geometry of Covariance Estimation for the Global Minimum-Variance Portfolio under Heavy Tails

The global minimum-variance portfolio (GMVP) is the canonical decision built from an estimated covariance matrix, yet covariance estimators are universally evaluated by matrix-norm loss, which is not the object the decision depends on. We characterise exactly how covariance-estimation error maps into GMVP suboptimality. We prove an exact regret identity and a non-asymptotic bound showing decision regret depends on the estimation error only through its action on the portfolio weights, scaled by portfolio concentration and the conditioning of the true covariance. From this we derive the decision geometry: GMVP regret is invariant to a (p-1)-dimensional projection of the p^2-dimensional error matrix, with invariance to the covariance-scale direction as an exact special case. We then apply the framework to heavy-tailed returns (tail index kappa in (2,4)), establishing the regret convergence rate implied by the centred operator-norm rate, and confirm the theory on a skew-t/t-copula simulation design with pre-registered analysis. The decision-focused advantage is a sharper constant and a concentration discount rather than a faster rate; we report an honest high-conditioning boundary of the rate prediction. The results complement recent decision-focused learning approaches by supplying the exact estimation geometry and consistency theory they lack.

Xavier Fonseca

Academy for AI, Games and Media, Breda University of Applied Sciences, Breda, The Netherlands.

Regret Minimization Portfolio Optimization Heavy-Tailed Distributions Covariance Estimation

Jun 24, 2026cs.LG

High-Probability PL-SGD with Markovian Noise: Optimal Mixing and Tail Dependence

We study first-order methods for smooth objectives satisfying the Polyak-Łojasiewicz (PL) condition when gradient samples are generated by an exogenous Markov chain. In the light-tailed setting, prior uniform-in-time high-probability bounds for ordinary Stochastic Gradient Descent (SGD) under a standard growth envelope scale as

\widetilde{O}(t_{mix}^2/k)

, leaving a gap with the

\widetilde{O}(t_{mix}/k)

expectation bounds. We close this gap using a lag-blocking argument to establish a uniform high-probability guarantee with a leading stochastic term of

\widetilde{O}(t_{mix}/(k+K_0))

under geometric mixing. We prove this linear dependence on the mixing time is optimal via a matching

Ω(σ^2 t_{mix}/k)

lower bound on a quadratic objective driven by a persistent two-state chain. We then extend this framework to heavy-tailed Markovian gradients satisfying a stationary finite-

p

-moment condition,

p \in (1,2]

. We design an all-samples clipped block method that uses every Markov transition while mitigating Markovian bias. Under a transition budget

T

, this algorithm achieves a high-probability stochastic error of

\widetilde{O}(σ_p^2(t_{mix}/T)^{2(p-1)/p})

. We establish a matching lower bound by reducing PL optimization to heavy-tailed mean estimation for a sticky Markov chain. Ultimately, this work tightly characterizes the optimal polynomial dependence on mixing time for light-tailed PL-SGD, and the optimal heavy-tail exponent and effective-sample-size dependence in the robust regime.

Dhruv Sarkar, Aprameyo Chakrabartty, Vaneet Aggarwal

Indian Institute of Technology Kharagpur · Mohamed bin Zayed University of Artificial Intelligence · Purdue University

Stochastic Optimization Optimization Convergence Analysis Heavy-Tailed Noise Stochastic Optimization Convergence Heavy-Tailed Distributions Heavy-Tailed Stochastic Optimization+1

Jun 22, 2026cs.LG

Open Problem: Is AdamW Effective Under Heavy-Tailed Noise?

AdamW is the de facto optimizer for training large language models (LLMs), yet the theory behind it still lives mostly in finite-variance regimes. This is increasingly unsatisfying, as empirical evidence indicates that stochastic gradient noise in LLM pretraining is typically heavy-tailed. Recent work shows that sign-based optimizers such as Lion and Muon achieve sharp heavy-tailed rates, and that AdaGrad can also converge under heavy-tailed noise. However, no rigorous convergence theory for AdamW has yet been established in this regime. Can AdamW converge under the same heavy-tailed assumptions, or does its second-moment accumulator create a genuine obstruction? We formulate this as an open problem, prove a positive weighted-metric benchmark, and give a corridor lower-bound mechanism showing how denominator memory can hide large gradients.

Dingzhi Yu, Hongyi Tao, Yuanyu Wan +2

1Nanjing University · 2Zhejiang University · 3Fudan University

Stochastic Optimization Heavy-Tailed Noise Stochastic Optimization Convergence Heavy-Tailed Distributions Adaptive Gradient Methods Heavy-Tailed Stochastic Optimization

Jun 15, 2026cs.LG

Tail-Shape Estimation in LLM Evaluation Is Fragile: A Protocol for Diagnosing False Positives

Recent work motivates moving large language model (LLM) evaluation from mean-based to tail-aware metrics, including conditional value-at-risk and tail-index estimates of reward-model error. We ask whether the canonical extreme-value-theory tail-index parameter, which isolates how heavy a tail is from how large the tail mass is, adds discriminative information beyond the mean and a standard tail-magnitude statistic in LLM evaluation. We pre-register a protocol covering admissibility, goodness-of-fit, threshold-stability, and effect-size requirements for any positive tail-shape claim. The protocol is the contribution of this paper; the empirical study below is a demonstration of what its gates catch. Applied to a standard LLM toxicity-evaluation setup under two structurally different scorer families, the protocol catches three distinct modes of false positives that a naive analysis would have published, and rejects the headline tail-shape claim on both scorers. We conclude that tail-shape estimation in the LLM toxicity-evaluation setups we examined is more fragile than the recent literature suggests, and recommend the protocol as a starting point for tail-index claims in similar setups.

Luca Zhou

Sapienza University of Rome

LLM Evaluation Language Model Safety Evaluation Heavy-Tailed Distributions Extreme Value Theory

Jun 12, 2026stat.ML

Gradient boosting for extremes: sampling theory and application to insurance

We develop a statistical learning theory for gradient boosting applied to the estimation of covariate-dependent Generalized Pareto (GP) distributions in the context of Peaks-over-Threshold modeling. After an orthogonal reparametrization of the GP likelihood that diagonalizes its Fisher information matrix, we cast the estimation problem within the Empirical Risk Minimization (ERM) framework and derive non-asymptotic error bounds for the boosting estimator. Our analysis accounts for three distinct sources of error in the process: statistical fluctuations, the approximation bias inherent to the asymptotic nature of the GP model-controlled under second-order regular variation-and the approximation error associated with the finite number of boosting iterates, making explicit the resulting bias-variance trade-off. We illustrate the practical benefits of the reparametrization through simulations, showing that it significantly reduces gradient correlation during training and improves convergence stability. The methodology is applied to a medical malpractice insurance dataset from the Texas Department of Insurance, comprising over 18 000 closed claims. The gradient boosting approach yields a good fit for the tail of settlement cost distributions and reveals that the number of days to settlement is the dominant predictor of tail heaviness, consistent with earlier findings in the reserving literature.

Stéphane Lhaut, Olivier Lopez

1CREST, CNRS, Ecole polytechnique, Groupe ENSAE-ENSAI, ENSAE Paris, Institut Polytechnique de Paris, Palaiseau, France

Gradient-Boosted Decision Trees Heavy-Tailed Distributions Statistical Learning Theory Extreme Value Theory

Jun 4, 2026quant-ph

Quantum enhanced rare event discovery and sampling

Financial crashes, cascading failures in infrastructure, and critical errors in AI systems are frequently triggered by events that occur with extremely small probability. Efficiently discovering and sampling events with probability below a threshold is therefore of critical interest. Yet this task is highly non-trivial using existing classical or quantum methods. Being rare, such events require an immense sampling overhead to collect sufficient data samples. Moreover, because the rare events are not known in advance, they cannot be flagged for amplification using standard techniques. Here, we introduce a quantum algorithm for rare-event discovery and sampling without first learning which events are rare. The algorithm achieves the optimal quantum scaling with the rarity threshold. We further demonstrate that this can achieve a quadratic speedup for heavy-tailed systems whose tail has nonvanishing total mass, and translates into a robust polynomial speedup for stationary stochastic processes, with the exponent determined by its entropy-rate structure.

Naixu Guo, Po-Wei Huang, Qisheng Wang +4

Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore · Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom · School of Computer Science, Shanghai Jiao Tong University, Shanghai 200240, China +4

Heavy-Tailed Distributions Rare-Event Estimation

Jun 2, 2026cs.LG

Variance Reduction for Heavy-Tailed Monetization Metrics in Ranking Experiments via Post-Stratification

Online evaluation of ranking and retrieval systems often relies on downstream monetization metrics such as app revenue or creator earnings. These metrics are typically heavy-tailed, with a small fraction of users dominating both mean and variance, leading to low statistical power and unreliable conclusions in A/B experiments -- especially under limited traffic. We present a practical framework for variance reduction in online experiments by combining post-stratification with CUPED. Our approach leverages pre-experiment covariates to improve the sensitivity of monetization experiments without requiring additional traffic. Deployed at ShareChat across ranking-driven monetization experiments, the method substantially reduces variance and improves decision stability, achieving equivalent statistical confidence with ~45% less traffic than standard metrics. We further discuss practical design choices, guardrails, and limitations, providing guidance on when post-stratification is appropriate for real-world information retrieval and Recommendation systems.

Neeti Pokharna, Olivier Jeunen, Yatharth Saraf +1

ShareChat · Bengaluru, Karnataka, India · Aampe +3

Causal Effect Estimation A/b Testing Heavy-Tailed Distributions IR Evaluation Variance Reduction

Jun 2, 2026math.OC

Bregman meets Lévy: Stochastic mirror descent with heavy-tailed noise in continuous and discrete time

We study the robustness of stochastic mirror descent (SMD) under heavy-tailed noise, focusing on whether the method retains its convergence guarantees when run with infinite-variance stochastic gradient input. To address this question in a principled manner, we begin by introducing a continuous-time model of SMD as a stochastic differential equation (SDE) driven by a centered Lévy noise process with finite

p

-th order moments,

1 < p \leq 2

. This scheme -- which we call the Lévy mirror flow (LMF) -- arises naturally as the scaling limit of SMD in the presence of heavy-tailed noise. In particular, when

p < 2

-- the heavy noise regime -- the trajectories of LMF generically exhibit jump discontinuities of arbitrary magnitude which, if frequent enough, lead to infinite variance. Nonetheless, despite this highly singular behavior, we show that LMF attains

ε

-optimality within

\mathcal{O}(ε^{-p/(p-1)})

time in the convex case, and within

\mathcal{\tilde O}(ε^{-1/(p-1)})

time for (relatively) strongly convex objectives. These guarantees provide a transparent characterization of the impact of frequent long jumps on the convergence of the process, and percolate to a series of matching discrete-time guarantees for several variants of SMD under heavy-tailed noise.

Pierre-Louis Cauvin, Panayotis Mertikopoulos

Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, LIG, 38000 Grenoble, France.

Stochastic Optimization Stochastic Differential Equations Heavy-Tailed Noise Mirror Descent Stochastic Optimization Convergence Heavy-Tailed Distributions+1

Jun 1, 2026stat.ML

Self-Regulating Annealing in Heavy-Tailed Diffusion Models

Diffusion models have emerged as a leading framework for deep generative modeling. While the standard Gaussian formulation is theoretically convenient, its suitability for heavy-tailed datasets remains unclear. To address this, heavy-tailed diffusion models (HTDMs) extend the standard formulation by replacing the Gaussian distribution with a Student's t-distribution, thereby improving tail fidelity on heavy-tailed datasets. Although stochastic differential equation (SDE)-based sampling is possible in HTDMs, it has not been fully explored. In this paper, we propose an SDE-based sampler for HTDMs that explicitly incorporates a state-dependent diffusion coefficient. This state dependence naturally induces a self-regulating annealing mechanism by adaptively modulating the effective noise scale. We theoretically explore this mechanism and experimentally verify its necessity for reproducing samples from a heavy-tailed distribution.

Keito Wakatsuki, Hideaki Shimazaki

Graduate School of Informatics Kyoto University, Kyoto, Japan

Stochastic Differential Equations Diffusion Sampling Heavy-Tailed Noise Generative Modeling Adaptive Sampling Heavy-Tailed Distributions+1

May 26, 2026stat.ML

Stop Suppressing the Tail: Causal Inference for Extreme Events

Estimating how an outcome responds to a continuous treatment (the Average Dose-Response Function, or ADRF) is a core causal-inference primitive. However, when outcomes possess heavy tails, standard robust double machine learning (DML) deliberately suppresses these extremes to stabilize the bulk average. In high-stakes settings, such as financial returns or climate losses, this omitted 1-in-1000 extreme event is the actual target quantity. Furthermore, current methods that read the tail from a model's residuals suffer from circular dependence, causing tail shape inferences to shift drastically based solely on whether the core estimator is switched between Huber and Welsch. The research proposes an ADRF estimator that emits a structured tail-shape output alongside the standard point estimate. Its tail diagnostic (PDHTE+JK) evaluates the per-treatment tail shape from the outcome centered by a pilot median, successfully breaking the circular dependence and rendering the diagnostic invariant to the choice of core method. The output encompasses four treatment-conditional quantities: tail shape

\hatξ(t)

, deep-tail return levels

\hat{Q}_α(t)

, conditional shortfalls

\hat{S}_α(t)

, the recovered mean ADRF, and an explicit refusal mechanism that declines extrapolation when extreme-value modeling is unsupported by the data. Compared to kernel-weighted quantile regression (QR), the proposed estimator reduces deep-tail (

α=0.001

) return-level MAE by 11% and conditional-shortfall MAE by 25.5% across a heavy-tailed panel. It also achieves a 20-29% MAE reduction in sample-scarce regimes (

n\le2000

). On freMTPL2 motor-insurance claims, it successfully triggered an explicit extrapolation refusal on the log-claim scale, which neither QR nor loss-only DML can produce.

Eichi Uehara

Causal Effect Estimation Causal Inference Heavy-Tailed Distributions Extreme Value Theory

May 20, 2026math.PR

Concentration of General Stochastic Approximation Under Heavy-Tailed Markovian Noise

We establish maximal concentration bounds for the iterates generated by stochastic approximation algorithms with general step sizes, where the noise has a finite-state Markovian component plus a Martingale-difference component. When the Martingale-difference noise is bounded, we show that the tail of the error can be sub-Gaussian, sub-Weibull, or something lighter than any Pareto but heavier than any Weibull, depending on the step size sequence and on whether the random operator is almost surely contractive, almost surely non-expansive, or expansive with positive probability. Our analysis relies on a novel Lyapunov function involving the moment-generating function of the solution to a Poisson equation, together with an auxiliary projected algorithm. We complement the upper bounds with worst-case examples showing that qualitatively sharper bounds are impossible. We further study the case of unbounded Martingale-difference noise when the average operator is contractive, and the step sizes are of order

1/k

. In this setting, we show that if the random operator is almost surely non-expansive, then the error tail is at most three times heavier than the noise tail, whereas if the random operator is expansive with positive probability, then the error may have substantially heavier tails. These results are obtained through a novel black-box truncation argument that reduces the unbounded-noise setting to the bounded-noise case.

Shubhada Agrawal, Siva Theja Maguluri, Martin Zubeldia

Indian Institute of Science · Georgia Institute of Technology · University of Minnesota

Heavy-Tailed Noise Stochastic Approximation Heavy-Tailed Distributions

May 19, 2026stat.ML

Tail Annealing for Heavy-Tailed Flow Matching

Standard generative models struggle with heavy-tailed data: Lipschitz architectures cannot produce power-law tails from Gaussian noise, and interpolating between heavy-tailed data and Gaussians is ill-posed. We propose a simple fix: apply the soft-log transform

φ(x) = \mathrm{sign}(x) \cdot \log(1 + |x|)

coordinate-wise to data before training, then exponentiate samples after generation. A Hill diagnostic decides per-coordinate whether to transform, leaving light-tailed margins untouched at no added complexity. This compresses heavy tails into a range where standard flow matching succeeds, without heavy-tailed base distributions or architectural modifications. We provide theoretical intuition for why this works: the log-transform maps Pareto tails to exponentials, and the induced dynamics implement a form of tail annealing via power transformations. On a 144-configuration multivariate benchmark (3 copulas,

d

up to 100, 4 tail indices), Log-FM dominates specialized baselines on

W_1

, CVaR

_{99}

, and extreme-quantile metrics, and is the only method with zero severe divergences across 2{,}880 runs.

Jean Pachebat

1CMAP, Ecole Polytechnique, Institut Polytechnique de Paris, France.

Flow Matching Generative Modeling Flow-Based Generative Modeling Heavy-Tailed Distributions

May 18, 2026math.OC

Can Adaptive Gradient Methods Converge under Heavy-Tailed Noise? A Case Study of AdaGrad

Many tasks in modern machine learning are observed to involve heavy-tailed gradient noise during the optimization process. To manage this realistic and challenging setting, new mechanisms, such as gradient clipping and gradient normalization, have been introduced to ensure the convergence of first-order algorithms. However, adaptive gradient methods, a famous class of modern optimizers that includes popular

\mathtt{Adam}

and

\mathtt{AdamW}

, often perform well even without any extra operations mentioned above. It is therefore natural to ask whether adaptive gradient methods can converge under heavy-tailed noise without any algorithmic changes. In this work, we take the first step toward answering this question by investigating a special case,

\mathtt{AdaGrad}

, the origin of adaptive gradient methods. We provide the first provable convergence rate for

\mathtt{AdaGrad}

in non-convex optimization when the tail index

p

satisfies

4/3<p\leq2

. Notably, this result is achieved without requiring any prior knowledge of

p

and is hence adaptive to the tail index. In addition, we develop an algorithm-dependent lower bound, suggesting that the existing minimax rate for heavy-tailed optimization is not attainable by

\mathtt{AdaGrad}

. Lastly, we consider

\mathtt{AdaGrad}\text{-}\mathtt{Norm}

, a popular variant of

\mathtt{AdaGrad}

in theoretical studies, and show an improved rate that holds for any

1<p\leq2

under an extra mild assumption.

Zijian Liu

Stern School of Business, New York University

Stochastic Optimization Nonconvex Stochastic Optimization Heavy-Tailed Noise Stochastic Optimization Convergence Heavy-Tailed Distributions Adaptive Gradient Methods+1

May 18, 2026stat.ML

Markov Chain Decoders Overcome the Heavy-Tail Limitations of Lipschitz Generative Models

Heavy-tailed distributions are prevalent in performance evaluation, network traffic, and risk modeling. This behavior poses a fundamental challenge for modern deep generative models. Standard Variational Autoencoders (VAEs) employ Gaussian decoder likelihoods and Lipschitz-constrained neural networks, a combination that is structurally incapable of producing heavy-tailed outputs: the Gaussian tail decays exponentially, and Lipschitz continuity prevents the decoder from amplifying rare events from the latent space input to sufficiently overcome this decay. We provide both a theoretical characterization of this limitation and a controlled empirical demonstration using synthetic Pareto data across a grid of tail indices

α

\in

{2, 3, 5, 30} and dimensions d

\in

{1, 5, 10}. As a solution, we replace the Gaussian decoder with a Phase-Type (PH) distribution based on Markov chains, while keeping the encoder, latent space, and training procedure identical. PH distributions allow for arbitrarily precise approximations of any positive-valued distributions, including heavy-tailed families. Experiments showed that the PH-based model reduces tail Kolmogorov-Smirnov distance by up to x6 and extreme quantile error by up to x10 compared to the Gaussian baseline for heavy-tailed data. These results demonstrate that integrating Markov chain-based distributions into the decoder of a generative model institutes a principled and practically effective solution to the heavy-tail generation problem.

Abdelhakim Ziani, Andras Horvath, Paolo Ballarini

Université Paris Saclay, Lab. MICS, CentraleSupélec, Gif-sur-Yvette, France · Università di Torino, Torino, Italy

Variational Autoencoders Generative Modeling Heavy-Tailed Distributions

May 18, 2026stat.ML

On Stability and Decomposition of Sample Quantiles under Heavy-Tailed Distributions

We study sample quantiles of distributions indexed by estimated parameters, with a on Value-at-Risk related to linear projections of financial returns that whose underlying probability law is heavy-tailed. In this setting, the projection direction and the empirical quantile threshold are estimated from the data, so the standard Bahadur representation under a fixed distribution does not separate the distinct sources of instability. A canonical starting point is Bahadur's representation, which expresses the sample quantile through the empirical distribution function plus a remainder term \cite{bahadur1966}. Empirical-process theory provides a usable scaffolding through the mechanics of half-spaces, symmetric differences, and Glivenko--Cantelli uniform convergence. They yield stability bounds, but absorb changes in projection direction and changes in quantile threshold into a single symmetric-difference measure. Interestingly, a global uniform-convergence requirement is imposed on what is intrinsically a local quantile-stability problem. This paper introduces a Q-Q orthogonality formulation for separating projection-direction and quantile-threshold effects. The object of interest is the difference between the empirical quantile computed using the estimated projection direction and the population quantile computed at the reference projection direction. We decompose this difference into three terms,

\hat q_α(\hat w)-q_α(w_0)=D_1+D_2+D_3

. Here,

D_1

measures the population quantile movement induced by perturbing the projection direction,

D_2

measures the empirical quantile fluctuation with the projection direction held fixed, and

D_3

is the Bahadur-type remainder.

Choudur Lakshminarayan

School of Business, Stevens Institute of Technology

Quantitative Finance Heavy-Tailed Distributions

May 13, 2026cs.LG

Do Heavy Tails Help Diffusion? On the Subtle Trade-off Between Initialization and Training

Recent works have proposed incorporating heavy-tailed (HT) noise into diffusion- and flow-based generative models, with the goals of better recovering the tails of target distributions and improving generative diversity. This motivation is intuitive: if the data are heavy-tailed, HT noise may appear better matched than light-tailed (LT) Gaussian noise. However, replacing Gaussian noise by HT noise also changes the underlying estimation problem. In this paper, we revisit this paradigm through a combined theoretical and empirical study, establishing sampling-error bounds for two representative diffusion models driven by HT and LT noise. We show that HT noise makes the statistical estimation problem harder, leading to less favorable sampling-error bounds. We support these findings with experiments on synthetic and real-world datasets, empirically recovering the predicted error trade-off. Our results call into question a growing design trend in generative modeling and challenge the use of HT noise to improve rare-region exploration.

Hamza Cherkaoui, Hélène Halconruy, Antonio Ocello

SAMOVAR, Télécom SudParis, Institut Polytechnique de Paris, Palaiseau, France · Modal’X, Université Paris Nanterre, Nanterre, France · CREST, ENSAE Paris, Institut Polytechnique de Paris, Palaiseau, France

Diffusion Model Sampling Heavy-Tailed Noise Heavy-Tailed Distributions Diffusion Models