Conditional Value-at-Risk (CVaR) is a widely used risk metric in applications such as finance. We derive concentration bounds for CVaR estimates, considering separately the cases of light-tailed and heavy-tailed distributions. In the light-tailed case, we use a classical CVaR estimator based on the empirical distributi…
Algorithm distinguishes light-tailed from non-light-tailed distributions.
problem Characterize the tail of a distribution using hazard rate.
method Careful bucketing scheme based on hazard rate.
result Polynomial number of samples required for success.
This work extends diffusion models to handle heavy-tailed targets, improving score estimation and sampling guarantees.
problem Score estimation and sampling guarantees for heavy-tailed targets in diffusion models.
method Kernel density estimation and minimax rates analysis for score estimation and sampling guarantees.
result Sharp minimax rates for score estimation and sampling guarantees for heavy-tailed targets, revealing qualitative differences between exponential and polynomial tails.
SS-GEN simulates rare events in heavy and light-tailed data.
problem Estimating probabilities of extreme events in multivariate data.
method Self-Similar Generative Estimation (SS-GEN) decomposes tail distribution into radial and angular components.
result SS-GEN generates representative extreme scenarios and estimates rare-event probabilities beyond observed data.
Study heavy-tailed weights' impact on neural network's spectral distribution.
problem Analyzing spectral distribution of conjugate kernel matrices with heavy-tailed weights.
method Computed limiting eigenvalue distribution through moments, considering heavy-tailed distributions and nonlinear activation functions.
result Heavy-tailed weights induce strong correlations, leading to fundamentally different spectral behavior.
We analyze systems of agents sharing light-tailed risky claims issued by different financial objects. Assuming exponentially distributed claims, we obtain that both agents' and system's losses follow generalized exponential mixture distributions. We show that this leads to qualitatively different results on individual …
Deep models can't generate heavy-tailed samples well.
problem Understanding the limitations of deep generative models in generating samples with heavy tails.
method Unified framework using concentration of measure and convex geometry, Gromov-Levy inequality.
result Deep generative models are not universal generators and can only produce concentrated samples with light tails.
New algorithm SELECT minimizes satisficing regret in bandits.
problem Minimizing regret in bandit optimization with satisficing arms.
method SELECT algorithm for satisficing regret minimization.
result SELECT achieves constant expected satisficing regret.
Privacy-preserving SGD with heavy-tailed noise achieves differential privacy guarantees.
problem Privacy preservation in noisy SGD with heavy-tailed noise.
method Differential privacy guarantees for SGD with heavy-tailed noise.
result SGD with heavy-tailed perturbations achieves (0,O(1/n))-DP. Study on Goodhart's law without independence assumptions, finds new patterns in optimisation.
problem Understanding when and how optimisation of a proxy metric leads to over-optimisation of the intended goal.
method Formalized Goodhart's law without independence and paradigm assumptions, studied different cases of goal and discrepancy tailness.
result Dependence between proxy metric and goal does not change Goodhart's effect for light-tailed cases, but over-optimisation occurs in heavy-tailed discrepancy cases.
New policy optimizes risk and optimality in stochastic bandits.
problem Optimizing risk in stochastic bandits with heavy-tailed risk.
method Designing policies with worst-case optimality for expected regret and light-tailed risk distribution.
result Achieves worst-case optimality for expected regret and light-tailed risk distribution.
New RDP guarantees for heavy-tailed SDEs and SGD.
problem Characterizing differential privacy for heavy-tailed noise in learning algorithms.
method Rényi flow computations and fractional Poincaré inequalities.
result First RDP guarantees for heavy-tailed SDEs with weaker dependence on dimension.
This paper analyzes bias-variance trade-off for clipped SFOMs, improving complexity guarantees for heavy-tailed noise.
problem Improving complexity guarantees for stochastic optimization methods with heavy-tailed noise.
method Novel analysis of bias-variance trade-off in gradient clipping for clipped SFOMs.
result Improved complexity guarantees for clipped SFOMs across various tail indices, including infinite mean noise.
Paper provides tail bounds for stochastic mirror descent in heavy-tailed noise.
problem Optimizing convex and Lipschitz functions with heavy-tailed noise.
method Develops tail bounds for optimization error of Stochastic Mirror Descent.
result Tail bounds extend to heavier-tailed noise regimes without diameter constraints.
Optimizes regret distribution in stochastic bandits for risk balance.
problem Balancing regret expectation and tail risk in stochastic bandits.
method Characterizes optimal regret tail probability for any threshold, proposes new policies.
result Discovers an intrinsic gap in optimal tail rate based on time horizon uncertainty.
Gaussian random vectors exhibit the loss of dimension phenomena, which relate to their joint survival tail behaviour. Besides, the fact that the components of such vectors are light-tailed complicates the approximations of various multivariate risk measures significantly. In this contribution we derive precise approxim…
Paper supports robust estimation in regression with heavy-tailed errors.
problem Support estimation in high-dimensional heteroscedastic mean regression.
method Use of Huber loss function and adaptive LASSO penalty for robust estimation.
result Sign-consistency and optimal rates of convergence in ℓ∞ norm. New PAC-Bayes bounds for heavy-tailed losses using supermartingales.
problem Extending PAC-Bayes bounds to heavy-tailed losses.
method Using supermartingales and bounded variance assumption.
result PAC-Bayes generalization bounds for heavy-tailed losses.
Study improves self-normalized bounds for vector-valued processes beyond sub-Gaussianity.
problem Limited understanding of self-normalized concentration for vector-valued processes outside sub-Gaussian frameworks.
method Developed concentration inequalities for self-normalized processes with light tails (e.g., Bennett, Bernstein bounds) for vector-valued data.
result Provided new insights and bounds for self-normalized processes with non-sub-Gaussian distributions.
Paper tackles robust batched bandits for heavy-tailed rewards.
problem Clinical trials and other applications with heavy-tailed rewards.
method Proposes robust batched bandit algorithms for heavy-tailed rewards in finite-arm and Lipschitz-continuous settings.
result Heavier-tailed rewards require fewer batches for near-optimal regret in the instance-independent regime and Lipschitz setting.
Paper develops a TR-SSQP method for noisy optimization with heavy-tailed noise.
problem Optimization problems with stochastic objectives and heavy-tailed noise.
method Trust-Region Stochastic Sequential Quadratic Programming (TR-SSQP) method.
result Achieves high-probability first-order and second-order stationarity bounds for heavy-tailed noise.
A simple log-transform fixes heavy-tailed data for generative models.
problem Standard generative models struggle with heavy-tailed data.
method Apply the soft-log transform to data before training and exponentiate samples after generation.
result Log-FM outperforms specialized baselines on multivariate benchmarks.
The tail of the distribution of a sum of a random number of independent and identically distributed nonnegative random variables depends on the tails of the number of terms and of the terms themselves. This situation is of interest in the collective risk model, where the total claim size in a portfolio is the sum of a …
Study on hidden units in finite Bayesian neural networks and their tail properties.
problem Understanding the behavior of hidden units in finite Bayesian neural networks.
method Introduced a generalized Weibull-tail property to describe hidden units tails.
result Unit priors become heavier-tailed going deeper, providing insights into finite Bayesian neural networks.
New algorithms handle heavy-tailed rewards in reinforcement learning.
problem Learning from heavy-tailed rewards in reinforcement learning.
method Robust mean estimation techniques for constructing algorithms.
result Near-optimal regret bounds achieved in heavy-tailed reward settings.
COMET Flows model multivariate extremes with heavy tails and asymmetric dependence.
problem Normalizing flows struggle with multivariate extremes and asymmetric tail dependence.
method COMET Flows decomposes modeling into marginal and copula parts; uses tail belief and kernel density for marginals, and low-dimensional manifold for tail dependence.
result COMET Flows outperform other models in capturing heavy-tailed marginals and asymmetric tail dependence.
New insights into why neural networks generalize well.
problem Understanding why neural networks generalize well despite heavy-tailed weight distributions.
method Developed a simple model to analyze the emergence of heavy-tailed empirical spectral densities (ESDs) in two-layer neural networks without gradient noise.
result Learning rates play a crucial role in shaping the ESDs of two-layer neural networks, leading to better generalization.
In this note, we study the ultimate ruin probabilities of a real-valued L{é}vy process X with light-tailed negative jumps. It is well-known that, for such L{é}vy processes, the probability of ruin decreases as an exponential function with a rate given by the root of the Laplace exponent, when the initial value goes to …
The standard deviation and Gini mean difference order based on tail behavior.
problem Ordering between standard deviation and Gini mean difference for real-valued risks.
method Analysis of the mean excess function of the pairwise difference ∣X−X′∣. result Dominance regimes of SD and GMD are determined by tail behavior of the distribution.
New bounds link SGD's generalization to heavy tails without topological assumptions.
problem Linking SGD's generalization error to heavy tails without additional assumptions.
method Developed Wasserstein stability bounds for heavy-tailed SDEs and their discretizations, converting to generalization bounds.
result Generalization bounds for a broader class of objective functions, including non-convex functions, without topological assumptions.
Deep neural networks with heavy-tailed weights converge to stable distributions.
problem Understanding the convergence of heavy-tailed weights in infinitely-wide neural networks.
method Analyzing infinitely-wide multi-layer perceptrons with i.i.d. symmetric α-stable weight distributions. result The vector of pre-activation values converges to i.i.d. symmetric α-stable distributions. New algorithm tackles multi-agent bandits with heavy-tailed data.
problem Maximizing system performance in multi-agent settings with heavy-tailed data.
method Algorithm exploits hub-like structures and synchronization among clients.
result Regret bound of O(M1−α1logT) for homogeneous settings, O(MlogT) for heterogeneous. Efficiently estimates sparse mean from heavy-tailed data.
problem Robustly estimating sparse mean from heavy-tailed distributions.
method Stability-based approach adapted for heavy-tailed data.
result Optimal sample complexity with logarithmic dependence on dimension.
Study Hilbert's projective metric for bounded growth functions leading to Sinkhorn's algorithm convergence.
problem Optimal transport in unbounded settings with heavy-tailed distributions.
method Hilbert's projective metric for integrable functions of bounded growth, kernel integral operators as contractions.
result Exponential convergence of Sinkhorn's algorithm for light-tailed marginal distributions.
Study shows how heavy-tailed Hawkes processes can model rough volatility in financial markets.
problem Modeling rough volatility in financial markets with heavy-tailed Hawkes processes.
method Established weak convergence of Hawkes process with power-law kernel, derived scaling limit for financial market model.
result Price-volatility process converges weakly to a rough Heston model after rescaling.
We study the problems related to the estimation of the Gini index in presence of a fat-tailed data generating process, i.e. one in the stable distribution class with finite mean but infinite variance (i.e. with tail index α∈(1,2)). We show that, in such a case, the Gini coefficient cannot be reliably estimated usin…
New algorithm detects changes in heavy-tailed data streams.
problem Detecting changes in heavy-tailed data streams.
method Clipped Stochastic Gradient Descent (SGD) combined with union bound.
result First algorithm with finite-sample false-positive rate guarantees for heavy-tailed data.
Stochastic optimization's success linked to heavy-tailed noise.
problem Understanding stochastic optimization's success mechanisms.
method Modeling stochastic optimization as random recurrence relations, analyzing multiplicative noise and heavy-tailed behavior.
result Multiplicative noise leads to heavy-tailed stationary behavior in optimization parameters.
New algorithms improve stopping time for best arm identification.
problem Efficiently identifying the best alternative in experiments.
method Proposed algorithms with exponential-tailed stopping time.
result Proved that some algorithms never stop, leading to new methods.
Studied how heavy-tailed behavior affects SGD's generalization in quadratic optimization.
problem Link between heavy-tailed behavior and generalization in SGD.
method Used heavy-tailed stochastic differential equation and proved stability bounds.
result Stability of SGD depends on the loss function's tail behavior.
We investigate the asymptotic behavior as time goes to infinity of Hawkes processes whose regression kernel has L1 norm close to one and power law tail of the form x−(1+α), with α∈(0,1). We in particular prove that when α∈(1/2,1), after suitable rescaling, their law converges to that of a kind of integr…
Financial market dynamics is rigorously studied via the exact generalized Langevin equation. Assuming market Brownian self-similarity, the market return rate memory and autocorrelation functions are derived, which exhibit an oscillatory-decaying behavior with a long-time tail, similar to empirical observations. Individ…
Develops a framework to test excessive influence of small data subsets.
problem Identifying when small data subsets significantly impact model conclusions.
method Formalizes the concept of most influential sets, deriving influence formulas and extreme value distributions.
result Allows rigorous hypothesis testing for excessive influence, resolving contested findings.
New MCMC methods map high-dimensional problems to spheres for better mixing.
problem Mixing issues in high-dimensional distributions, especially heavy-tailed ones.
method Stereographic Markov Chain Monte Carlo (MCMC) methods that map high-dimensional problems to spheres.
result Uniformly ergodic samplers for various distributions, including heavy-tailed ones, with faster convergence in higher dimensions.
PPO's gradients are heavy-tailed, affecting learning; a robust estimator improves performance.
problem Heavy-tailedness of PPO gradients causing learning issues.
method Characterized heavy-tailed gradients, identified likelihood ratios and advantages as sources, proposed GMOM as a robust estimator.
result GMOM improves PPO performance without clipping tricks.
Bayesian method improves extreme quantile estimation with zero coverage error.
problem Estimating extreme quantiles with zero coverage error in small samples.
method Bayesian quantile estimation using Jeffreys prior.
result Bayesian method results in zero coverage error, unlike maximum likelihood.
Proves generalization bounds for SGD using Feller processes and Hausdorff dimension.
problem Characterizing generalization properties of SGD in deep learning.
method Proves generalization bounds for SGD under Feller process approximation, linking generalization error to the Hausdorff dimension of trajectories.
result Generalization error controlled by the Hausdorff dimension of trajectories, which is linked to the tail behavior of the driving process.
When maximum likelihood estimation is infeasible, one often turns to score matching, contrastive divergence, or minimum probability flow to obtain tractable parameter estimates. We provide a unifying perspective of these techniques as minimum Stein discrepancy estimators, and use this lens to design new diffusion kerne…