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.
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…
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.
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 …
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.
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.
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.
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.
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.
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. 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.
In this paper a quantitative analysis of the ruin probability in finite time of discrete risk process with proportional reinsurance and investment of finance surplus is focused on. It is assumed that the total loss on a unit interval has a light-tailed distribution -- exponential distribution and a heavy-tailed distrib…
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.
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.
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 …
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 …
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. 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 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.
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. SMTM improves MCMC sampling in high dimensions with multiple proposals and stereographic integration.
problem Improving MCMC performance in high-dimensional sampling.
method Integrating multiple-try Metropolis with stereographic MCMC framework.
result SMTM outperforms classical MTM and other methods in high-dimensional sampling.
We study the problem of high-dimensional sparse mean estimation in the presence of an ε-fraction of adversarial outliers. Prior work obtained sample and computationally efficient algorithms for this task for identity-covariance subgaussian distributions. In this work, we develop the first efficient algorithms for rob…
Generative Adversarial Networks improve robust statistics for various distributions.
problem Estimating unknown parameters in adversarially corrupted samples.
method Designing GANs with specific loss functions for robust estimation.
result Extends robust estimation to broader families of distributions.
Paper shows MoM is optimal under adversarial contamination for certain distributions.
problem Optimality of MoM under adversarial contamination.
method Upper and lower bounds for MoM's error under adversarial contamination.
result MoM is (minimax) optimal for distributions with finite variance and infinite variance with finite absolute moments.
Along with the recent advances in scalable Markov Chain Monte Carlo methods, sampling techniques that are based on Langevin diffusions have started receiving increasing attention. These so called Langevin Monte Carlo (LMC) methods are based on diffusions driven by a Brownian motion, which gives rise to Gaussian proposa…
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.
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…
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.
Basel II and Solvency 2 both use the Value-at-Risk (VaR) as the risk measure to compute the Capital Requirements. In practice, to calibrate the VaR, a normal approximation is often chosen for the unknown distribution of the yearly log returns of financial assets. This is usually justified by the use of the Central Limi…
New algorithms for best arm identification in bandits robust to misspecified parameters.
problem Inconsistent learning performance of traditional MAB algorithms when parameters are misspecified.
method Proposes two classes of asymptotically near-optimal algorithms for statistically robust MAB under fixed-budget pure exploration.
result Establishes fundamental performance limits and proposes algorithms that are asymptotically near-optimal.
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 solves nonparametric contextual bandit with unbounded contexts.
problem Sequential decision making with unbounded context distributions.
method Two nearest neighbor methods combined with UCB exploration.
result Achieves minimax optimal regret under weak margin condition and light-tailed 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 transforms Gaussian data to simulate various target distributions.
problem Generating observations from different target distributions given a single Gaussian observation.
method Designs computationally efficient procedures to approximate target distributions.
result Establishes reduction-based computational lower bounds for high-dimensional statistical models.
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.
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.
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.
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.
Consider the problem of finding a population or a probability distribution amongst many with the largest mean when these means are unknown but population samples can be simulated or otherwise generated. Typically, by selecting largest sample mean population, it can be shown that false selection probability decays at an…
Efficient algorithm learns mixture models of heavy-tailed distributions.
problem Learning mixture models of heavy-tailed distributions.
method Efficient high-dimensional sparse Fourier transforms.
result Algorithm succeeds for heavy-tailed distributions, including Laplace but excluding Gaussians.
New guarantees for VI in symmetric cases, extending previous results.
problem Symmetry in variational inference for complex distributions.
method Analysis of f-divergences and their stationary points under symmetry. result Symmetry-matching principles ensure recovery of mean and correlation matrix.
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…
Standard results in stochastic convex optimization bound the number of samples that an algorithm needs to generate a point with small function value in expectation. More nuanced high probability guarantees are rare, and typically either rely on "light-tail" noise assumptions or exhibit worse sample complexity. In this …
New method for estimating sparse means in noisy data.
problem Estimating the mean of a sparse distribution in the presence of outliers.
method Difference-of-Pairs Filtering technique for list-decodable sparse mean estimation.
result First sample and computationally efficient algorithm for list-decodable sparse mean estimation.
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.
Upper bounds for CV errors apply to lasso and other models.
problem Bounding CV errors for lasso and similar models.
method Rademacher complexity and Orlicz-Ψν norm. result Upper bounds are tight and stable for lasso.
Understanding efficiency in high dimensional linear models is a longstanding problem of interest. Classical work with smaller dimensional problems dating back to Huber and Bickel has illustrated the benefits of efficient loss functions. When the number of parameters p is of the same order as the sample size n, $p \…