arXiv subjects

Bart P. G. van Parys

Publications and source records attributed to Bart P. G. van Parys.

6 recordsLinked to original sources

arXiv2026

A stochastic subgradient method with optimal failure exponent

Fix a target accuracy $\varepsilon$, a gradient-noise level $s$, and a horizon $N$. We wish to design algorithms which minimize the probability of observing a suboptimality gap which exceeds the target accuracy, i.e., $\mathcal{E}_N = -\log \sup_{f,P} \mathbb{P}_P(f(x_A) - f_\star \ge \varepsilon)$, with the noise law $P$ known only to be sub-Gaussian. We single out a uniformly averaged schedule which is harmonic (of the form $h_k = R^2/(\varepsilon (N+m-k))$) and prove, via an optimized exponential supermartingale argument, that it attains the optimal exponent $\mathcal{E}_N^\star = \varepsilon^2 N (1+o(1))/(2R^2 s^2)$. Optimality is certified by a matching impossibility result: under Gaussian noise, a gradient-masking change of measure caps the exponent of every algorithm at the same leading order. In a small-noise limit, our setting degenerates into the adversarial-error model of Gösgens and van Parys (2025) for subgradient methods.

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arXiv2026

Large and Moderate Deviations for Conservative Tail-Index Estimation

To design systems that are protected against events much rarer than the observational record, extreme-value methods are needed to extrapolate distribution tails. Tail-index estimators such as the Hill estimator are central to this extrapolation, but overestimating the tail exponent can lead to substantial underestimation of rare-event probabilities. Motivated by this, we derive large- and moderate-deviation asymptotics for the Hill estimator and use them to construct estimators whose probability of exceeding the true tail index decays at a controlled exponential rate (the decay rate). In the large-deviations regime, we show that a simple rescaled version of the Hill estimator achieves an optimal balance between bias and decay rate among scale-invariant estimators based on the same top $k$ order statistics. Under a second-order condition, we quantify the effect of the Hill bias, analyze a bias-corrected estimator, and identify sufficient conditions for moderate deviations in the boundary case where the second-order parameter $ρ$ equals zero.

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arXiv2026

Decision-Centric Large Deviations for Data-Driven Capital Buffers in Ruin Models

We consider an insurance risk model with a random walk structure in which the underlying probability law is unknown. A decision maker observes a statistic $Q_n$ computed from $n$ historical observations and then needs to choose a capital buffer of the form $C_n=n f(c,Q_n)$, where $c=\log(1/δ)/n$ balances the amount of data and the tolerated ruin probability $δ$. The classical safe capital buffer is inversely proportional to the adjustment coefficient $γ$, the exponential rate at which the ruin probability decays; when the law is unknown, this coefficient must be inferred from the data. The profile $f$ couples the statistical cost of observing an atypical historical statistic with the future ruin exponent induced by the resulting decision. We study the joint ex ante probability (over both the historical sample and an independent future risk process) that the future maximum exceeds the data-dependent buffer. We show that the naive plug-in rule fails to achieve the prescribed logarithmic decay exponent $c$, illustrating the adverse impact of model uncertainty when making decisions under rare-event constraints. We then identify a profile $f^*=f^*(c,Q_n)$ with the following appealing properties: its lower semicontinuous majorants are safe, while regular continuous rules that fall strictly below $f^*$ are, under mild conditions, unsafe. We illustrate the potential applicability of our framework by developing three parametric examples. In nonparametric settings, we show that a single exponential envelope leads to degenerate capital buffers, whereas a two-level exponential envelope yields a nondegenerate capital buffer which we prove to be safe.

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arXiv2026

Optimal Estimators for Heavy-Tailed Mean Estimation via Convex Analysis

We study optimal estimation of the location parameter of a distribution known only to lie in a symmetric moment class $\mathcal C_0$: the mean-zero distributions with bounded moment $\intϕ\, d\mathbb P\le B$ for a fixed even $ϕ$. Our main result concerns the fixed-margin regime, where the error margin $Δ$ is fixed as $n\to\infty$: we give an exact large-deviation characterization of the smallest worst-case probability $β_n(Δ)$ of an error exceeding $Δ$ that any measurable estimator can guarantee with $n$ observations. Its exponential rate is exactly a two-point Hellinger exponent over the class shifted to means $\pmΔ$, $r(Δ)=-\log\sup_{\mathbb P_{\pmΔ}\in\mathcal C_{\pmΔ}}\int\sqrt{d\mathbb P_{-Δ}\, d\mathbb P_Δ}$, achieved non-asymptotically, $β_n(Δ)\le e^{-nr(Δ)}$, by a monotone $M$-estimator synthesized from a two-parameter convex program. Lagrangian duality collapses the infinite-dimensional search over estimating functions to two multipliers, which determine a pair of envelopes characterizing the optimal estimating functions; the sandwich shape posited ad hoc in prior constructions emerges naturally. For bounded variance ($ϕ(x)=x^2$, $B=σ^2$) the exponent is $r(Δ)=\tfrac12\log(1+Δ^2/σ^2)$. In the fixed-confidence regime, holding $β$ fixed and letting the optimal margin $Δ_n(β)$ shrink with $n$, the same synthesis stays optimal to leading order for several concrete classes. As $β\downarrow0$ it attains the sharp constant $\sqrt2$ of Catoni for bounded variance and the constant $L(α)$ of Lee and Bhatt et al. for bounded $α$-moments, $α\in(1,2)$, thereby shown tight; for slowly varying $ϕ$ it is leading-order minimax at every fixed $β$. The least-favorable distributions are simple, supported on at most three atoms.

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arXiv2026

Robust Mean Estimation for Optimization: The Impact of Heavy Tails

We consider the problem of constructing a least conservative estimator of the expected value $μ$ of a non-negative heavy-tailed random variable. We require that the probability of overestimating the expected value $μ$ is kept appropriately small; a natural requirement if its subsequent use in a decision process is anticipated. In this setting, we show it is optimal to estimate $μ$ by solving a distributionally robust optimization (DRO) problem using the Kullback-Leibler (KL) divergence. We further show that the statistical properties of KL-DRO compare favorably with other estimators based on truncation, variance regularization, or Wasserstein DRO.

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arXiv2026

Globalized Adversarial Regret Optimization: Robust Decisions with Uncalibrated Predictions

Optimization problems routinely depend on uncertain parameters that must be predicted before a decision is made. Classical robust and regret formulations are designed to handle erroneous predictions and can provide statistical error bounds in simple settings. However, when predictions lack rigorous error bounds (as is typical of modern machine learning methods) classical robust models often yield vacuous guarantees, while regret formulations can paradoxically produce decisions that are more optimistic than even a nominal solution. We introduce Globalized Adversarial Regret Optimization (GARO), a decision framework that controls adversarial regret, defined as the gap between the worst-case cost and the oracle robust cost, uniformly across all possible uncertainty set sizes. By design, GARO delivers absolute or relative performance guarantees against an oracle with full knowledge of the prediction error, without requiring any probabilistic calibration of the uncertainty set. We show that GARO equipped with a relative rate function generalizes the classical adaptation method of Lepski to downstream decision problems. We derive exact tractable reformulations for problems with affine worst-case cost functions and polyhedral norm uncertainty sets, and provide a discretization and a constraint-generation algorithm with convergence guarantees for general settings. Finally, experiments demonstrate that GARO yields solutions with a more favorable trade-off between worst-case and mean out-of-sample performance, as well as stronger global performance guarantees.

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