A Sharp Joint Bias-Energy Envelope for Radial Clipping

Abstract

Radial clipping does two things at once: it cuts off the part of a vector outside a ball and leaves a bounded vector inside the ball. The part cut off defines the residual. For a random vector, we study the magnitude of the bias in the mean and the energy, defined as the centered second moment of the clipped vector. Controlling these effects is important in stochastic optimization with heavy tails, where the choice of the clipping threshold affects convergence bounds for algorithms. In the deterministic problem, for a fixed threshold $\tau >0$, $1\le p\le 2$, and nonnegative coefficients $\alpha ,\beta$, we find the smallest constant that simultaneously bounds the residual and energy in terms of $\Vert x{\Vert }^p$. This same constant remains sharp in the stochastic counterpart for random vectors in a Hilbert space. We also find the exact supremum of the corresponding linear combination of normalized bias and energy for arbitrary $\alpha ,\beta \in \mathbb{R}$. We then give a geometric picture in the energy-bias plane. Varying the weights, we describe the smallest closed convex region containing all normalized pairs and the attainable set itself, identifying unattainable boundary values. Every attainable pair at a fixed threshold can be realized by a real-valued random variable taking at most two values. We also consider all thresholds simultaneously. For any integrable random vector in a Hilbert space, we construct a single ${\mathbb{R}}^3$-valued random vector preserving its norm distribution, bias magnitude, and energy simultaneously at every threshold $\tau >0$. Finally, we apply the joint bound to risk-adjusted reinsurance costs for a prescribed retention threshold and to the logarithmic shortfall of a clipped portfolio strategy relative to any constant-fraction strategy. AI was used in preparing this paper (see Section 8).

arXiv comment61 pages, 14 figures (7 in each language). Complete English version followed by Russian version. Revised and expanded; includes simultaneous three-dimensional realization for all thresholds and two applications