Sequential Euclidean connections with exponential memory: distributional performance and adversarial robustness
Abstract
Points in the unit ball of ℝd are processed sequentially. Each new point pi is connected to a state xi−1 that summarizes earlier observations, after which xi=γxi−1+(1−γ)pi, with 0≤γ≤1. The cost is the sum of the α-powers of the connection lengths. This constant-gain rule interpolates between the input-order path and the star centered at the initial point. For independent uniform points, we establish the stationary insertion-length distribution and prove that it decreases in stochastic order as γ increases. If d+α>2, or if (d,α)=(1,1), the optimal constant parameter satisfies 1−γN*=Θ(N−1/2), with an explicit asymptotic constant and closed bounds. For α=1, its leading expected tree length equals that of the center star and is eventually smaller than the expected lengths of both endpoint constructions. For α=2, the optimizer is unique and characterized exactly. For the same N, choosing 1−γN as a fixed positive multiple of N−1/2 gives a sharp two-term expansion of the expected uniform-input cost and a maximal adversarial mean cost of 1+O(N−1/2). For every fixed 0≤γ<1 and 0<α≤3, the exact asymptotic adversarial value is (2/(1+γ))α. When d≥2, exponential weighting is within a factor smaller than 1.161α of the best fixed nonnegative weighted rule with the same average look-back, for 0<α≤3. Comparison with the running mean highlights its time-homogeneous update, stationary coefficient profile, and fixed effective memory.
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BibTeX
@misc{castro2026sequential,
title = {{Sequential Euclidean connections with exponential memory: distributional performance and adversarial robustness}},
author = {Pedro M. M. de Castro},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.25298},
eprint = {2608.25298},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25298},
}
RIS
TY - UNPB
TI - Sequential Euclidean connections with exponential memory: distributional performance and adversarial robustness
AU - Pedro M. M. de Castro
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.25298
UR - https://arxiv.org/abs/2608.25298
AB - Points in the unit ball of ℝ^d are processed sequentially. Each new point p_i is connected to a state x_i−1 that summarizes earlier observations, after which x_i=γx_i−1+(1−γ)p_i, with 0≤γ≤1. The cost is the sum of the α-powers of the connection lengths. This constant-gain rule interpolates between the input-order path and the star centered at the initial point. For independent uniform points, we establish the stationary insertion-length distribution and prove that it decreases in stochastic order as γ increases. If d+α>2, or if (d,α)=(1,1), the optimal constant parameter satisfies 1−γ_N^*=Θ(N^−1/2), with an explicit asymptotic constant and closed bounds. For α=1, its leading expected tree length equals that of the center star and is eventually smaller than the expected lengths of both endpoint constructions. For α=2, the optimizer is unique and characterized exactly. For the same N, choosing 1−γ_N as a fixed positive multiple of N^−1/2 gives a sharp two-term expansion of the expected uniform-input cost and a maximal adversarial mean cost of 1+O(N^−1/2). For every fixed 0≤γ<1 and 0<α≤3, the exact asymptotic adversarial value is (2/(1+γ))^α. When d≥2, exponential weighting is within a factor smaller than 1.161^α of the best fixed nonnegative weighted rule with the same average look-back, for 0<α≤3. Comparison with the running mean highlights its time-homogeneous update, stationary coefficient profile, and fixed effective memory.
ER -