带指数内存的欧氏树序列构建：分布性能与最坏情况保证
Sequential Euclidean connections with exponential memory: distributional performance and adversarial robustness
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中文总结 AI 辅助
该研究针对$\mathbb R^d$单位球点序列，提出指数加权内存的欧氏树构建方法，推导其分布性能与最坏情况代价的理论界，在$d\geq2$时给出最优参数的渐近表达式，证明其性能优于路径结构且接近最佳对抗值。
中文摘要 AI 辅助
设$p_0,p_1,\ldots,p_N$为$\mathbb R^d$单位球中的点，按规定顺序处理。我们研究插入代价$\sum_{i=1}^N\lVert p_i-x_{i-1}\rVert^\alpha$，其中每个$x_{i-1}$由先前观测到的点计算得到。输入顺序路径对输入分布敏感，但在对抗性输入下可能反复付出直径代价；中心星结构具有可控的最坏情况规模，但忽略观测序列。我们通过$x_0=p_0$和$x_i=\gamma x_{i-1}+(1-\gamma)p_i$（$0\leq\gamma\leq1$）将过去压缩为一个点，因此$x_i$是输入的指数加权记忆，用一个$d$维工作状态点维护。对于独立均匀分布的点，平稳插入长度在通常的随机序中随$\gamma$增大而非递增。若$d\geq2$且$\alpha>0$，$N$次插入的每个最优常数参数满足$1-\gamma_N^*=\Theta(N^{-1/2})$。我们确定其渐近常数和由此产生的$\sqrt N$修正，给出关于$d$和$\alpha$的显式界。对于$\alpha=1$，主导期望树长等于中心星的树长，且严格小于端点构造的树长；对于$\alpha=2$，当$N\geq2$时最优参数唯一，满足$1-\gamma_N^*=N^{-1/2}-\frac{1}{2}N^{-1}+O(N^{-3/2})$。对于任意输入序列和固定的$0\leq\gamma<1$，当$0<\alpha\leq3$时，最大渐近平均代价为$(2/(1+\gamma))^\alpha$，在$\gamma>0$时严格小于路径值。在固定非负加权规则中，若贡献点与最近点的输入顺序平均距离相同，则指数加权在至少二维空间中与最佳对抗值的比值小于$1.161^\alpha$，且该比值随平均距离增大趋于1。
英文摘要
Points in the unit ball of $\mathbb R^d$ 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 $0\leqγ\leq1$. 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 $0\leqγ<1$ and $0<α\leq3$, the exact asymptotic adversarial value is $(2/(1+γ))^α$. When $d\geq2$, 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 $0<α\leq3$. Comparison with the running mean highlights its time-homogeneous update, stationary coefficient profile, and fixed effective memory.
发表机构
- Centro de Informática, Universidade Federal de Pernambuco（巴西联邦伯南布哥大学信息学院）
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