Uniform Short-Interval BDH Bounds

• Uniform short-interval BDH bounds are explicit high-probability concentration inequalities tailored for low-probability, rare-event regimes.
• They exploit conditioning, refined symmetrization, and effective complexity at scale n·p₀ to achieve a √(p₀/n) deviation rate.
• These bounds provide sharper finite-sample guarantees over classical VC methods, benefiting applications like anomaly detection and extreme value theory.

Uniform short-interval BDH bounds constitute a class of explicit, high-probability uniform concentration inequalities that sharpen classical Vapnik–Chervonenkis (VC)–type bounds in the rare-event (small-probability) regime. These results characterize the maximal deviation between the empirical measure and the underlying probability law over classes of Borel sets constrained to lie within a low-probability region. Uniform short-interval BDH–type bounds exploit vanishing set probabilities to yield a √(p₀/n) concentration rate, with effective complexity measured at scale n p₀, offering substantial improvement over traditional VC rates when p₀ ≪ 1 (Lhaut et al., 2021).

1. Foundational Definitions and Setting

Let $X_1, ..., X_n$ denote i.i.d. samples from a law $P$ on $\mathbb{R}^d$. For a class $\mathcal{C}$ of Borel subsets of $\mathbb{R}^d$, the empirical measure is defined as

$$P_n(A) = \frac{1}{n} \sum_{i=1}^n \mathbf{1}_{X_i \in A}.$$

The VC dimension of $\mathcal{C}$, denoted $d = VC\dim(\mathcal{C})$, is assumed finite. The shattering coefficient, $S_{\mathcal{C}}(m)$, gives the maximal number of distinct intersections with any $m$-point subset. Uniform short-interval BDH bounds require all sets $A \in \mathcal{C}$ to be contained within a “low-probability” region $\Delta$, i.e., there exists $P(\Delta) = p_0 \ll 1$ with $A \subseteq \Delta$ for all $A \in \mathcal{C}$. Equivalently, $\sup_{A \in \mathcal{C}}P(A) \le p_0$ (Lhaut et al., 2021).

2. Principal Uniform Short-Interval Bounds

Three principal uniform concentration inequalities with explicit constants describe the BDH–type control over

$\sup_{A \in \mathcal{C}}|P_n(A)-P(A)|.$

Symmetrize after conditioning (Theorem 3.1): Given $n p_0 \ge 4 \log(4/\delta)$, with probability at least $1-\delta$,

$$\sup_{A \in \mathcal{C}}|P_n(A)-P(A)| \le \frac{2}{3n}\log\frac{4}{\delta} + \sqrt{ \frac{p_0}{n} \sqrt{ 2 \log\frac{4}{\delta} + 2\sqrt{\log\frac{8}{\delta}+\log S_{\mathcal{C}}(4 n p_0) } + 1 }}.$$

Symmetrize before conditioning (Theorem 3.2): If $n p_0 \ge 2 \log(8/\delta)$, with probability at least $1-\delta$,

$$\sup_{A \in \mathcal{C}}|P_n(A)-P(A)| \le \sqrt{ \frac{2 p_0}{n} \left[ 2 \sqrt{\log\frac{8}{\delta}+\log S_{\mathcal{C}}(8 n p_0)} + 1 \right] }.$$

Expectation + McDiarmid + Sauer’s lemma (Corollary 4.4): If $d < \infty$, for all $\delta \in (0,1)$, with probability at least $1-\delta$,

$$\sup_{A \in \mathcal{C}} |P_n(A) - P(A)| \le \frac{2}{3n} \log \frac{1}{\delta} + \sqrt{ \frac{2 p_0}{n} \sqrt{ 2\log\frac{1}{\delta} + \sqrt{ \log 2 + d\log(2 n p_0 + 1) } + \frac{\sqrt{2}}{2} } }.$$

For comparison, the classical relative VC inequality (e.g., Anthony & Bartlett, Lugosi–Mendelson) bounds

$$\sup_{A \in \mathcal{C}} |P_n(A)-P(A)| \le 2 \sqrt{ \frac{2 p_0}{n} \left[ \log\frac{12}{\delta}+\log S_{\mathcal{C}}(2n)\right] }$$

when $n p_0 \ge \tfrac{8}{3} \log(3/\delta)$ (Lhaut et al., 2021).

3. Asymptotic Behavior and Regime Comparison

When $p_0 \to 0$ with $n$ such that $n p_0$ grows (e.g., $p_0 = O(1/n^\alpha)$), all bounds scale as $O\left(\sqrt{p_0 d \log (n p_0)/n}\right)$. This reflects a $\sqrt{p_0}$ gain over the classical $O\left(\sqrt{d \log n/n}\right)$ rate, more pronounced as the event probability shrinks. In the important case of tail probabilities—$\mathcal{C} = \{(-\infty, t] : t \le F^{-1}(p_0)\}$—with $S_{\mathcal{C}}(m) = m+1$, the dimension dependence drops out, reducing the leading term to $\sqrt{\log(n p_0)}$ (Lhaut et al., 2021).

In numerical illustrations ($p_0=10^{-3}$, $\delta=10^{-2}$), the new “expectation+McDiarmid” bound is consistently an order of magnitude below the classical relative VC bound across $n \in [10^3, 10^8]$. Theorems 3.1 and 3.2 outperform the classical bound as soon as $n p_0 \gtrsim 10^3$, plateauing close to the expectation-based bound at larger $n$ (Lhaut et al., 2021).

4. Methodological Innovations

Uniform short-interval BDH bounds rely on several technical innovations:

• Conditioning trick (Lemma 3.1): Condition on the number $K$ of points landing in $\Delta$ (i.e., $K\sim\text{Bin}(n,p_0)$), then represent the empirical process over $\mathcal{C}$ as a rescaled empirical process based on $K$ i.i.d. samples from $P(\cdot|\Delta)$.
• Symmetrization refinement: Applying a refined symmetrization lemma (Appendix A.2) yields improved constants.
• Complexity at effective scale: Instead of controlling complexity at scale $n$, the bounds evaluate the shattering coefficient at effective mass $n p_0$, capturing the reduced “effective complexity” in the rare-event domain.
• McDiarmid and Sauer’s lemma: For expectation-based arguments, the combination leads to the clean $\sqrt{d \log(n p_0)/n}$ scaling.
• Comparison to classical single-set Bernstein/Bennett/Hoeffding: While one-set bounds depend on fixed $p_0$, uniform short-interval bounds reflect the supérmum over $\mathcal{C}$ using $S_{\mathcal{C}}(n p_0)$, which is advantageous as $p_0 \to 0$ (Lhaut et al., 2021).

5. Special Cases and Applications

The refinements are especially impactful in settings where only rare event/short-interval deviations are of interest. For tail probabilities (distribution function estimation in the far tail), $\mathcal{C}$ as a family of left intervals yields $S_{\mathcal{C}}(m) = m+1$, so complexity scaling is directly tied to the size of the short interval and decouples from ambient dimension.

In practice, these bounds facilitate sharper finite-sample guarantees for VC classes in the low-probability regime, benefiting applications in anomaly detection, extreme value theory, and statistical learning where small-mass events are central (Lhaut et al., 2021).

6. Impact and Context within Empirical Process Theory

The methodology underlying uniform short-interval BDH bounds marks an explicit shift from classical VC theory, which evaluates complexity at the full sample size, to a paradigm that dynamically adapts complexity to the measure of the rare event region. This results in concrete, numerically useful bounds with explicit, interpretable constants. The deployment of the shattering coefficient at $n p_0$ scale, alongside refined concentration analysis, constitutes a significant theoretical advance for understanding uniform laws of large numbers under vanishing mass constraints (Lhaut et al., 2021).

7. Relationship to Related Gaussian Approximation Results

While uniform short-interval BDH bounds pertain to empirical process deviations for indicator functions of rare events, analogous “short-interval” bounds for maxima of Gaussian processes—specifically fractional Brownian motion—have been developed. For example, Borovkov–Mishura–Novikov–Zhitlukhin derived upper and lower bounds for the approximation error in discrete maxima, capturing rates of order $n^{-H}\sqrt{\ln n}$ for Hurst parameter $H \in (0,1/2)$ and elucidating asymptotics in the fine partition regime (Borovkov et al., 2016). A plausible implication is that both BDH bounds and Gaussian short-interval bounds exploit reduced effective complexity or variance in rare/short-interval regimes, reflecting corresponding phenomena in empirical and Gaussian process theory.

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Key references: "Uniform concentration bounds for frequencies of rare events" (Lhaut et al., 2021); "New and refined bounds for expected maxima of fractional Brownian motion" (Borovkov et al., 2016).