Closed-form empirical Bernstein confidence sequences for scalars and matrices

TL;DR

The paper develops a new closed-form, variance-adaptive confidence sequence for tracking the time-varying average conditional mean of bounded observations, addressing nonstationarity and providing practical, scalable bounds. Central to the approach is a novel empirical Bernstein supermartingale built via a modified Fan inequality and the method of mixtures, which yields a time-uniform bound that is then approximated in closed form by $C_t^{\textnormal{apx}}$. The authors show that, under iid data, the half-width scales as $O(\sqrt{\log t / t})$ with a variance-dependent constant and discuss asymptotic sharpness relative to existing bounds, including a stitching variant that achieves an iterated-log rate. They also extend the framework to random matrices with bounded eigenvalues, deriving a matrix CS with analogous properties. The results offer a practical, tight, time-uniform alternative to prior bounds for sequential estimation and monitoring in bounded settings.

Abstract

We derive a new closed-form variance-adaptive confidence sequence (CS) for estimating the average conditional mean of a sequence of bounded random variables. Empirically, it yields the tightest closed-form CS we have found for tracking time-varying means, across sample sizes up to $\approx 10^6$. When the observations happen to have the same conditional mean, our CS is asymptotically tighter than the recent closed-form CS of Waudby-Smith and Ramdas [38]. It also has other desirable properties: it is centered at the unweighted sample mean and has limiting width (multiplied by $\sqrt{t/\log t}$) independent of the significance level. We extend our results to provide a CS with the same properties for random matrices with bounded eigenvalues.

Figures (6)

• Figure 1: Comparison of the width of our closed-form CS $C_t^{\textnormal{apx}}$ (Theorem \ref{['thm:closed-form']}) with that of the closed-form CS of waudby2024estimating (WSR) and that of howard2021time (HRMS) under the assumption of a constant conditional mean. We fix $\kappa = 0.25$ and $\alpha=0.05$ for all experiments and plot the oracle Bernstein CS of howard2021time for comparison. The middle column plots the width of the CSs with the y-axis on a log-scale. The final column plots the x-axis on log-scale.
• Figure 2: The performance $C_t^{\textnormal{apx}}$ (Theorem \ref{['thm:closed-form']}), $C_t^\textnormal{\tiny HRMS}$ and $C_t^\textnormal{\tiny WSR}$ under a time-varying mean (dotted black line). For the left plot we draw the observations iid from $\textnormal{Ber}(0.8)$ for $t\leq N/10$, and then for $\textnormal{Ber}(0.2)$ thereafter. The second plot generates $p_t$ to target a sinusoidal-like conditional mean, and then draws $X_t \sim \textnormal{Ber}(p_t)$. Here $\alpha=0.05$ and we use Theorem \ref{['thm:closed-form']} with $\kappa=0.25$.
• Figure 3: (a) Comparison of the exact CS $C_t^{\textnormal{mix}}$ defined in Proposition \ref{['prop:mixture-cs-tn']} and the relaxation $C_t^{\textnormal{apx}}$ defined in Theorem \ref{['thm:closed-form']}. The differences between the two bounds are negligible. We also plot $t_0$, the time at which $C_t^{\textnormal{apx}}$ becomes valid (vertical dotted line). Here the observations were drawn independently as $X_t \sim \text{Ber(0.5)}$. (b) The growth of $U_t$ under various distributions. We plot the line $y=x$ for reference, demonstrating that $U_t$ scales as $ct$ for some $c<1$ in each case.
• Figure 4: Comparison of Theorem \ref{['thm:stitching']} and $C_t^\textnormal{\tiny HRMS}$, the two CSs which have iterated logarithm rates. Theorem \ref{['thm:stitching']} is implemented with $h(j) = \zeta(s)(j+1)^s$ and we use $\eta=2$, $s=1.4$ for both bounds. While $C_t^\textnormal{stch}$ is tighter than $C_t^\textnormal{\tiny HRMS}$ at smaller sample sizes, $C_t^\textnormal{\tiny HRMS}$ is tighter in the limit as discussed in the text.
• Figure 5: Comparison of $C_t^{\textnormal{apx}}$ against the oracle Bernstein Bern on the two point distribution discussed in Appendix \ref{['app:psiE-vs-sigma']} which maximizes the ratio $\mathbb{E}\psi_E(|-\mu|)/\sigma^2$. We use $\epsilon=0.01$ and $\alpha=0.05$. For $C_t^{\textnormal{apx}}$ we use $\kappa=0.25$ as usual.

Theorems & Definitions (26)

• Lemma 2.1: Modified Fan's Inequality
• proof
• Lemma 2.2: Modified Howard's inequality
• proof
• Proposition 3.1
• Remark 3.2
• Theorem 3.3
• Lemma 3.4
• Theorem 4.1
• Lemma 5.1