Time-Uniform Confidence Sequences

• Time-uniform confidence sequences are data-dependent intervals that ensure simultaneous coverage across all time points, making them robust to optional stopping.
• They are constructed via nonnegative supermartingales, employing tools such as Ville's inequality, mixture martingales, and stitching techniques to uniformly control Type I error.
• These methods are crucial for applications in sequential analysis, bandit algorithms, and adaptive procedures in statistics and machine learning.

A time-uniform confidence sequence (CS) is a sequence of data-dependent confidence intervals (or sets, bands, or regions) for an estimand or parameter of interest that is designed to contain the true parameter simultaneously at all time points with a user-specified coverage probability (such as $1-\alpha$). Unlike classical fixed-sample confidence intervals, which guarantee coverage only at a predetermined sample size, a time-uniform confidence sequence guarantees validity at arbitrary or adaptively chosen stopping times—rendering it fundamentally robust to optional stopping and sequential analysis. Time-uniformity is achieved through probabilistic tools rooted in nonnegative (super)martingales and their associated maximal inequalities, such as Ville's inequality, and through constructions that often blend and extend classical concentration methods, boundary-crossing problems, sequential tests, and sequential mixture techniques.

1. Core Definition and Uniform Coverage Principle

A (one-parameter) time-uniform confidence sequence is a sequence of sets $CI_t$ (often intervals for real parameters) produced based on data $X_1, \dots, X_t$ such that, for a parameter $\theta^*$,

$$\mathbb{P}\left(\forall t \geq 1:\; \theta^* \in CI_t \right) \geq 1 - \alpha.$$

This assertion is strong: the coverage is uniform over the potentially infinite sequence of times $t$ and hence at any stopping time (possibly data-dependent, random, and unbounded). In contrast, a family of classical CIs of level $1-\alpha$ at each $t$ would not control family-wise error over repeated peeking (the probability that $\theta^*$ ever falls outside some $CI_t$ can be much greater than $CI_t$0). Thus, CSs are immune to data-dependent stopping and "peeking," a property referred to as "anytime-validity" (Gnettner et al., 14 Feb 2025, Orabona et al., 2021, Howard et al., 2018).

2. General Construction Frameworks

The time-uniform property of confidence sequences is underpinned by the construction of nonnegative supermartingales (often called test martingales or e-processes) with respect to a suitable filtration. Ville's inequality states that for any nonnegative supermartingale $CI_t$1 initialized at $CI_t$2,

$CI_t$3

which ensures the time-uniform control of Type I error. By constructing $CI_t$4 for each candidate $CI_t$5, inverting the acceptance region of the test (i.e., the set of $CI_t$6 where $CI_t$7), and intersecting these sets over $CI_t$8, one obtains a CS with coverage $CI_t$9 (Howard et al., 2018, Aolaritei et al., 15 Dec 2025, Karampatziakis et al., 2021, Arnold et al., 2024).

Universal classes of CSs are constructed via:

• Mixture martingales ("mixture boundaries"), integrating the likelihood ratio martingale over a prior or mixing distribution (Howard et al., 2018);
• Peeling/stitching, union-bounding over geometrically spaced epochs to control the maximal crossing probability (enabling bounds of law-of-iterated-logarithm (LIL) type) (Howard et al., 2018, Orabona et al., 2021);
• Self-normalized processes, adapting to unknown variance using empirical-Bernstein methods (Howard et al., 2018, Chugg et al., 2023, Mineiro, 2022);
• Universal-betting and portfolio-based approaches, connecting online wealth processes with regret-optimal martingales (Orabona et al., 2021, Ryu et al., 2022);
• PAC-Bayesian mixtures for vector-parametric settings (Chugg et al., 2023, Lee et al., 2024).

3. Regimes, Exact and Asymptotic Results, and Key Theorems

General Sub-$X_1, \dots, X_t$0 Martingale Setting

For (real-valued or vector-valued) data adapted to a filtration $X_1, \dots, X_t$1, and a centered sum (martingale) $X_1, \dots, X_t$2 with intrinsic time $X_1, \dots, X_t$3 (predictable variance proxy or similar), $X_1, \dots, X_t$4 is $X_1, \dots, X_t$5-sub-$X_1, \dots, X_t$6 if, for each $X_1, \dots, X_t$7,

$X_1, \dots, X_t$8

with $X_1, \dots, X_t$9 (Howard et al., 2018).

This structure yields:

• Linear ("Chernoff") Boundary: for fixed $\theta^*$0

$\theta^*$1

is a time-uniform high-probability upper bound for $\theta^*$2 with probability $\theta^*$3 (Howard et al., 2018).

• Stitched (LIL) Boundary: using a union bound over epochs in $\theta^*$4, one derives curved (log-log) boundaries achieving minimax iterated-log rates

$\theta^*$5

for polynomial stitching (see Eq. 2.2 in (Howard et al., 2018)), yielding $\theta^*$6 rate, i.e., LIL-optimal shrinking.

• Mixture Boundaries: for any mixing distribution $\theta^*$7 over $\theta^*$8,

$\theta^*$9

is a valid time-uniform boundary (Howard et al., 2018). This recovers explicit closed-form bounds in sub-Gaussian, sub-exponential, sub-Poisson settings.

Empirical-Bernstein Confidence Sequences

For bounded or heavy-tailed observations, self-normalized concentration is employed by replacing a deterministic variance proxy with empirical variance (or analogous quadratic forms) (Howard et al., 2018, Chugg et al., 2023, Mineiro, 2022). These yield boundaries that shrink at $$\mathbb{P}\left(\forall t \geq 1:\; \theta^* \in CI_t \right) \geq 1 - \alpha.$$0 rate and remain nonasymptotic and time-uniform.

Multivariate and General Functionals

Extension to means of random vectors relies on PAC-Bayesian mixtures over shrinking balls or ellipsoids (Chugg et al., 2023), generalizing the LIL and empirical-Bernstein bounds to confidence sphere sequences (CSSs) and to functionals such as covariance matrices, quantiles, and more (Howard et al., 2018, Howard et al., 2019, Mineiro et al., 2023).

Connections to Classical Principles

• Cramér-Chernoff: The uniform Chernoff boundary generalizes exponential concentration from fixed-$$\mathbb{P}\left(\forall t \geq 1:\; \theta^* \in CI_t \right) \geq 1 - \alpha.$$1 to the time-uniform setting (Howard et al., 2018).
• Law of the Iterated Logarithm (LIL): "Stitching" constructs CSs with radius $$\mathbb{P}\left(\forall t \geq 1:\; \theta^* \in CI_t \right) \geq 1 - \alpha.$$2, removing the necessity of asymptotic or moment conditions for LIL-type shrinkage (Howard et al., 2018).
• Sequential Probability Ratio Test (SPRT): The mixture boundary via exponential martingales coincides with SPRT in one-parameter exponential families; time-uniform CSs correspond to always-valid sequential tests with power one (Howard et al., 2018).
• Self-normalization: Empirical-Bernstein boundaries require only boundedness and enable adaptivity to variance in the sequential context (Howard et al., 2018, Mineiro, 2022).

4. Applications and Special Cases

Time-uniform confidence sequences have found application in an array of problems:

• Stochastic Optimization (SGD): Anytime-valid CSs for online suboptimality certificates and stopping rules without reliance on strong convexity or smoothness (Aolaritei et al., 15 Dec 2025).
• Sequential Estimation of Quantiles and CDFs: Time-uniform CSs for quantiles, simultaneous CSs for the entire distribution function, and bandit/A/B-testing of quantile differences (Howard et al., 2019, Mineiro et al., 2023).
• Model Selection and Comparison: Sequential model confidence sets with nonasymptotic Type I error control, replacing fixed-sample MCS with time-uniform, order-adaptive inference (Arnold et al., 2024).
• Bandit and RL Applications: Time-uniform bounds for off-policy evaluation, generalized linear models, and regret minimization in bandit algorithms (Karampatziakis et al., 2021, Lee et al., 2024).

5. Theoretical Properties and Rates of Shrinkage

The nonasymptotic width of CSs scales as

$$\mathbb{P}\left(\forall t \geq 1:\; \theta^* \in CI_t \right) \geq 1 - \alpha.$$3

where typically $$\mathbb{P}\left(\forall t \geq 1:\; \theta^* \in CI_t \right) \geq 1 - \alpha.$$4 and $$\mathbb{P}\left(\forall t \geq 1:\; \theta^* \in CI_t \right) \geq 1 - \alpha.$$5 grows as $$\mathbb{P}\left(\forall t \geq 1:\; \theta^* \in CI_t \right) \geq 1 - \alpha.$$6 for stitched (LIL-type) sequences, yielding

$$\mathbb{P}\left(\forall t \geq 1:\; \theta^* \in CI_t \right) \geq 1 - \alpha.$$7

Some constructions allow adaptive rate control, trading early conservativeness for late sharpness (Gnettner et al., 14 Feb 2025, Xie et al., 2024). In heavy-tailed regimes, rate-optimal adaptation is achievable through robust test martingale mixtures (Mineiro, 2022).

Special cases include:

• Sub-Gaussian: Stitched or mixture CSs match the minimax LIL rate.
• Self-normalized/Bernstein: Empirical-Bernstein CSs automatically adapt to the variance structure, with sharper widths than Hoeffding-type bounds when variance is small (Howard et al., 2018, Chugg et al., 2023).
• Matrix-valued processes: Noncommutative martingale techniques yield operator-norm CSs for covariance estimation and principal component analysis (Howard et al., 2018).
• Functionals in potential outcomes models: Sequential estimation of average treatment effects via CSs for influence-function means (Howard et al., 2018).

6. Implementation Considerations and Limitations

Time-uniform CSs can almost always be implemented online, with per-round complexity determined by the required sufficient statistics (often only means and variances). Mixture-based and self-normalized boundaries are computationally tractable and have been implemented in public software.

Limitations and trade-offs:

• Conservativeness: Nonasymptotic CSs (finite-sample valid) may be wider than fixed-time CIs, especially at small $$\mathbb{P}\left(\forall t \geq 1:\; \theta^* \in CI_t \right) \geq 1 - \alpha.$$8, but shrink optimally as $$\mathbb{P}\left(\forall t \geq 1:\; \theta^* \in CI_t \right) \geq 1 - \alpha.$$9.
• Assumptions: Generalized sub-$t$0 processes require martingale difference structure and perhaps boundedness or moment assumptions; empirical-Bernstein requires bounded observations, though heavy-tailed adaptations exist (Mineiro, 2022, Howard et al., 2018).
• Tuning and optimality: Peeling/stitching boundaries and mixtures require parameter tuning to minimize constants in nonasymptotic settings. Explicit formulae exist for most canonical families; for more general or adaptive settings, discrete mixture boundaries or numerical inversion may be used (Howard et al., 2018, Gnettner et al., 14 Feb 2025).

7. Summary Table of Fundamental Construction Principles

Principle	Key Construction	Reference(s)
Martingale Boundary	Ville's inequality, stitching, mixture	(Howard et al., 2018, Aolaritei et al., 15 Dec 2025)
Self-Normalization	Empirical-Bernstein, variance-adaptive	(Howard et al., 2018, Chugg et al., 2023, Mineiro, 2022)
Mixture Martingales	Conjugate mixtures, discrete mixtures	(Howard et al., 2018, Gnettner et al., 14 Feb 2025)
PAC-Bayes/Vector Extension	Mixtures over shrinking balls, convex hull	(Chugg et al., 2023, Lee et al., 2024)
Universal-Betting/Portfolio	Wealth processes, regret minimization	(Orabona et al., 2021, Ryu et al., 2022)
Sequential Testing	Test inversion, SPRT, model confidence sets	(Arnold et al., 2024, Shin et al., 2020)

All valid time-uniform confidence sequences employ nonnegative supermartingale constructions or reductions to maximally time-uniform tests. The integrative framework treats CIs, CSs, and sequential $t$1-values/tests as dual objects, unifying modern sequential inference.

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References:

• (Howard et al., 2018) Howard, R., Ramdas, A., et al. (2018). Time-uniform, nonparametric, nonasymptotic confidence sequences.
• (Aolaritei et al., 15 Dec 2025) "Stopping Rules for Stochastic Gradient Descent via Anytime-Valid Confidence Sequences" (2025).
• (Orabona et al., 2021) "Tight Concentrations and Confidence Sequences from the Regret of Universal Portfolio" (2021).
• (Chugg et al., 2023) "Time-Uniform Confidence Spheres for Means of Random Vectors" (2023).
• (Ryu et al., 2022) "On Confidence Sequences for Bounded Random Processes via Universal Gambling Strategies" (2022).
• (Mineiro, 2022) "A lower confidence sequence for the changing mean of non-negative right heavy-tailed observations with bounded mean" (2022).
• (Howard et al., 2019) "Sequential estimation of quantiles with applications to A/B-testing and best-arm identification" (2019).
• (Lee et al., 2024) "A Unified Confidence Sequence for Generalized Linear Models, with Applications to Bandits" (2024).
• (Arnold et al., 2024) "Sequential model confidence sets" (2024).
• (Karampatziakis et al., 2021) "Off-policy Confidence Sequences" (2021).
• (Shin et al., 2020) "Nonparametric iterated-logarithm extensions of the sequential generalized likelihood ratio test" (2020).
• (Mineiro et al., 2023) "Time-uniform confidence bands for the CDF under nonstationarity" (2023).
• (Gnettner et al., 14 Feb 2025) "A new and flexible class of sharp asymptotic time-uniform confidence sequences" (2025).

See (Howard et al., 2018) for foundational development and detailed proofs covering most settings described above.