# Time-uniform, nonparametric, nonasymptotic confidence sequences

arXiv: https://arxiv.org/abs/1810.08240 · HTML: https://mlsanity.com/paper/1810.08240/

## Key facts

- Cited by 229 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Oct 2018.

## Built on by (most cited first, 30 of 229)

- [Dueling RL: Reinforcement Learning with Trajectory Preferences](https://mlsanity.com/paper/2111.04850/) (2021)
- [DoG is SGD's Best Friend: A Parameter-Free Dynamic Step Size Schedule](https://mlsanity.com/paper/2302.12022/) (2023)
- [High-probability Bounds for Non-Convex Stochastic Optimization with Heavy Tails](https://mlsanity.com/paper/2106.14343/) (2021)
- [Making SGD Parameter-Free](https://mlsanity.com/paper/2205.02160/) (2022)
- [Tight Concentrations and Confidence Sequences from the Regret of Universal Portfolio](https://mlsanity.com/paper/2110.14099/) (2021)
- [Tracking the risk of a deployed model and detecting harmful distribution shifts](https://mlsanity.com/paper/2110.06177/) (2021)
- [Anytime-valid off-policy inference for contextual bandits](https://mlsanity.com/paper/2210.10768/) (2022)
- [On the Theory of Reinforcement Learning with Once-per-Episode Feedback](https://mlsanity.com/paper/2105.14363/) (2021)
- [E-detectors: a nonparametric framework for sequential change detection](https://mlsanity.com/paper/2203.03532/) (2022)
- [Guarantees for Epsilon-Greedy Reinforcement Learning with Function Approximation](https://mlsanity.com/paper/2206.09421/) (2022)
- [SGD with AdaGrad Stepsizes: Full Adaptivity with High Probability to Unknown Parameters, Unbounded Gradients and Affine Variance](https://mlsanity.com/paper/2302.08783/) (2023)
- [Time-uniform central limit theory and asymptotic confidence sequences](https://mlsanity.com/paper/2103.06476/) (2021)
- [Parameter-free Regret in High Probability with Heavy Tails](https://mlsanity.com/paper/2210.14355/) (2022)
- [Proximal Point Imitation Learning](https://mlsanity.com/paper/2209.10968/) (2022)
- [On the Sublinear Regret of GP-UCB](https://mlsanity.com/paper/2307.07539/) (2023)
- [E-Values Expand the Scope of Conformal Prediction](https://mlsanity.com/paper/2503.13050/) (2025)
- [A unified recipe for deriving (time-uniform) PAC-Bayes bounds](https://mlsanity.com/paper/2302.03421/) (2023)
- [Semiparametric Efficient Inference in Adaptive Experiments](https://mlsanity.com/paper/2311.18274/) (2023)
- [Protected Test-Time Adaptation via Online Entropy Matching: A Betting Approach](https://mlsanity.com/paper/2408.07511/) (2024)
- [Neural Active Learning with Performance Guarantees](https://mlsanity.com/paper/2106.03243/) (2021)
- [Fully Adaptive Composition in Differential Privacy](https://mlsanity.com/paper/2203.05481/) (2022)
- [Near-optimal inference in adaptive linear regression](https://mlsanity.com/paper/2107.02266/) (2021)
- [More PAC-Bayes bounds: From bounded losses, to losses with general tail behaviors, to anytime validity](https://mlsanity.com/paper/2306.12214/) (2023)
- [Almost Optimal Variance-Constrained Best Arm Identification](https://mlsanity.com/paper/2201.10142/) (2022)
- [Sequential Kernelized Independence Testing](https://mlsanity.com/paper/2212.07383/) (2022)
- [Prediction-Powered E-Values](https://mlsanity.com/paper/2502.04294/) (2025)
- [Off-policy Confidence Sequences](https://mlsanity.com/paper/2102.09540/) (2021)
- [Weak Signal Asymptotics for Sequentially Randomized Experiments](https://mlsanity.com/paper/2101.09855/) (2021)
- [Is Your Imitation Learning Policy Better than Mine? Policy Comparison with Near-Optimal Stopping](https://mlsanity.com/paper/2503.10966/) (2025)
- [Bregman Deviations of Generic Exponential Families](https://mlsanity.com/paper/2201.07306/) (2022)

## Abstract (arXiv)

A confidence sequence is a sequence of confidence intervals that is uniformly valid over an unbounded time horizon. Our work develops confidence sequences whose widths go to zero, with nonasymptotic coverage guarantees under nonparametric conditions. We draw connections between the Cram'er-Chernoff method for exponential concentration, the law of the iterated logarithm (LIL), and the sequential probability ratio test -- our confidence sequences are time-uniform extensions of the first; provide tight, nonasymptotic characterizations of the second; and generalize the third to nonparametric settings, including sub-Gaussian and Bernstein conditions, self-normalized processes, and matrix martingales. We illustrate the generality of our proof techniques by deriving an empirical-Bernstein bound growing at a LIL rate, as well as a novel upper LIL for the maximum eigenvalue of a sum of random matrices. Finally, we apply our methods to covariance matrix estimation and to estimation of sample average treatment effect under the Neyman-Rubin potential outcomes model.