Time-Uniform Confidence Spheres for Means of Random Vectors
Authors: Ben Chugg, Hongjian Wang, Aaditya Ramdas
TMLR 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Figure 1: Left: Comparison of Theorem 2.2 and its stitched version, Theorem 2.3, against the results of Hsu et al. (2012). ... Right: A comparison of estimators with iterated logarithm rates. ... Simulation details can be found in Appendix D. Figure 2: Left: Comparison between our sequential Catoni-Giulini estimator (Theorem 3.7), geometric median-of-means (GMo M) (Minsker, 2015), and tournament median-of-means (TMo M) (Lugosi and Mendelson, 2019a). ... Simulation details can be found in Appendix D. |
| Researcher Affiliation | Academia | Ben Chugg EMAIL Hongjian Wang EMAIL Aaditya Ramdas EMAIL Departments of Statistics and Machine Learning Carnegie Mellon University |
| Pseudocode | No | The paper describes mathematical proofs and theorems without providing any structured pseudocode or algorithm blocks. Methods are presented using mathematical notation and theoretical derivations. |
| Open Source Code | Yes | Code can be found at https://github.com/bchugg/confidence-spheres. |
| Open Datasets | No | Figure 3: Left: The width of our empirical Bernstein CI as n , which approaches its asymptotic width W . We use α = 0.05, d = 2, and random vectors comprised of two Beta(10,10) distributed random variables. Right: Performance of our empirical Bernstein bound compared to the multivariate (non-empirical) Bernstein bound baseline, with oracle access to the true variance. Shaded areas provide the standard deviation across 100 trials. The distributions are mixtures of betas and binomials. |
| Dataset Splits | No | The paper describes generating synthetic data for simulations and plotting but does not mention any specific training, test, or validation splits. The data is generated for direct comparison and visualization of bounds. |
| Hardware Specification | No | The 'Simulation Details' section describes the parameters and data generation for plotting but does not specify any hardware (e.g., GPU, CPU models, or cloud resources) used for these simulations. |
| Software Dependencies | No | The paper provides a link to its code repository but does not list any specific software dependencies with version numbers (e.g., Python, PyTorch versions, or specific libraries). |
| Experiment Setup | Yes | Section 2.1. The bound in Theorem 2.2 is implemented with the parameters 2 Tr(Σ) log(1/α) β + 2 log(1/α) ( Σ + Tr(Σ)/β)t log(t + 10e4). Theorem 2.3 is implemented directly as stated. We compute the bound of Hsu et al. (2012) as in (9), making it time-uniform via one of two methods: Either a union bound (taking αt = α/(t2 + t) at each timestep t), or by the doubling method of Duchi and Haque (2024) above. In the left hand side of Figure 1 we use Tr Σ2 = Tr(Σ) = Σ = 1 (eg distributions with identity covariance matrix). In the figure on the right hand side, we fix Σ = 1, Tr Σ2 = 10 and vary Tr(Σ). |