Convergence and optimality of matrix-valued confidence sequences

Establish a convergence rate for the sizes of the online-learning-based confidence sequences for matrix means and prove their optimality under an explicit criterion.

Background

The paper constructs confidence sequences for the mean of a matrix-valued random variable by reducing confidence-sequence validity to online learning of quantum states. Numerical experiments indicate that the construction can produce smaller confidence sets than several competing methods.

Despite this empirical evidence, the paper does not provide a theoretical convergence rate for the sizes of the resulting confidence sets and does not establish optimality under any criterion. These unresolved issues concern the theoretical advantages of the proposed online-learning-based construction.

References

Second, although our confidence sequence appears competitive empirically, we have not established any convergence rate in terms of the sizes of its confidence sets, nor have we proved its optimality under any criterion.

— Regret Lower Bound for Online Learning of Quantum States with Logarithmic Loss, with Applications to Quantum State Tomography and Confidence Sequences for Matrix Means  (2610.21273 - Li et al., 15 Oct 2026) in Concluding Remarks