Stanford's SLE-UCB algorithm hits optimal regret-stability trade-off, resolving open question
New proof: bandit algorithms can't beat T^(3/2) regret-times-variance bound. SLE-UCB matches it.
Kaifei Wang, Yinyu Ye, and Han Zhong introduce SLE-UCB, a tunable multi-armed bandit algorithm that combines a running lower-envelope index with a decreasing pull-count stabilizer. They study the trade-off between worst-case regret and instability, where instability is the largest standard deviation of a terminal pull count across K arms and T rounds. They prove a finite-time lower bound on the product of regret and instability that scales as T^(3/2), with a constant independent of K and T, under a finite-time regret condition and without the regularity assumptions used in prior asymptotic analysis. SLE-UCB achieves a product that is O(T^(3/2) log K), matching the lower bound exactly in T and within a logarithmic factor in K. This resolves the open question about the arm-dependent regret–instability frontier.
- SLE-UCB guarantees regret × instability = O(T^(3/2)·log K), matching the new lower bound R·S ≥ C·T^(3/2) in the time horizon T
- Proves the trade-off holds for any algorithm with finite-time regret, without prior asymptotic regularity assumptions
- The new offline top-prefix representation plus Efron-Stein inequality controls pull-count variance, resolving a published open question about the arm-dependent frontier
Why It Matters
Gives ML engineers a way to choose between average performance and run-to-run consistency in bandit-based systems like A/B testing and personalized recommendations.