Does LinUCB Need Valid Confidence Sets?

Ahmed Ben Yahmed, Marc Abeille, David Janz, Clément Calauzènes

We show that replacing LinUCB's usual confidence width $β$ by any fixed $α>0$ preserves sublinear regret when the action set is $M$-smooth and $m$-strongly convex. Our high-probability regret bound for this generalisation, $α$-LinUCB, establishes that valid confidence sets are not necessary for the success of LinUCB. We identify the mechanism behind this result: action-set curvature drives $α$-LinUCB to explore every direction, even when optimism is not guaranteed. To analyse this mechanism, we introduce a logarithmically capped spectral potential that, unlike the standard log-determinant potential, caps contributions from well-explored directions. This yields lower bounds on the smallest eigenvalue of the design matrix. At the standard confidence width $α=β\asymp\sqrt{d\log T}$, our $T$-round regret bound in $d$ dimensions is $O\bigl(\frac{M}{m}d\sqrt{T\log T}\bigr)$, up to iterated-logarithmic factors, improving the horizon dependence of the classical LinUCB guarantee by a factor of $\sqrt{\log T}$.