Low-Rank Bandits via Tight Two-to-Infinity Singular Subspace Recovery
This work addresses the problem of improving efficiency and optimality in low-rank bandit settings for reinforcement learning and decision-making applications, though it is incremental by building on existing spectral methods.
The paper tackles contextual bandits with low-rank reward matrices by developing efficient algorithms for policy evaluation, best policy identification, and regret minimization, achieving near-minimax optimal sample complexities, such as scaling as r(m+n)/ε² log(1/δ) for policy evaluation, and improving regret bounds to r^{7/4}(m+n)^{3/4}√T.
We study contextual bandits with low-rank structure where, in each round, if the (context, arm) pair $(i,j)\in [m]\times [n]$ is selected, the learner observes a noisy sample of the $(i,j)$-th entry of an unknown low-rank reward matrix. Successive contexts are generated randomly in an i.i.d. manner and are revealed to the learner. For such bandits, we present efficient algorithms for policy evaluation, best policy identification and regret minimization. For policy evaluation and best policy identification, we show that our algorithms are nearly minimax optimal. For instance, the number of samples required to return an $\varepsilon$-optimal policy with probability at least $1-δ$ typically scales as ${r(m+n)\over \varepsilon^2}\log(1/δ)$. Our regret minimization algorithm enjoys minimax guarantees typically scaling as $r^{7/4}(m+n)^{3/4}\sqrt{T}$, which improves over existing algorithms. All the proposed algorithms consist of two phases: they first leverage spectral methods to estimate the left and right singular subspaces of the low-rank reward matrix. We show that these estimates enjoy tight error guarantees in the two-to-infinity norm. This in turn allows us to reformulate our problems as a misspecified linear bandit problem with dimension roughly $r(m+n)$ and misspecification controlled by the subspace recovery error, as well as to design the second phase of our algorithms efficiently.