Linear Bandit Algorithms with Sublinear Time Complexity
This work addresses efficiency challenges in online learning applications like recommendation systems, where arm sets are large, but it is incremental as it builds on existing approximate MIPS solvers.
The paper tackles the problem of high computational complexity in linear bandit algorithms for large arm sets by proposing two algorithms with sublinear per-step time complexity, achieving a more than 72 times speedup while maintaining similar regret performance.
We propose two linear bandits algorithms with per-step complexity sublinear in the number of arms $K$. The algorithms are designed for applications where the arm set is extremely large and slowly changing. Our key realization is that choosing an arm reduces to a maximum inner product search (MIPS) problem, which can be solved approximately without breaking regret guarantees. Existing approximate MIPS solvers run in sublinear time. We extend those solvers and present theoretical guarantees for online learning problems, where adaptivity (i.e., a later step depends on the feedback in previous steps) becomes a unique challenge. We then explicitly characterize the tradeoff between the per-step complexity and regret. For sufficiently large $K$, our algorithms have sublinear per-step complexity and $\tilde O(\sqrt{T})$ regret. Empirically, we evaluate our proposed algorithms in a synthetic environment and a real-world online movie recommendation problem. Our proposed algorithms can deliver a more than 72 times speedup compared to the linear time baselines while retaining similar regret.