Sublinear quantum algorithms for training linear and kernel-based classifiers
This provides a significant speedup for classification tasks in machine learning, potentially impacting applications that rely on large-scale data processing, though it is incremental as it builds on existing quantum techniques.
The paper tackles the problem of training linear and kernel-based classifiers with constant margin by designing sublinear quantum algorithms that achieve a quadratic speedup, reducing runtime from classical $ ilde{O}(n+d)$ to $ ilde{O}(\sqrt{n} + \sqrt{d})$.
We investigate quantum algorithms for classification, a fundamental problem in machine learning, with provable guarantees. Given $n$ $d$-dimensional data points, the state-of-the-art (and optimal) classical algorithm for training classifiers with constant margin runs in $\tilde{O}(n+d)$ time. We design sublinear quantum algorithms for the same task running in $\tilde{O}(\sqrt{n} +\sqrt{d})$ time, a quadratic improvement in both $n$ and $d$. Moreover, our algorithms use the standard quantization of the classical input and generate the same classical output, suggesting minimal overheads when used as subroutines for end-to-end applications. We also demonstrate a tight lower bound (up to poly-log factors) and discuss the possibility of implementation on near-term quantum machines. As a side result, we also give sublinear quantum algorithms for approximating the equilibria of $n$-dimensional matrix zero-sum games with optimal complexity $\tildeΘ(\sqrt{n})$.