Learning rotations with little regret

We describe online algorithms for learning a rotation from pairs of unit vectors in $$\mathbb {R}^n$$Rn. We show that the expected regret of our online algorithm compared to the best fixed rotation chosen offline over T iterations is $$\sqrt{nT}$$nT. We also give a lower bound that proves that this expected regret bound is optimal within a constant factor. This resolves an open problem posed in COLT 2008. Our online algorithm for choosing a rotation matrix is essentially an incremental gradient descent algorithm over the set of all matrices, with specially tailored projections. We also show that any deterministic algorithm for learning rotations has $$\varOmega (T)$$Ω(T) regret in the worst case.

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