Private online learning through lazy algorithms

We study the problem of private online learning, specifically, online expert prediction (OPE) and online convex optimization (OCO). We propose a new transformation that transforms lazy online learning algorithms into private algorithms. We apply our transformation for differentially private OPE and OCO using existing lazy algorithms for these problems. Our final algorithms obtain regret, which significantly improves regret in the high privacy regime. my≪1<annotation encoding="application/x-tex”>\varepsilon\ll 1my≪1getting trecord⁡d+t1/3record⁡(d)/my2/3<annotation encoding="application/x-tex”>\sqrt{T \log d} + T^{1/3} \log(d)/\varepsilon^{2/3}tlookgramd​+t1/3lookgram(d)/my23 for DP-OPE and t+t1/3d/my2/3<annotation encoding="application/x-tex”>\sqrt{T} + T^{1/3} \sqrt{d}/\varepsilon^{2/3}t​+t1/3d​/my23 for DP-OCO. We also complement our results with a lower bound for DP-OPE, showing that these rates are optimal for a natural family of low-switching private algorithms.