  We study the task of $(\epsilon, \delta)$-differentially private online
convex optimization (OCO). In the online setting, the release of each distinct
decision or iterate carries with it the potential for privacy loss. This
problem has a long history of research starting with Jain et al. [2012] and the
best known results for the regime of {\epsilon} not being very small are
presented in Agarwal et al. [2023]. In this paper we improve upon the results
of Agarwal et al. [2023] in terms of the dimension factors as well as removing
the requirement of smoothness. Our results are now the best known rates for
DP-OCO in this regime.
  Our algorithms builds upon the work of [Asi et al., 2023] which introduced
the idea of explicitly limiting the number of switches via rejection sampling.
The main innovation in our algorithm is the use of sampling from a strongly
log-concave density which allows us to trade-off the dimension factors better
leading to improved results.
