  We study unconstrained Online Linear Optimization with Lipschitz losses.
Motivated by the pursuit of instance optimality, we propose a new algorithm
that simultaneously achieves ($i$) the AdaGrad-style second order gradient
adaptivity; and ($ii$) the comparator norm adaptivity also known as "parameter
freeness" in the literature. In particular,
  - our algorithm does not employ the impractical doubling trick, and does not
require an a priori estimate of the time-uniform Lipschitz constant;
  - the associated regret bound has the optimal $O(\sqrt{V_T})$ dependence on
the gradient variance $V_T$, without the typical logarithmic multiplicative
factor;
  - the leading constant in the regret bound is "almost" optimal.
  Central to these results is a continuous time approach to online learning. We
first show that the aimed simultaneous adaptivity can be achieved fairly easily
in a continuous time analogue of the problem, where the environment is modeled
by an arbitrary continuous semimartingale. Then, our key innovation is a new
discretization argument that preserves such adaptivity in the discrete time
adversarial setting. This refines a non-gradient-adaptive discretization
argument from (Harvey et al., 2023), both algorithmically and analytically,
which could be of independent interest.
