  Regret minimization methods are a powerful tool for learning approximate Nash
equilibrium (NE) in two-player zero-sum imperfect information extensive-form
games (IIEGs). We consider the problem in the interactive bandit-feedback
setting where we don't know the dynamics of the IIEG. In general, only the
interactive trajectory and the reached terminal node value $v(z^t)$ are
revealed. To learn NE, the regret minimizer is required to estimate the
full-feedback loss gradient $\ell^t$ by $v(z^t)$ and minimize the regret. In
this paper, we propose a generalized framework for this learning setting. It
presents a theoretical framework for the design and the modular analysis of the
bandit regret minimization methods. We demonstrate that the most recent bandit
regret minimization methods can be analyzed as a particular case of our
framework. Following this framework, we describe a novel method SIX-OMD to
learn approximate NE. It is model-free and extremely improves the best existing
convergence rate from the order of $O(\sqrt{X B/T}+\sqrt{Y C/T})$ to $O(\sqrt{
M_{\mathcal{X}}/T} +\sqrt{ M_{\mathcal{Y}}/T})$. Moreover, SIX-OMD is
computationally efficient as it needs to perform the current strategy and
average strategy updates only along the sampled trajectory.
