Given a strongly convex function $f(x)$ with Lipschitz continuous gradient and Lipschitz Hessian, i.e.,
$$\mu I \leq \nabla ^2 f(x)\leq LI,\quad ||\nabla^2f(x)-\nabla f^2(y)||\leq M||x-y||^2,$$
and if we use the BFGS to approximate the curvature information, i.e., $$B_{t+1} = B_t+\frac{y_ty_t^\top}{y_t^\top s_t}-\frac{B_ts_ts_t^\top B_t}{s_t^\top B_ts_t},$$
where $y_t=\nabla f_{t+1}-\nabla f_t$ and $s_t=x_{t+1}-x_t$, can we bound the difference between $B_t$ and $\nabla^2 f_t$ other than the triangle inequality ? It seems a intuitive question to ask but I couldn't find anything on this. $$||B_t-\nabla^2 f_t||\leq \,\,??$$ You can also assume $\gamma I\leq B_t\leq \Gamma I$ if that helps. Feel free to put on more "meaningful" assumptions.