To add to the linked answers: if the upper bound is obtained by taking a convex relaxation (e.g. a semidefinite relaxation), you could strengthen the formulation by running a cutting-plane method to solve the continuous relaxation and applying the cuts generated from the relaxation before branching. If you do this your initial upper bound should match the relaxation's bound, and the bound will be informative (while applying $\theta \leq UB$ wouldn't help the solver decide how to branch). That said, solving the relaxation via a cutting-plane method might be time consuming, especially if there isn't a "nice" way of cheaply generating a cut from the relaxation, so this might do more harm than good (it depends on the problem).
This idea has been explored by Fischetti et. al. (see section 4.2 of https://pubsonline.informs.org/doi/10.1287/mnsc.2016.2461), who showed that it works really well for facility location problems.