  Recent development of Deep Reinforcement Learning (DRL) has demonstrated
superior performance of neural networks in solving challenging problems with
large or even continuous state spaces. One specific approach is to deploy
neural networks to approximate value functions by minimising the Mean Squared
Bellman Error (MSBE) function. Despite great successes of DRL, development of
reliable and efficient numerical algorithms to minimise the MSBE is still of
great scientific interest and practical demand. Such a challenge is partially
due to the underlying optimisation problem being highly non-convex or using
incomplete gradient information as done in Semi-Gradient algorithms. In this
work, we analyse the MSBE from a smooth optimisation perspective and develop an
efficient Approximate Newton's algorithm.
  First, we conduct a critical point analysis of the error function and provide
technical insights on optimisation and design choices for neural networks. When
the existence of global minima is assumed and the objective fulfils certain
conditions, suboptimal local minima can be avoided when using over-parametrised
neural networks. We construct a Gauss Newton Residual Gradient algorithm based
on the analysis in two variations. The first variation applies to discrete
state spaces and exact learning. We confirm theoretical properties of this
algorithm such as being locally quadratically convergent to a global minimum
numerically. The second employs sampling and can be used in the continuous
setting. We demonstrate feasibility and generalisation capabilities of the
proposed algorithm empirically using continuous control problems and provide a
numerical verification of our critical point analysis. We outline the
difficulties of combining Semi-Gradient approaches with Hessian information. To
benefit from second-order information complete derivatives of the MSBE must be
considered during training.
