# Analysis and Optimisation of Bellman Residual Errors with Neural Function Approximation

Recent development of Deep Reinforcement Learning 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 function. Despite great successes of Deep Reinforcement Learning, development of reliable and efficient numerical algorithms to minimise the Bellman Error 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 incorrect gradient information as done in Semi-Gradient algorithms. In this work, we analyse the Mean Squared Bellman Error from a smooth optimisation perspective combined with a Residual Gradient formulation. Our contribution is two-fold. First, we analyse critical points of the error function and provide technical insights on the optimisation procure and design choices for neural networks. When the existence of global minima is assumed and the objective fulfils certain conditions we can eliminate suboptimal local minima when using over-parametrised neural networks. We can construct an efficient Approximate Newton's algorithm based on our analysis and confirm theoretical properties of this algorithm such as being locally quadratically convergent to a global minimum numerically. Second, 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 short coming of Semi-Gradients. To benefit from an approximate Newton's algorithm complete derivatives of the Mean Squared Bellman error must be considered during training.

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