Stochastic Optimal Control and Estimation with Multiplicative and Internal Noise

A pivotal brain computation relies on the ability to sustain perception-action loops. Stochastic optimal control theory offers a mathematical framework to explain these processes at the algorithmic level through optimality principles. However, incorporating a realistic noise model of the sensorimotor system -- accounting for multiplicative noise in feedback and motor output, as well as internal noise in estimation -- makes the problem challenging. Currently, the algorithm that is commonly used is the one proposed in the seminal study in (Todorov, 2005). After discovering some pitfalls in the original derivation, i.e., unbiased estimation does not hold, we improve the algorithm by proposing an efficient gradient descent-based optimization that minimizes the cost-to-go while only imposing linearity of the control law.