Tailored max-out networks for learning convex PWQ functions
This provides an incremental improvement for learning-based control by offering a specific network architecture to represent convex PWQ functions exactly, addressing a known bottleneck in topology selection.
The paper tackles the problem of efficiently learning convex piecewise quadratic functions in control applications by proposing a tailored neural network topology that can exactly describe these functions, resulting in a max-out network with only one hidden layer and two neurons.
Convex piecewise quadratic (PWQ) functions frequently appear in control and elsewhere. For instance, it is well-known that the optimal value function (OVF) as well as Q-functions for linear MPC are convex PWQ functions. Now, in learning-based control, these functions are often represented with the help of artificial neural networks (NN). In this context, a recurring question is how to choose the topology of the NN in terms of depth, width, and activations in order to enable efficient learning. An elegant answer to that question could be a topology that, in principle, allows to exactly describe the function to be learned. Such solutions are already available for related problems. In fact, suitable topologies are known for piecewise affine (PWA) functions that can, for example, reflect the optimal control law in linear MPC. Following this direction, we show in this paper that convex PWQ functions can be exactly described by max-out-NN with only one hidden layer and two neurons.