My hope is that we will eventually have computationally feasible algorithms that satisfy provable (or at least conjectured) infra-Bayesian regret bounds for some sufficiently rich hypothesis space. Currently, even in the Bayesian case, we only have such algorithms for poor hypothesis spaces, such as MDPs with a small number of states. We can also rule out such algorithms for some large hypothesis spaces, such as short programs with a fixed polynomial-time bound. In between, there should be some hypothesis space which is small enough to be feasible and rich enough to be useful. Indeed, it seems to me that the existence of such a space is the simplest explanation for the success of deep learning (that is, for the ability to solve a diverse array of problems with relatively simple and domain-agnostic algorithms). But, at present I only have speculations about what this space looks like.
My hope is that we will eventually have computationally feasible algorithms that satisfy provable (or at least conjectured) infra-Bayesian regret bounds for some sufficiently rich hypothesis space. Currently, even in the Bayesian case, we only have such algorithms for poor hypothesis spaces, such as MDPs with a small number of states. We can also rule out such algorithms for some large hypothesis spaces, such as short programs with a fixed polynomial-time bound. In between, there should be some hypothesis space which is small enough to be feasible and rich enough to be useful. Indeed, it seems to me that the existence of such a space is the simplest explanation for the success of deep learning (that is, for the ability to solve a diverse array of problems with relatively simple and domain-agnostic algorithms). But, at present I only have speculations about what this space looks like.