Ideas from: The Complexity of Dynamic Programming; Chow, Tsitsiklis

  1. Is concerned with multidimensional continuous state and action spaces
  2. Can just discretize s,a dimensions and get guarantees as to upper bounds on the approximation error.
  3. Lipshitz smoothness is assumed.
  4. They define a problem instance in terms of <T, R, γ, ε>, s.t. given these things, the result should be an estimated value function that is ε close to optimal.
  5. Assume the program partitions the space into hypercubes of side length h, so they have volume h^n for a dimension n.
  6. Concerned with the computational complexity, and not the sample complexity, as the MDP is assumed known already so alot of it isn’t so relevant – I won’t be paying close attention to the computational issues.
  7. Their bounds are tight, they do this by having two different MDPs that are as different as possible given the assumptions and data already gathered.
  8. They split cells into those that are well sampled, and those that are badly sampled, like Rmax.
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