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MachineLearning-Lecture17
Instructor (Andrew Ng) :Okay, good morning. Welcome back. So I hope all of you had a good Thanksgiving break. After the problem sets, I suspect many of us needed one. Just one quick announcement so as I announced by email a few days ago, this afternoon we’ll be doing another tape ahead of lecture, so I won’t physically be here on Wednesday, and so we’ll be taping this Wednesday’s lecture ahead of time. If you’re free this afternoon, please come to that; it’ll be at 3:45 p.m. in the Skilling Auditorium in Skilling 193 at 3:45. But of course, you can also just show up in class as usual at the usual time or just watch it online as usual also.
Okay, welcome back. What I want to do today is continue our discussion on Reinforcement Learning in MDPs. Quite a long topic for me to go over today, so most of today’s lecture will be on continuous state MDPs, and in particular, algorithms for solving continuous state MDPs, so I’ll talk just very briefly about discretization. I’ll spend a lot of time talking about models, assimilators of MDPs, and then talk about one algorithm called fitted value iteration and two functions which builds on that, and then hopefully, I’ll have time to get to a second algorithm called, approximate policy iteration
Just to recap, right, in the previous lecture, I defined the Reinforcement Learning problem and I defined MDPs, so let me just recap the notation. I said that an MDP or a Markov Decision Process, was a ? tuple, comprising those things and the running example of those using last time was this one right, adapted from the Russell and Norvig AI textbook. So in this example MDP that I was using, it had 11 states, so that’s where S was. The actions were compass directions: north, south, east and west.
The state transition probability is to capture chance of your transitioning to every state when you take any action in any other given state and so in our example that captured the stochastic dynamics of our robot wondering around [inaudible], and we said if you take the action north and the south, you have a .8 chance of actually going north and .1 chance of veering off, so that .1 chance of veering off to the right so said model of the robot’s noisy dynamic with a [inaudible]and the reward function was that +/-1 at the absorbing states and -0.02 elsewhere. This is an example of an MDP, and that’s what these five things were. Oh, and I used a discount factor G of usually a number slightly less than one, so that’s the 0.99. And so our goal was to find the policy, the control policy and that’s at ?, which is a function mapping from the states of the actions that tells us what action to take in every state, and our goal was to find a policy that maximizes the expected value of our total payoff. So we want to find a policy. Well, let’s see. We define value functions Vp (s) to be equal to this. We said that the value of a policy ? from State S was given by the expected value of the sum of discounted rewards, conditioned on your executing the policy ? and you’re stating off your [inaudible] to say in the State S, and so our strategy for finding the policy was sort of comprised of two steps. So the goal is to find a good policy that maximizes the suspected value of the sum of discounted rewards, and so I said last time that one strategy for finding the [inaudible]of a policy is to first compute the optimal value function which I denoted V*(s) and is defined like that. It’s the maximum value that any policy can obtain, and for example, the optimal value function for that MDP looks like this. So in other words, starting from any of these states, what’s the expected value of the sum of discounted rewards you get, so this is V*. We also said that once you’ve found V*, you can compute the optimal policy using this.
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