1.19 Machine learning lecture 20  (Page 44/14)

Instructor (Andrew Ng) :Okay. Good morning. Just one quick announcement before I start. Poster session, next Wednesday, 8:30 as you already know, and poster boards will be made available soon, so the poster boards we have are 20 inches by 30 inches in case you want to start designing your posters. That’s 20 inches by 30 inches. And they will be available this Friday, and you can pick them up from Nicki Salgudo who’s in Gates 187, so starting this Friday. I’ll send out this information by e-mail as well, in case you don’t want to write it down.

For those you that are SCPD students, if you want to show up here only on Wednesday for the poster session itself, we’ll also have blank posters there, or you’re also welcome to buy your own poster boards. If you do take poster boards from us then please treat them well. For the sake of the environment, we do ask you to give them back at the end of the poster session. We’ll recycle them from year to year. So if you do take one from us, please don’t cut holes in it or anything. So welcome to the last lecture of this course. What I want to do today is tell you about one final class of reinforcement learning algorithms. I just want to say a little bit about POMDPs, the partially observable MDPs, and then the main technical topic for today will be policy search algorithms. I’ll talk about two specific algorithms, essentially called reinforced and called Pegasus, and then we’ll wrap up the class. So if you recall from the last lecture, I actually started to talk about one specific example of a POMDP, which was this sort of linear dynamical system. This is sort of LQR, linear quadratic revelation problem, but I changed it and said what if we only have observations YT. And what if we couldn’t observe the state of the system directly, but had to choose an action only based on some noisy observations that maybe some function of the state?

So our strategy last time was that we said that in the fully observable case, we could choose actions – AT equals two, that matrix LT times ST. So LT was this matrix of parameters that [inaudible] describe the dynamic programming algorithm for finite horizon MDPs in the LQR problem. And so we said if only we knew what the state was, we choose actions according to some matrix LT times the state. And then I said in the partially observable case, we would compute these estimates. I wrote them as S of T given T, which were our best estimate for what the state is given all the observations. And in particular, I’m gonna talk about a Kalman filter which we worked out that our posterior distribution of what the state is given all the observations up to a certain time that was this.

So this is from last time. So that given the observations Y one through YT, our posterior distribution of the current state ST was Gaussian would mean ST given T sigma T given T. So I said we use a Kalman filter to compute this thing, this ST given T, which is going to be our best guess for what the state is currently. And then we choose actions using our estimate for what the state is, rather than using the true state because we don’t know the true state anymore in this POMDP. So it turns out that this specific strategy actually allows you to choose optimal actions, allows you to choose actions as well as you possibly can given that this is a POMDP, and given there are these noisy observations. It turns out that in general finding optimal policies with POMDPs – finding optimal policies for these sorts of partially observable MDPs is an NP-hard problem. Just to be concrete about the formalism of the POMDP – I should just write it here – a POMDP formally is a tuple like that where the changes are the set Y is the set of possible observations, and this O subscript S are the observation distributions. And so at each step, we observe – at each step in the POMDP, if we’re in some state ST, we observe some observation YT drawn from the observation distribution O subscript ST, that there’s an index by what the current state is. And it turns out that computing the optimal policy in a POMDP is an NP-hard problem. For the specific case of linear dynamical systems with the Kalman filter model, we have this strategy of computing the optimal policy assuming full observability and then estimating the states from the observations, and then plugging the two together.