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0:05
Welcome to deep reinforcement learning or CS224R.
0:10
I'm really excited to be teaching it again this quarter. My name is Chelsea. I am the instructor of the course. I'm an assistant professor here at Stanford. I do research on reinforcement learning, robotics, and language models, and a variety of other topics. So the plan for today. So we're going to be covering the course goals and some logistics. Hopefully this will be fairly short because all this information is available online. After that, we'll be talking about why
0:34
study deep reinforcement learning and actually get into some technical content on modeling behavior and reinforcement learning. The key learning goals for today is what even is deep reinforcement learning? How do we go about actually representing behavior in a way that we can optimize behavior? And how do we formulate a reinforcement learning problem? So what do we mean by deep reinforcement learning? So first, for reinforcement learning, we mean decision-making problems.
1:02
And these are problems where a system needs to make multiple decisions based on a stream of information. So it'll observe and then take an action based on that observation, and then observe again and take another action. And we'll be studying those of problems as well as the solutions to those problems, and that includes things like imitation learning, model-free and model-based reinforcement learning, offline and online reinforcement learning, multi-task
1:27
and meta reinforcement learning, as well as a little bit of a extra focus on reinforcement learning for large language models, and reinforcement learning for robots. So that's the crux of reinforcement learning aspect of the course. And we'll be emphasizing solutions that scale to deep neural networks. And we'll be spending a lot less time on solutions that don't pertain to deep neural networks. Now, how does deep RL differ from other machine learning topics?
1:54
So hopefully a lot of you have some experience with machine learning. And oftentimes in machine learning, you'll have some input/output pair x, y. And you'll have a bunch of these in some data set, and then your goal is to learn something that can map from x to y. And another thing is that we typically assume in these data sets is that these data points are sampled from some distribution and are sampled independent and identical way from that distribution.
2:32
And this is actually where reinforcement learning differs quite a bit. So in supervised learning we have labeled data. We learn this function. We're directly told what to output. We're pulled what the answer is for that given input, and the inputs are i.i.d sampled. Now, in reinforcement learning, the picture is a little bit different. So ultimately our goal is to learn behavior. And we'll actually use different notation to refer to inputs and outputs.
3:03
So specifically our goal will be to map from states to actions. And we'll refer to the function that we do that mapping as pi. And unlike in supervised learning, we're not going to be directly told what the answer is. Our goal will actually be to learn this behavior from much more indirect feedback. And so in reinforcement learning, the goal is to learn this behavior pi of a given s. And this will be from experience, and actually experience often drawn from the policy that you're learning
3:35
or this function that you're learning. And the data is not going to be sampled from a single distribution. It's actually going to be sampled in a way that actually depends on the policy or the function that you're currently learning. So these are the two big differences. And what I mean by behavior is this could include motor control like controlling your robot. It could include a chatbot that is interacting with a person. It could be playing a game. It could be driving a car.
4:03
It could be an agent that's interacting with the web and a wide range of other domains. So that's how reinforcement learning differs from typical machine learning. We don't have the ability to cover everything in deep reinforcement learning, and so we're going to try to focus on the core concepts behind deep reinforcement learning methods, so that you can then use those core concepts to understand more advanced methods, should you choose to dive deeper into the topic.
4:31
We'll also focus on implementation of algorithms to get to the point where you feel comfortable implementing algorithms yourself. And we'll give examples in robotics, control, and language models, especially in lecture and in projects, but a lot of the techniques will generalize more broadly, and also just to focus on topics that we think are the most useful and the most exciting to cover. If you're interested in more theory and other applications that are beyond some of these, CS234 can also
5:01
be an excellent choice. And then the core goal is really to be able to understand and implement existing and emerging methods in deep reinforcement. Cool. So why study deep reinforcement learning?
5:16
I'll list out a number of things here, and then we'll dive into each one a little bit more. So one we already talked about is going beyond supervised example. And there's a lot of scenarios where AI model predictions have consequences. And if we want to take into account those consequences, then we need a framework for thinking about those consequences and how to optimize the behavior with respect to those consequences. So for example, if you listen to music on Spotify, for example,
5:45
and the Spotify algorithm is recommending a song for you to play or even actually just playing it directly for you, that recommendation has consequences such that the next time it recommends a song, it probably shouldn't recommend the same song, or it might want to also stay within the same genre, and so forth. And so if we want to think about developing AI systems that can take into account those consequences, that's what deep reinforcement learning allows us to do.
6:10
There's also scenarios where direct supervision isn't available. So for example, if you have a coding assistant like Cursor or Copilot, and you want to give it feedback saying, did a great job there or did a poor job there, that supervision is not the output. It's a more indirect supervision. And learning from that indirect supervision or really any objective can be formulated as a reinforcement learning problem. So it's a lot more flexible than supervised learning.
6:42
Reinforcement learning is also widely used and deployed in a number of performant AI systems.
6:49
Also learning from experience, in my opinion, seems fundamental to intelligence. And so just from a philosophical level in terms of building intelligent machines, figuring out how to actually develop algorithms that can learn from their own experience seems pretty fundamental to that. And it's fundamental to be able to actually make discoveries of new things rather than just mimicking the data. And then lastly, there's also some pretty exciting open research problems as well, and so
7:18
it makes it a pretty exciting area to do research on. And then lastly, last weekend, I was telling a friend that I was making slides on why study deep reinforcement learning, and how I was making several slides on it, and they were like, this question doesn't seem very hard. The answer is just one million in total compensation from OpenAI. So maybe if you're not motivated by 1 through 4, this can be some motivation.
7:48
So diving into beyond supervised learning. Decision-making problems are everywhere and can't be directly formulated supervised learning. Any AI agent is captured by this. Also if you want your AI system to interact with people like chatbots or recommenders or scenarios where decisions will affect future outcomes or observations, these feedback loops, these are reinforcement learning problems. And then yeah, like I mentioned before, if you don't have labels in your objective, isn't just accuracy
8:16
or it's nondifferentiable. These are scenarios where reinforcement learning is useful.
8:23
Now, I mentioned that deep RL is used in a variety of different performance systems. What am I talking about there? Two examples of legged robots that have learned behavior with reinforcement learning, in this case, reinforcement learning and simulation and then transferred to the real world. So you can get some cool stuff like puppy robots and dancing humanoids. And these are actually real robots in the real world. Policies that are trained with deep reinforcement learning.
8:50
Beyond legged robots and dancing, you can also get robots that can manipulate with arms to do tasks using, in this case, imitation learning, but still within the scope of this course. So the left is a robot that is unloading laundry and folding the laundry. On the right is a robot that can fold some simple origami. So some pretty cool stuff you can do with these methods.
9:19
You can also learn to play complex games. So several years ago, DeepMind used deep reinforcement learning to train a system to play the game of Go, and it actually defeated Lee Sedol, which was really impressive in its own right. And another really cool thing about it was coming back to this discovery challenge of AI systems. There was this move that surprised everyone that was an expert in Go. It was called Move 37, and it discovered a strategy that humans previously hadn't thought of before.
9:56
And so this is an example of how reinforcement learning can be a tool to discover new solutions. And this might be useful beyond games. But if we want to, for example, discover curious diseases or discover things that require experimentation, it provides a tool for doing those things.
10:14
And it's not just robots and games, so nearly all modern language models use some form of RL for post-training, especially for more advanced reasoning as well. There's also research on traffic control using reinforcement learning and finding ways to insert cars into human-driven cars in order to increase the throughput on different roads. Research on training generative models with reinforcement learning. In this case, for example, trying to make the generations adhere to a particular prompt.
10:48
So what these are showing is on the left. If you ask the base generative model to make a picture of a dolphin riding a bike, it'll give you this, which is not a very good picture. And then you can optimize it through reinforcement learning to get a picture that is actually following the prompt more effectively, and likewise for other prompts as well.
11:08
And this also is being used for Chip design. This is a paper that designed chips that are used in Google's production TPU, which is also pretty useful. Yeah. Great. So those are a few examples. There's a number of others that I didn't cover as well. The next thing that I mentioned in terms of studying deep RL seems somewhat fundamental to intelligence. And for this, I want to tell a little bit of a story which is that right after I graduated undergrad,
11:42
I joined a lab that was looking at reinforcement learning for robots. And right next to where I was sitting was this robot here. And oftentimes there was a post-doc in the lab who was running experiments with reinforcement learning, and so I would look over at the robot and it would be doing things like this. And the thing that really fascinated me about watching this robot is you can actually see it learning over time. And at the beginning it knew nothing about how
12:11
to, for example, insert and assemble these two parts of a toy airplane, and at the end, it's actually able to do it quite performantly. So this is really, really cool. You can actually see it learning from its own experience in the real world. So essentially reinforcement learning enables this ability to get better with practice that isn't present in other machine learning systems. Now, there is one caveat to this, which is that this particular robot had its eyes closed.
12:37
It wasn't using a camera, essentially, and that seemed like a really fundamental limitation to me. And so I actually then worked with the postdoc on a follow-up project with cameras, and it was able to, in this case, then, learn in the real world how to insert the red block into the shape sorting cube. And again here, with experience is able to get better and better as it practices the task over and over again.
13:07
And then after the learning process, it's still learning, still improving. It ends up getting a success rate. I think around 95% or 100% success at the end. And then I can hold the block out, or at least a young version of me can hold the block out for the robot, and the robot can then put the red block into the same currency. Now, these are very simple tasks. This is in some ways a long time ago, and now we have robots that can fold laundry and do dancing and so forth.
13:43
But one of the reasons why I like showing these videos is they really show the reinforcement learning process in the real world and the ability to improve with practice. At the same time, there's also a ton of exciting research problems that are open. And so if you're someone who's excited about doing research, there's challenges like, how does this robot learn to represent what is good for the task. Or how does an agent generalize its behavior,
14:06
not just to one scenario, but to many different scenarios? And can we apply such a system at scale with large, diverse data sets and transfer from other tasks and other goals? And can we use reinforcement learning to learn long horizon tasks like cooking a meal or solving a really hard math problem? And can robots practice fully autonomously? And for this last point, I showed you some videos of reinforcement learning. That's not actually always what it looks like.
14:30
So sometimes if you run reinforcement learning on robot, it looks more like this, where the robot is trying to hit the hockey puck into the goal. And every time the robot attempts the task, it then needs to start over and re-attempt it, and the robot doesn't actually know how to undo the task. And this is my friend Yevgen. You might notice that Yevgen is actually doing more work than what the robot is doing, which doesn't quite seem like the right way of doing things.
14:54
And it's not practical to collect a lot of data this way.
14:59
And along all these topics, these are some of the topics that we'll be covering in this course as well. So we'll be talking about reward learning and how to represent different rewards. We talk about offline RL, multitask RL, and meta-RL, hierarchy, and reasoning, and reset-free RL as well. Cool. So any questions on why deep RL is interesting to study? Yeah. So the question was, how do you learn a reward when the robot is learning in the real world?
15:30
And in that particular example, we represented the reward as the distance between-- or the negative distance between the position of the object and the goal position of the object. And you might have noticed that in the case with the shape cube, the robot was actually holding the cube, and so that meant that we could actually measure using the robot arm position where the cube was as well. Now, you can't do this in practice. In many scenarios it's specific to that task.
16:00
And in the hockey example, there was actually a mocap marker on the hockey puck that measured where it is in-- yeah, where it is in the room. And in general, it's very hard. And that's why reward learning is necessary in order to solve the problem in a more general way. Good question.
16:23
So now let's dive into modeling behavior and reinforcement learning. So we need to figure out how to represent experience as data. So as the robot was practicing, we need to think about how we can actually represent what was captured in that video as data to learn from. And so here's another example of a robot learning in the real world. We might also want to think about how to represent text from a chatbot, for example. In either of these cases we need to think about experience
16:56
that the agent collect. How do we actually write that down as data? So the way that we do that, typically in reinforcement learning, is first we'll think about the state of the world. And we'll use S to denote the current state of the world.
17:13
And for example, in the case of the robot, the state of the world might be the position of the arm and the position of the towel and the position of the hook. Or if you don't have access to information about the towel directly, it might just be an image captured from the robot's camera and the position of the joints of the arm if you have access to that. Now, you can't always have direct access to the state of the world, and so sometimes we'll refer to an observation O, and use
17:49
observations instead of states. And the difference between observations and states is that with observations it may not capture all of the information you need to make a decision. And you might actually need to think about past observations that you've seen before. So for example, for a chatbot, if you're interacting with it, you can think of the observation at a given time step as the most recent message that you sent to that chatbot. And then that most recent message
18:16
doesn't tell the chatbot everything it needs to know about what the next response should be. And so what you need is you actually need to give it the history of past observation, not just the current observation, in order to make a decision. And we'll talk a little bit more about this state observation distinction in a couple slides. So this is capturing what the agent is observing. Now, we also need to capture how it's taking actions, how it's actually making decisions.
18:45
And we'll refer to those decisions as actions. And we'll use the letter a to denote actions. And then in terms of a given interaction with the environment, we will think of that as a trajectory. So that will be a sequence of states and actions or a sequence of observations and actions. So this is one way to represent behavior, and this will then result in a data set of trajectories rather than a data set of xy pairs. And then we need to think about what is actually
19:25
the goal of the task, and that's where reward functions come in. And reward functions are essentially telling us how good is a given state and action. In many cases, the reward function will only depend on the state. And for example, it might tell you how close is the towel to being hung, or is the towel hung or not. Or in the case of a teapot it might be, how happy is the user? Or did they give me a thumbs up or a thumbs down? And then this experience is the trajectory,
19:59
and that trajectory might be augmented with the rewards.
20:05
So each time you roll out the policy, you'll get some data in this form. Let's talk a little bit more about states and observations first. We can then think about states and actions, is in some ways a lot like a Markov chain. So they will have the first state of the world. This is, for example, in the robot case, let's say the towel is on the table and there's a hook, and the goal is to put the towel on the hook. This first state is like the robot in some neutral position
20:40
and the towel on the table. And then the robot will take some action. This is like moving its arm to a particular location. And that action and the current state will inform what the next state is. And then from there-- maybe the towel is in the same position if the robot was just moving in free space, or maybe the towel is moved. And the robot will then take another action, and it will move its arm again. And the states and actions will then inform the next state.
21:19
Now, one crucial thing here is that we can think of how the environment is changing under what's called a dynamics function. So you have P of S3 given S2 and a2. This is the dynamics of the world. This is, does the arm actually move in space, and how does the towel move in space? as a function of where it was before and what the robot did. And the crucial thing to notice here is that S3 only depends on S2 and a2, and it doesn't depend on S1. And so there's no S1 in this probability distribution.
22:01
And essentially, S3 is going to be independent of S1 if you condition on S2. And this independence property is called the Markov property. And it's one of the things that actually ends up being very useful in reinforcement learning. So it allows you to break up the problem into these scenarios where you don't have to worry about all the dependencies that come before it. Now, if you have observations instead of states, then things look a little bit different.
22:42
And the way that we can think about observations is the observation is a function of this current space, and it may not contain all the information that's in the state. And so the observation at any given point in time. For completeness we'll draw each of them here. And then if you don't have access, if you can't observe the space, that means now that O2-- or sorry, O3, it's not just a function of O2 and a2, it's actually also a function of O1 if you cannot observe the same.
23:22
Cool. So on the slide, we can think about this model where we have these dynamics. We don't know how the world is going to evolve. We can't predict the future. Under these dynamics are unknown. And these dynamics are independent of St minus 1 , in this case, and that's known as the Markov property. Any questions on that? Yeah. So the question is, in practice, how many past observations do we need to infer the state? It is heavily domain dependent.
23:58
Yeah. Heavily domain dependent. In a lot of robotics examples, you can often get away with just using the current sensor observations. But it depends a lot on what the sensors are like. If you have tactile sensors, for example, those oftentimes don't contain as much information as vision observation, for example. You have no idea when you're not making contact. And then once you make contact you know there's a lot more partially observable than an image, for example.
24:29
But then if you have the images in addition to that, then that gives you a more complete representation. Yeah. So why is the reward function a function of both the state and the action and not just the state? One example of this is-- we can go back to a robotics example. So say that you are a humanoid robot, and you're trying to learn how to walk, and maybe one natural reward function is just how much forward progress have you made.
24:52
So where is your body, and that would be dependent on the state. One scenario where you might want the action is that scenario where if you don't want to be using too much energy because you don't want to be like tensing up all the motors, for example, or I guess-- muscles make more sense. But if you don't want to be pushing too much energy into your motors, then you'll want to depend on the actions as well, and you could basically have some penalty on how much torque you're
25:18
putting through the motors. So that's one example. In many cases, you can just depend on the state. But there are some examples where you want to penalize taking too much action. Yeah. So the question is, if the observation is only part of the state, then why does a future observation not independent of past observations? And the reason for that is that the-- we can actually go back to the tactile example. So say that you make contact and then you stop making contact
25:48
and then you make contact again, in that case, the most immediate past observation doesn't actually tell you that much information about the current state, whereas this observation actually tells you a lot more information about the state. And so then O3, you can actually figure out more about O3 when you don't just condition on O2, but also condition on O1. And there's also a way for folks familiar with graphical models to actually think about. If you marginalize out these variables,
26:23
you'll actually see that in the graphical model. I don't want to go into that today, but that's another way that you can see this. Correct. So yeah. If you were to estimate and predict O3, then you need to use past observations. But the actual observation in this graphical model is dependent on the state which might be unobserved or unknown. Cool. So let's walk through a couple examples of what this might look like. So in the robotics case, the state could include three different things.
26:57
One would be the RGB images from any cameras on the robot, the positions of all the joints of the robot, and the velocities of the joints. And so this is a fairly common state representation for a robot system. And the images would be actually a different dimensionality than the positions, and so you might need to actually configure your neural network in a way that can accept those different dimensionalities. The action would then be the commanded next joint position.
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So how are you going to move your joint? If your joint's currently at 90 degrees, maybe you'll try to move it to 92 degrees. The trajectory then would be a sequence of states and actions. So a sequence of videos-- or a sequence of images, which is essentially a video, a sequence of joint positions, and a sequence of command and joint positions.
27:48
Now, what does that sequence actually entail? Typically what we do is we'll discretize time and the sequence will then, for example, be at 20 Hertz. So your video will be 20 frames per second. And you'll have joint observations 20 times per second. And you'll also have commanded actions 20 times per second. I always found this a little bit unintuitive when I first learned it because, as a person, I don't really think about making a decision 20 times
28:16
every second, but it is oftentimes the most practical way to represent data and behavior on physical systems, like robots. And then for the reward function, there's a few things you could use here. One could be one, if the towel is on the hook in state S and 0. Otherwise you could also have some sort of penalty on the torques that you're applying on the action or something like that. So that's a robot example. Now let's go to a chatbot example. So here the observation-- we likely
28:52
wouldn't have a compact state representation, so we be using observations. And the observation could be the user's most recent message. The action would be the chatbots next message. So this would be potentially a sentence or a paragraph. It potentially could actually be very large. And then you can think of the trajectory as the conversation trace, so the history of the current conversations. And unlike in the robotics example, this won't necessarily be at a fixed 20 Hertz frequency.
29:25
It might take longer for a user to respond or something like that. So this is an example of how trajectories might vary in terms of how time is represented. So in the first case, time is represented as a 20th of a second. Whereas in the second case, time is represented as when the user responds next.
29:45
And then as an example reward. Maybe it's 1 if the user gives an upvote, negative 10 if the user gives a downvote, and 0 if there's no user feedback.
29:57
These are examples of how we can set up reinforcement learning. Now I would like to ask you to formulate a reinforcement learning problem. And in particular, I'd like you to define the state or observation, the action trajectory and reward, for either one of these three examples or choose your own example. Yeah. So in this example, basically the question is, why would this be an observation persistent state? It could actually be formulated both ways, potentially.
30:30
It depends on the details of the sensor measurement. So as one example, if your sensors are just looking like a LiDAR, like the current depth around you, and a picture of what's in front of you, maybe something along those lines, one thing that that would be missing is how fast the cars are moving around you, the velocity of the cars. And that's probably quite important for what's going to happen in the future if it stops versus moving at 60 miles per hour.
31:03
Now, you could potentially try to include a history of LiDAR information to get that velocity information and then represent your state as a three-second history, for example. And this will represent a theme which is typically to handle observations. We'll use a little bit of history and then that can form your state representation. So it's a little bit ambiguous in this particular case. And yeah, it comes down to the details. Yeah. So the question is, in the robot example,
31:34
if the reward is 1, if it's on the hook and 0 otherwise, how do you actually learn? because if you don't actually ever do that at any point, then you'll have very little supervision signal. You'll just have 0 reward all the time. And so in that particular video, we actually showed the robot how to do it a couple times as initialization, and that helped it with what we'll call exploration. And if you have very sparse rewards, rewards that have very little signal
32:07
and are not nicely shaped towards the goal, then usually that data might be important for jumpstarting the reinforcement learning problem. Cool. Yeah. So if folks couldn't hear that, you'd have the state being something like the HTML and the URL, information about the web page. And where you're at, the action would be interacting with it, and the reward would be whether or not some task was completed. Yeah. So the question is, is basically everything in observation
32:34
because sensors may not be perfect? So you could always choose to model something as an observation if you're worried about sensor noise or something like that. There are, in some ways, potentially large downsides to it if you need to add a lot of history. And so sometimes we will approximate something as the state, even if it might be lacking information. Maybe, I don't know, once every 10,000 times the sensor fails and gives you a black image or something like that.
33:09
Even though that would technically be an observation case, you might still use it as a state representation because of how it affects the algorithms. So if I understand correctly, you're asking-- for observations, it could either be everything currently or it could be a delta from the previous observation.
33:28
Yeah. It would come down to different trade-offs oftentimes. If you are going to be including a lot of history, then compressing it by using deltas could potentially make a lot of sense. But once we get into policies, maybe it'll also become more clear. So now that we've talked about formulating the reinforcement learning problem, let's talk about actually representing behavior. So we want to map typically from states or observations to actions. And to do that, we'll use what's called a policy.
34:01
And this policy will often use pi to denote a policy or basically almost always use pi. And this can just be a neural network. In this case, it's a convolutional neural network. It could be a transformer. It could be a simple fully connected neural network. It could be a linear function. And this is going to be mapping from the states or observations, or the sequence of observations to actions. We'll use typically theta to refer to the parameters of that neural network
34:28
that you're trying to learn. And so then when you're actually using a policy to take actions, we'll then observe a state, take an action, and observe a next state. And when we take actions we're actually going to be sampling from our neural network. So we'll actually be running a forward pass through the neural network to select an action, and then execute that action in the real world and observe the next state.
34:56
Cool. And so the result of this process is a trajectory. And when we're using a policy to generate this trajectory, we often refer to it as a rollout or an episode. You can think of it as-- one episode of learning is like one attempt and then trying it again and again. And I guess the terminology rollout is in the sense that you're sequentially going policy dynamics, and so you're rolling out this process to create a trajectory.
35:28
Great. And then if you only have observations and you don't have states, the solution is to give your policy memory, and so you can pass in multiple observations. Whereas if you only give it a state, you actually don't have to give it any memory, and you can just give it as input state. This is very convenient because if you have a large amount of memory, it will take up a lot of memory on your computer or on your GPU. But if you only have observations, then you have, in some ways, no choice but
36:00
to provide that memory to the policy. And for this you may use sequence models, for example. In language models, you're going to be using sequence models that use the previous observations. Yeah. So the question is, is T like the current time step or is it general from the beginning to the end? So this capital T, I'm using to denote the end of the trajectory, so the length, the number of states and actions in a trajectory. And then t, I'm referring to a given time step
36:27
and generalizing it so that you can think about this as happening at all time steps from lowercase t equals 1 to lowercase t equals capital T. Is it ever useful to condition it on both the observation and the state? So in this case, the state has strictly more information than the current observation, and so it isn't useful to add the observation if the state is truly the state that captures everything. So if we go back to this diagram, if we have states,
37:00
then we only need to provide the current state. Whereas if we have observations, that's where adding history is helpful. So it's a distinction between states and observation. Why is it having history of observation, [INAUDIBLE]? Yeah. So basically the past observations will help give you more information about what S1 is. And so if you only have observations and you don't know what the underlying state of the world is-- as another example in language modeling,
37:29
if you are having a conversation with a user, each thing that the user says is a different observation, and the past observations will actually tell you a lot about what the current state of the world is. So if their first question was like, can you write this piece of code to me? And then you maybe wrote some piece of code and they gave you some feedback, it's actually very important to include all those past observations to actually capture the full state of the world.
37:50
I mean, that might not be true in all cases. But in many, many cases, the previous observations will also be useful for the current state. So there also could be scenarios where you might want a two-stage process, one where you first infer the state, and then a second process where you then predict the action from that state. If you have some idea for a good representation for the underlying state, then that information could be useful in trying to model the underlying state.
38:22
In other cases, it's simple as just to go straight from observations side. OK. Moving on a little bit. So we've talked about states actions policies. What is actually the goal of reinforcement learning? So our goal is to maximize reward. And so if we have a trajectory, we don't want to just optimize for the reward at one time step, but we want to optimize for having reward across the entire trajectory. And so we can represent this as maximizing the sum of rewards.
38:57
Now, one thing that came up is that this is actually not a deterministic quantity. So if you have a policy and you're then executing that policy in the world, the reward that you get isn't always going to be the same every time you run your policy. And so a question for you. Why is this not a deterministic quantity? And what are the sources of variability that might cause the sum of rewards to differ from one trajectory to another trajectory?
39:30
So one source of variability is that the state doesn't just depend on the action in the state, but there's also some randomness that can affect the next state. Other sources of variability. So your policy may not be deterministic either. So there's two really key sources. Here's a good question. You might have a stochastic reward function. So typically we actually model the reward-- is a very interesting point. We often typically model it as a function and have the source of randomness
39:59
come from states and actions. Now, if you don't know the state, then it could actually be viewed as stochastic if you only have the observations. So if you're doing reward learning, the question is, is there stochasticity there? Typically you'll still model it as a reward function. Well, you might not have full certainty over exactly what the reward is, and perhaps you might want to leverage that uncertainty in some ways in the learning process, that's maybe something to take offline.
40:34
There's some considerations there. So the other thing is your goals might change over time. And in that case, the reward function will change. We'll actually often model that as a whole different decision process at that point if the rewards change rather than having it being within the same one.
40:55
There are scenarios where you might want a nonstationary decision process, I guess. And you can actually often put that randomness into the state instead of putting it in the reward function. Maybe there's some underlying goal that you can't observe that's in the state, And then the reward function remains deterministic. And so yeah, generally without losing generality, you can push that into the state if needed. Cool. So the two main sources of variability
41:25
that you have to worry about are the world is stochastic so environment is dynamic. This is a distribution. It's not a deterministic function. And the second is that the policy is also not deterministic. And so in driving, for example, other cars are going to behave in ways that are random, and your car might not make the same decision every time.
41:50
Cool. And so if we actually want to think about maximizing rewards, we can then add in the policy into this diagram where, if you have access to state, there would be an error from here to here that is modeling this distribution, and likewise from here to here. Now, if we then wanted to actually write down what rewards look like, we need to think about a probability distribution over trajectories. You need to think about what is actually the distribution over futures that we'll see.
42:32
And the way that we can think about that is we'll have some initial state. There may actually also be randomness here that we haven't talked about. The initial conditions of the world might change. And if you're like playing a game, for example, how the cards are dealt out will vary. And then after the state, you're going to be making a decision with your policy. And then once you make that decision, there is randomness that arises from the dynamic.
43:06
And then if we want to think about the probability of a given trajectory, is going to be the probability of the first state, then the probability of picking the next action, then the probability of observing the next state, then the probability of picking the action after that, and then the next state, and so on. And so then we can think about the probability of a given trajectory as being the initial state probability plus the product of the probabilities from the policy and the dynamics where
43:40
here t equals 1 to capital T. And then once we formulate this probability distribution over trajectories, then we can think about maximizing reward as actually maximizing our expected rewards under this probability distribution. So we'll think about the expectation of trajectories sampled from this distribution of the sum of state and actions in this trajectory of the reward of that state and action. And then this is our goal. Our goal is to maximize this.
44:28
And our goal is to find a policy pi that maximizes these expected sum of rewards, and so that is written right here.
44:46
Now, note that, unlike in supervised learning, this doesn't look like something that you can--
44:56
pi actually doesn't even appear in this equation, but the way that it comes in is how it's affecting the trajectory. So this distribution of trajectories is a function of pi right here. And you actually want to find the policy that will lead to the highest reward. And a lot of this class will be figuring out how do we actually optimize behavior under this particular objective. Now the question is, does capital T have to be fixed? It does not have to be fixed. It can be variable.
45:28
It's a little bit easier to write down if it's fixed. But yeah, you can also think about it as basically the length of the message. Yeah. So one other topic that will come up in depth a lot more is when you think about rewards, this is weighting the rewards over the entire trajectory equally, and it's saying that you care about immediate reward just as much as you care about future rewards. And in some cases, you might care about immediate reward more than future reward.
46:02
So for example, if this T is like 1,000 years, maybe you don't care so much about your rewards 1,000 years from now, then you care about your rewards today. And so another way that you can actually formulate the reinforcement learning objective is not just this vanilla sum of rewards, but you can actually weight your rewards by what's called a discount factor, where that discount factor actually depends on T. So one of the ways we often do that is adding a discount factor that's
46:33
raised to the power of t. And this means that if a discount factor-- usually this is some value that's greater than 0 and less than or equal to 1. If this is 1, it's the same as the original formulation. And as you make it lower and lower, you're being more greedy. You're carrying a lot more about near-term rewards than about far rewards. We'll talk a little bit more about discount factors in a couple lectures.
47:08
Cool. As an aside, we've talked about how our policy is a distribution, and so it's not deterministic. It's stochastic. Why might we want to do that? There's a couple reasons why we might want our policy to not be deterministic. The first is that if you want to learn, you probably have to try different things. If you're learning how to play tennis, you probably shouldn't try the same exact technique over and over again. You need to actually explore and try different strategies in order to do it.
47:42
And so that can be represented as a stochastic policy that's trying a few different strategies and trying to figure out which strategy is best. And then in other cases, there's scenarios where you collect data from people or from different systems, and those people are taking different strategies. And if you want to model the way that they're approaching a problem, you may need your policies to be stochastic as well. And for this, we can also leverage tools
48:10
from generative models where you're actually thinking about not just a neural network that maps from A to B, but actually a generative model over actions given states and observations. I think this is the last thing we'll touch on and introduce, which is, how do we decide how good-- well, OK, second to last thing. How do we decide how good a policy is? And this is really useful to think about because if you don't know how good your policy is then
48:37
you're not going to be able to make your policy better. And the terminology that we use to think about this is referred to as a value function. So we often think about the value function for a policy of a given state, and we denote it typically as V pi of S. And essentially what this means is, it's the future expected reward if you start at S and then follow pi from then onward. And so it's basically this equation except starting at S under your current policy.
49:18
And similarly to this, another thing that we'll start talking about, not in Friday's lecture but starting next week, is Q value functions as well. And this is very similar to the value function. But instead of actually starting from S, then taking action a, and then following pi, where action a might be different from what policy pi might have taken. I'm not going to go into depth on what these two functions are today, but I wanted to introduce them because we'll be using them quite a lot.
49:50
And so it's good to start wrapping your head around what they are as we start to use them more in future lectures.
49:59
And then the types of algorithms we'll cover. The idea behind imitation learning is to mimic a policy that achieves high reward. We'll then next week start talking about policy gradients that try to directly differentiate this objective and actually learn from practice and experience. From there, we'll talk about actor critic algorithms that estimate the value of the current policy and use that to make policy better. We'll talk about value-based methods that
50:26
try to estimate the value of the optimal policy, and then use that to back out a good policy. And then lastly, we'll talk about model-based methods that actually learn how to model the dynamics of the world, try to predict the future, and then use that to improve the policy or for planning.
50:44
Now, you might be wondering, why do we have so many different algorithms for optimizing this objective? In supervised learning, we just have gradient descent. It works. It's great. It's simple. In reinforcement learning, things are less simple. Algorithms make different trade-offs and thrive under different assumptions. And so it's going to depend on, for example, how easy or cheap is it to collect data. And if you're in a simulator, for example, it's really, really cheap to collect
51:09
lots of data versus if you're handwriting responses or actually interacting with a person that's writing things, that's going to be more expensive. Also, how cheap and easy are different forms of supervision? Do you have demonstrations? Do you have detailed rewards? How important is stability and ease of use versus data efficiency? Maybe you have more time to change some of the hyperparameters if it doesn't need massive amounts of data. There's also some considerations with regard
51:40
to what your actions look like. How high-dimensional are they? Are they continuous or discrete? Is it easy to learn the dynamics model? So yeah, these are some of the reasons why we have a lot of different algorithms. And we'll be talking about different algorithms and where they're applied in the later parts of this course. So to recap, there's a lot of terminology and a lot of notation in reinforcement learning, and wanted to start introducing them today because we'll be starting to use them
52:08
quite a bit in the following lectures. So we talked about states. We talked about observations. We talked about actions, reward functions, and also initial state distribution and dynamics. These form what's called a Markov decision process. Or if you only have observations, it's called a partially-observable Markov decision process or an MDP or a POMDP. I didn't want to bring in this notation until the very end because it doesn't actually really matter what it's called, but it's useful.
52:35
In the long term if people are referring to MDPs, now you know what they're referring to. They're referring to this formulation of a reinforcement learning problem. And then we also talked about trajectories, policies, the objective of reinforcement learning, value functions, and Q functions briefly.
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