From 60a3b372d1268430ed5e390d61b032ccbe41b3a5 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Tue, 2 Sep 2025 11:22:30 -0500 Subject: [PATCH] update --- content/CSE510/CSE510_L3.md | 240 +++++++++++++++++++++++++++ content/CSE510/_meta.js | 1 + public/CSE510/MDP-optimal-policy.png | Bin 0 -> 73013 bytes 3 files changed, 241 insertions(+) create mode 100644 content/CSE510/CSE510_L3.md create mode 100644 public/CSE510/MDP-optimal-policy.png diff --git a/content/CSE510/CSE510_L3.md b/content/CSE510/CSE510_L3.md new file mode 100644 index 0000000..f1c6a22 --- /dev/null +++ b/content/CSE510/CSE510_L3.md @@ -0,0 +1,240 @@ +# CSE510 Lecture 3 + +## Introduction and Definition of MDPs + +### Definition and Examples + +#### Reinforcement Learning + +A computational framework for behavior +learning through reinforcement + +- RL is for an agent with the capacity to act +- Each action influences the agent’s future observation +- Success is measured by a scalar reward signal +- Goal: find a policy that maximizes expected total rewards + +Mathematical Model: Markov Decision Processes (MDP) + +#### Markov Decision Processes (MDP) + +A Finite MDP is defined by: + +- A finite set of states $s \in S$ +- A finite set of actions $a \in A$ +- A transition function $T(s, a, s')$ +- Probability that a from s leads to s', i.e., +$P(s'| s, a)$ +- Also called the model or the dynamics +- A reward function $R(s)$ ( Sometimes $R(s,a)$ or $R(s, a, s')$ ) +- A start state +- A start state +- Maybe a terminal state + +A model for sequential decisionmaking problem under uncertaint + +#### States + +- **Stat is a snapshot of everything that matters for the next decision** +- _Experience_ is a sequence of observations, actions, and rewards. +- _Observation_ is the raw input of the agent's sensors +- The state is a summary of the experience. + +$$ +s_t=f(o_1, r_1, a_1, \ldots, a_{t-1}, o_t, r_t) +$$ + +- The state can **include immediate "observations," highly processed observations, and structures built up over time from sequences of observations, memories** etc. +- In a fully observed environment, $s_t= f(o_t)$ + +#### Action + +- **Action = choice you make now** +- They are used by the agent to interact with the world. +- They can have many different temporal granularities and abstractions. +- Actions can be defined to be + - The instantaneous torques on the gripper + - The instantaneous gripper translation, rotation, opening + - Instantaneous forces applied to the objects + - Short sequences of the above + +#### Rewards + +- **Reward = score you get as a result** +- They are scalar values provided by the environment to the agent that indicate whether goals have been achieved, + - e.g., 1 if goal is achieved, 0 otherwise, or -1 for overtime step the goal is not achieved +- Rewards specify what the agent needs to achieve, not how to achieve it. +- The simplest and cheapest form of supervision, and surprisingly general. +- **Dense rewards are always preferred if available** + - e.g., distance changes to a goal. + +#### Dynamics or the Environment Model + +- **Transition = dice roll** the world makes after your choice. +- How the state change given the current state and action + +$$ +P(S_{t+1}=s'|S_t=s_t, A_t=a_t) +$$ + +- Modeling the uncertainty + - Everyone has their own "world model", capturing the physical laws of the world. + - Human also have their own "social model", by their values, beliefs, etc. +- Two problems: + - Planning: the dynamics model is known + - Reinforcement learning: the dynamics model is unknown + +#### Assumptions we have for MDP + +**First-Order Markovian dynamics** (history independence) + +- Next state only depend on current state and current action + +$$ +P(S_{t+1}=s'|S_t=s_t,A_t=a_t,S_1,A_1,\ldots,S_{t-1},A_{t-1}) = P(S_{t+1}=s'|S_t=s_t,A_t=a_t) +$$ + +**State-dependent** reward + +- Reward is a deterministic function of current state + +**Stationary dynamics**: do not depend on time + +$$ +P(S_{t+1}=s'|A_t,S_t) = P(S_{k+1}=s'|A_k,S_k),\forall t,k +$$ + +**Full observability** of the state + +- Though we can't predict exactly which state we will reach when we execute an action, after the action is executed, we know the new state. + +### Examples + +#### Atari games + +- States: raw RGB frames (one frame is not enough, so we use a sequence of frames, usually 4 frames) +- Action: 18 actions in joystick movement +- Reward: score changes + +#### Go + +- States: features of the game board +- Action: place a stone or resign +- Reward: win +1, lose -1, draw 0 + +#### Autonomous car driving + +- States: speed, direction, lanes, traffic, weather, etc. +- Action: steer, brake, throttle +- Reward: +1 for reaching the destination, -1 for honking from surrounding cars, -100 for collision (exmaple) + +#### Grid World + +A maze-like problem + +- The agent lives in a grid + +- States: position of the agent +- Noisy actions: east, south, west, north +- Dynamics: actions not always go as planned + - 80% of the time, the action North takes the agent north (if there is a wall, it stays) + - 10% of the time, the action North takes the agent west and 10% of the time, the action North takes the agent east +- Reward the agent receives each time step + - Small "living" reward each step (can be negative) + - Big reward for reaching the goal + +> [!NOTE] +> +> Due to the noise in the actions, it is insufficient to just output a sequence of actions to reach the goal. + +### Solution and its criterion + +### Solution to an MDP + +- Actions have stochastic effects, so the state we end up in is uncertain +- This means that we might end up in states where the remainder of the action sequence doesn't apply or is a bad choice +- A solution should tell us what the best action is for any possible situation/state that might arise + +### Policy as output to an MDP + +A stationary policy is a mapping from states to actions + +- $\pi: S \to A$ +- $\pi(s)$ is the action to take in state $s$ (regardless of the time step) +- Specifies a continuously reactive controller + +We don't want to output just any policy + +We want to output a good policy + +One that accumulates a lot of rewards + +### Value of a policy + +Value function + +$V:S\to \mathbb{R}$ associates value with each state + +$$ +\begin{aligned} +V^\pi(s) &= \mathbb{E}\left[\sum_{t=0}^\infty \gamma^t R(s_t)|s_0=s,a_t=\pi(s_t), s_{t+1}|s_t,a_t\sim P\right] \\ +&= \mathbb{E}\left[R(s_t) + \gamma \sum_{t=1}^\infty \gamma^{t-1} R(s_{t+1})|s_0=s,a_t=\pi(s_t), s_{t+1}|s_t,a_t\sim P\right] \\ +&= R(s) + \gamma \sum_{s'\in S} P(s'|s,\pi(s)) V^\pi(s') +\end{aligned} +$$ + +Future rewards "discounted" by $\gamma$ per time step + +We value the state by the expected total rewards from this state onwards, discounted by $\gamma$ for each time step. + +> A small $\gamma$ means model would short-sighted and reduce computation complexity. + +#### Bellman Equation + +Basically, it gives one step lookahead value of a policy. + +$$ +V^\pi(s) = R(s) + \gamma \sum_{s'\in S} P(s'|s,\pi(s)) V^\pi(s') +$$ + +Today's value = Today's reward + discounted future value + +### Optimal Policy and Bellman Optimality Equation + +The goal for a MDP is to compute or learn an optimal policy. + +- An optimal policy is one that achieves the highest value at any state + +$$ +\pi^* = \arg\max_\pi V^\pi(s) +$$ + +We define the optimal value function suing Bellman Optimality Equation (Proof left as an exercise) + +$$ +V^*(s) = R(s) + \gamma \max_{a\in A} \sum_{s'\in S} P(s'|s,a) V^*(s') +$$ + +The optimal policy is + +$$ +\pi^*(s) = \arg\max_{a\in A} \sum_{s'\in S} P(s'|s,a) V^*(s') +$$ + +![Optimal Policy](https://notenextra.trance-0.com/CSE510/MDP-optimal-policy.png) + +> [!NOTE] +> +> When $R(s)$ is small, the agent will prefer to take actions that avoids punishment in short term. + +### The existence of the optimal policy + +Theorem: for any Markov Decision Process + +- There exists an optimal policy +- There can be many optimal policies, but all optimal policies achieve the same optimal value function +- There is always a deterministic optimal policy for any MDP + +## Value Iteration + +## Policy Iteration diff --git a/content/CSE510/_meta.js b/content/CSE510/_meta.js index 6ca2d87..491ae8c 100644 --- a/content/CSE510/_meta.js +++ b/content/CSE510/_meta.js @@ -5,4 +5,5 @@ export default { }, CSE510_L1: "CSE510 Deep Reinforcement Learning (Lecture 1)", CSE510_L2: "CSE510 Deep Reinforcement Learning (Lecture 2)", + CSE510_L3: "CSE510 Deep Reinforcement Learning (Lecture 3)", } \ No newline at end of file diff --git a/public/CSE510/MDP-optimal-policy.png b/public/CSE510/MDP-optimal-policy.png new file mode 100644 index 0000000000000000000000000000000000000000..f58d93c73488cd7e04878b8ab60009d863373ae5 GIT binary patch literal 73013 zcmeFZc|6qb`}aMiQlF%zg|bJXY}vP%e2}shvTunIvW|UCDua+PvQ7w1wipZsL-sw{ z_p*&$V;IXYu2bLN@4xGD-P_~-=f3X#F=d$ZeJ;m&9M9M5DEyJO+PQyN{((Rs=N>*# 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