notes RL ayush.pdf

Chapter 1: Introduction 1.1 What is Deep Reinforcement Learning? Deep Reinforcement Learning (DRL): The integration of deep learning and reinforcement learning aimed at learning optimal actions to maximize rewards in various states of an environment. 1....

Chapter 1: Introduction 1.1 What is Deep Reinforcement Learning? Deep Reinforcement Learning (DRL): The integration of deep learning and reinforcement learning aimed at learning optimal actions to maximize rewards in various states of an environment. 1. Deep Learning: Purpose: Approximates functions for high-dimensional, complex problems where exact solutions are infeasible with tabular methods. Techniques: Utilizes deep neural networks. Applications: Image recognition, speech recognition, and pedestrian recognition in images. 2. Reinforcement Learning (RL): Purpose: Learns optimal actions through feedback from the environment. Mechanism: Operates via trial and error, learning from the outcomes of actions. Applications: Solving sequential decision problems like playing games, autonomous driving, and controlling robotic systems. Interaction: The agent interacts with complex, high-dimensional environments. Feedback Loop: Actions are taken, feedback is received, and the agent learns to optimize its actions based on this feedback. 1.1.1 Deep Learning Function Approximation: Uses neural networks to approximate complex functions in high-dimensional spaces. Progress: Significant advancements in recognizing images and understanding spoken language. 1.1.2 Reinforcement Learning Learning Paradigm: RL is distinct from supervised learning and unsupervised learning as it learns from the consequences of actions rather than from static datasets. Exploration and Exploitation: Balances exploring new actions and exploiting known successful actions to optimize rewards. 1.1.3 Applications of Deep Reinforcement Learning Autonomous Driving: Systems learn to navigate and make driving decisions. Game Playing: Achieving superhuman performance in games like Atari, Go, poker, and StarCraft. Molecular Recombination: Designing molecules for pharmaceuticals. Robotics: Teaching robots to perform complex tasks and maneuvers. 1.1.4 Four Related Fields 1. Psychology: Studies human learning processes which inspire DRL methodologies. Explores how humans learn through interaction and feedback. 2. Mathematics: Provides the theoretical foundation for algorithms used in DRL. Involves areas like probability, statistics, and optimization. 3. Engineering: Focuses on practical applications of DRL technologies.  Develops systems and machines that implement DRL algorithms. 4. Biology: Examines biological learning processes that influence DRL. Investigates how living organisms adapt and learn from their environment. 1.2 Three Machine Learning Paradigms 1.2.1 Supervised Learning Definition: Learning from labeled data where the correct output is provided. Example: Image classification where each image is labeled with the object it contains. 1.2.2 Unsupervised Learning Definition: Learning patterns from unlabeled data. Example: Clustering algorithms that group similar data points together without predefined labels. 1.2.3 Reinforcement Learning Definition: Learning from the consequences of actions by receiving rewards or penalties. Example: An agent playing a game learns to maximize its score by trying different moves and learning from wins and losses.  Questions ML (Machine Learning) 1 Questions and Answers 1. What is Intelligence? Answer: Intelligence is the ability to learn, understand, and apply knowledge to solve prob- lems and adapt to new situations. 2. What is Machine Learning? Answer: Machine Learning is a subset of artificial intelligence that involves training algo- rithms to make predictions or decisions based on data. 3. What is Accuracy? Answer: Accuracy is the measure of correctly predicted instances out of the total instances in a dataset. 4. What is the Confusion Matrix? Answer: A Confusion Matrix is a table used to evaluate the performance of a classification model by showing the true positives, true negatives, false positives, and false negatives. 5. What is Overfitting? Answer: Overfitting occurs when a model learns the training data too well, including noise and outliers, resulting in poor performance on new, unseen data. 6. What is Generalization? Answer: Generalization is the ability of a machine learning model to perform well on new, unseen data by learning the underlying patterns rather than memorizing the training data. 7. Explain Bias-Variance. Answer: The Bias-Variance trade-off is the balance between two sources of error in machine learning models. High bias can cause underfitting, while high variance can cause overfitting. 8. What is Regularization? Answer: Regularization is a technique used to prevent overfitting by adding a penalty to the loss function for large coefficients in the model. 9. What is End-to-end learning? Answer: End-to-end learning involves training a model to directly map input data to the desired output in a single process, typically using deep neural networks. 10. What is Loss? Answer: Loss is a measure of how well a machine learning model’s predictions match the actual target values. It is used to guide the optimization of the model. 11. What is CNN? Answer: CNN (Convolutional Neural Network) is a type of deep learning model designed to process and analyze visual data by using convolutional layers to extract features. 1 12. What is RNN? Answer: RNN (Recurrent Neural Network) is a type of neural network designed for sequential data, where connections between nodes form a directed graph along a temporal sequence. 13. What is LSTM? Answer: LSTM (Long Short-Term Memory) is a type of RNN architecture that is capable of learning long-term dependencies and avoiding the vanishing gradient problem. 14. What is ImageNet? Answer: ImageNet is a large-scale visual database used for benchmarking image classification and object detection algorithms. 15. What is PyTorch? Answer: PyTorch is an open-source deep learning framework that provides flexible and efficient tools for building and training neural networks. 2  Notes on Chapter 2: Tabular Value-Based Reinforcement Learning 1 Core Concepts Agent: An entity that interacts with the environment, making decisions and learning from the outcomes. Environment: The world in which the agent operates, providing states and rewards based on the agent’s actions. MDP (Markov Decision Process): A mathematical framework for modeling decision-making with states, actions, transitions, rewards, and policies. State: A representation of the current situation in the environment. Action: A decision made by the agent that affects the environment. Reward: Feedback from the environment based on the action taken. Value: The expected cumulative reward of a state or state-action pair. Policy: A strategy used by the agent to determine actions based on states. Exploration and Exploitation: The balance between exploring new actions to find better re- wards and exploiting known actions that provide good rewards. 2 Core Problem Objective: Learn a policy from interactions with the environment to maximize cumulative reward. 3 Core Algorithms Value Iteration: Iteratively updates the value of each state based on possible actions and rewards to compute the optimal policy. Temporal Difference Learning: Updates value estimates based on the difference between esti- mated and actual rewards. Q-learning: An off-policy algorithm that learns the value of state-action pairs and updates values based on the best possible actions. 4 Finding a Supermarket Example Scenario: Moving to a new city and finding a supermarket without a map. Actions: Exploring the city to find the supermarket and noting the route. Learning: Follow the same route (exploitation) or try a new route (exploration) to find a shorter path. 1 5 Sequential Decision Problems Definition: Problems where the agent makes a sequence of decisions to maximize cumulative future rewards. Examples: Grid worlds, mazes, and box puzzles where the agent navigates to reach a goal. 5.1 Grid Worlds Description: A rectangular grid where the agent moves to reach a goal while avoiding obstacles. Goal: Find the sequence of actions to reach the goal state from the start state. 5.2 Mazes and Box Puzzles Mazes: Complex environments requiring trajectory planning. Box Puzzles (e.g., Sokoban): Puzzles where the agent pushes boxes to specific locations, with irreversible actions. 6 Tabular Value-Based Agents 6.1 Agent and Environment Agent: Learns from interacting with the environment. Environment: Provides states, rewards, and transitions based on the agent’s actions. Interaction: The agent takes actions, receives new states and rewards, and updates its policy based on the rewards received. 6.2 Markov Decision Process (MDP) Formalism: Defined as a 5-tuple (S, A, Ta , Ra , γ): S: Finite set of states. A: Finite set of actions. Ta : Transition probabilities between states. Ra : Reward function for state transitions. γ: Discount factor for future rewards. 6.2.1 State S Representation: The configuration of the environment. Types: Deterministic Environment: Each action leads to a specific state. Stochastic Environment: Actions can lead to different states based on probabilities. 6.2.2 State Representation Description: How states are defined and represented in the environment. 6.2.3 Deterministic and Stochastic Environment Deterministic: Predictable outcomes for actions. Stochastic: Probabilistic outcomes for actions. 2 6.2.4 Action A Types: Discrete: Finite set of actions (e.g., moving in a grid). Continuous: Infinite set of actions (e.g., robot movements). 6.2.5 Irreversible Environment Action Definition: Actions that cannot be undone once taken. 6.2.6 Discrete or Continuous Action Space Discrete: Limited number of actions. Continuous: Unlimited range of actions. 6.2.7 Transition Ta Function: Describes how actions change states. Graph View: States and actions as nodes, transitions as edges. 6.2.8 Graph View of the State Space Description: Visual representation of states and transitions. 6.2.9 Trial and Error, Down and Up Method: Learning through experimentation and feedback. 6.2.10 Reward Ra Definition: Feedback from the environment based on the action taken. Sparse Rewards: Rewards are only given for specific states (e.g., terminal states in chess). Dense Rewards: Rewards are given for every state. 6.2.11 Discount Factor γ Purpose: Balances the importance of future rewards versus immediate rewards. Range: [0, 1] γ = 0: Only considers immediate rewards. γ = 1: Treats all future rewards equally. 6.2.12 Policy π Definition: A strategy that maps states to actions. Objective: Find the optimal policy π ∗ that maximizes the cumulative reward. 6.3 MDP Objective 6.3.1 Trace τ Definition: A sequence of state-action pairs. Return R: The cumulative reward of a trace. 6.3.2 State Value V Definition: Expected cumulative reward starting from a state s and following policy π. Equation: "∞ # X π i V (s) = Eτ γ · rt+i | st = s i=0 3 6.3.3 State-Action Value Q Definition: Expected cumulative reward starting from a state s, taking action a, and following policy π. Equation: "∞ # X Qπ (s, a) = Eτ γ i · rt+i | st = s, at = a i=0 6.3.4 Reinforcement Learning Objective Goal: Find the policy that maximizes the expected cumulative reward. 6.3.5 Bellman Equation State Value: ! X ′ ′ V (s) = max R(s, a) + γ Ta (s, s )V (s ) a s′ State-Action Value: X Q(s, a) = R(s, a) + γ Ta (s, s′ ) max ′ Q(s′ , a′ ) a s′ 6.4 MDP Solution Methods 6.4.1 Hands On: Value Iteration in Gym OpenAI Gym: A toolkit for developing and comparing reinforcement learning algorithms. Taxi Example: Solve the taxi problem using value iteration by updating state values based on the Bellman equation. 6.4.2 OpenAI Gym Description: A platform for developing and comparing reinforcement learning algorithms. Installation: Use pip install gym to install the Gym environment. 6.4.3 Taxi Example with Value Iteration Scenario: Taxi problem where the agent picks up and drops off passengers. Implementation: Uses value iteration to solve the MDP. Steps: 1. Initialize Environment: Set up the Taxi environment. 2. Define Value Function: Initialize the value function. 3. Iterate Value Function: Update the value function using Bellman updates until convergence. 4. Build Policy: Derive the policy from the value function. 5. Execute Policy: Apply the policy to the environment to evaluate performance. 6.4.4 Model-Free Learning Monte Carlo Sampling: Learning from complete episodes by averaging returns. Temporal Difference Learning: Learning from incomplete episodes by bootstrapping. Bias-Variance Trade-off : Monte Carlo methods have high variance and low bias, while temporal difference methods have low variance and high bias. 6.4.5 Monte Carlo Sampling Description: Generates random episodes and uses returns to update the value function. 4 6.4.6 Temporal Difference Learning Description: Updates value estimates based on differences between successive state values. 6.4.7 Exploration Bandit Theory: Balances exploration and exploitation. -greedy Exploration: Chooses a random action with probability , and the best-known action with probability 1-. 6.4.8 Bandit Theory Description: Studies strategies for choosing actions to maximize rewards with minimal trials. 6.4.9 -greedy Exploration Description: Introduces randomness in action selection to ensure exploration. 6.4.10 Off-Policy Learning On-Policy SARSA: Updates policy based on the actions taken by the current policy. Off-Policy Q-Learning: Updates policy based on the best possible actions, not necessarily those taken by the current policy. 6.4.11 On-Policy SARSA Description: Uses the state-action value of the current policy to update. 6.4.12 Off-Policy Q-Learning Description: Uses the maximum state-action value to update, regardless of the action taken. 6.4.13 Sparse Rewards and Reward Shaping Sparse Rewards: Rewards given only at specific states, making learning more difficult. Reward Shaping: Modifying the reward function to make learning easier by providing more frequent feedback. 6.4.14 Hands On: Q-learning on Taxi Scenario: Apply Q-learning to the Taxi problem. Implementation: 1. Initialize Environment: Set up the Taxi environment. 2. Define Q-Table: Initialize Q-values for all state-action pairs. 3. Run Episodes: For each episode, reset the environment and take actions based on -greedy policy. 4. Update Q-Values: Use Q-learning update rule to adjust Q-values. 5. Evaluate Performance: Assess the learned policy by its ability to complete tasks efficiently. 6.4.15 Tuning Learning Rate Description: Adjusting hyperparameters like exploration parameter (), discount factor (), and learning rate () to improve performance. 5 7 Classic Gym Environments 7.1 Mountain Car and Cartpole Mountain Car: The agent must drive a car up a steep hill by building momentum through back-and-forth movements. Cartpole: The agent must balance a pole on a moving cart by applying forces. 7.1.1 Mountain Car Objective: Drive the car to the top of the mountain using momentum. Challenge: The car’s engine is not strong enough to climb the mountain in one go. 7.1.2 Cartpole Objective: Balance a pole on a moving cart. Challenge: Apply forces to keep the pole balanced and prevent it from falling over. 7.2 Path Planning and Board Games Path Planning: Finding the optimal route in a grid or maze. Board Games: Strategy games like chess and checkers. 7.2.1 Path Planning Description: Solving navigation tasks in mazes or environments with obstacles. Example: Sokoban, where the agent pushes boxes to specific locations. 7.2.2 Board Games Description: Planning and strategy games. Example: Chess and checkers, where the agent learns optimal moves. 8 Bootstrapping in Temporal Difference Learning Bootstrapping: The process of refining an estimate using successive updates. In TD learning, boot- strapping involves updating value estimates using other learned estimates rather than waiting for the final outcome. Process: TD Learning updates the value of a state based on the value of the next state and the reward received: V (s) ← V (s) + α [r + γV (s′ ) − V (s)] Here, s is the current state, s′ is the next state, r is the reward received after moving from s to s′ , α is the learning rate, and γ is the discount factor. Example: Consider an agent navigating a grid world. When it moves from state s to state s′ and receives a reward r, it immediately updates the value of s using the estimated value of s′. 9 Bias-Variance Trade-off Definition: The bias-variance trade-off is a key concept in machine learning, balancing the error intro- duced by bias and variance. Bias: The error introduced by approximating a real-world problem, which may be complex, by a simplified model. Variance: The error introduced by the model’s sensitivity to small fluctuations in the training set. Trade-off : - High Bias: Simplistic models (e.g., linear models) might not capture the underlying pattern (underfitting). - High Variance: Complex models (e.g., deep neural networks) might capture noise in the training data as if it were a true pattern (overfitting). 6  In Reinforcement Learning: - Monte Carlo Methods: Low bias, high variance. They wait until the end of an episode to update the values, leading to high variance because each complete return can vary significantly. - Temporal Difference Methods: High bias, low variance. They update the values after each step using bootstrapping, leading to more stable updates but introducing bias as they rely on current estimates. Example: - Monte Carlo: A robot learning a task by simulating entire episodes. It gathers complete returns, resulting in high variance but low bias as it does not bootstrap. - TD Learning: The same robot using TD learning updates its value estimates at each step, reducing variance but introducing bias due to bootstrapping. 10 Summary and Further Reading Summary: Reinforcement learning solves sequential decision problems by learning policies through interaction with the environment. Further Reading: Explore advanced topics and algorithms in reinforcement learning. 11 Exercises We will end with questions on key concepts, with programming exercises to build up more experience. Questions The questions below are meant to refresh your memory, and should be answered with yes, no, or short answers of one or two sentences. 1. In reinforcement learning, the agent can choose which training examples are gener- ated. Why is this beneficial? What is a potential problem? Answer: This is beneficial because it allows the agent to explore different parts of the state space, leading to a more comprehensive understanding of the environment. A potential prob- lem is that the agent might spend too much time exploring suboptimal parts of the state space, leading to inefficient learning. 2. What is Grid world? Answer: Grid world is a type of environment used in reinforcement learning, where an agent navigates a rectangular grid to reach a goal while avoiding obstacles. 3. Which five elements does an MDP have to model reinforcement learning problems? Answer: The five elements are States (S), Actions (A), Transition probabilities (Ta ), Rewards (Ra ), and Discount factor (γ). 4. In a tree diagram, is successor selection of behavior up or down? Answer: Successor selection of behavior is down. 5. In a tree diagram, is learning values through backpropagation up or down? Answer: Learning values through backpropagation is up. 6. What is τ ? Answer: τ is a trace, which is a sequence of state-action pairs. 7. What is π(s)? Answer: π(s) is a policy, which defines the action taken by the agent in state s. 8. What is V (s)? Answer: V (s) is the state value, representing the expected cumulative reward starting from state s and following policy π. 7  9. What is Q(s, a)? Answer: Q(s, a) is the state-action value, representing the expected cumulative reward start- ing from state s, taking action a, and following policy π. 10. What is dynamic programming? Answer: Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems, using the principle of optimality. 11. What is recursion? Answer: Recursion is a method of solving problems where the solution depends on solutions to smaller instances of the same problem. 12. Do you know a dynamic programming method to determine the value of a state? Answer: Yes, value iteration is a dynamic programming method used to determine the value of a state. 13. Is an action in an environment reversible for the agent? Answer: Not always. Some environments have irreversible actions. 14. Mention two typical application areas of reinforcement learning. Answer: Two typical application areas are game playing (e.g., chess, Go) and robotics (e.g., robotic control). 15. Is the action space of games typically discrete or continuous? Answer: The action space of games is typically discrete. 16. Is the action space of robots typically discrete or continuous? Answer: The action space of robots is typically continuous. 17. Is the environment of games typically deterministic or stochastic? Answer: The environment of games can be either deterministic or stochastic, but many classic board games have deterministic environments. 18. Is the environment of robots typically deterministic or stochastic? Answer: The environment of robots is typically stochastic due to the unpredictability of real-world conditions. 19. What is the goal of reinforcement learning? Answer: The goal of reinforcement learning is to learn a policy that maximizes the cumulative reward. 20. Which of the five MDP elements is not used in episodic problems? Answer: The discount factor (γ) is typically less emphasized in episodic problems where episodes have a clear termination. 21. Which model or function is meant when we say “model-free” or “model-based”? Answer: ”Model-free” refers to methods that do not use a model of the environment’s dy- namics (e.g., Q-learning), while ”model-based” refers to methods that use a model of the environment to make decisions (e.g., value iteration). 22. What type of action space and what type of environment are suited for value-based methods? Answer: Value-based methods are suited for discrete action spaces and environments where the state and action spaces are not excessively large. 8 23. Why are value-based methods used for games and not for robotics? Answer: Value-based methods are used for games because games often have discrete action spaces and clearly defined rules. In contrast, robotics often involves continuous action spaces and highly dynamic, unpredictable environments that are better suited for policy-based or model-based methods. 24. Name two basic Gym environments. Answer: Two basic Gym environments are Mountain Car and Cartpole. 12 In Class RL Questions 1. What is the biological name of RL? Operant Conditioning. 2. What two elements are central to RL Interaction? Agent and Environment. 3. What problems is RL applied to? Sequential decision-making problems. 4. What is the main problem of assigning reward? Defining a reward function that accurately reflects long-term objectives without unintended side effects. 5. How can we model sequential decision problems? Using Markov Decision Processes (MDPs). 6. Name the 5 MDP elements. States (S), Actions (A), Transition probabilities (Ta ), Rewards (Ra ), Discount factor (γ). 7. Which of these are in the agent, which are in the environment? Agent: Actions, Policy; Environment: States, Transition probabilities, Rewards, Discount factor. 8. What goes up, what goes down? Rewards go up, actions go down. 9. What is the name of the recursion relation central to the value function? Bellman Equation. 10. What is the name of an algorithm that computes it? Value Iteration or Policy Iteration. 11. What is model-free? Methods that allow the agent to learn directly from raw experience without a model of the environment dynamics (e.g., Q-learning). 12. What is model-based? Methods that use a model of the environment to make decisions (e.g., Dyna-Q). 13. Why do we need Temporal Difference? 9  To combine ideas from Monte Carlo methods and dynamic programming, updating value estimates based on partial returns that allows agent to learn directly from raw experience without a model of the environment dynamics. 14. Why do we need Exploration and Exploitation? Mention a simple algorithm. To balance learning new information and utilizing known information; ϵ-greedy. 15. How can we compute minimal regret? No regret is the difference between the best Using regret minimization algorithms like the Boltzmann policy. action reward and the action took 16. SARSA is X and Q-learning is Y SARSA is on-policy, and Q-learning is off-policy. 17. What is Gym? OpenAI Gym is a toolkit for developing and comparing reinforcement learning algorithms. 10  Notes on Chapter 3: Deep Learning 1 Deep Learning 1.1 Core Concepts Deep Learning: A subset of machine learning that involves training neural networks with multiple layers (deep networks) to model complex patterns in data. Deep learning is particularly effective for tasks involving high-dimensional data such as images, audio, and text. Neural Networks: Computational models inspired by the human brain, consisting of intercon- nected layers of nodes (neurons) that process input data and learn patterns through training. 1.2 Core Problem Core Problem in Deep Learning: The main challenge in deep learning is to train deep neu- ral networks effectively to generalize well on unseen data. This involves optimizing the network parameters to minimize a loss function, ensuring stability and convergence during training, and dealing with issues such as overfitting and vanishing gradients. 1.3 Core Algorithm Gradient Descent: A key optimization algorithm used in deep learning to minimize the loss function by iteratively updating the network parameters in the direction of the negative gradient of the loss. θ ← θ − α∇θ J(θ) where θ are the parameters, α is the learning rate, and J(θ) is the loss function. 1.4 End-to-end Learning End-to-end Learning: A training approach where raw input data is directly mapped to the desired output through a single, integrated process, typically using deep neural networks. 2 Large, High-Dimensional Problems Large, high-dimensional problems are characterized by vast and complex state and action spaces, which are common in applications such as video games and real-time strategy games. 2.1 Atari Arcade Games Atari Games: These games serve as a benchmark in deep reinforcement learning research. They present a variety of tasks that are challenging for AI due to their high-dimensional state spaces (e.g., raw pixel inputs) and complex dynamics. 2.2 Real-Time Strategy and Video Games Real-Time Strategy (RTS) Games: These games involve managing resources, strategic plan- ning, and real-time decision-making, making them more complex than arcade games. They feature larger state and action spaces, requiring sophisticated AI techniques. 1 3 Deep Value-Based Agents Deep value-based agents use deep learning to approximate value functions, enabling them to handle large and high-dimensional state spaces. 3.1 Generalization of Large Problems with Deep Learning Generalization is crucial for deep learning models to perform well on unseen data, especially in large, high-dimensional problems. 3.1.1 Minimizing Supervised Target Loss Supervised Target Loss: In supervised learning, the loss function measures the difference be- tween predicted outputs and actual targets. Common loss functions include Mean Squared Error (MSE) for regression tasks and Cross-Entropy Loss for classification tasks. n 1X MSE = (yi − ŷi )2 n i=1 where yi are the true values and ŷi are the predicted values. 3.1.2 Bootstrapping Q-Values Q-Learning: A reinforcement learning algorithm that updates Q-values using the Bellman equa- tion. Bootstrapping refers to using current estimates to update future estimates.   Q(s, a) ← Q(s, a) + α r + γ max ′ Q(s′ , a′ ) − Q(s, a) a 3.1.3 Deep Reinforcement Learning Target-Error Target-Error: In deep reinforcement learning, target-error refers to the difference between pre- dicted Q-values and target Q-values used for training the network. Reducing this error is essential for stable learning. 3.2 Three Challenges Deep value-based agents face three main challenges: coverage, correlation, and convergence. 3.2.1 Coverage Coverage: Ensuring that the agent explores all relevant parts of the state space to learn a com- prehensive policy. Inadequate coverage can lead to poor generalization and suboptimal policies. 3.2.2 Correlation Correlation: In sequential decision problems, consecutive states are often correlated, leading to inefficient learning and convergence issues. Techniques like experience replay help decorrelate the training data. 3.2.3 Convergence Convergence: Ensuring that the learning algorithm converges to an optimal policy. This involves addressing issues like the deadly triad, which includes function approximation, bootstrapping, and off-policy learning. 3.2.4 Deadly Triad Deadly Triad: The combination of function approximation, bootstrapping, and off-policy learning can lead to instability and divergence in reinforcement learning algorithms. 2 3.3 Stable Deep Value-Based Learning To achieve stable learning in deep value-based agents, several techniques are used, such as decorrelat- ing states, infrequent updates of target weights, and hands-on practice with examples like DQN and Breakout. 3.3.1 Decorrelating States Experience Replay: A technique where the agent stores past experiences (state, action, reward, next state) in a replay buffer and samples from this buffer to break correlations in the training data. 3.3.2 Infrequent Updates of Target Weights Target Network: A network used to provide stable target values in Q-learning. The weights of the target network are updated less frequently than the main network, which helps in stabilizing learning. 3.3.3 Hands On: DQN and Breakout Gym Example DQN (Deep Q-Network): Combines Q-learning with deep neural networks to handle high- dimensional state spaces, such as raw pixels from games. 3.4 Improving Exploration Exploration is crucial in reinforcement learning to ensure that the agent can discover optimal policies. Various methods help improve exploration. 3.4.1 Overestimation Overestimation: A problem in Q-learning where the estimated Q-values can be overly optimistic. This can be mitigated using techniques like Double Q-learning. 3.4.2 Prioritized Experience Replay Prioritized Experience Replay: An extension of experience replay where transitions are sam- pled based on their TD error, giving priority to experiences that are more surprising or informative. 3.4.3 Advantage Function Advantage Function: A measure of the relative value of an action compared to the average value of all actions in that state. It helps reduce variance in policy gradient methods. A(s, a) = Q(s, a) − V (s) 3.4.4 Distributional Methods Distributional RL: An approach where the distribution of possible future rewards is modeled rather than just the expected value. This provides a richer representation of uncertainty. 3.4.5 Noisy DQN Noisy DQN: An extension of DQN where noise is added to the parameters of the network to encourage exploration. 4 Atari 2600 Environments Atari 2600 games are commonly used for benchmarking reinforcement learning algorithms due to their diverse and challenging environments. 3 4.1 Network Architecture Network Architecture: The structure of the neural network used in deep reinforcement learning, typically involving convolutional layers for processing visual inputs from Atari games. 4.2 Benchmarking Atari Benchmarking: Evaluating the performance of reinforcement learning algorithms on a standard set of Atari 2600 games to compare effectiveness and efficiency. 5 Conclusion 6 Summary and Further Reading 6.1 Summary Deep learning and reinforcement learning can be combined to solve large, high-dimensional prob- lems. Techniques like experience replay, target networks, and prioritized experience replay are essential for stable and efficient learning. 6.2 Further Reading Explore additional resources on deep reinforcement learning, such as research papers, books, and online courses to gain a deeper understanding of the field. 7 Exercise 1. What is Gym? Answer: Gym is a toolkit for developing and comparing reinforcement learning algorithms, providing environments for training and testing RL agents. 2. What are the Stable Baselines? Answer: Stable Baselines are a set of reliable implementations of reinforcement learning algorithms in Python, designed to provide stable and efficient learning. 3. The loss function of DQN uses the Q-function as target. What is a consequence? Answer: A consequence is that it can lead to overestimation bias in the Q-values, potentially resulting in suboptimal policies. 4. Why is the exploration/exploitation trade-off central in reinforcement learning? Answer: It is central because the agent needs to balance exploring new actions to discover better rewards and exploiting known actions to maximize rewards. 5. Name one simple exploration/exploitation method. Answer: ϵ-greedy is a simple exploration/exploitation method. 6. What is bootstrapping? Answer: Bootstrapping is a method in reinforcement learning where current estimates are used to update future estimates. 7. Describe the architecture of the neural network in DQN. 4  Answer: The neural network in DQN typically consists of convolutional layers followed by fully connected layers to process high-dimensional input like raw pixel data from games. 8. Why is deep reinforcement learning more susceptible to unstable learning than deep supervised learning? Answer: Deep reinforcement learning is more susceptible due to the combination of function approximation, bootstrapping, and the use of sequentially correlated data. 9. What is the deadly triad? Answer: The deadly triad refers to the combination of function approximation, bootstrap- ping, and off-policy learning that can lead to instability and divergence in reinforcement learning algorithms. 10. How does function approximation reduce stability of Q-learning? Answer: Function approximation can introduce estimation errors that accumulate over time, reducing the stability and leading to divergence. 11. What is the role of the replay buffer? Answer: The replay buffer stores past experiences to break correlations in the training data and improve learning stability. 12. How can correlation between states lead to local minima? Answer: Correlation between states can cause the agent to get stuck in suboptimal policies by repeatedly reinforcing similar experiences. 13. Why should the coverage of the state space be sufficient? Answer: Sufficient coverage ensures the agent explores all relevant parts of the state space to learn a comprehensive and optimal policy. 14. What happens when deep reinforcement learning algorithms do not converge? Answer: When they do not converge, the agent’s performance becomes unstable and unpre- dictable, failing to learn a useful policy. 15. How large is the state space of chess estimated to be? 1047 , 10170 , or 101685 ? Answer: The state space of chess is estimated to be 1047. 16. How large is the state space of Go estimated to be? 1047 , 10170 , or 101685 ? Answer: The state space of Go is estimated to be 10170. 17. How large is the state space of StarCraft estimated to be? 1047 , 10170 , or 101685 ? Answer: The state space of StarCraft is estimated to be 101685. 18. What does the rainbow in the Rainbow paper stand for, and what is the main message? Answer: The ”Rainbow” stands for combining several improvements to the DQN algorithm. The main message is that integrating multiple enhancements can significantly boost perfor- mance in reinforcement learning. 19. Mention three Rainbow improvements that are added to DQN. Answer: Three Rainbow improvements are Double Q-learning, Prioritized Experience Re- play, and Dueling Network Architectures. 5 8 In Class Questions 1. What is Deep Reinforcement Learning? Answer: Deep Reinforcement Learning combines reinforcement learning (RL) and deep learning (DL), using deep neural networks to approximate value functions or policies in high- dimensional state and action spaces. 2. What is the Curse of Dimensionality? Answer: The Curse of Dimensionality refers to the exponential increase in computational complexity and data requirements as the number of dimensions (features) in the input space grows, making it difficult to learn and generalize. 3. What is ALE? Answer: ALE (Arcade Learning Environment) is a platform used for evaluating the perfor- mance of reinforcement learning algorithms using Atari 2600 games. 4. What is End-to-end in DRL for Atari? Answer: End-to-end in DRL for Atari means training a deep neural network directly from raw pixel inputs to game actions without manually designing features. 5. What is the biggest challenge in DRL for Atari? Answer: The biggest challenge in DRL for Atari is handling the high-dimensional input space and learning effective policies from raw pixel data. 6. What is the deadly triad? Answer: The deadly triad in reinforcement learning refers to the combination of function approximation, bootstrapping, and off-policy learning, which can lead to instability and di- vergence. 7. What is the Convergence problem in DRL? Answer: The convergence problem in DRL refers to the difficulty in ensuring that learning algorithms converge to an optimal policy, especially in the presence of the deadly triad and unstable training dynamics. 8. How does DQN solve the stability problem? Answer: DQN solves the stability problem using techniques like experience replay and target networks to break correlations in the data and provide stable target values. 9. What is Rainbow? Name three approaches. Answer: Rainbow is an integrated approach combining several improvements to DQN. Three approaches included in Rainbow are Double Q-learning, Prioritized Experience Replay, and Dueling Network Architectures. 10. What is Mujoco? Answer: Mujoco (Multi-Joint dynamics with Contact) is a physics engine used for simulating complex robotic and biomechanical systems, often used in reinforcement learning research. 11. What are the Stable Baselines? 6  Answer: Stable Baselines are a set of reliable implementations of reinforcement learning algorithms in Python, designed to provide stable and efficient learning. 12. Gym = X and Stable Baselines = Y Answer: Gym = Environments for training and testing RL agents, Stable Baselines = Implementations of RL algorithms. 13. What is the Zoo? Answer: The Zoo typically refers to a collection or repository of pre-trained reinforcement learning models or a suite of environments and benchmarks for evaluating RL algorithms. 7  Notes on Chapter 4: Policy-Based Reinforcement Learning 1 Core Concepts Policy-Based Reinforcement Learning: A method that directly optimizes the policy that the agent follows, without explicitly using a value function. This approach is particularly effective in continuous action spaces where value-based methods may become unstable. 2 Core Problem Core Problem in Policy-Based RL: Finding the optimal policy in environments with continuous action spaces, such as robotics, self-driving cars, and real-time strategy games. The challenge lies in efficiently exploring and exploiting the action space to improve the policy. 3 Core Algorithms Policy Gradient Methods: Optimize the policy by following the gradient of expected reward with respect to the policy parameters. Examples include REINFORCE and Actor-Critic methods. 4 Jumping Robots Example: Jumping robots in continuous action spaces illustrate the need for policy-based methods to handle the infinite possibilities of action values. 5 Continuous Problems Continuous Problems: These are problems where the action space is continuous rather than discrete. Examples include robotic control and real-time strategy games where actions can take any value within a range. 5.1 Continuous Policies Definition: Policies that can take on any value within a range, suitable for environments where actions are not discrete but continuous. Equation: The policy π(a|s, θ) represents the probability of taking action a in state s parameter- ized by θ. Example: A robotic arm moving over arbitrary angles. 5.2 Stochastic Policies Definition: Policies that introduce randomness in action selection to encourage exploration and prevent premature convergence to suboptimal actions. Equation: The stochastic policy is often modeled by a probability distribution, e.g., π(a|s, θ) = N (µ(s, θ), σ(s, θ)) for a Gaussian distribution. Example: A self-driving car deciding between multiple safe paths. 1 5.3 Environments: Gym and MuJoCo Gym: A toolkit for developing and comparing reinforcement learning algorithms with a wide variety of environments. MuJoCo: A physics engine for simulating continuous control tasks, often used in reinforcement learning research for robotics. 5.3.1 Robotics Application: Using policy-based methods to control robotic arms, drones, and other mechanical systems with continuous actions. 5.3.2 Physics Models Application: Simulating realistic physical interactions in environments to train agents for tasks like walking, jumping, or manipulating objects. 5.3.3 Games Application: Training agents in complex games with continuous actions, such as strategy games and simulations. 6 Policy-Based Agents Policy-Based Agents: Agents that use policy gradient methods to optimize their actions. These agents directly learn the policy that maps states to actions. 6.1 Policy-Based Algorithm: REINFORCE Algorithm 1 REINFORCE Algorithm 1: Initialize policy parameters θ 2: for each episode do 3: Generate episode using policy π(a|s, θ) 4: for each step in episode do 5: Compute return R 6: Update policy parameters: θ ← θ + α∇θ log π(a|s, θ)R 7: end for 8: end for Equation: The policy gradient update rule is θ ← θ + α∇θ log π(a|s, θ)R where R is the return. Example: Training a policy to maximize the reward in a game by directly updating the policy parameters based on the observed returns. 6.2 Online and Batch Online: Updating the policy parameters after every episode. Batch: Accumulating gradients over multiple episodes and then updating the parameters. 6.3 Bias-Variance Trade-Off in Policy-Based Methods Explanation: Balancing the trade-off between bias (error due to approximations) and variance (error due to randomness) to ensure stable and efficient learning. 2 6.4 Actor Critic Bootstrapping Algorithm 2 Actor-Critic Algorithm 1: Initialize actor parameters θ and critic parameters ϕ 2: for each episode do 3: Generate episode using policy π(a|s, θ) 4: for each step in episode do 5: Compute TD error: δ = r + γV (s′ , ϕ) − V (s, ϕ) 6: Update critic parameters: ϕ ← ϕ + βδ∇ϕ V (s, ϕ) 7: Update actor parameters: θ ← θ + α∇θ log π(a|s, θ)δ 8: end for 9: end for Temporal Difference Bootstrapping: Using estimates of future rewards to update the value function incrementally. Example: Using Actor-Critic methods to train a robot to navigate through an environment by continuously updating both the policy and value functions based on observed rewards and states. 6.5 Temporal Difference Bootstrapping Definition: A method that updates value estimates based on estimates of future values. It combines the ideas of dynamic programming and Monte Carlo methods. Equation: The update rule for the state-value function V is: V (st ) ← V (st ) + α [rt+1 + γV (st+1 ) − V (st )] where rt+1 is the reward received after taking action at in state st , γ is the discount factor, and α is the learning rate. Example: Using TD learning to update the value estimates in a game of tic-tac-toe by bootstrap- ping the current estimate with the next state’s estimated value. 6.6 Baseline Subtraction with Advantage Function Advantage Function: Measures how much better an action is compared to the average action in a given state. Equation: A(s, a) = Q(s, a) − V (s) Example: Improving policy updates by subtracting a baseline value from the observed rewards to reduce variance and make learning more stable. 6.7 Generic Policy Gradient Formulation Equation: ∇θ J(θ) = E[∇θ log π(a|s, θ)A(s, a)] 6.8 Asynchronous Advantage Actor Critic Algorithm: A3C (Asynchronous Advantage Actor-Critic) runs multiple agents in parallel to up- date the policy and value functions asynchronously. Example: Speeding up learning by having multiple agents explore the environment simultaneously and asynchronously updating the global policy and value functions. 3 Algorithm 3 Trust Region Policy Optimization (TRPO) 1: Initialize policy parameters θ 2: for each iteration do 3: Sample trajectories using current policy πθk 4: Compute policy gradient ĝ 5: Compute step direction dˆ using conjugate gradient 6: Update policy parameters: θk+1 = θk + αdˆ within trust region 7: end for 6.9 Trust Region Optimization Explanation: TRPO ensures policy updates are within a trust region to maintain stability and prevent large, destabilizing updates. 6.10 Entropy and Exploration Entropy: Adding an entropy term to the objective function to encourage exploration by preventing premature convergence. Soft Actor Critic: An algorithm that incorporates entropy maximization into the policy update to balance exploration and exploitation. Example: Encouraging an agent to explore more by adding an entropy term to the objective function, preventing it from getting stuck in suboptimal policies. 6.11 Deterministic Policy Gradient Algorithm: DDPG (Deep Deterministic Policy Gradient) combines DQN and policy gradients for environments with continuous actions. Equation: The deterministic policy gradient is ∇θ J(θ) = E[∇a Q(s, a)∇θ µ(s, θ)] where µ is the policy. Example: Training a robotic arm to precisely control its movements using deterministic policy gradients. 6.12 Hands On: PPO and DDPG MuJoCo Examples PPO: Proximal Policy Optimization, a method that simplifies TRPO while retaining performance. DDPG: Applying DDPG to control tasks in MuJoCo to demonstrate policy learning in continuous action spaces. Example: Implementing PPO and DDPG in a MuJoCo environment to train agents for various control tasks. 7 Locomotion and Visuo-Motor Environments Locomotion and Visuo-Motor Environments: These environments focus on training agents to move and interact with their surroundings using visual inputs and motor actions. 7.1 Locomotion Application: Training agents to move and navigate in environments, such as walking, running, or flying. Example: Using policy-based methods to train a bipedal robot to walk or a drone to fly through an obstacle course. 4 7.2 Visuo-Motor Interaction Application: Combining visual perception with motor control to interact with objects and envi- ronments. Example: Training an agent to play table tennis by integrating visual input to track the ball and motor control to hit it accurately. 7.3 Benchmarking Explanation: Using standardized tasks and environments to evaluate and compare the perfor- mance of reinforcement learning algorithms. Example: Evaluating different policy-based methods on common benchmarks like MuJoCo loco- motion tasks or Atari games to compare their effectiveness. 8 Conclusion 9 Summary and Further Reading 9.1 Summary Policy-based reinforcement learning directly optimizes the policy, making it suitable for continuous action spaces. Various algorithms like REINFORCE, Actor-Critic, TRPO, and PPO are used to handle the challenges in these environments. 9.2 Further Reading Suggested literature and resources for a deeper understanding of policy-based reinforcement learn- ing and its applications in complex environments. Questions 1. Why are value-based methods difficult to use in continuous action spaces? Value-based methods are difficult to use in continuous action spaces because they require discretizing the action space or finding the maximum value action, which is computationally infeasible. 2. What is MuJoCo? Can you name a few example tasks? MuJoCo is a physics engine used for simulating continuous control tasks. Examples include humanoid locomotion, robotic arm manipulation, and bipedal walking. 3. What is an advantage of policy-based methods? Policy-based methods can directly optimize the policy for continuous action spaces without requiring discretization or explicit value function estimation. 4. What is a disadvantage of full-trajectory policy-based methods? Full-trajectory policy-based methods can have high variance in their gradient estimates, which can slow down learning and make it unstable. 5. What is the difference between actor critic and vanilla policy-based methods? Actor-Critic methods combine policy-based (actor) and value-based (critic) approaches to reduce variance and improve learning stability, whereas vanilla policy-based methods use only the policy gradient. 6. How many parameter sets are used by actor critic? How can they be represented in a neural network? 5  Actor-Critic methods use two parameter sets: one for the actor (policy) and one for the critic (value function). They can be represented in a neural network with separate layers or heads for the actor and critic. 7. Describe the relation between Monte Carlo REINFORCE, n-step methods, and tem- poral difference bootstrapping. Monte Carlo REINFORCE uses full episode returns for updates, n-step methods use returns over n steps, and temporal difference bootstrapping uses a single step with a bootstrapped estimate of future rewards. 8. What is the advantage function? The advantage function measures the relative value of an action compared to the average value of all actions in a given state, defined as A(s, a) = Q(s, a) − V (s). 9. Describe a MuJoCo task that methods such as PPO can learn to perform well. PPO can learn to control a bipedal robot to walk or run efficiently in a MuJoCo environment by optimizing the policy for continuous and complex movements. 10. Give two actor critic approaches to further improve upon bootstrapping and advantage functions, that are used in high-performing algorithms such as PPO and SAC. Two approaches are using clipped surrogate objectives in PPO and entropy regularization in SAC to encourage exploration and stabilize learning. 11. Why is learning robot actions from image input hard? Learning robot actions from image input is hard due to the high-dimensionality of images, requiring complex feature extraction and representation learning, and the challenge of associ- ating visual input with motor actions. In class Questions Policy-Based 1. When do you use value-based, when policy-based? Use value-based methods in environments with discrete actions where you can estimate the value of actions precisely. Use policy-based methods in continuous action spaces or when the policy needs to be directly optimized. 2. What is policy-based? Policy-based reinforcement learning directly optimizes the policy that the agent follows, with- out explicitly using a value function. 3. Name a famous policy-based algorithm. REINFORCE. 4. What is a modern hybrid approach? Actor-Critic methods, which combine value-based and policy-based approaches. 5. What is a modern AC algorithm? Asynchronous Advantage Actor-Critic (A3C). 6. What approaches are there for improving AC algorithms? Approaches include using experience replay, prioritized experience replay, entropy regular- ization, trust region optimization (TRPO), proximal policy optimization (PPO), and asyn- chronous updates (A3C). 6  Notes on Chapter 5: Model-Based Reinforcement Learning 1 Core Concepts Model-Based Reinforcement Learning (MBRL): A method that involves creating a model of the environment’s dynamics and using it for planning and decision-making. This contrasts with model-free methods, which learn policies or value functions directly from experience. Transition Model: Represents the dynamics of the environment, mapping current states and actions to next states and rewards. Planning: Using the transition model to simulate future states and rewards to determine the best actions to take. Sample Efficiency: The ability of a reinforcement learning algorithm to learn effectively from a limited amount of data. 2 Core Problem Learning the Environment’s Dynamics: The main challenge in MBRL is accurately learning the transition model and effectively using this model for planning. This involves handling high- dimensional state spaces, dealing with uncertainty, and ensuring sample efficiency. 3 Core Algorithms Core Algorithms: Model-based methods typically involve two main steps: 1. Learning the Model: This involves learning the transition model that maps states and actions to next states and rewards. 2. Using the Model for Planning: This involves using the learned model to simulate future states and rewards to plan the best actions. 4 Building a Navigation Map Example: Building a navigation map to illustrate model-based reinforcement learning. This in- volves learning the transitions between different locations and using this knowledge to plan optimal routes. 5 5.1 Dynamics Models of High-Dimensional Problems Transition Model and Knowledge Transfer: The transition model captures the dynamics of the environment, allowing knowledge transfer to new but related tasks, improving learning and planning efficiency. Sample Efficiency: MBRL can be more sample-efficient than model-free methods because it leverages the learned model to simulate and plan, reducing the need for extensive interaction with the real environment. 1 6 5.2 Learning and Planning Agents Tabular Imagination: Using a table-based representation of states and transitions for planning. This method is straightforward but scales poorly with high-dimensional state spaces. Hands On: Imagining Taxi Example: A practical example using the taxi environment to demonstrate model-based planning by simulating trajectories and updating policies based on imag- ined experiences. Reversible Planning and Irreversible Learning: Planning can involve reversible steps (trying different paths and backtracking), whereas learning often involves irreversible updates to the model or policy. Four Types of Model-Based Methods: 1. Learning the Model: Focusing on accurately modeling the environment’s dynamics. 2. Planning with the Model: Using the learned model to plan and make decisions. 3. End-to-End Methods: Combining learning and planning in a single framework. 4. Hybrid Methods: Integrating model-based and model-free approaches. 6.1 5.2.1 Learning the Model Modeling Uncertainty: Addressing the uncertainty in the learned model by incorporating prob- abilistic models or ensembles of models to capture the variability in the environment’s dynamics. Latent Models: Using latent variable models to represent the underlying structure of the envi- ronment. These models can capture complex dependencies and reduce the dimensionality of the state space. 6.2 5.2.2 Planning with the Model Trajectory Rollouts and Model-Predictive Control: Using the model to simulate trajectories and optimize actions over a finite horizon. Model-Predictive Control (MPC): An algorithmic approach where the model is used to predict future states and rewards, optimizing actions over a short horizon and updating the plan as new information becomes available. Algorithm 1 Model-Predictive Control (MPC) 1: Initialize model parameters θ 2: for each time step t do 3: Observe current state st 4: for each action a do 5: Predict future state st+1 ∼ p(st+1 |st , a; θ) 6: Evaluate cost c(st , a) 7: end for 8: Select action at = arg mina c(st , a) 9: Execute action at 10: Update model parameters θ using observed transition (st , at , st+1 ) 11: end for End-to-End Learning and Planning-by-Network: Integrating learning and planning into a single neural network architecture that can learn to predict dynamics and optimize policies simultaneously. 2 7 5.3 High-Dimensional Environments Overview of Model-Based Experiments: Discusses various experiments and applications of MBRL in high-dimensional environments. Small Navigation Tasks: Application of MBRL to simple navigation tasks to illustrate the principles and benefits of model-based approaches. Robotic Applications: Using MBRL for controlling robotic systems, where precise modeling of dynamics and planning is crucial for effective operation. Atari Games Applications: Application of MBRL to Atari games, demonstrating its ability to handle complex, high-dimensional state spaces. Algorithm 2 Model-Based Learning and Planning 1: Initialize model parameters θ 2: Initialize policy parameters ϕ 3: for each episode do 4: Generate trajectories using current policy πϕ 5: Update model parameters θ using observed transitions 6: Plan using the learned model to improve policy πϕ 7: Update policy parameters ϕ based on planned trajectories 8: end for 8 Hands On: PlaNet Example PlaNet Example: A detailed example using the PlaNet algorithm, which combines probabilistic models and planning for effective learning in high-dimensional environments. 9 Summary and Further Reading Summary: A recap of the key points covered in the chapter, emphasizing the benefits and chal- lenges of MBRL. Further Reading: Suggested literature and resources for a deeper understanding of MBRL and its applications in various domains. Quick Questions 1. What is the advantage of model-based over model-free methods? Model-based methods can achieve higher sample efficiency by using a learned model to simu- late and plan actions, reducing the need for extensive interactions with the real environment. 2. Why may the sample complexity of model-based methods suffer in high-dimensional problems? In high-dimensional problems, accurately learning the transition model requires a large number of samples, which can lead to increased sample complexity. 3. Which functions are part of the dynamics model? The dynamics model typically includes the transition function T (s, a) = s′ and the reward function R(s, a). 4. Mention four deep model-based approaches. Four deep model-based approaches are PlaNet, Model-Predictive Control (MPC), World Mod- els, and Dreamer. 3  5. Do model-based methods achieve better sample complexity than model-free? Yes, model-based methods generally achieve better sample complexity because they can use the learned model to simulate experiences and plan actions efficiently. 6. Do model-based methods achieve better performance than model-free? Model-based methods can achieve better performance in terms of sample efficiency, but model- free methods may achieve better asymptotic performance in some cases due to more accurate direct policy learning. 7. In Dyna-Q the policy is updated by two mechanisms: learning by sampling the envi- ronment and what other mechanism? The policy in Dyna-Q is updated by learning from simulated experiences generated by the model. 8. Why is the variance of ensemble methods lower than of the individual machine learning approaches that are used in the ensemble? Ensemble methods average the predictions of multiple models, which reduces variance and improves robustness compared to individual models. 9. What does model-predictive control do and why is this approach suited for models with lower accuracy? Model-predictive control (MPC) optimizes actions over a short horizon and frequently re- plans based on new observations, making it well-suited for models with lower accuracy by continuously correcting errors. 10. What is the advantage of planning with latent models over planning with actual mod- els? Planning with latent models can reduce computational complexity and capture essential fea- tures of the environment, making planning more efficient and scalable. 11. How are latent models trained? Latent models are typically trained using variational autoencoders (VAEs) or other unsuper- vised learning techniques to learn compact representations of the state space. 12. Mention four typical modules that constitute the latent model. Four typical modules of a latent model are the encoder, decoder, dynamics model, and reward model. 13. What is the advantage of end-to-end planning and learning? End-to-end planning and learning can jointly optimize model learning and policy learning, leading to better integration and performance. 14. Mention two end-to-end planning and learning methods. Two end-to-end planning and learning methods are Dreamer and PlaNet. In class Question 1. Why Model-based? Model-based methods are used because they can achieve higher sample efficiency by learning and utilizing a model of the environment’s dynamics, which allows for better planning and decision-making with fewer interactions with the real environment. 2. What is the “Model”? 4  The “Model” refers to a representation of the environment’s dynamics, typically including a transition function that predicts future states based on current states and actions, and a reward function that predicts the rewards received. 3. What is the difference between model-free and model-based? Model-free methods learn policies or value functions directly from experience without ex- plicitly modeling the environment, whereas model-based methods first learn a model of the environment’s dynamics and use this model for planning and decision-making. 4. Does it work? Yes, model-based methods have been shown to work effectively in various tasks, particularly when sample efficiency is crucial, although they may face challenges in highly complex or high-dimensional environments. 5. Is Dyna Hybrid? How is it Hybrid? Yes, Dyna is a hybrid approach that combines model-free learning (learning from real experi- ences) and model-based learning (learning from simulated experiences generated by a model) to improve sample efficiency and planning. 6. What is the difference between planning and learning? Planning involves using a model to simulate and optimize future actions before execution, while learning involves updating the model or policy based on actual experiences from interacting with the environment. 7. What is the weakness of Model-based? The primary weakness of model-based methods is that they can suffer from model inaccura- cies, especially in complex or high-dimensional environments, which can lead to suboptimal planning and decision-making. 8. Name two ways in which the weakness can be improved Using ensemble models to capture uncertainty and reduce the impact of model inaccuracies. Integrating model-free methods to refine policies based on real experiences, complementing the model-based approach. 9. Name two ways in which the model can be improved Incorporating probabilistic or Bayesian approaches to better handle uncertainty in the model. Using deep learning techniques to create more expressive models that can capture complex dynamics. 10. Name two ways in which the planning can be improved Employing Model-Predictive Control (MPC) to iteratively re-plan actions based on new ob- servations, which helps correct errors in the model. Using trajectory optimization techniques to generate more accurate plans over longer horizons. 11. What is the biggest drawback of MuZero? The biggest drawback of MuZero is its high computational complexity and resource require- ments, which can make it challenging to implement and scale for real-world applications. 12. What is wonderful about MuZero? MuZero’s ability to learn both the model and the policy end-to-end without prior knowledge of the environment’s dynamics is a significant advancement, allowing it to perform exceptionally well across a variety of tasks, including complex games. 5  Notes on Chapter 6: Self-Play and Two-Agent Zero-Sum Problems 1 Core Concepts Self-Play: A training method where an agent learns by playing against itself. This technique is particularly effective in two-agent zero-sum games. Two-Agent Zero-Sum Game: A game where one player’s gain is exactly equal to the other player’s loss. Examples include chess, Go, and backgammon. Perfect Information Game: A game where all players have access to all information about the game state. 2 Core Problem Core Problem: The main challenge is to develop agents that can learn to play complex two-agent zero-sum games from scratch, without relying on human knowledge or heuristics. This involves creating algorithms that can effectively explore, learn, and plan in these environments. 3 Core Algorithms Core Algorithms: Key algorithms used in self-play include Minimax, Monte Carlo Tree Search (MCTS), Policy and Value Networks, and various curriculum learning techniques. 4 Self-Play in Games Example: Learning to play backgammon through self-play. This involves the agent playing against itself, gradually improving its strategy through repeated games. 5 6.1 Two-Agent Zero-Sum Problems Zero-Sum Perfect-Information Games: Games where one player’s gain is another player’s loss, and all information is available to both players. 5.1 6.1.1 The Difficulty of Playing Go Playing Strength in Go: Go is a highly complex game with a vast number of possible moves, making it difficult for traditional algorithms to achieve high playing strength. 5.2 6.1.2 AlphaGo Achievements AlphaGo Achievements: AlphaGo was the first program to defeat a world champion Go player, using a combination of deep neural networks and Monte Carlo Tree Search (MCTS). 1 6 6.2 Tabula Rasa Self-Play Agents Cycle of Virtuous Improvement: A self-reinforcing cycle where self-play leads to continuous improvement in an agent’s playing ability. AlphaGo Zero Self-Play in Detail: AlphaGo Zero learned to play Go from scratch using only self-play, without any human input or heuristics. Algorithm 1 AlphaGo Zero Self-Play Cycle 1: Initialize neural network θ 2: for each iteration do 3: Generate self-play games using current policy πθ 4: Train neural network θ using self-play data 5: Update policy πθ based on new neural network parameters 6: end for 6.1 6.2.1 Move-Level Self-Play Minimax: A decision rule used for minimizing the possible loss for a worst-case scenario. In games, it is used to find the optimal move by assuming the opponent also plays optimally. Equation: Minimax Minimax(s) = max min ′ Utility(s′ ) a s Monte Carlo Tree Search (MCTS): An algorithm that uses random sampling of the search space to make decisions in game playing. It balances exploration of new moves and exploitation of known good moves. 6.1.1 6.2.1.2 Monte Carlo Tree Search The Four MCTS Operations: 1. Selection: Starting from the root, recursively select optimal child nodes until a leaf node is reached. 2. Expansion: If the leaf node is not terminal, expand it by adding one or more child nodes. 3. Simulation: Run a simulation from the new nodes to obtain an outcome. 4. Backpropagation: Update the values of all nodes on the path from the leaf to the root based on the simulation result. Algorithm 2 Monte Carlo Tree Search (MCTS) 1: Initialize root node N0 2: for each iteration do 3: Nl ← Selection(N0 ) 4: Nc ← Expansion(Nl ) 5: z ← Simulation(Nc ) 6: Backpropagation(Nc , z) 7: end for 8: return arg maxa Q(N0 , a) Policies: In MCTS, policies guide the selection and expansion steps. The Upper Confidence bounds applied to Trees (UCT) is a common policy used. Equation: UCT Selection s Q(s, a) ln N (s) U CT = +c N (s, a) N (s, a) 2  P-UCT: A variant of UCT that incorporates prior probabilities from a neural network to guide the search. Equation: P-UCT p Q(s, a) N (s) P − U CT = + c · P (a|s) · N (s, a) 1 + N (s, a) Exploration/Exploitation: Balancing the exploration of new actions with the exploitation of known rewarding actions. Applications: MCTS has been successfully applied in games like Go, Chess, and Shogi, particu- larly in programs like AlphaGo and AlphaZero. 6.2 6.2.2 Example-Level Self-Play Policy and Value Network: Neural networks used to approximate the policy (probability dis- tribution over actions) and the value function (expected outcome from a state). Equation: Policy Network π(a|s; θ) Equation: Value Network V (s; θ) Stability and Exploration: Ensuring stable learning through regularization and encouraging exploration to avoid local minima. 6.3 6.2.3 Tournament-Level Self-Play Curriculum Learning: A method where the agent learns tasks in a sequence of increasing difficulty, which helps in better generalization and faster learning. Algorithm 3 Self-Play Curriculum Learning 1: Initialize curriculum C with tasks of increasing difficulty 2: for each task c ∈ C do 3: Train agent on task c using self-play 4: end for Self-Play Curriculum Learning: Gradually increasing the difficulty of self-play tasks to improve the agent’s performance. Supervised and Reinforcement Curriculum Learning: Combining supervised learning with reinforcement learning in a curriculum to leverage the strengths of both methods. Procedural Content Generation: Automatically generating tasks or environments to train the agent. Active Learning: Allowing the agent to choose the most informative examples to learn from. Single-Agent Curriculum Learning: Applying curriculum learning techniques in a single-agent context to improve performance. 7 6.3 Self-Play Environments How to Design a World Class Go Program?: Discussing the architectural and algorithmic components needed to create a top-performing Go program. AlphaGo Zero Performance: Highlighting the performance metrics and achievements of Al- phaGo Zero, which learned to play Go from scratch using self-play. 3  AlphaZero: A generalization of AlphaGo Zero that achieved superhuman performance in Chess, Shogi, and Go using the same architecture. General Game Architecture: The architecture used in AlphaZero and similar programs, com- bining neural networks with MCTS. Open Self-Play Frameworks: Discussing open frameworks and tools for developing self-play agents. Hands On: Hex in Polygames Example: A practical example using the Polygames framework to create a self-play agent for the game of Hex. 8 Summary and Further Reading Summary: A recap of the key points covered in the chapter, emphasizing the benefits and chal- lenges of self-play in two-agent zero-sum games. Further Reading: Suggested literature and resources for a deeper understanding of self-play methods and their applications. Questions 1. What are the differences between AlphaGo, AlphaGo Zero, and AlphaZero? AlphaGo used supervised learning from human games and reinforcement learning, AlphaGo Zero learned purely from self-play without human data, and AlphaZero generalized the approach to Chess and Shogi, learning from self-play alone. 2. What is MCTS? Monte Carlo Tree Search (MCTS) is a search algorithm that uses random sampling of the search space to make decisions, balancing exploration and exploitation. 3. What are the four steps of MCTS? The four steps of MCTS are Selection, Expansion, Simulation, and Backpropagation. 4. What does UCT do? Upper Confidence bounds applied to Trees (UCT) selects actions in MCTS by bal- ancing the value of known rewards with the exploration of less-visited actions. 5. Give the UCT formula. How is P-UCT different? UCT Formula: s Q(s, a) ln N (s) U CT = +c N (s, a) N (s, a) P-UCT incorporates prior probabilities from a neural network: p Q(s, a) N (s) P − U CT = + c · P (a|s) · N (s, a) 1 + N (s, a) 6. Describe the function of each of the four operations of MCTS. Selection: Select the optimal child node recursively until a leaf node is reached. Expansion: Add one or more child nodes to the leaf node if it is not terminal. Simulation: Run a simulation from the new nodes to obtain an outcome. Backpropagation: Update the values of all nodes on the path from the leaf to the root based on the simulation result. 4  7. How does UCT achieve trading off exploration and exploitation? Which inputs does it use? UCT achieves trade-off by using the formula that balances the average reward (exploitation) with the exploration term that favors less-visited actions. It uses the number of visits to the state-action pair N (s, a) and the total number of visits to the state N (s). 8. When Cp is small, does MCTS explore more or exploit more? When Cp is small, MCTS tends to exploit more. 9. For small numbers of node expansions, would you prefer more exploration or more exploitation? For small numbers of node expansions, more exploration is generally preferred to gather more information about the search space. 10. What is a double-headed network? How is it different from regular actor critic? A double-headed network has two output heads, one for the policy (action probabilities) and one for the value (expected return). It differs from regular actor-critic which typically has separate networks for policy and value functions. 11. Which three elements make up the self-play loop? (You may draw a picture.) The three elements are Self-Play, Training the Neural Network, and Updating the Policy. 12. What is tabula rasa learning? Tabula rasa learning refers to learning from scratch without any prior knowledge or data. 13. How can tabula rasa learning be faster than reinforcement learning on top of super- vised learning of grandmaster games? Tabula rasa learning can be faster as it avoids the constraints of biased data and explores the search space more freely, potentially discovering novel strategies. 14. What is curriculum learning? Curriculum learning is a method where an agent learns tasks in a sequence of increasing difficulty to improve generalization and learning speed. In class Question 1. In what year did AlphaGo beat Lee Sedol? AlphaGo beat Lee Sedol in 2016. 2. How many AlphaGos are there? There are three main versions: AlphaGo, AlphaGo Zero, and AlphaZero. 3. What is the difference between AlphaGo and AlphaGo Zero? AlphaGo used supervised learning from human games and reinforcement learning, while Al- phaGo Zero learned purely from self-play without any human data. 4. What are the two architectural elements of AlphaGo Zero? The two architectural elements are the policy network and the value network. 5. What is self-play? Self-play is a training method where an agent learns by playing against itself. 6. How many levels of self-play are there in AlphaGo Zero? 5  There are three levels: move-level, example-level, and tournament-level self-play. 7. What is Minimax? Minimax is a decision rule used for minimizing the possible loss for a worst-case scenario in zero-sum games. 8. What is the estimated size of the state in Go? The estimated size of the state space in Go is 10170. 9. What are the two architectural elements of conventional Chess programs? The two architectural elements are the search algorithm and the evaluation function. 10. How does MCTS work? MCTS works by selecting nodes to explore based on a balance of exploration and exploitation, expanding new nodes, simulating outcomes from those nodes, and backpropagating the results to update the tree. 11. What does the UCT formula determine? The UCT formula determines the action selection in MCTS by balancing the average reward and the exploration term. 12. Explain the AlphaGo Zero self-play loop. The AlphaGo Zero self-play loop involves generating self-play games using the current policy, training the neural network with self-play data, and updating the policy based on the new neural network parameters. 13. What is the biggest problem that was overcome in AlphaGo Zero? The biggest problem overcome was the reliance on human knowledge and heuristics, allowing the system to learn entirely from self-play. 14. Why is AlphaGo Zero faster than AlphaGo? AlphaGo Zero is faster because it does not rely on human data, which simplifies the learning process and reduces the computational overhead associated with supervised learning. 15. What is curriculum learning? Curriculum learning is a method where an agent learns tasks in a sequence of increasing difficulty to improve generalization and learning speed. 16. What is AlphaZero? AlphaZero is a generalization of AlphaGo Zero that achieved superhuman performance in Chess, Shogi, and Go using the same architecture and learning from self-play alone. 6  Notes on Chapter 7: Multi-Agent Reinforcement Learning 1 Core Concepts Multi-Agent Reinforcement Learning (MARL): A subfield of reinforcement learning involv- ing multiple agents that interact within an environment, each learning to optimize its strategy based on the interactions with other agents. Game Theory: The study of mathematical models of strategic interactions among rational decision-makers. It provides frameworks for analyzing competitive, cooperative, and mixed be- haviors in multi-agent settings. 2 Core Problem Core Problem: Developing algorithms that enable multiple agents to learn and interact effec- tively in a shared environment. This includes dealing with challenges such as partial observability, nonstationary environments, and large state spaces. 3 Core Algorithms Core Algorithms: Key algorithms in MARL include: – Counterfactual Regret Minimization (CFR): An algorithm that minimizes regret by considering counterfactual scenarios. – Deep Counterfactual Regret Minimization (Deep CFR): A variant of CFR using deep learning. – Centralized Training/Decentralized Execution: Training agents together but allowing independent execution. – Opponent Modeling: Predicting and responding to other agents’ actions. – Communication Strategies: Enabling agents to share information. – Evolutionary Algorithms: Inspired by natural selection. – Swarm Computing: Inspired by social animals’ collective behavior. 4 Self-Driving Car Example Example: Coordination among self-driving cars at intersections. Each car must learn to navigate without collisions while optimizing its route. This requires understanding the dynamics of other cars (agents) and reacting accordingly. 5 Multi-Agent Problems 5.1 Game Theory Game Theory: The study of strategic interactions where the outcome for each participant depends on the actions of all others. It includes concepts such as Nash equilibrium and Pareto efficiency. – Nash Equilibrium: A situation where no agent can benefit by changing its strategy while the others keep theirs unchanged. 1  – Pareto Efficiency: An allocation where no agent can be made better off without making another worse off. 5.2 Stochastic Games and Extensive-Form Games Stochastic Games: Games with probabilistic transitions between states. Agents must plan over an uncertain future. Extensive-Form Games: Represented by a game tree, showing the sequential nature of decisions and information available at each decision point. – Nodes: Represent states or decision points. – Edges: Represent actions taken by the agents. 5.3 Competitive, Cooperative, and Mixed Strategies Competitive Strategies: Agents aim to maximize their individual rewards, often at the expense of others. – Example: Chess, where each player tries to checkmate the opponent. Cooperative Strategies: Agents work together to achieve a common goal. – Example: Collaborative robotics where robots work together to complete a task. Mixed Strategies: Agents may exhibit both competitive and cooperative behaviors. – Example: Capture the Flag, where team members cooperate within teams and compete against the opposing team. 6 Competitive Behavior Competitive Behavior: Strategies where agents aim to outperform each other, often leading to adversarial relationships. This involves actions like bluffing, deception, and counter-strategies. 7 Cooperative Behavior Cooperative Behavior: Strategies where agents coordinate their actions to achieve a common objective. This involves sharing information, planning joint actions, and aligning goals. 7.1 Multi-Objective Reinforcement Learning Multi-Objective Reinforcement Learning: Involves optimizing multiple objectives simulta- neously, often requiring trade-offs between competing goals. – Example: A robot that needs to minimize energy consumption while maximizing task com- pletion. 8 Mixed Behavior Mixed Behavior: Agents exhibit both competitive and cooperative strategies, depending on the context and their objectives. – Example: A marketplace where businesses compete for customers but may cooperate in industry standards. 2 8.1 Iterated Prisoner’s Dilemma Iterated Prisoner’s Dilemma: A repeated game where agents choose to cooperate or defect, illustrating the tension between individual rationality and collective benefit. – Cooperation: Leads to mutual benefit. – Defection: Leads to individual benefit at the cost of the other. 9 Challenges 9.1 Partial Observability Partial Observability: Agents have incomplete information about the environment or other agents, making it difficult to make optimal decisions. – Example: Poker, where players cannot see each other’s cards. 9.2 Nonstationary Environments Nonstationary Environments: The environment changes over time, which can alter the strate- gies and behaviors that are effective. – Example: Financial markets where stock prices change based on various factors. 9.3 Large State Space Large State Space: The complexity of the state space makes learning and planning computa- tionally intensive and challenging. – Example: Go, with an estimated state space of 10170. 10 Multi-Agent Reinforcement Learning Agents 10.1 Competitive Behavior 10.1.1 Counterfactual Regret Minimization (CFR) CFR: An algorithm for decision-making in games that minimizes regret by considering counter- factual scenarios where different decisions could have been made. Algorithm 1 Counterfactual Regret Minimization (CFR) 1: Initialize regret and strategy for each information set 2: for each iteration do 3: Traverse game tree to compute regrets 4: Update strategy based on regrets 5: end for 10.1.2 Deep Counterfactual Regret Minimization (Deep CFR) Deep CFR: A variant of CFR that uses deep learning to handle large state and action spaces, enabling more complex decision-making. 10.2 Cooperative Behavior 10.2.1 Centralized Training/Decentralized Execution Centralized Training/Decentralized Execution: Training agents together with shared in- formation but allowing them to act independently during execution. This approach is practical in environments where agents must operate autonomously but can benefit from shared learning experiences. 3 10.2.2 Opponent Modeling Opponent Modeling: Predicting and responding to the actions of other agents to improve strate- gic decision-making. – Example: In competitive games like chess, modeling the opponent’s strategy can provide a significant advantage. 10.2.3 Communication Communication: Developing strategies that enable agents to share information and coordinate their actions effectively. – Example: In a team-based game, agents can communicate to synchronize their actions and plan joint strategies. 10.2.4 Psychology Psychology: Understanding the mental models and strategies of other agents to enhance cooper- ation and predict behavior. – Example: In negotiation scenarios, understanding the opponent’s preferences and goals can lead to better outcomes. 11 Mixed Behavior 11.0.1 Evolutionary Algorithms Evolutionary Algorithms: Optimization algorithms inspired by natural selection, used to evolve agent strategies over time. – Example: Genetic algorithms that evolve solutions by combining and mutating successful strategies. 11.0.2 Swarm Computing Swarm Computing: Algorithms inspired by the collective behavior of social animals, such as ants or bees, used for distributed problem-solving. – Example: Ant colony optimization for finding optimal paths. 11.0.3 Population-Based Training Population-Based Training: Training multiple agents with different strategies and allowing them to evolve through a process similar to natural selection. – Example: Training multiple versions of a neural network and selecting the best performers for further training. 11.0.4 Self-Play Leagues Self-Play Leagues: Systems where agents continuously play against each other to improve their strategies, simulating a competitive league environment. – Example: AI agents in games like Dota 2 or StarCraft playing thousands of matches against each other to improve. 12 Multi-Agent Environments 12.1 Competitive Behavior: Poker Poker: A competitive game where agents must balance bluffing and strategic decision-making to succeed against other players. The complexity of hidden information and probabilistic outcomes makes it a challenging testbed for MARL. 4 12.2 Cooperative Behavior: Hide and Seek Hide and Seek: A cooperative game where agents must work together to find optimal hiding or seeking strategies. This involves learning to coordinate actions and share information effectively. 12.3 Mixed Behavior: Capture the Flag and StarCraft 12.3.1 Capture the Flag Capture the Flag: A game that combines competitive and cooperative elements, requiring teams to strategize to capture the opponent’s flag while defending their own. – Example: Agents must coordinate their movements and strategies within their team while competing against the opposing team. 12.3.2 StarCraft StarCraft: A real-time strategy game that involves both competitive and cooperative strategies, requiring complex decision-making and coordination. – Example: Managing resources, building units, and executing strategies in real-time against opponents. 12.4 Hands On: Hide and Seek in the Gym Example Gym Example: A practical implementation of multi-agent hide and seek in the OpenAI Gym environment. This example illustrates cooperative behaviors and how agents can learn to hide and seek effectively through MARL. 13 Multiplayer Environments Multiplayer Environments: Environments where multiple agents interact, such as online games or simulated ecosystems. These environments require robust multi-agent strategies to handle dy- namic interactions and competition. 14 Summary and Further Reading 14.1 Summary Summary: Recap of the key points, emphasizing the importance and challenges of MARL, includ- ing competitive, cooperative, and mixed strategies. Highlighting the complexities of multi-agent interactions and the algorithms developed to handle these challenges. 14.2 Further Reading Further Reading: Suggested resources for a deeper understanding of MARL and related topics. This includes academic papers, textbooks, and online resources that provide more in-depth coverage of the theories and applications discussed in the chapter. Questions and Answers 1. Why is there so much inte