# Learn AI Series (#104) - Dynamic Programming ![variant-a-12-green.png](https://images.hive.blog/DQmbC2LKb7BLTH4egg4if975nSnX46BrtSKKfa1JDatAaFq/variant-a-12-green.png) ### What will I learn - You will learn the Bellman equation: the recursive relationship that ties the value of a state to the value of the states it leads to; - policy evaluation: how to compute the value of any given policy by iterating the Bellman equation until it settles; - policy improvement: turning a value function into a better policy with a single greedy step, and why that step is garanteed to help; - policy iteration: alternating evaluation and improvement untill the policy stops changing; - value iteration: collapsing the whole thing into one update with the Bellman optimality equation; - where dynamic programming works beautifully, and exactly where it falls apart. ### Requirements - A working modern computer running macOS, Windows or Ubuntu; - An installed Python 3(.11+) distribution; - The ambition to learn AI and machine learning. ### Difficulty - Beginner ### Curriculum (of the `Learn AI Series`): - [Learn AI Series (#1) - What Machine Learning Actually Is](https://hive.blog/hive-196387/@scipio/learn-ai-series-1-what-machine-learning-actually-is) - [Learn AI Series (#2) - Setting Up Your AI Workbench - Python and NumPy](https://hive.blog/hive-196387/@scipio/learn-ai-series-2-setting-up-your-ai-workbench-python-and-numpy) - [Learn AI Series (#3) - Your Data Is Just Numbers - How Machines See the World](https://hive.blog/hive-196387/@scipio/learn-ai-series-3-your-data-is-just-numbers-how-machines-see-the-world) - [Learn AI Series (#4) - Your First Prediction - No Math, Just Intuition](https://hive.blog/hive-196387/@scipio/learn-ai-series-4-your-first-prediction-no-math-just-intuition) - [Learn AI Series (#5) - Patterns in Data - What "Learning" Actually Looks Like](https://hive.blog/hive-196387/@scipio/learn-ai-series-5-patterns-in-data-what-learning-actually-looks-like) - [Learn AI Series (#6) - From Intuition to Math - Why We Need Formulas](https://hive.blog/hive-196387/@scipio/learn-ai-series-6-from-intuition-to-math-why-we-need-formulas) - [Learn AI Series (#7) - The Training Loop - See It Work Step by Step](https://hive.blog/hive-196387/@scipio/learn-ai-series-7-the-training-loop-see-it-work-step-by-step) - [Learn AI Series (#8) - The Math You Actually Need (Part 1) - Linear Algebra](https://hive.blog/hive-196387/@scipio/learn-ai-series-8-the-math-you-actually-need-part-1-linear-algebra) - [Learn AI Series (#9) - The Math You Actually Need (Part 2) - Calculus and Probability](https://hive.blog/hive-196387/@scipio/learn-ai-series-9-the-math-you-actually-need-part-2-calculus-and-probability) - [Learn AI Series (#10) - Your First ML Model - Linear Regression From Scratch](https://hive.blog/hive-196387/@scipio/learn-ai-series-10-your-first-ml-model-linear-regression-from-scratch) - [Learn AI Series (#11) - Making Linear Regression Real](https://hive.blog/hive-196387/@scipio/learn-ai-series-11-making-linear-regression-real) - [Learn AI Series (#12) - Classification - Logistic Regression From Scratch](https://hive.blog/hive-196387/@scipio/learn-ai-series-12-classification-logistic-regression-from-scratch) - [Learn AI Series (#13) - Evaluation - How to Know If Your Model Actually Works](https://hive.blog/hive-196387/@scipio/learn-ai-series-13-evaluation-how-to-know-if-your-model-actually-works) - [Learn AI Series (#14) - Data Preparation - The 80% Nobody Talks About](https://hive.blog/hive-196387/@scipio/learn-ai-series-14-data-preparation-the-80-nobody-talks-about) - [Learn AI Series (#15) - Feature Engineering and Selection](https://hive.blog/hive-196387/@scipio/learn-ai-series-15-feature-engineering-and-selection) - [Learn AI Series (#16) - Scikit-Learn - The Standard Library of ML](https://hive.blog/hive-196387/@scipio/learn-ai-series-16-scikit-learn-the-standard-library-of-ml) - [Learn AI Series (#17) - Decision Trees - How Machines Make Decisions](https://hive.blog/hive-196387/@scipio/learn-ai-series-17-decision-trees-how-machines-make-decisions) - [Learn AI Series (#18) - Random Forests - Wisdom of Crowds](https://hive.blog/hive-196387/@scipio/learn-ai-series-18-random-forests-wisdom-of-crowds) - [Learn AI Series (#19) - Gradient Boosting - The Kaggle Champion](https://hive.blog/hive-196387/@scipio/learn-ai-series-19-gradient-boosting-the-kaggle-champion) - [Learn AI Series (#20) - Support Vector Machines - Drawing the Perfect Boundary](https://hive.blog/hive-196387/@scipio/learn-ai-series-20-support-vector-machines-drawing-the-perfect-boundary) - [Learn AI Series (#21) - Mini Project - Predicting Crypto Market Regimes](https://hive.blog/hive-196387/@scipio/learn-ai-series-21-mini-project-predicting-crypto-market-regimes) - [Learn AI Series (#22) - K-Means Clustering - Finding Groups](https://hive.blog/hive-196387/@scipio/learn-ai-series-22-k-means-clustering-finding-groups) - [Learn AI Series (#23) - Advanced Clustering - Beyond K-Means](https://hive.blog/hive-196387/@scipio/learn-ai-series-23-advanced-clustering-beyond-k-means) - [Learn AI Series (#24) - Dimensionality Reduction - PCA](https://hive.blog/hive-196387/@scipio/learn-ai-series-24-dimensionality-reduction-pca) - [Learn AI Series (#25) - Advanced Dimensionality Reduction - t-SNE and UMAP](https://hive.blog/hive-196387/@scipio/learn-ai-series-25-advanced-dimensionality-reduction-t-sne-and-umap) - [Learn AI Series (#26) - Anomaly Detection - Finding What Doesn't Belong](https://hive.blog/hive-196387/@scipio/learn-ai-series-26-anomaly-detection-finding-what-doesnt-belong) - [Learn AI Series (#27) - Recommendation Systems - "Users Like You Also Liked..."](https://hive.blog/hive-196387/@scipio/learn-ai-series-27-recommendation-systems-users-like-you-also-liked) - [Learn AI Series (#28) - Time Series Fundamentals - When Order Matters](https://hive.blog/hive-196387/@scipio/learn-ai-series-28-time-series-fundamentals-when-order-matters) - [Learn AI Series (#29) - Time Series Forecasting - Predicting What Comes Next](https://hive.blog/hive-196387/@scipio/learn-ai-series-29-time-series-forecasting-predicting-what-comes-next) - [Learn AI Series (#30) - Natural Language Processing - Text as Data](https://hive.blog/hive-196387/@scipio/learn-ai-series-30-natural-language-processing-text-as-data) - [Learn AI Series (#31) - Word Embeddings - Meaning in Numbers](https://hive.blog/hive-196387/@scipio/learn-ai-series-31-word-embeddings-meaning-in-numbers) - [Learn AI Series (#32) - Bayesian Methods - Thinking in Probabilities](https://hive.blog/hive-196387/@scipio/learn-ai-series-32-bayesian-methods-thinking-in-probabilities) - [Learn AI Series (#33) - Ensemble Methods Deep Dive - Stacking and Blending](https://hive.blog/hive-196387/@scipio/learn-ai-series-33-ensemble-methods-deep-dive-stacking-and-blending) - [Learn AI Series (#34) - ML Engineering - From Notebook to Production](https://hive.blog/hive-196387/@scipio/learn-ai-series-34-ml-engineering-from-notebook-to-production) - [Learn AI Series (#35) - Data Ethics and Bias in ML](https://hive.blog/hive-196387/@scipio/learn-ai-series-35-data-ethics-and-bias-in-ml) - [Learn AI Series (#36) - Mini Project - Complete ML Pipeline](https://hive.blog/hive-196387/@scipio/learn-ai-series-36-mini-project-complete-ml-pipeline) - [Learn AI Series (#37) - The Perceptron - Where It All Started](https://hive.blog/hive-196387/@scipio/learn-ai-series-37-the-perceptron-where-it-all-started) - [Learn AI Series (#38) - Neural Networks From Scratch - Forward Pass](https://hive.blog/hive-196387/@scipio/learn-ai-series-38-neural-networks-from-scratch-forward-pass) - [Learn AI Series (#39) - Neural Networks From Scratch - Backpropagation](https://hive.blog/hive-196387/@scipio/learn-ai-series-39-neural-networks-from-scratch-backpropagation) - [Learn AI Series (#40) - Training Neural Networks - Practical Challenges](https://hive.blog/hive-196387/@scipio/learn-ai-series-40-training-neural-networks-practical-challenges) - [Learn AI Series (#41) - Optimization Algorithms - SGD, Momentum, Adam](https://hive.blog/hive-196387/@scipio/learn-ai-series-41-optimization-algorithms-sgd-momentum-adam) - [Learn AI Series (#42) - PyTorch Fundamentals - Tensors and Autograd](https://hive.blog/hive-196387/@scipio/learn-ai-series-42-pytorch-fundamentals-tensors-and-autograd) - [Learn AI Series (#43) - PyTorch Data and Training](https://hive.blog/hive-196387/@scipio/learn-ai-series-43-pytorch-data-and-training) - [Learn AI Series (#44) - PyTorch nn.Module - Building Real Networks](https://hive.blog/hive-196387/@scipio/learn-ai-series-44-pytorch-nnmodule-building-real-networks) - [Learn AI Series (#45) - Convolutional Neural Networks - Theory](https://hive.blog/hive-196387/@scipio/learn-ai-series-45-convolutional-neural-networks-theory) - [Learn AI Series (#46) - CNNs in Practice - Classic to Modern Architectures](https://hive.blog/hive-196387/@scipio/learn-ai-series-46-cnns-in-practice-classic-to-modern-architectures) - [Learn AI Series (#47) - CNN Applications - Detection, Segmentation, Style Transfer](https://hive.blog/hive-196387/@scipio/learn-ai-series-47-cnn-applications-detection-segmentation-style-transfer) - [Learn AI Series (#48) - Recurrent Neural Networks - Sequences](https://hive.blog/hive-196387/@scipio/learn-ai-series-48-recurrent-neural-networks-sequences) - [Learn AI Series (#49) - LSTM and GRU - Solving the Memory Problem](https://hive.blog/hive-196387/@scipio/learn-ai-series-49-lstm-and-gru-solving-the-memory-problem) - [Learn AI Series (#50) - Sequence-to-Sequence Models](https://hive.blog/hive-196387/@scipio/learn-ai-series-50-sequence-to-sequence-models) - [Learn AI Series (#51) - Attention Mechanisms](https://hive.blog/hive-196387/@scipio/learn-ai-series-51-attention-mechanisms) - [Learn AI Series (#52) - The Transformer Architecture (Part 1)](https://hive.blog/hive-196387/@scipio/learn-ai-series-52-the-transformer-architecture-part-1) - [Learn AI Series (#53) - The Transformer Architecture (Part 2)](https://hive.blog/hive-196387/@scipio/learn-ai-series-53-the-transformer-architecture-part-2) - [Learn AI Series (#54) - Vision Transformers](https://hive.blog/hive-196387/@scipio/learn-ai-series-54-vision-transformers) - [Learn AI Series (#55) - Generative Adversarial Networks](https://hive.blog/hive-196387/@scipio/learn-ai-series-55-generative-adversarial-networks) - [Learn AI Series (#56) - Mini Project - Building a Transformer From Scratch](https://hive.blog/hive-196387/@scipio/learn-ai-series-56-mini-project-building-a-transformer-from-scratch) - [Learn AI Series (#57) - Language Modeling - Predicting the Next Word](https://hive.blog/hive-196387/@scipio/learn-ai-series-57-language-modeling-predicting-the-next-word) - [Learn AI Series (#58) - GPT Architecture - Decoder-Only Transformers](https://hive.blog/hive-196387/@scipio/learn-ai-series-58-gpt-architecture-decoder-only-transformers) - [Learn AI Series (#59) - BERT and Encoder Models](https://hive.blog/hive-196387/@scipio/learn-ai-series-59-bert-and-encoder-models) - [Learn AI Series (#60) - Training Large Language Models](https://hive.blog/hive-196387/@scipio/learn-ai-series-60-training-large-language-models) - [Learn AI Series (#61) - Instruction Tuning and Alignment](https://hive.blog/hive-196387/@scipio/learn-ai-series-61-instruction-tuning-and-alignment) - [Learn AI Series (#62) - Prompt Engineering - Getting the Most from LLMs](https://hive.blog/hive-196387/@scipio/learn-ai-series-62-prompt-engineering-getting-the-most-from-llms) - [Learn AI Series (#63) - Embeddings and Vector Search](https://hive.blog/hive-196387/@scipio/learn-ai-series-63-embeddings-and-vector-search) - [Learn AI Series (#64) - Retrieval-Augmented Generation (RAG) - Basics](https://hive.blog/hive-196387/@scipio/learn-ai-series-64-retrieval-augmented-generation-rag-basics) - [Learn AI Series (#65) - RAG - Advanced Techniques](https://hive.blog/hive-196387/@scipio/learn-ai-series-65-rag-advanced-techniques) - [Learn AI Series (#66) - Working with LLM APIs](https://hive.blog/hive-196387/@scipio/learn-ai-series-66-working-with-llm-apis) - [Learn AI Series (#67) - Building AI Agents (Part 1) - Foundations](https://hive.blog/hive-196387/@scipio/learn-ai-series-67-building-ai-agents-part-1-foundations) - [Learn AI Series (#68) - Building AI Agents (Part 2) - Advanced Patterns](https://hive.blog/hive-196387/@scipio/learn-ai-series-68-building-ai-agents-part-2-advanced-patterns) - [Learn AI Series (#69) - Fine-Tuning Language Models](https://hive.blog/hive-196387/@scipio/learn-ai-series-69-fine-tuning-language-models) - [Learn AI Series (#70) - Running Local Models](https://hive.blog/hive-196387/@scipio/learn-ai-series-70-running-local-models) - [Learn AI Series (#71) - Text Generation Techniques](https://hive.blog/hive-196387/@scipio/learn-ai-series-71-text-generation-techniques) - [Learn AI Series (#72) - Tokenization Deep Dive](https://hive.blog/hive-196387/@scipio/learn-ai-series-72-tokenization-deep-dive) - [Learn AI Series (#73) - LLM Evaluation](https://hive.blog/hive-196387/@scipio/learn-ai-series-73-llm-evaluation) - [Learn AI Series (#74) - The Hugging Face Ecosystem](https://hive.blog/hive-196387/@scipio/learn-ai-series-74-the-hugging-face-ecosystem) - [Learn AI Series (#75) - Multimodal Models - Text Meets Vision](https://hive.blog/hive-196387/@scipio/learn-ai-series-75-multimodal-models-text-meets-vision) - [Learn AI Series (#76) - Mini Project - Your Own AI Assistant](https://hive.blog/hive-196387/@scipio/learn-ai-series-76-mini-project-your-own-ai-assistant) - [Learn AI Series (#77) - Image Processing Fundamentals](https://hive.blog/hive-196387/@scipio/learn-ai-series-77-image-processing-fundamentals) - [Learn AI Series (#78) - Object Detection (Part 1) - Foundations](https://hive.blog/hive-196387/@scipio/learn-ai-series-78-object-detection-part-1-foundations) - [Learn AI Series (#79) - Object Detection (Part 2) - Modern Approaches](https://hive.blog/hive-196387/@scipio/learn-ai-series-79-object-detection-part-2-modern-approaches) - [Learn AI Series (#80) - Image Segmentation](https://hive.blog/hive-196387/@scipio/learn-ai-series-80-image-segmentation) - [Learn AI Series (#81) - Pose Estimation and Tracking](https://hive.blog/hive-196387/@scipio/learn-ai-series-81-pose-estimation-and-tracking) - [Learn AI Series (#82) - Optical Character Recognition](https://hive.blog/hive-196387/@scipio/learn-ai-series-82-optical-character-recognition) - [Learn AI Series (#83) - Video Understanding](https://hive.blog/hive-196387/@scipio/learn-ai-series-83-video-understanding) - [Learn AI Series (#84) - Generative Images - Diffusion Models (Part 1)](https://hive.blog/hive-196387/@scipio/learn-ai-series-84-generative-images-diffusion-models-part-1) - [Learn AI Series (#85) - Generative Images - Diffusion Models (Part 2)](https://hive.blog/hive-196387/@scipio/learn-ai-series-85-generative-images-diffusion-models-part-2) - [Learn AI Series (#86) - Image-to-Image and Editing](https://hive.blog/hive-196387/@scipio/learn-ai-series-86-image-to-image-and-editing) - [Learn AI Series (#87) - 3D Vision](https://hive.blog/hive-196387/@scipio/learn-ai-series-87-3d-vision) - [Learn AI Series (#88) - Face Analysis](https://hive.blog/hive-196387/@scipio/learn-ai-series-88-face-analysis) - [Learn AI Series (#89) - Medical and Scientific Imaging](https://hive.blog/hive-196387/@scipio/learn-ai-series-89-medical-and-scientific-imaging) - [Learn AI Series (#90) - Self-Supervised Learning for Vision](https://hive.blog/hive-196387/@scipio/learn-ai-series-90-self-supervised-learning-for-vision) - [Learn AI Series (#91) - Mini Project - Building a Visual AI System](https://hive.blog/hive-196387/@scipio/learn-ai-series-91-mini-project-building-a-visual-ai-system) - [Learn AI Series (#92) - Audio Fundamentals for AI](https://hive.blog/hive-196387/@scipio/learn-ai-series-92-audio-fundamentals-for-ai) - [Learn AI Series (#93) - Speech Recognition](https://hive.blog/hive-196387/@scipio/learn-ai-series-93-speech-recognition) - [Learn AI Series (#94) - Text-to-Speech (TTS)](https://hive.blog/hive-196387/@scipio/learn-ai-series-94-text-to-speech-tts) - [Learn AI Series (#95) - Audio Classification](https://hive.blog/hive-196387/@scipio/learn-ai-series-95-audio-classification) - [Learn AI Series (#96) - Music Generation](https://hive.blog/hive-196387/@scipio/learn-ai-series-96-music-generation) - [Learn AI Series (#97) - Speaker Recognition and Diarization](https://hive.blog/hive-196387/@scipio/learn-ai-series-97-speaker-recognition-and-diarization) - [Learn AI Series (#98) - Natural Language Understanding for Voice](https://hive.blog/hive-196387/@scipio/learn-ai-series-98-natural-language-understanding-for-voice) - [Learn AI Series (#99) - Audio Enhancement](https://hive.blog/hive-196387/@scipio/learn-ai-series-99-audio-enhancement) - [Learn AI Series (#100) - Multimodal Audio-Visual Models](https://hive.blog/hive-196387/@scipio/learn-ai-series-100-multimodal-audio-visual-models) - [Learn AI Series (#101) - Mini Project: Voice-Controlled AI Assistant](https://hive.blog/hive-196387/@scipio/learn-ai-series-101-mini-project-voice-controlled-ai-assistant) - [Learn AI Series (#102) - What Is Reinforcement Learning?](https://hive.blog/hive-196387/@scipio/learn-ai-series-102-what-is-reinforcement-learning) - [Learn AI Series (#103) - Multi-Armed Bandits](https://hive.blog/hive-196387/@scipio/learn-ai-series-103-multi-armed-bandits) - [Learn AI Series (#104) - Dynamic Programming](https://hive.blog/hive-196387/@scipio/learn-ai-series-104-dynamic-programming) (this post) # Learn AI Series (#104) - Dynamic Programming ### Solutions to Episode #103 Exercises **Exercise 1:** Multi-armed bandit tournament -- run five algorithms (EpsilonGreedy at 0.1 and 0.01, DecayingEpsilonGreedy, UCB with c=2, and Gaussian Thompson Sampling) across 20 randomly-generated 10-armed environments and tally average regret, optimal-pull percentage, and per-environment wins. ```python import numpy as np class BanditEnvironment: def __init__(self, k=10, seed=42): self.rng = np.random.RandomState(seed) self.q_true = self.rng.normal(0, 1, size=k) self.k = k self.best_arm = int(np.argmax(self.q_true)) def pull(self, arm): return self.q_true[arm] + np.random.normal(0, 1) class EpsilonGreedy: def __init__(self, k, epsilon=0.1): self.k, self.epsilon = k, epsilon self.q = np.zeros(k) self.n = np.zeros(k) def choose_arm(self): if np.random.random() < self.epsilon: return np.random.randint(self.k) return int(np.argmax(self.q)) def update(self, arm, reward): self.n[arm] += 1 self.q[arm] += (reward - self.q[arm]) / self.n[arm] class DecayingEpsilonGreedy(EpsilonGreedy): def __init__(self, k, initial_eps=1.0, decay=0.999): super().__init__(k, initial_eps) self.decay = decay def update(self, arm, reward): super().update(arm, reward) self.epsilon *= self.decay class UCB: def __init__(self, k, c=2.0): self.k, self.c = k, c self.q = np.zeros(k) self.n = np.zeros(k) self.t = 0 def choose_arm(self): self.t += 1 for arm in range(self.k): if self.n[arm] == 0: return arm bonus = self.c * np.sqrt(np.log(self.t) / self.n) return int(np.argmax(self.q + bonus)) def update(self, arm, reward): self.n[arm] += 1 self.q[arm] += (reward - self.q[arm]) / self.n[arm] class GaussianThompson: def __init__(self, k): self.k = k self.mu = np.zeros(k) self.lam = np.ones(k) self.alpha = np.ones(k) self.beta = np.ones(k) def choose_arm(self): samples = np.zeros(self.k) for i in range(self.k): tau = np.random.gamma(self.alpha[i], 1.0 / self.beta[i]) sigma = 1.0 / np.sqrt(tau * self.lam[i]) samples[i] = np.random.normal(self.mu[i], sigma) return int(np.argmax(samples)) def update(self, arm, reward): old_mu, old_lam = self.mu[arm], self.lam[arm] self.lam[arm] += 1 self.mu[arm] = (old_lam * old_mu + reward) / self.lam[arm] self.alpha[arm] += 0.5 self.beta[arm] += (0.5 * old_lam * (reward - old_mu) ** 2 / self.lam[arm]) def make_agent(name, k): return { "eps-0.1": lambda: EpsilonGreedy(k, 0.1), "eps-0.01": lambda: EpsilonGreedy(k, 0.01), "decay": lambda: DecayingEpsilonGreedy(k), "ucb": lambda: UCB(k, c=2.0), "thompson": lambda: GaussianThompson(k), }[name]() class BanditTournament: NAMES = ["eps-0.1", "eps-0.01", "decay", "ucb", "thompson"] def __init__(self, n_envs=20, n_steps=10000, k=10): self.n_envs, self.n_steps, self.k = n_envs, n_steps, k def run(self): regret = {n: [] for n in self.NAMES} opt = {n: [] for n in self.NAMES} wins = {n: 0 for n in self.NAMES} gaps = [] for seed in range(self.n_envs): env = BanditEnvironment(self.k, seed=seed) sq = np.sort(env.q_true) gaps.append((seed, sq[-1] - sq[-2])) this_env = {} for name in self.NAMES: np.random.seed(1000 + seed) agent = make_agent(name, self.k) optimal = env.q_true[env.best_arm] total, hits = 0.0, 0 for _ in range(self.n_steps): arm = agent.choose_arm() r = env.pull(arm) agent.update(arm, r) total += optimal - r hits += (arm == env.best_arm) regret[name].append(total) opt[name].append(100 * hits / self.n_steps) this_env[name] = total wins[min(this_env, key=this_env.get)] += 1 print(f"{'algorithm':12s}{'avg regret':>12s}" f"{'optimal%':>10s}{'wins':>6s}") for name in self.NAMES: print(f"{name:12s}{np.mean(regret[name]):12.1f}" f"{np.mean(opt[name]):9.1f}%{wins[name]:6d}") hardest = min(gaps, key=lambda g: g[1]) print(f"\nHardest env: seed {hardest[0]} (gap={hardest[1]:.3f})") BanditTournament(n_envs=20, n_steps=10000).run() ``` The key insight: averaged over 20 environments, Thompson Sampling wins the most of them and posts the lowest mean regret, with UCB a close second. Both crush the fixed-epsilon agents, because a fixed epsilon keeps burning pulls on random arms forever -- even after it already knows the answer. EpsilonGreedy(0.01) does well when the arms are well-separated (barely needs to explore) but occasionally gets unlucky and never finds the best arm, which inflates its average regret. **Exercise 2:** Contextual bandit news recommender -- LinUCB over 5 article categories with 6-dimensional user features, learning online from sigmoid click feedback, compared against a random baseline. ```python import numpy as np class LinUCB: def __init__(self, n_arms, d, alpha=1.0): self.n_arms, self.alpha = n_arms, alpha self.A = [np.eye(d) for _ in range(n_arms)] self.b = [np.zeros(d) for _ in range(n_arms)] def choose_arm(self, ctx): scores = np.zeros(self.n_arms) for a in range(self.n_arms): A_inv = np.linalg.inv(self.A[a]) theta = A_inv @ self.b[a] scores[a] = ctx @ theta + self.alpha * np.sqrt(ctx @ A_inv @ ctx) return int(np.argmax(scores)) def update(self, arm, ctx, reward): self.A[arm] += np.outer(ctx, ctx) self.b[arm] += reward * ctx class NewsRecommender: CATS = ["tech", "sports", "politics", "cooking", "finance"] def __init__(self, seed=42): self.rng = np.random.RandomState(seed) self.d = 6 # age, gender, morning, weekday, mobile, returning self.weights = self.rng.randn(len(self.CATS), self.d) self.agent = LinUCB(len(self.CATS), self.d, alpha=0.5) def click_prob(self, arm, ctx): return 1.0 / (1.0 + np.exp(-(self.weights[arm] @ ctx))) def run(self, n=10000): first, last = [], [] counts = np.zeros(len(self.CATS), dtype=int) rand_clicks = 0 for t in range(n): ctx = self.rng.randn(self.d) arm = self.agent.choose_arm(ctx) click = int(self.rng.random() < self.click_prob(arm, ctx)) self.agent.update(arm, ctx, click) counts[arm] += 1 if t < 1000: first.append(click) if t >= n - 1000: last.append(click) rand_arm = self.rng.randint(len(self.CATS)) rand_clicks += int(self.rng.random() < self.click_prob(rand_arm, ctx)) print(f"CTR first 1000: {np.mean(first):.3f}") print(f"CTR last 1000: {np.mean(last):.3f}") print(f"Random CTR: {rand_clicks / n:.3f}") print("Per-article selection counts:") for name, c in zip(self.CATS, counts): print(f" {name:9s}: {c}") NewsRecommender().run() ``` The CTR in the final 1000 visits edges above the first 1000 -- LinUCB learns fast, so it's already doing well early -- but the headline number is the gap against the random baseline, which sits stuck around 0.50 while the agent runs well above 0.80. That gap is the whole point: random ignores the context, LinUCB reads the user features and matches each visitor to the article most likely to land. LinUCB is the algorithm behind a huge chunk of real recommendation traffic, so this little demo is closer to production than it looks ;-) **Exercise 3:** Non-stationary bandit detector -- a sliding-window epsilon-greedy agent that watches for change-points and temporarily boosts exploration when an arm's mean shifts, compared to a fixed-epsilon agent with no detection. ```python import numpy as np def make_q(phase, k=10): """One clearly-best arm; which arm is best rotates each phase.""" q = np.full(k, -0.5) q[phase % k] = 2.0 return q class FixedEpsilon: """Full-history epsilon-greedy -- the baseline that cannot forget.""" def __init__(self, k, epsilon=0.1): self.k, self.epsilon = k, epsilon self.q = np.zeros(k) self.n = np.zeros(k) def choose_arm(self, t): if np.random.random() < self.epsilon: return np.random.randint(self.k) return int(np.argmax(self.q)) def update(self, arm, reward, t): self.n[arm] += 1 self.q[arm] += (reward - self.q[arm]) / self.n[arm] class ChangeDetector: """Sliding-window epsilon-greedy with a two-window change test and a temporary exploration boost when a shift is detected.""" def __init__(self, k, window=50, base_eps=0.1): self.k, self.window, self.base_eps = k, window, base_eps self.history = [[] for _ in range(k)] self.boost_until = -1 self.detections = [] def epsilon(self, t): return 0.5 if t < self.boost_until else self.base_eps def choose_arm(self, t): if np.random.random() < self.epsilon(t): return np.random.randint(self.k) est = np.full(self.k, np.inf) for a in range(self.k): recent = self.history[a][-self.window:] if recent: est[a] = np.mean(recent) return int(np.argmax(est)) def update(self, arm, reward, t): h = self.history[arm] h.append(reward) # Compare this arm's most recent window against the one before it if len(h) >= 2 * self.window and t % 20 == 0: recent = np.mean(h[-self.window:]) older = np.mean(h[-2 * self.window:-self.window]) if abs(recent - older) > 1.0: self.detections.append((t, arm)) self.boost_until = t + 100 def simulate(agent, n_steps=5000, k=10, shift=1000): rng = np.random.RandomState(0) total_regret = 0.0 for t in range(n_steps): q = make_q(t // shift, k) # best arm rotates every `shift` steps arm = agent.choose_arm(t) reward = q[arm] + rng.normal(0, 0.5) agent.update(arm, reward, t) total_regret += np.max(q) - q[arm] return total_regret det = ChangeDetector(10) det_regret = simulate(det) fixed_regret = simulate(FixedEpsilon(10)) print("Detected changes (step, arm):") for step, arm in det.detections[:8]: print(f" step {step}, arm {arm}") print(f"\nSliding-window + detection regret: {det_regret:.1f}") print(f"Fixed full-history regret: {fixed_regret:.1f}") ``` The full-history agent is a disaster here: once it's convinced arm 0 is best, its sample mean is anchored by hundreds of old pulls, so even after arm 0 stops being good it keeps going back -- racking up roughly four times the regret. The sliding-window agent forgets old data, and the two-window change test fires right after each shift (you'll see detections at ~1040, ~2020, ~3020, ~4040), boosting epsilon to 0.5 for 100 steps so it re-locks onto the new best arm fast. The threshold of 1.0 is a knob: too low and you re-explore on noise, too high and you miss real changes. Tuning that knob is the whole game in non-stationary settings. ### On to today's episode Right, episode 104. Last time we built up the whole **bandit** toolkit -- epsilon-greedy, UCB, Thompson Sampling -- and I promised we'd start putting the mathematical framework on a firmer footing. Bandits were RL with the difficulty dial turned all the way down: one decision, immediate reward, no states. _But the real world has states._ Where you are determines what you can do, and what you do now determines where you end up next. That sequential structure is exactly what we stripped out for bandits, and now we're going to put it back. Back in episode #102 we defined the **Markov Decision Process** -- states, actions, transition probabilities, rewards, and a discount factor gamma. We even built a little grid world and let a Q-learning agent stumble its way to the goal. Today we're going to solve that same kind of problem, but properly, with full knowledge of the rules. This is **dynamic programming** (DP), and it's where the core ideas of RL crystalize. Here's the catch, and it's a big one: DP assumes you know _everything_ about the environment. Every state, every action, every transition probability, every reward. Think of it as the "open-book exam" version of RL -- you've got the answer key (the model) right in front of you, and the only question is how to use it. That sounds unrealistic, and for a lot of real problems it is. But almost every algorithm we build after this is, at heart, a clever way of doing dynamic programming _without_ the answer key. So you really want to understand the open-book version first ;-) ## The Bellman equation Everything in RL value theory hangs off one idea, and it's so simple it almost feels like cheating. The value of where you are equals the reward you're about to get, plus the (discounted) value of where you end up next. That's it. That recursive self-reference is the **Bellman equation**, named after Richard Bellman who coined the term "dynamic programming" in the 1950s (partly, the story goes, because the name sounded impressive enough to keep his research funding safe -- a man after my own heart). For a policy pi, the **state value function** V^pi(s) satisfies: ``` V^pi(s) = SUM_a pi(a|s) * SUM_s' P(s'|s,a) * [R(s,a,s') + gamma * V^pi(s')] ``` In words: stand in state s, average over the actions your policy might take (weighted by pi(a|s)), then average over the states you might land in (weighted by the transition probabilities P), and for each of those add the immediate reward plus the discounted value of continuing onward from s'. The value of _here_ is defined in terms of the value of _there_. It looks circular, and it sort of is -- the trick is that we can solve the circularity by iterating, which is exactly what we'll do in a moment. Let's make it concrete with a 4x4 grid world. The two opposite corners are terminal (think of them as the exits), every step costs you -1, and the agent just wants out as fast as possible: ```python import numpy as np class GridWorldMDP: """4x4 grid world with fully known dynamics.""" def __init__(self, size=4): self.size = size self.n_states = size * size self.n_actions = 4 # up, right, down, left self.terminal_states = {0, self.n_states - 1} # opposite corners self.action_deltas = {0: (-1, 0), 1: (0, 1), 2: (1, 0), 3: (0, -1)} def state_to_pos(self, s): return s // self.size, s % self.size def pos_to_state(self, row, col): return row * self.size + col def get_transition(self, state, action): """Returns (next_state, reward, done). Deterministic transitions to keep things readable.""" if state in self.terminal_states: return state, 0.0, True row, col = self.state_to_pos(state) dr, dc = self.action_deltas[action] new_row = max(0, min(self.size - 1, row + dr)) new_col = max(0, min(self.size - 1, col + dc)) next_state = self.pos_to_state(new_row, new_col) reward = -1.0 # every step hurts a little done = next_state in self.terminal_states return next_state, reward, done ``` Bumping into a wall just leaves you where you are (the `max`/`min` clamp the position), and you still pay your -1 for the privilege. The optimal play is obvious to a human -- walk the shortest path to the nearest exit -- but the agent doesn't know that yet. It only knows the rules. Our job is to turn the rules into a value for every cell. ## Policy evaluation: how good is a policy? Given a fixed policy pi, **policy evaluation** answers the question "what is V^pi(s) for every state?" We don't solve the linear system directly (you _could_, it's just a system of linear equations) -- in stead we iterate. Start with V(s) = 0 everywhere, then sweep through the states applying the Bellman equation again and again. Each sweep pushes value information one step further outward, and the whole thing is garanteed to converge to the true V^pi: ```python def policy_evaluation(mdp, policy, gamma=1.0, theta=1e-6): """Compute V^pi by iterating the Bellman equation. policy: array (n_states, n_actions) of action probabilities theta: convergence threshold """ V = np.zeros(mdp.n_states) while True: delta = 0.0 for s in range(mdp.n_states): if s in mdp.terminal_states: continue v_old = V[s] v_new = 0.0 for a in range(mdp.n_actions): next_s, reward, done = mdp.get_transition(s, a) next_value = 0.0 if done else V[next_s] v_new += policy[s, a] * (reward + gamma * next_value) V[s] = v_new delta = max(delta, abs(v_old - v_new)) if delta < theta: break return V mdp = GridWorldMDP(size=4) random_policy = np.ones((mdp.n_states, mdp.n_actions)) / mdp.n_actions V = policy_evaluation(mdp, random_policy) print("Values under the random policy:") for row in range(mdp.size): for col in range(mdp.size): s = mdp.pos_to_state(row, col) print(f"{V[s]:6.1f}", end=" ") print() ``` Under the random policy -- shuffle off in a random direction with 25% probability each -- the cells far from any exit end up deeply negative. That makes sense: a drunkard's walk from the middle of the grid takes a _long_ time to stumble into a corner, and every wasted step costs -1. Cells right next to an exit sit close to zero, because escape is one lucky step away. The `theta` parameter is just our patience: once no state's value changes by more than 0.000001 in a full sweep, we call it converged. Notice the `delta = max(delta, abs(...))` line -- that's how we measure "did anything still move?" across the whole sweep. Nota bene: with gamma=1.0 (no discounting) this only converges because the episode is garanteed to terminate; with continuing tasks you'd need gamma < 1 to keep the sum finite, exactly as we saw with the discount factor in episode #102. ## Policy improvement: making it better Knowing V^pi is nice, but the point of having values is to _act better_. **Policy improvement** does the obvious thing: in each state, instead of blindly following pi, look one step ahead and ask "which action gives me the best immediate-reward-plus-next-value?" Pick that one. This is the **greedy policy** with respect to V: ```python def policy_improvement(mdp, V, gamma=1.0): """Build the greedy policy from a value function.""" new_policy = np.zeros((mdp.n_states, mdp.n_actions)) for s in range(mdp.n_states): if s in mdp.terminal_states: new_policy[s] = 1.0 / mdp.n_actions # irrelevant at exits continue action_values = np.zeros(mdp.n_actions) for a in range(mdp.n_actions): next_s, reward, done = mdp.get_transition(s, a) next_value = 0.0 if done else V[next_s] action_values[a] = reward + gamma * next_value best_action = int(np.argmax(action_values)) new_policy[s, best_action] = 1.0 # all probability on the winner return new_policy ``` The **policy improvement theorem** is the quiet hero here: the greedy policy is _always_ at least as good as the one you started with, in every state, garanteed. No hand-waving. If V^pi says you're sitting on a value of -10 and stepping right lands you in a cell worth -5, then stepping right is worth -1 + gamma*(-5) = -6, which beats -10. Greedy notices that and grabs it. Do this in every state at once and you can only improve (or, if you were already optimal, stay put). _That_ guarantee is what makes the next algorithm work. ## Policy iteration: the full algorithm Now we just bolt the two pieces together. **Policy iteration** alternates evaluation and improvement and refuses to stop untill the policy stops changing: 1. Start with any policy (random is fine); 2. **Evaluate** it -- compute V^pi; 3. **Improve** it -- go greedy with respect to V^pi; 4. If the policy changed, back to step 2; 5. If it didn't change, you're done -- and what you have is _optimal_. ```python def policy_iteration(mdp, gamma=1.0): """Find the optimal policy by alternating evaluation and improvement.""" policy = np.ones((mdp.n_states, mdp.n_actions)) / mdp.n_actions iteration = 0 while True: iteration += 1 V = policy_evaluation(mdp, policy, gamma) new_policy = policy_improvement(mdp, V, gamma) if np.allclose(policy, new_policy): print(f"Policy iteration converged in {iteration} iterations") break policy = new_policy return policy, V optimal_policy, optimal_V = policy_iteration(mdp) arrows = ["^", ">", "v", "<"] print("\nOptimal policy:") for row in range(mdp.size): for col in range(mdp.size): s = mdp.pos_to_state(row, col) if s in mdp.terminal_states: print(" T ", end=" ") else: best_a = int(np.argmax(optimal_policy[s])) print(f" {arrows[best_a]} ", end=" ") print() print("\nOptimal values:") for row in range(mdp.size): for col in range(mdp.size): s = mdp.pos_to_state(row, col) print(f"{optimal_V[s]:6.1f}", end=" ") print() ``` Here's the part that surprised me the first time I saw it: policy iteration converges in a _tiny_ number of outer loops -- typically 3 to 5 for a grid world, and that count barely grows as the grid gets bigger. The expensive bit is hidden inside policy evaluation (which itself runs many Bellman sweeps). The improvement step is cheap and decisive -- a couple of greedy passes and the policy locks into the optimum. The arrows you print should all point sensibly toward the nearest exit, and the values should read like a distance map: -1 next to a corner, more negative as you move away. ## Value iteration: the shortcut Policy iteration has a slightly wasteful smell to it -- we run policy evaluation _all the way to convergence_ just to throw most of that precision away on the next greedy step. _Why fully evaluate a policy you're about to replace?_ **Value iteration** says: don't. Take a single Bellman backup, but instead of averaging over the policy's actions, take the **max** over actions. That one change folds evaluation and improvement into a single update: ```python def value_iteration(mdp, gamma=1.0, theta=1e-6): """Find optimal values directly via the Bellman optimality equation.""" V = np.zeros(mdp.n_states) iteration = 0 while True: iteration += 1 delta = 0.0 for s in range(mdp.n_states): if s in mdp.terminal_states: continue v_old = V[s] action_values = np.zeros(mdp.n_actions) for a in range(mdp.n_actions): next_s, reward, done = mdp.get_transition(s, a) next_value = 0.0 if done else V[next_s] action_values[a] = reward + gamma * next_value V[s] = np.max(action_values) # the max IS the improvement delta = max(delta, abs(v_old - V[s])) if delta < theta: print(f"Value iteration converged in {iteration} iterations") break policy = policy_improvement(mdp, V, gamma) return policy, V optimal_policy_vi, optimal_V_vi = value_iteration(mdp) ``` The difference is one operator. Policy evaluation computes `V(s) = SUM_a pi(a|s) * [R + gamma*V(s')]` -- an _expectation_ under the current policy. Value iteration computes `V(s) = max_a [R + gamma*V(s')]` -- the _best_ you could do. That max is the **Bellman optimality equation**, and because it picks the best action every sweep, it improves the policy and evaluates it in the same breath. You never store an explicit policy until the very end, when you read it off the converged values. Having said that, the two methods compute the same answer -- run both on this grid and `optimal_V` and `optimal_V_vi` agree to within `theta`. Value iteration usually wins on big problems because it skips the expensive inner evaluation loop; policy iteration sometimes wins when evaluation is cheap and you want the policy to lock in after a handful of decisive improvements. Horses for courses. ## Why does any of this converge? A fair question, because we keep waving the word "garanteed" around. The reason is that the Bellman operator is a **contraction mapping** with factor gamma. In plain terms: every time you apply a full Bellman backup, the distance between your current value estimate and the true value function shrinks by at least a factor of gamma. Shrink something repeatedly by a constant factor below 1 and it collapses toward a single fixed point -- and that fixed point is the value function we're after. It's the same reason compound interest in reverse eats a number down to nothing. With gamma=1.0 we lean on guaranteed termination instead, but the moment you set gamma < 1 the contraction argument gives you convergence for free, no matter how you initialize V. That's not just trivia. The contraction property is _the_ theoretical guarantee underneath value-based RL, and it keeps mattering long after we leave the open-book setting -- it's why bootstrapped updates from incomplete experience still pull estimates in the right direction. ## When does DP work (and when doesn't it)? Dynamic programming needs the full model: every state, every transition probability, every reward, all known in advance. That confines it to problems where two things hold: - the state space is small enough to enumerate (or at least to represent in a table); - the dynamics are known -- you have a simulator or a clean mathematical model of how the world responds. For a 4x4 grid (16 states), DP is instant. For 100x100 (10,000 states), still fine. But point it at an Atari frame -- on the order of 10^8 possible pixel configurations per screen -- and it falls flat on its face. You cannot enumerate that table, let alone sweep it. And for most interesting problems you don't even _have_ the transition probabilities; the universe doesn't hand you P(s'|s,a) on a plate. The methods we'll build next are precisely the escape from these two limitations. **Monte Carlo methods** drop the need for a model entirely -- they let the agent play out whole episodes and average the actual returns it sees, no transition probabilities required. **Temporal difference methods** go a step further and update value estimates from a _single_ step of experience, blending Monte Carlo's learning-from-reality with DP's bootstrapping-from-its-own-estimates. And eventually we swap the lookup table for a neural network, so the very same Bellman ideas can cope with state spaces too vast to ever write down. But -- and this is the whole reason DP comes first -- every one of those methods is doing dynamic programming in disguise. The Bellman equation, policy evaluation, policy improvement, the optimality operator: they all carry straight over. The algorithms change to cope with missing models and giant state spaces; the underlying ideas do not budge an inch. ## Wrapping up - the **Bellman equation** ties the value of a state to the value of its successors -- V(s) = reward + gamma*V(s') -- and that one recursive relationship is the backbone of every RL algorithm we'll meet; - **policy evaluation** iterates the Bellman equation to compute V^pi(s) for a fixed policy, and the contraction property garantees it converges; - **policy improvement** reads a value function and produces a greedy policy that, by the policy improvement theorem, is provably at least as good as the original; - **policy iteration** alternates the two untill the policy stops changing -- and it gets there in a surprisingly small handful of outer iterations; - **value iteration** fuses evaluation and improvement with the Bellman optimality equation V*(s) = max_a [R + gamma*V*(s')] -- one operator, often faster on large problems; - DP demands the full model (states, transitions, rewards), which is exactly the assumption the next batch of methods will tear down -- while keeping the Bellman ideas completely intact. ### Exercises **Exercise 1:** Build a **stochastic grid world solver**. Extend `GridWorldMDP` so that `get_transition` is probabilistic: the chosen action succeeds with probability `p=0.8`, and with probability `0.2` the agent slips to one of the two perpendicular directions (0.1 each). This means `get_transition` should return a _list_ of `(prob, next_state, reward, done)` tuples instead of a single outcome. Rewrite `policy_evaluation` to sum over those outcomes. Run value iteration with `gamma=0.9` on the 4x4 grid and compare the optimal values and policy against the deterministic version. The values should be more negative (slipping wastes steps) and the policy should sometimes steer away from walls to avoid slipping into them. **Exercise 2:** Build a **gamma sweep experiment**. Using the deterministic 4x4 grid world, run value iteration for `gamma` in `[0.5, 0.7, 0.9, 0.99, 1.0]`. For each gamma, print: (a) the number of iterations to convergence, (b) the optimal value of the start-adjacent cell, (c) whether the optimal policy (the arrows) changes at all. You should find the policy is identical across all gammas for this simple shortest-path problem (the optimal route doesn't depend on patience here), but the _values_ and the iteration counts shift -- lower gamma converges faster because the contraction is stronger. **Exercise 3:** Build a **policy iteration vs value iteration profiler**. Add a counter to both `policy_iteration` and `value_iteration` that tallies the total number of Bellman backups performed (one backup = one evaluation of `reward + gamma*next_value` for a single state-action pair). Run both on grid worlds of size 4, 8, and 16 (`gamma=0.9`) and print a table of total backups for each method at each size. Confirm that policy iteration needs few outer iterations but many backups per evaluation, while value iteration spreads its backups more evenly -- and see which one does fewer total backups as the grid grows. ### De groeten! @scipio

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