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Machine Learning & Data Mining CS/CNS/EE 155 Lecture 17: The Multi-Armed Bandit Problem 1Lecture 17: The Multi-Armed Bandit Problem

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Announcements Lecture Tuesday will be Course Review Final should only take a 4-5 hours to do – We give you 48 hours for your flexibility Homework 2 is graded – We graded pretty leniently – Approximate Grade Breakdown: – 64: A 61: A- 58: B+ 53: B 50: B- 47: C+ 42: C 39: C- Homework 3 will be graded soon Lecture 17: The Multi-Armed Bandit Problem2

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Today The Multi-Armed Bandits Problem – And extensions Advanced topics course on this next year Lecture 17: The Multi-Armed Bandit Problem3

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Recap: Supervised Learning Training Data: Model Class: Loss Function: Learning Objective: E.g., Linear Models E.g., Squared Loss Optimization Problem 4Lecture 17: The Multi-Armed Bandit Problem

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But Labels are Expensive! 5 Image Source: x “Crystal” “Needle” “Empty” … “Sports” “World News” “Science” y Cheap and Abundant! Expensive and Scarce! Human Annotation Running Experiment Lecture 17: The Multi-Armed Bandit Problem

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Solution? Let’s grab some labels! – Label images – Annotate webpages – Rate movies – Run Experiments – Etc… How should we choose? Lecture 17: The Multi-Armed Bandit Problem6

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Interactive Machine Learning Start with unlabeled data: Loop: – select x i – receive feedback/label y i How to measure cost? How to define goal? Lecture 17: The Multi-Armed Bandit Problem7

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Crowdsourcing Lecture 17: The Multi-Armed Bandit Problem8 “Mushroom” Unlabeled Labeled Initially Empty Labeled Initially Empty Repeat

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Aside: Active Learning Lecture 17: The Multi-Armed Bandit Problem9 “Mushroom” Unlabeled Labeled Initially Empty Labeled Initially Empty Goal: Maximize Accuracy with Minimal Cost Repeat Choose

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Passive Learning Lecture 17: The Multi-Armed Bandit Problem10 “Mushroom” Unlabeled Labeled Initially Empty Labeled Initially Empty Repeat Random

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Comparison with Passive Learning Conventional Supervised Learning is considered “Passive” Learning Unlabeled training set sampled according to test distribution So we label it at random – Very Expensive! Lecture 17: The Multi-Armed Bandit Problem11

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Aside: Active Learning Cost: uniform – E.g., each label costs $0.10 Goal: maximize accuracy of trained model Control distribution of labeled training data Lecture 17: The Multi-Armed Bandit Problem12

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Problems with Crowdsourcing Assumes you can label by proxy – E.g., have someone else label objects in images But sometimes you can’t! – Personalized recommender systems Need to ask the user whether content is interesting – Personalized medicine Need to try treatment on patient – Requires actual target domain Lecture 17: The Multi-Armed Bandit Problem13

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Personalized Labels Lecture 17: The Multi-Armed Bandit Problem14 Sports Unlabeled Labeled Initially Empty Labeled Initially Empty Choose Repeat What is Cost? Real System End User

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The Multi-Armed Bandit Problem Lecture 17: The Multi-Armed Bandit Problem15

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Formal Definition K actions/classes Each action has an average reward: μ k – Unknown to us – Assume WLOG that u 1 is largest For t = 1…T – Algorithm chooses action a(t) – Receives random reward y(t) Expectation μ a(t) Goal: minimize Tu 1 – (μ a(1) + μ a(2) + … + μ a(T) ) Lecture 17: The Multi-Armed Bandit Problem16 Basic Setting K classes No features Algorithm Simultaneously Predicts & Receives Labels If we had perfect information to start Expected Reward of Algorithm

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Sports # Shown Average Likes : 0 Interactive Personalization (5 Classes, No features) Lecture 17: The Multi-Armed Bandit Problem17

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# Shown Average Likes : 0 Interactive Personalization (5 Classes, No features) Lecture 17: The Multi-Armed Bandit Problem18 Sports

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# Shown Average Likes : 0 Interactive Personalization (5 Classes, No features) Lecture 17: The Multi-Armed Bandit Problem19 Politics

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# Shown Average Likes : 1 Interactive Personalization (5 Classes, No features) Lecture 17: The Multi-Armed Bandit Problem20 Politics

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# Shown Average Likes : 1 Interactive Personalization (5 Classes, No features) Lecture 17: The Multi-Armed Bandit Problem21 World

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# Shown Average Likes : 1 Interactive Personalization (5 Classes, No features) Lecture 17: The Multi-Armed Bandit Problem22 World

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# Shown Average Likes : 1 Interactive Personalization (5 Classes, No features) Lecture 17: The Multi-Armed Bandit Problem23 Economy

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# Shown Average Likes : 2 Interactive Personalization (5 Classes, No features) Lecture 17: The Multi-Armed Bandit Problem24 Economy …

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# Shown Average Likes : 24 What should Algorithm Recommend? Lecture 17: The Multi-Armed Bandit Problem25 Exploit:Explore: Best: Politics Economy Celebrity How to Optimally Balance Explore/Exploit Tradeoff? Characterized by the Multi-Armed Bandit Problem

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( ) Opportunity cost of not knowing preferences “no-regret” if R(T)/T 0 – Efficiency measured by convergence rate Regret: Time Horizon ( ) … …

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Recap: The Multi-Armed Bandit Problem K actions/classes Each action has an average reward: μ k – All unknown to us – Assume WLOG that u 1 is largest For t = 1…T – Algorithm chooses action a(t) – Receives random reward y(t) Expectation μ a(t) Goal: minimize Tu 1 – (μ a(1) + μ a(2) + … + μ a(T) ) Lecture 17: The Multi-Armed Bandit Problem27 Basic Setting K classes No features Algorithm Simultaneously Predicts & Receives Labels Regret

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The Motivating Problem Slot Machine = One-Armed Bandit Goal: Minimize regret From pulling suboptimal arms Lecture 17: The Multi-Armed Bandit Problem28 Each Arm Has Different Payoff

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Implications of Regret If R(T) grows linearly w.r.t. T: – Then R(T)/T constant > 0 – I.e., we converge to predicting something suboptimal If R(T) is sub-linear w.r.t. T: – Then R(T)/T 0 – I.e., we converge to predicting the optimal action Lecture 17: The Multi-Armed Bandit Problem29 Regret:

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Experimental Design How to split trials to collect information Static Experimental Design – Standard practice – (pre-planned) Lecture 17: The Multi-Armed Bandit Problem30 Treatment PlaceboTreatmentPlaceboTreatment …

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Sequential Experimental Design Adapt experiments based on outcomes Lecture 17: The Multi-Armed Bandit Problem31 Treatment PlaceboTreatment …

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Sequential Experimental Design Matters Lecture 17: The Multi-Armed Bandit Problem32

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Sequential Experimental Design MAB models sequential experimental design! Each treatment has hidden expected value – Need to run trials to gather information – “Exploration” In hindsight, should always have used treatment with highest expected value Regret = opportunity cost of exploration Lecture 17: The Multi-Armed Bandit Problem33 basic

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Online Advertising Lecture 17: The Multi-Armed Bandit Problem34 Largest Use-Case of Multi-Armed Bandit Problems

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The UCB1 Algorithm Lecture 17: The Multi-Armed Bandit Problem35

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# Shown Average Likes Confidence Intervals Lecture 17: The Multi-Armed Bandit Problem36 ** Maintain Confidence Interval for Each Action – Often derived using Chernoff-Hoeffding bounds (**) = [0.1, 0.3]= [0.25, 0.55] Undefined

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UCB1 Confidence Interval Lecture 17: The Multi-Armed Bandit Problem37 Expected Reward Estimated from data Total Iterations so far (70 in example below) # Shown Average Likes #times action k was chosen

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The UCB1 Algorithm At each iteration – Play arm with highest Upper Confidence Bound: Lecture 17: The Multi-Armed Bandit Problem # Shown Average Likes

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Balancing Explore/Exploit “Optimism in the Face of Uncertainty” Lecture 17: The Multi-Armed Bandit Problem39 Exploitation Term Exploration Term # Shown Average Likes

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Analysis (Intuition) Lecture 17: The Multi-Armed Bandit Problem40 With high probability (**): ** Proof of Theorem 1 in Value of Best Arm Upper Confidence Bound of Best Arm The true value is greater than the lower confidence bound. Bound on regret at time t+1

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Lecture 17: The Multi-Armed Bandit Problem Iterations2000 Iterations 5000 Iterations Iterations

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How Often Sub-Optimal Arms Get Played An arm never gets selected if: The number of times selected: – Prove using Hoeffding’s Inequality Lecture 17: The Multi-Armed Bandit Problem42 Bound grows slowly with time Shrinks quickly with #trials Theorem 1 in

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Regret Guarantee With high probability: – UCB1 accumulates regret at most: Lecture 17: The Multi-Armed Bandit Problem43 Theorem 1 in #Actions Gap between best & 2 nd best ε = μ 1 – μ 2 Time Horizon

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Recap: MAB & UCB1 Interactive setting – Receives reward/label while making prediction Must balance explore/exploit Sub-linear regret is good – Average regret converges to 0 Lecture 17: The Multi-Armed Bandit Problem44

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Extensions Contextual Bandits – Features of environment Dependent-Arms Bandits – Features of actions/classes Dueling Bandits Combinatorial Bandits General Reinforcement Learning Lecture 17: The Multi-Armed Bandit Problem45

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Contextual Bandits K actions/classes Rewards depends on context x: μ(x) For t = 1…T – Algorithm receives context x t – Algorithm chooses action a(t) – Receives random reward y(t) Expectation μ(x t ) Goal: Minimize Regret Lecture 17: The Multi-Armed Bandit Problem46 K classes Best class depends on features Algorithm Simultaneously Predicts & Receives Labels Bandit multiclass prediction

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Linear Bandits K actions/classes – Each action has features x k – Reward function: μ(x) = w T x For t = 1…T – Algorithm chooses action a(t) – Receives random reward y(t) Expectation μ a(t) Goal: regret scaling independent of K Lecture 17: The Multi-Armed Bandit Problem47 K classes Linear dependence Between Arms Algorithm Simultaneously Predicts & Receives Labels Labels can share information to other actions

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Example Treatment of spinal cord injury patients – Studied by Joel Burdick’s Multi-armed bandit problem: – Thousands of arms Lecture 17: The Multi-Armed Bandit Problem Images from Yanan Sui UCB1 Regret Bound: Want regret bound that scales independently of #arms E.g., linearly in dimensionality of features x describing arms

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Dueling Bandits K actions/classes – Preference model P(a k > a k’ ) For t = 1…T – Algorithm chooses actions a(t) & b(t) – Receives random reward y(t) Expectation P(a(t) > b(t)) Goal: low regret despite only pairwise feedback Lecture 17: The Multi-Armed Bandit Problem49 K classes Can only measure pairwise preferences Algorithm Simultaneously Predicts & Receives Labels Only pairwise rewards

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Example in Sensory Testing (Hypothetical) taste experiment: vs – Natural usage context Experiment 1: Absolute Metrics 3 cans 2 cans1 can 5 cans 3 cans Total: 8 cansTotal: 9 cans Very Thirsty!

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Example in Sensory Testing (Hypothetical) taste experiment: vs – Natural usage context Experiment 1: Relative Metrics All 6 prefer Pepsi

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Example Revisited Treatment of spinal cord injury patients – Studied by Joel Burdick’s Dueling Bandits Problem! Lecture 17: The Multi-Armed Bandit Problem Images from Yanan Sui Patients cannot reliably rate individual treatments Patients can reliably compare pairs of treatments

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Combinatorial Bandits Sometimes, actions must be selected from combinatorial action space: – E.g., shortest path problems with unknown costs on edges aka: Routing under uncertainty If you knew all the parameters of model: – standard optimization problem Lecture 17: The Multi-Armed Bandit Problem53

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General Reinforcement Learning Bandit setting assumes actions do not affect the world – E.g., sequence of experiments does not affect the distribution of future trials Lecture 17: The Multi-Armed Bandit Problem54 Treatment PlaceboTreatmentPlaceboTreatment …

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Markov Decision Process M states K actions Reward: μ(s,a) – Depends on state For t = 1…T – Algorithm (approximately) observes current state s t Depends on previous state & action taken – Algorithm chooses action a(t) – Receives random reward y(t) Expectation μ(s t,a(t)) Lecture 17: The Multi-Armed Bandit Problem55 ** Example: Personalized Tutoring [Emma Brunskill et al.] (**)

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Summary Interactive Machine Learning – Multi-armed Bandit Problem – Basic result: UCB1 – Surveyed Extensions Advanced Topics in ML course next year Next lecture: course review – Bring your questions! Lecture 17: The Multi-Armed Bandit Problem56

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