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CMU 15-896 Noncooperative games 2: Learning and minimax Teacher: Ariel Procaccia Reminder: The Minimax Theorem Theorem [von Neumann, 1928]: Every 2-player zero-sum game has a unique value such that: Player 1 can guarantee value at o

  1. CMU 15-896 Noncooperative games 2: Learning and minimax Teacher: Ariel Procaccia

  2. Reminder: The Minimax Theorem • Theorem [von Neumann, 1928]: Every 2-player zero-sum game has a unique value such that: Player 1 can guarantee value at o least Player 2 can guarantee loss at o most • We will prove the theorem via no-regret learning 15896 Spring 2016: Lecture 18 2

  3. How to reach your spaceship • Each morning pick one of possible routes • Then find out how long each route took • Is there a strategy for picking routes that does almost as well as the best 53 minutes fixed route in hindsight? 47 minutes ⋯ 15896 Spring 2016: Lecture 18 3

  4. The model • View as a matrix (maybe infinite #columns) Adversary Algorithm • Algorithm picks row, adversary column • Alg pays cost of (row,column) and gets column as feedback • Assume costs are in 15896 Spring 2016: Lecture 18 4

  5. The model • Define average regret in time steps as (average per-day cost of alg) (average per-day cost of best fixed row in hindsight) • No-regret algorithm: regret as • Not competing with adaptive strategy, just the best fixed row 15896 Spring 2016: Lecture 18 5

  6. Example • Algorithm 1: Alternate between U and D • Poll 1: What is algorithm 1’s worst-case average regret? 1. Adversary 2. Algorithm 1 0 3. 0 1 4. 15896 Spring 2016: Lecture 18 6

  7. Example • Algorithm 2: Choose action that has lower cost so far • Poll 2: What is algorithm 2’s worst-case average regret? 1. Adversary 2. Algorithm 1 0 3. 0 1 4. 15896 Spring 2016: Lecture 18 7

  8. What can we say more generally about deterministic algorithms? 15896 Spring 2016: Lecture 18 8

  9. Using expert advice • Want to predict the stock market • Solicit advice from experts Expert = someone with an opinion o Day Expert 1 Expert 2 Expert 3 Charlie Truth 1 � � � � � 2 � � � � � ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ • Can we do as well as best in hindsight? 15896 Spring 2016: Lecture 18 9

  10. Simpler question • One of the experts never makes a mistake • We want to find out which one • Algorithm 3: Take majority vote over experts that have been correct so far • Poll 3: What is algorithm 3’s worst-case number of mistakes? Θ 1 1. Θ log � 2. Θ��� 3. ∞ 4. 15896 Spring 2016: Lecture 18 10

  11. What if no expert is perfect? • Idea: Run algorithm 3 until all experts are crossed off, then repeat • Makes at most mistakes per mistake of the best expert • But this is wasteful: we keep forgetting what we’ve learned 15896 Spring 2016: Lecture 18 11

  12. Weighted Majority • Intuition: Making a mistake doesn’t disqualify an expert, just lowers its weight • Weighted Majority Algorithm: Start with all experts having weight 1 o Predict based on weighted majority vote o Penalize mistakes by cutting weight in o half 15896 Spring 2016: Lecture 18 12

  13. Expert 1 Expert 2 Expert 3 Charlie Alg Truth 1 1 1 1 Weight 1 � � � � � � Prediction 1 0.5 1 1 1 Weight 2 � � � � � � Prediction 2 0.5 1 0.5 0.5 Weight 3 Wrong, 1 Right, 1.5 Wrong, 2 Right, 3 15896 Spring 2016: Lecture 18 13

  14. Weighted Majority: Analysis #mistakes we’ve made so far • #mistakes of best expert so far • total weight (starts at ) • • For each mistake, drops by at least 25% � after mistakes: � • Weight of best expert is � � � � 15896 Spring 2016: Lecture 18 14

  15. Randomized Weighted Majority • Randomized Weighted Majority Algorithm: Start with all experts having weight 1 o Predict proportionally to weights: the total o weight of is � and the total weight of � � is � , predict with probability � � �� � and � � with probability � � �� � Penalize mistakes by removing fraction of o weight 15896 Spring 2016: Lecture 18 15

  16. Randomized Weighted Majority Idea: smooth out the worst case Wrong, 1 The worst-case is What about 90-10? ∼ 50-50: now we have We’re very likely to a 50% chance of agree with the getting it right majority 15896 Spring 2016: Lecture 18 16

  17. Analysis • At time we have a fraction � of weight on experts that made a mistake • Prob. � of making a mistake, remove � fraction of total weight • ����� � � • ����� � � � � ln 1 � � � �� (next slide) 15896 Spring 2016: Lecture 18 17

  18. Analysis � � � ln�1 � �� � � � �� 15896 Spring 2016: Lecture 18 18

  19. Analysis � • Weight of best expert is ���� • ����� ���� ��� � • By setting and solving, we get � • Since , • Average regret is 15896 Spring 2016: Lecture 18 19

  20. More generally • Each expert is an action with cost in • Run Randomized Weighted Majority Choose expert with probability � o Update weights: � � � o • Same analysis applies: Our expected cost: � � � o Fraction of weight removed: � � � o So, fraction removed (our cost) o 15896 Spring 2016: Lecture 18 20

  21. Proof of the minimax thm • Suppose for contradiction that zero-sum game has � � such that: If column player commits first, there is a row o that guarantees row player at least � If row player commits first, there is a column o that guarantees row player at most � • Scale matrix so that payoffs to row player are in , and let � � 15896 Spring 2016: Lecture 18 21

  22. Proof of the minimax thm • Row player plays RWM, and column player responds optimally to current mixed strategy • After steps ALG � best row in hindsight �2 � log � o Best row in hindsight � � ⋅ � � o ALG � � ⋅ � � o • It follows that � � contradiction for large • enough 15896 Spring 2016: Lecture 18 22

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