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Regret Minimization: Algorithms and Applications Yishay Mansour Tel Aviv Univ. Many thanks for my co-authors: A. Blum, N. Cesa-Bianchi, and G. Stoltz

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3 Weather Forecast Sunny: Rainy: No meteorological understanding! –using other web sites forecastWeb site CNN BBC weather.com OUR Goal: Nearly the most accurate forecast

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4 Route selection Challenge: Partial Information Goal: Fastest route

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5 Rock-Paper-Scissors (1,-1)(-1,1) (0, 0) (-1,1)(0, 0)(1,-1) (0, 0)(1,-1)(-1,1)

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6 Rock-Paper-Scissors Play multiple times –a repeated zero-sum game How should you “learn” to play the game?! –How can you know if you are doing “well” –Highly opponent dependent –In retrospect we should always win … (1,-1)(-1,1) (0, 0) (-1,1)(0, 0)(1,-1) (0, 0)(1,-1)(-1,1)

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7 Rock-Paper-Scissors The (1-shot) zero-sum game has a value –Each player has a mixed strategy that can enforce the value Alternative 1: Compute the minimax strategy –Value V= 0 –Strategy = (⅓, ⅓, ⅓) Drawback: payoff will always be the value V –Even if the opponent is “weak” (always plays ) (1,-1)(-1,1) (0, 0) (-1,1)(0, 0)(1,-1) (0, 0)(1,-1)(-1,1)

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8 Rock-Paper-Scissors Alternative 2: Model the opponent –Finite Automata Optimize our play given the opponent model. Drawbacks: –What is the “right” opponent model. –What happens if the assumption is wrong. (1,-1)(-1,1) (0, 0) (-1,1)(0, 0)(1,-1) (0, 0)(1,-1)(-1,1)

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9 Rock-Paper-Scissors Alternative 3: Online setting –Adjust to the opponent play –No need to know the entire game in advance –Payoff can be more than the game’s value V Conceptually: –Have a set of comparison class of strategies. –Compare performance in hindsight (1,-1)(-1,1) (0, 0) (-1,1)(0, 0)(1,-1) (0, 0)(1,-1)(-1,1)

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10 Rock-Paper-Scissors Comparison Class H: –Example : A={,, } –Other plausible strategies: Play what your opponent played last time Play what will beat your opponent previous play Goal: Online payoff near the best strategy in the class H Tradeoff: –The larger the class H, the difference grows. (1,-1)(-1,1) (0, 0) (-1,1)(0, 0)(1,-1) (0, 0)(1,-1)(-1,1)

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11 Rock-Paper-Scissors: Regret Consider A={,, } –All the pure strategies Zero-sum game: Given any mixed strategy σ of the opponent, there exists a pure strategy a ∊ A whose expected payoff is at least V Corollary: For any sequence of actions (of the opponent) We have some action whose average value is V

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12 Rock-Paper-Scissors: Regret payoffopponentwe Average payoff -1/3 payoffopponentplay New average payoff 1/3

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13 Rock-Paper-Scissors: Regret More formally: After T games, Û = our average payoff, U(h) = the payoff if we play using h regret(h) = U(h)- Û External regret: max h ∊ H regret(h) Claim: If for every a ∊ A we have regret(a) ≤ ε, then Û ≥ V- ε

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14 [Blum & M] and [Cesa-Bianchi, M & Stoltz]

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15 Regret Minimization: Setting Online decision making problem (single agent) At each time, the agent: –selects an action –observes the loss/gain Goal: minimize loss (or maximize gain) Environment model: –stochastic versus adversarial Performance measure: –optimality versus regret

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16 Regret Minimization: Model Actions A={1, …,N} Number time steps: t ∊ { 1, …, T} At time step t: –The agent selects a distribution p i t over A –Environment returns costs c i t ∊ [0,1] –Online loss: l t = Σ i c i t p i t Cumulative loss : L online = Σ t l t Information Models: –Full information: observes every action’s cost –Partial information: observes only its own cost

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17 Stochastic Environment Costs: for each action i, c i t are i.i.d. r.v. (for diff. t) –Assuming an oblivious opponent Full information Greedy Algorithm: –selects the action with the lowest average cost. Analysis: –Assume two actions –Boolean cost: Pr[c i =1]=p i and p 1 < p 2 –Concentration bound: Pr[avg 1 t > avg 2 t ] < exp{-(p 1 -p 2 ) 2 t} –Expected regret: E[Σ (c 2 t+1 -c 1 t+1 ) I(avg 1 t > avg 2 t )] –sum over t=1, … Σ t (p 2 -p 1 ) exp{-(p 1 -p 2 ) 2 t}= O( 1/ (p 2 -p 1 ) )

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18 Stochastic Environment Partial information Tradeoff: Exploration versus Exploitation Simple approximate solution: –sample each action O(ε -2 log N/δ) times –Afterwards, select the best observed action Gittin’s Index –Simple optimal selection rule Assuming a Bayesian prior and discounting

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19 Competitive Analysis Costs: c i t are generated adversarially, –might depend on the online algorithm decisions inline with our game theory applications Online Competitive Analysis: –Strategy class = any dynamic policy –too permissive Always wins rock-paper-scissors Performance measure: –compare to the best strategy in a class of strategies

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20 External Regret Static class –Best fixed solution Compares to a single best strategy (in H) The class H is fixed beforehand. –optimization is done with respect to H Assume H=A –Best action: L best = MIN i {Σ t c i t } –External Regret = L online – L best Normalized regret is divided by T

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21 External regret: Bounds Average external regret goes to zero –No regret –Hannan [1957] Explicit bounds –Littstone & Warmuth ‘94 –CFHHSW ‘97 –External regret = O(√Tlog N)

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23 External Regret: Greedy Simple Greedy: –Go with the best action so far. For simplicity loss is {0,1} Loss can be N times the best action –holds for any deterministic online algorithm Can not be worse: –L online < N L best

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24 External Regret: Randomized Greedy Randomized Greedy: –Go with a random best action. Loss is ln(N) times the best action Analysis: best Expected loss during the time that best increases from k to k+1 is 1/N + 1/(N-1) + … ≈ ln(N)

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25 External Regret: PROD Algorithm Regret is √Tlog N PROD Algorithm: –plays sub-best actions –Uses exponential weights w a = (1-η) L a –Normalize weights Analysis: –W t = weights of all actions at time t –F t = fraction of weight of actions with loss 1 at time t Also, expected loss: L ON = ∑ F t W t+1 = W t (1-ηF t ) FtFt WtWt W t+1 (1-η)

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26 External Regret: Bounds Derivation Bounding W T Lower bound: W T > (1-η) L min Upper bound: W T = W 1 Π t (1-ηF t ) ≤ W 1 Π t exp{-ηF t } = W 1 exp{-η L ON } using 1-x ≤ e -x Combined bound: (1-η) L min ≤ W 1 exp{-η L ON } Taking logarithms: L min log(1- η) ≤ log(W 1 ) -ηL ON Final bound: L ON ≤ L min + ηL min +log(N)/η Optimizing the bound: η = √log(N)/T L ON ≤ L min +2√ T log(N)

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27 External Regret: Summary We showed a bound of 2√T log N More refined bounds √Q log N where Q = Σ t (c t best ) 2

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28 External Regret: Summary How surprising are the results … –Near optimal result in online adversarial setting very rear … –Lower bound: stochastic model stochastic assumption does not help … –Models an “improved” greedy –An “automatic” optimization methodology Find the best fixed setting of parameters

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29 External Regret and classification Connections to Machine Learning: H – the hypothesis class cost – an abstract loss function –no need to specify in advance Learning setting – online –learner: observes point, predicts, observes loss Regret guarantee: –compares to the best classifier in H.

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31 Types of Regret (Internal) Comparison is based on online’s decisions. –Depends on the actions of the online algorithm –modify a single decision (consistently) Each time action A was done do action B Comparison class is not well define in advanced. Scope: –Stronger then External Regret –Weaker then competitive analysis.

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32 Internal Regret Assume action sequence A=a 1 … a T –Modified input (b → d ) : Change every a i t =b to a i t =d, and create actions seq. B. L(b→d) is the cost of B –using the same costs c i t Internal regret L online –min {b,d} L online (b→d) = max {b,d} Σ t (c b t –c d t )p b t Swap Regret: –Change action i to action F(i)

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33 Internal regret No regret bounds –Foster & Vohra –Hart & Mas-Colell –Based on the approachability theorem Blackwell ’56 –Cesa-Bianchi & Lugasi ’03 Internal regret = O(log N + √T log N)

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34 Regret: Example External regret –You should have bought S&P 500 –Match boy i to girl i Internal regret –Each time you bought IBM you should have bought SUN –Stable matching

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35 Internal Regret Intuition: –considers swapping two actions (consistently) –A more complex benchmark: depends on the online actions selected! Game theory applications: –Avoiding dominated actions –Correlated equilibrium Reduction from External Regret [Blum & M]

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36 Dominated actions Action a i is dominated by b i if for every a -i we have u i (a i, a -i ) < u i (b i, a -i ) Clearly, we like to avoid dominated action –Remark: an action can be also dominated by a mixed action Q: can we guarantee to avoid dominated actions?! a b

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37 Dominated Actions & Internal Regret How can we test it?! –in retrospect a i is dominates b i –Every time we played a i we do better with b i Define internal regret –swapping a pair of actions No internal regret no dominated actions Internal Regret (a b) Modified Payoff (a b) Our Payoff our actions 2-1=121a b c 5-2=352a d 9-3=693a b d 1-0=110a

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38 Dominated actions & swap regret Swap regret –An action sequence σ = σ 1, …, σ t –Modification function F:A A –A modified sequence σ(F) = F(σ 1 ), …, F(σ t ) Swap_regret = max F V(σ (F)) - V(σ ) Theorem: If Swap_regret < R then in at most R/ε steps we play ε-dominated actions. σ(F)σ ba cb cc ba bd ba cb bd ba

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39 Calibration Model: each step predict a probability and observe outcome Goal: prediction calibrated with outcome –During time steps where the prediction is p the average outcome is (approx) p predictionsoutcome Calibration:.3.5 1/3 1/2 Predict prob. of rain

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Calibration to Regret Reduction to Swap/Internal regret: –Discrete Probabilities Say: {0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0} –Loss of action x at time t: (x – c t ) 2 –y*(x)= argmax y Regret(x,y) y*(x)=avg(c t |x) –Consider Regret(x,y*(x))

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41 Correlated Equilibrium Q a distribution over joint actions Scenario: –Draw joint action a from Q, –player i receives action a i and no other information Q is a correlated Eq if: –for every player i, the recommended action a i is a best response given the induced distribution. Q a1a1 a2a2 a3a3 a4a4

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42 Swap Regret & Correlated Eq. Correlated Eq NO Swap regret Repeated game setting Assume swap_regret ≤ ε –Consider the empirical distribution A distribution Q over joint actions –For every player it is ε best response –Empirical history is an ε correlated Eq.

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43 External Regret to Internal Regret: Generic reduction Input: –N (External Regret) algorithms –Algorithm A i, for any input sequence : L A i ≤ L best,i + R i A1A1 AiAi AnAn

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44 External to Internal: Generic reduction General setting (at time t): –Each A i outputs a distribution q i A matrix Q –We decide on a distribution p –Adversary decides on costs c= –We return to A i some cost vector A1A1 AiAi AnAn q1q1 qiqi qnqn Q pc

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45 Combining the experts Approach I: –Select an expert A i with probability r i –Let the “selected” expert decide the outcome –action distribution p=Qr Approach II: –Directly decide on p. Our approach: make p=r –Find a p such that p=Qp Q p

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46 Distributing loss Adversary selects costs c= Reduction: –Return to A i cost vector c i = p i c –Note: Σ c i = c A1A1 AiAi AnAn c

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47 External to Internal: Generic reduction Combination rule: –Each A i outputs a distribution q i Defines a matrix Q –Compute p such that p=Qp p j = Σ i p i q i,j –Adversary selects costs c= –Return to A i cost vector p i c Q p

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48 Motivation Dual view of p: –p i is the probability of selecting action i –p i is the probability of selecting algo. A i Then use A i probability, namely q i Breaking symmetry: –The feedback to A i depends on p i

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49 Proof of reduction: Loss of A i (from its view) – = p i Regret guarantee (for any action i) w.r.t. action j: –L i = Σ t p i t ≤ Σ t p i t c j t + R i Online loss: –L online = Σ t = Σ t = Σ t Σ i p i t =Σ i L i For any swap function F: –L online ≤ L online,F + Σ i R i

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50 Swap regret Corollary: For any swap F: Improved bound: –Note that i L max,i T worse case: all L max,i are equal. –Improved bound:

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52 Reductions between Regrets External Regret Full Information External Regret Partial Information Internal Regret Full Information Internal Regret Partial Information [AM] [BM] [CMS]

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53 More elaborate regret notions Time selection functions [Blum & M] –determines the relevance of the next time step –identical for all actions –multiple time-selection functions Wide range regret [Lehrer, Blum & M] –Any set of modification functions mapping histories to actions

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54 Conclusion and Open problems Reductions –External to Internal Regret full information partial information SWAP regret Lower Bound –poly in N= |A| lower bounds [Auer] Wide Range Regret –Applications …

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55 Many thanks for my co-authors: A. Blum, N. Cesa-Bianchi, and G. Stoltz

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56 Partial Information Partial information: –Agent selects action i –Observes the loss of action i –No information regarding the loss of other actions How can we handle this case?

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57 Reductions between Regrets External Regret Full Information External Regret Partial Information Internal Regret Full Information Internal Regret Partial Information [AM, podc 03] [BM, colt 05] [CMS, colt 05]

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58 Information: Full versus Partial Importance Sampling: –maintain weights as before. –update the selected action k by loss c k t /p k t –Expectation is maintain –Need to argue directly on the algorithm. Used in: [ACFS], [HM] and others –Regret Bound about T 1/2

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59 Information: Full versus Partial Partial information Works in blocks of size B –explore each action with small probability with high probability explores each action once –Otherwise uses FI action distribution At the end of B: Feeds the explored action to FI Regrets: –Regret of FI on T/B time steps (each of size B) –Exploration regret T/B

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60 Information model: Full versus Partial Full Information Regret Algorithm Partial Information exploit sample

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61 Information: Full versus Partial Anslysis: –Regret of FI on T/B time steps (each of size B) Regret ~ sqrt{ B T} –Exploration regret O(N log N) in block Regret ~ T/B Optimizing B ~ T 1/3 Performance guarantee: Regret ~ T 2/3 Benefit: Simpler and wider applicability.

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62 More elaborate regret notions Time selection functions [Blum & M] –determines the relevance of the next time step –identical for all actions –multiple time-selection functions Wide range regret [Lehrer, Blum & M] –Any set of modification functions mapping histories to actions

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64 Routing Model: select a path for each sessions Loss: The latency on the path Performance Goal: Achieve the best latencies

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65 Financial Markets Model: Select a portfolio each day. Gain: The changes in stock values. Performance Goal: Compare well with the best busy and hold policy.

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66 Parameter Tuning Model: Multiple parameters. Optimization: any Performance Goal: guarantee the performance of (almost) the best set of parameters

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67 General Motivation Development Cycle –develop product (software) –test performance –tune parameters –deliver “tuned” product Challenge: can we combine –testing –tuning –runtime

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68 Challenges Large action spaces –Regret is logarithmic –For d dimension regret ~ d Computational issue: –How to aggregate weights efficiently Some existing results [KV,KKL]

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69 Summary Regret bounds –simple algorithms –impressive bounds near optimal Applications –methodological –algorithmic Great potential

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70 Many thanks for my co-authors: A. Blum, N. Cesa-Bianchi, and G. Stoltz

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