Online Algorithms: Learning & Optimization with No Regret.

Transcription

1 Online Algorithms: Learning & Optimization with No Regret. Daniel Golovin 1

2 The Setup Optimization: Model the problem (objective, constraints) Pick best decision from a feasible set. Learning: Model the problem (objective, hypothesis class) Pick best hypothesis from a feasible set. 2

3 Online Learning/Optimization Choose an action x t 2 X Get f t (x t ) and feedback f t : X! [0; 1] Same feasible set X in each round t Different Reward Models: Stochastic, Arbitrary but Oblivious, Adaptive and Arbitrary 3

4 Concrete Example: Commuting Pick a path x t from home to school. Pay cost f t (x t ) := P e2x t c t (e) Then see all edge costs for that round. Dealing with Limited Feedback: later in the course. 4

5 Other Applications Sequential decision problems Streaming algorithms for optimization/learning with large data sets Combining weak learners into strong ones ( boosting ) Fast approximate solvers for certain classes of convex programs Playing repeated games 5

6 Binary prediction with a perfect expert n hypotheses ( experts ) h 1 ; h 2 ; : : : ; h n Guaranteed that some hypothesis is perfect. Each round, get a data point pt and classifications h i (p t ) 2 f0; 1g Output binary prediction xt, observe correct label Minimize # mistakes Any Suggestions? 6

7 A Weighted Majority Algorithm Each expert votes for it's classification. Only votes from experts who have never been wrong are counted. Go with the majority # mistakes M log 2 (n) Weights w it = I(h i correct on rst t rounds ). W t = P i w it. W 0 = n, W T 1 Mistake on round t implies W t+1 W t =2 So 1 W T W 0 =2 M = n=2 M 7

8 Weighted Majority [Littlestone & Warmuth '89] What if there's no perfect expert? Each expert i has a weight w(i), votes for it's classification in {-1, 1}. Go with the weighted majority, predict sign( i w i x i ). Halve weights of wrong experts. Let m = # mistakes of best expert. How many mistakes M do we make? Weights w it = (1=2) ( # mistakes by i on rst t rounds). Let W t := P i w it. Note W 0 = n, W T (1=2) m Mistake on round t implies W t W t So (1=2) m W T W 0 (3=4) M = n (3=4) M Thus (4=3) M n 2 m and M 2:41(m + log 2 (n)). 8

9 Can we do better? M 2:41(m + log 2 (n)) Experts\Time e e 1 1 No deterministic algorithm can get M < 2m. What if there are more than 2 choices? 9

10 Notation: Define loss or cost functions ct and define the regret of x 1, x 2,..., x T as R T = TX c t (x t ) t=1 Regret Maybe all one can do is hope to end up with the right regrets. Arthur Miller TX t=1 c t (x ) where x = argmin x2x P T t=1 c t(x) A sequence has \no-regret" if R T = o(t ). Questions: How can we improve Weighted Majority? What is the lowest regret we can hope for? 10

11 The Hedge/WMR Algorithm* [Freund & Schapire '97] Hedge(²) p t (i) := w it = X w jt Initialize w i0 = 1 for all i. j In each round t: Choose expert e t from categorical distribution p t Select x t = x(e t ; t), the advice/prediction of e t. For each i, set w i;t+1 = w it (1 ²) c t(x(e i ;t)) How does this compare to WM? * Pedantic note: Hedge is often called Randomized Weighted Majority, and abbreviated WMR, though WMR was published in the context of binary classification, unlike Hedge. 11

12 The Hedge/WMR Algorithm Hedge(²) p t (i) := w it = X w jt Initialize w i0 = 1 for all i. j In each round t: Choose expert e t from categorical distribution p t Select x t = x(e t ; t), the advice/prediction of e t. For each i, set w i;t+1 = w it (1 ²) c t(x(e i ;t)) Randomization Influence shrinks exponentially with cumulative loss. Intuitively: Either we do well on a round, or total weight drops, and total weight can't drop too much unless every expert is lousy. 12

13 t=1 Hedge Performance Theorem: Let x 1 ; x 2 ; : : : be the choices of Hedge(²). Then " T # X µ 1 E c t (x t ) O PT T + ln(n) 1 ² ² P T where O PT T := min i t=1 c t(x(e i ; t)). ³ pln(n)=opt If ² =, the regret is ( p O PT ln(n)) 13

14 Hedge Analysis Intuitively: Either we do well on a round, or total weight drops, and total weight can't drop too much unless every expert is lousy. Let W t := P i w it. Then W 0 = n and W T +1 (1 ²) OPT. W t+1 = X i w it (1 ²) c t(x it ) (1) = X W t p t (i)(1 ²) c t(x it ) [def of pt(i)] (2) i X W t p t (i) (1 ² c t (x it )) [Bernoulli's ineq] (3) i If x > 1; r 2 (0; 1) then (1 + x) r 1 + rx = W t (1 ² E [c t (x t )]) (4) W t exp ( ² E [c t (x t )]) [1 x e x ] (5) 14

15 E t=1 Hedge Analysis W T +1 =W 0 exp W 0 =W T +1 exp ln(n) ² Ã Ã ² ² + OPT 1 ² TX t=1 TX t=1 E [c t (x t )] E [c t (x t )] Recall W 0 = n and W T +1 (1 ²) OPT. " X T # c t (x t ) 1 µ ² ln W0 ln(n) O PT ln(1 ²) ² ² W T +1!! 15

16 If ² = Lower Bound ³ pln(n)=opt, the regret is ( p O PT ln(n)) Can we do better? Let c t (x)» Bernoulli(1/2) for all x and t. Let Z i := P T t=1 c t(x(e i ; t)). Then Z i» Bin(T; 1=2) is roughly normally distributed, with ¾ = 2p 1 T. P [Z i ¹ k¾] = exp (k 2 ) We get about ¹ = T=2, best choice is likely to get ¹ ( p T ln(n)) = ¹ ( p O PT ln(n)). 16

17 What have we shown? Simple algorithm that learns to do nearly as well as best fixed choice. Hedge can exploit any pattern that the best choice does. Works for Adaptive Adversaries. Suitable for playing repeated games. Related ideas appearing in Algorithmic Game Theory literature. 17

18 Related Questions Optimize and get no-regret against richer classes of strategies/experts: All distributions over experts All sequences of experts that have K transitions [Auer et al '02] Various classes of functions of input features [Blum & Mansour '05] E.g., consider time of day when choosing driving route. Arbitrary convex set of experts, metric space of experts, etc, with linear, convex, or Lipschitz costs. [Zinkevich '03, Kleinberg et al '08] All policies of a K-state initially unknown Markov Decision Process that models the world. [Auer et al '08] Arbitrary sets of strategies in R n with linear costs that we can optimize offline. [Hannan'57, Kalai & Vempala '02] 18

19 Related Questions Other notions of regret (see e.g., [Blum & Mansour '05]) Time selection functions: get low regret on mondays, rainy days, etc. Sleeping experts: if rule if(p) then predict Q is right 90% of the time it applies, be right 89% of the time P applies. Internal regret & swap regret: If you played x1,..., xt then have no regret against g(x 1 ),..., g(x T ) for every g:x X 19

20 Sleeping Experts if rule if(p) then predict Q is right 90% of the time it applies, be right 89% of the time P applies. Get this for every rule simultaneously. Idea: Generate lots of hypotheses that specialize on certain inputs, some good, some lousy, and combine them into a great classifier. Many applications: [Freund et al '97, Blum '97, Blum & Mansour '05] Document classification, Spam filtering, Adaptive Uis,... if ( physics in D) then classify D as science. Predicates can overlap. 20

21 Sleeping Experts Predicates can overlap E.g., predict college major given the classes C you're enrolled in? if(ml-101, CS-201 in C) then CS if(ml-101, Stats-201 in C) then Stats What do we predict for students enrolled in ML-101, CS-201, and Stats-201? 21

22 Sleeping Experts [Algorithm from Blum & Mansour '05] SleepingExperts(, E, F) Input: 2 (0; 1), experts E, time selection functions F Initialize we;f 0 = 1 for all e 2 E; f 2 F. In each round t: Let we t = P f f(t)wt e;f. Let W t = P e wt e. Let p t e = we=w t t. Choose expert e t from categorical distribution p t Select x t = x(e t ; t), the advice/prediction of e t. For each e 2 E; f 2 F w t+1 e;f = wt e;f f(t)(c t(e) E[c t (e t )]) 22

23 Sleeping Experts [Algorithm from Blum & Mansour '05] w t+1 e;f = wt e;f f(t)(c t(e) E[c t (e t )]) Ensures total sum of weights can never increase. X e;f w t e;f nm for all t w T e;f = Y t 0 f(t)(c t(e) E[c t (e t )]) = P t 0 [f(t)(c t(e) E[c t (e t )])] nm 23

24 Sleeping Experts Performance Let n = jej; m = jfj. Fix T 2 N. Let C(e; f) := P T t=1 f(t) c t(e) Let C alg (f) := P T t=1 f(t) c t(e t ) Then for all e 2 E; f 2 F E [C alg (f)] 1 If = 1 ² is close to 1, ³ C(e; f) + log 1= (nm) E [C alg (f)] = (1 + (²)) C(e; f) + Optimizing yields a regret bound of O( p C(e; f) log(nm) + log(nm)). µ log2 (nm) ² 24