Putting Bayes to sleep
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1 Putting Bayes to sleep Wouter M. Koolen Dmitry Adamskiy Manfred Warmuth Thursday 30 th August, 2012
2 Menu How a beautiful and intriguing piece of technology provides new insights in existing methods and improves the state of the art in expert tracking and online multitask learning Koolen (Royal Holloway) Putting Bayes to sleep 2 / 12
3 Point of departure We are trying to predict a sequence y 1, y 2,.... Definition A model issues a prediction each round. We denote the prediction of model m in round t by P(y t y <t, m). Say we have several models m = 1,..., M. How to combine their predictions? Koolen (Royal Holloway) Putting Bayes to sleep 3 / 12
4 Bayesian answer Place a prior P(m) on models. Bayesian predictive distribution P(y t y <t ) = M P(y t y <t, m)p(m y <t ) m=1 where posterior distribution is incrementally updated by P(m y t ) = P(y t y <t, m)p(m y <t ) P(y t y <t ) Bayes is fast: predict in O(M) time per round. Bayes is good: regret w.r.t. model m on data y T bounded by T ( ln P(y t y <t ) + ln P(y t y <t, m)) ln P(m). t=1 Koolen (Royal Holloway) Putting Bayes to sleep 4 / 12
5 Specialists Definition (FSSW97,CV09) A specialist may or may not issue a prediction. Prediction P(y t y <t, m) only available for awake m W t. Say we have several specialists m = 1,..., M. How to combine their predictions? Koolen (Royal Holloway) Putting Bayes to sleep 5 / 12
6 Specialists Definition (FSSW97,CV09) A specialist may or may not issue a prediction. Prediction P(y t y <t, m) only available for awake m W t. Say we have several specialists m = 1,..., M. How to combine their predictions? If we imagine a prediction for the sleeping specialists, we can use Bayes Choice: sleeping specialists predict with Bayesian predictive distribution That sounds circular. Because it is. $%!#? Koolen (Royal Holloway) Putting Bayes to sleep 5 / 12
7 Bayes for specialists Place a prior P(m) on models. Bayesian predictive distribution P(y t y <t ) = P(y t y <t, m)p(m y <t ) + P(y t y <t )P(m y <t ) m W t m / W t Koolen (Royal Holloway) Putting Bayes to sleep 6 / 12
8 Bayes for specialists Place a prior P(m) on models. Bayesian predictive distribution P(y t y <t ) = P(y t y <t, m)p(m y <t ) + P(y t y <t )P(m y <t ) m W t m / W t has solution P(y t y <t ) = m W t P(y t y <t, m)p(m y <t ). m W t P(m y <t ) Koolen (Royal Holloway) Putting Bayes to sleep 6 / 12
9 Bayes for specialists Place a prior P(m) on models. Bayesian predictive distribution P(y t y <t ) = P(y t y <t, m)p(m y <t ) + P(y t y <t )P(m y <t ) m W t m / W t has solution m W P(y t y <t ) = t P(y t y <t, m)p(m y <t ). m W t P(m y <t ) The posterior distribution is incrementally updated by P(m y t ) = P(y t y <t,m)p(m y <t) P(y t y <t) if m W t, P(y t y <t)p(m y <t) P(y t y <t) = P(m y <t ) if m / W t. Bayes is fast: predict in O(M) time per round. Bayes is good: regret w.r.t. model m on data y T bounded by ( ln P(y t y <t ) + ln P(y t y <t, m)) ln P(m). t : 1 t T and m W t Koolen (Royal Holloway) Putting Bayes to sleep 6 / 12
10 The specialists trick Specialists just a curiosity? Koolen (Royal Holloway) Putting Bayes to sleep 7 / 12
11 The specialists trick Specialists just a curiosity? Specialists trick: Start with input models Create many derived virtual specialists Control how they predict Control when they are awake Run Bayes on all these specialists Carefully choose prior Low regret w.r.t. class of intricate combinations of input models. Fast execution. Bayes on specialists collapses Koolen (Royal Holloway) Putting Bayes to sleep 7 / 12
12 Application 1: Freund s Problem In practice: Large number M of models Some are good some of the time Most are bad all the time We do not know in advance which models will be useful when. Goal: compare favourably on data y T with the best alternation of N M models with B T blocks. T outcomes A B C A C B C A partition model, N = 3 good models, B = 8 blocks Bayesian reflex: average all partition models. NP hard. Koolen (Royal Holloway) Putting Bayes to sleep 8 / 12
13 Application 1: sleeping solution Create partition specialists T outcomes zzzzzzzzz... A A A B B C C C Bayes is fast: O(M) time per trial, O(M) space Bayes is good: regret close to information-theoretic lower bound Koolen (Royal Holloway) Putting Bayes to sleep 9 / 12
14 Application 1: sleeping solution Create partition specialists T outcomes zzzzzzzzz... A A A B B C C C Bayes is fast: O(M) time per trial, O(M) space Bayes is good: regret close to information-theoretic lower bound Bayesian interpretation for Mixing Past Posteriors Faster algorithm Slightly improved regret bound Explain mysterious factor 2 Koolen (Royal Holloway) Putting Bayes to sleep 9 / 12
15 Application 2: online multitask learning In practice: Large number M of models Data y 1, y 2,... from K interleaved tasks Observe task label κ 1, κ 2,... before prediction We do not know in advance which tasks are similar. Goal: Compare favourably to best clustering of tasks with N K cells K tasks A B A C B A cluster model, N = 3 cells Bayesian reflex: average all cluster models. NP hard. Koolen (Royal Holloway) Putting Bayes to sleep 10 / 12
16 Application 2: sleeping solution Create cluster specialists A B zzzzzzzzz... K tasks A A B C Bayes is fast: O(M) time per trial, O(MK) space Bayes is good: regret close to information-theoretic lower bound Intriguing algorithm Regret independent of task switch count Koolen (Royal Holloway) Putting Bayes to sleep 11 / 12
17 Conclusion The specialists trick NP-hard offline problem Ditch Bayes on coordinated comparator models Instead create uncoordinated virtual specialists Craft prior so that Bayes collapses (efficient emulation) Small regret Koolen (Royal Holloway) Putting Bayes to sleep 12 / 12
18 Conclusion The specialists trick NP-hard offline problem Ditch Bayes on coordinated comparator models Instead create uncoordinated virtual specialists Craft prior so that Bayes collapses (efficient emulation) Small regret Thank you! Koolen (Royal Holloway) Putting Bayes to sleep 12 / 12
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princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 8: Decision-making under total uncertainty: the multiplicative weight algorithm Lecturer: Sanjeev Arora Scribe: (Today s notes below are