Factor Graphs, the Sum-Product Algorithm and TrueSkill TM

Transcription

1 Factor Graphs, the Sum-Product Algorithm and TrueSkill TM Kuhwan Jeong 1 1 Department of Statistics, Seoul National University, South Korea July, / 17

2 Introduction x i : a variable taking on values in some domain A i, i = 1,..., n g(x 1,..., x n) : a function of x 1,..., x n g i (x i ) : the marginal function w.r.t. x i g i (x i ) = g(x 1,..., x n) x 1 A 1 x n A n = x i g(x 1,..., x n) x i 1 A i 1 x i+1 A i+1 The sum-product algorithm is a efficient procedure for computing marginal functions that a) exploit the way in which the global function factors, and b) reuses intermediate values. It is a simple way to understand a large number of seemingly different algorithms that have been developed. 2 / 17

3 Factor Graphs Suppose that g(x 1,..., x n) factors into a product of several local functions, g(x 1,..., x n) = j J f j (X j ) where J is a discrete index set, X j is a subset of {x 1,..., x n}, and f j (X j ) is a function having the elements of X j as arguments. A factor graph is a bipartite graph that expresses the structure of the factorization. It has a variable node for each variable x i, a factor node for each local function f j, and an edge between a variable node x i and a factor node f j if and only if x i is an argument of f j. Example. g(x 1, x 2, x 3, x 4, x 5 ) = f A (x 1 )f B (x 2 )f C (x 1, x 2, x 3 )f D (x 3, x 4 )f E (x 3, x 5 ) 3 / 17

4 Computing a Single Marginal Function Example (continued). g(x 1, x 2, x 3, x 4, x 5 ) = f A (x 1 )f B (x 2 )f C (x 1, x 2, x 3 )f D (x 3, x 4 )f E (x 3, x 5 ) g 1 (x 1 ) = f A (x 1 ) f B (x 2 ) f C (x 1, x 2, x 3 ) f D (x 3, x 4 ) f E (x 3, x 5 ) x 2 x 3 x 4 x 5 = f A (x 1 ) f B(x 2 )f C (x 1, x 2, x 3 ) f D (x 3, x 4 ) f E (x 3, x 5 ) x 1 x 3 x 3 g 3 (x 3 ) = f A (x 1 )f B (x 2 )f C (x 1, x 2, x 3 ) f D (x 3, x 4 ) f E (x 3, x 5 ) x 3 x 3 x 3 4 / 17

5 Computing a Single Marginal Function Single-i Sum-Product algorithm. Take x i as the root vertex. Each leaf node sends a message to its parent. Each vertex waits for messages from all of its children before computing the message to be sent to its parent. A variable node simply sends the product of messages received from its children µ x f = h n(x)\f µ h x (x) where n(x) is the set of functions of which x is an argument. A factor node f with a parent x forms the product of f with the messages received from its children, and then operates the summation x on the result µ f x (x) = f (X) µ y f (y) x y n(f )\{x} where X = n(f ) is the set of arguments of f. g i (x i ) is obtained as the product of all messages received at x i. 5 / 17

6 Computing a Single Marginal Function Example (continued). µ x4 f D = µ x5 f E = 1, µ fd x 3 = f D (x 3, x 4 ), µ fe x 3 = f E (x 3, x 5 ), x 3 x 3 µ x3 f C = f D (x 3, x 4 ) f E (x 3, x 5 ), µ fb x 2 = µ x2 f C = f B (x 2 ), µ fc x 1 = f B(x 2 )f C (x 1, x 2, x 3 ) f D (x 3, x 4 ) f E (x 3, x 5 ), x 1 µ fa x 1 = f A (x 1 ), g 1 (x 1 ) = f A (x 1 ) f B(x 2 )f C (x 1, x 2, x 3 ) f D (x 3, x 4 ) f E (x 3, x 5 ). x 1 6 / 17

7 Computing a Single Marginal Function Example (continued). µ fa x 1 = µ x1 f C = f A (x 1 ), µ fb x 2 = µ x2 f C = f B (x 2 ), µ fc x 3 = f A (x 1 )f B (x 2 )f C (x 1, x 2, x 3 ), x 3 µ x4 f D = µ x5 f E = 1, µ fd x 3 = f D (x 3, x 4 ), µ fe x 3 = f E (x 3, x 5 ), x 3 x 3 g 3 (x 3 ) = f A (x 1 )f B (x 2 )f C (x 1, x 2, x 3 ) f D (x 3, x 4 ) f E (x 3, x 5 ). 7 / 17

8 Computing All Marginal Functions Sum-Product algorithm. As in the single-i algorithm, message passing is initiated at the leaves. Each vertex v remains idle until messages have arrived on all but one of the edges incident on v. 8 / 17

9 Example : TrueSkill TrueSkill ranking system is a skill based ranking system for Xbox Live developed at Microsoft Research. The purpose is to both identify and track the skills of gamers in order to be able to match them into competitive matches. TrueSkill ranking system only uses the final standings of all teams in a game in order to update the skill estimates of all gamers playing in this game. Ranking systems have been proposed for many sports but possibly the most prominent ranking system in use today is the Elo system. 9 / 17

10 Elo System In 1959, Arpad Elo developed a statistical rating system for Chess, which was adopted by the World Chess Federation FIDE in It models the probability of the possible game outcomes as a function of the two players skill ratings s 1 and s 2. In a game each player i exhibits performance p i N (s i, β 2 ). The probability that player 1 wins is given by ( s1 s 2 P(p 1 > p 2 s 1, s 2 ) = Φ ). 2β Let y = 1 if player 1 wins, y = 1 if player 2 wins and y = 0 if a draw occurs. After the game, the skill ratings s 1 and s 2 are updated by s 1 s 2 s 1 + y s 2 y where = αβ ( ( )) y + 1 s1 s 2 π Φ. 2 2β 10 / 17

11 TrueSkill Assume an independent normal prior p(s) = n i=1 N (s i; µ i, σ 2 i ). Each player i exhibits a performance p i N (p i ; s i, β 2 ). From among a population of n players {1,..., n} in a game let k teams compete a match. The team assignments are specified by k non-overlapping subsets A j {1,..., n}, A i A j = if i j. The performance t j of team j is modeled as the sum of the performances of its members t j = p i. i A j 11 / 17

12 TrueSkill The outcome r = (r 1,..., r k ) {1,..., k} k is specified by a rank r j for each team j. Disregarding draws, the probability of a game outcome r is modeled as P(r t 1,..., t k ) = P(t r(1) > t r(2) > > t r(k) ). If draws are permitted the wining outcome r (j) < r (j+1) requires t r(j) > t r(j+1) + ɛ and the draw outcome r (j) = r (j+1) requires t r(j) t r(j+1) ɛ, where ɛ is a draw margin. 12 / 17

13 TrueSkill Consider a game with 3 teams with A 1 = {1}, A 2 = {2, 3} and A 3 = 4. Assume that team 1 is the winner and that teams 2 and 3 draw, i.e., r = (1, 2, 2). The factor graph representing the joint distribution P(s, p, t r, A) is depicted. 13 / 17

14 TrueSkill The quantities of interest are the posterior distribution P(s i r, A). P(s i r, A) is calculated from the joint distribution integrating out the individual performances {p i } and the team performances {t i }, P(s i r, A) = P(s, p, t r, A)dpdt 14 / 17

15 Approximate Message Passing The message passing is characterized by the following equations: P(v k ) = µ f vk (v k ), f n(v k ) µ f vj (v j ) = f (v) µ vk f (v k ) = h n(v k )\{f } v i n(f )\{v j } µ h vk (v k ). m vi f (v i )dv j, The TrueSkill factor graph is acyclic and the majority of messages can be represented compactly as 1-dimensional Gaussians. However, messages from the comparison factors I( > ɛ) or I( ɛ) to the performance differences d i are non Gaussian. We approximate these messages by approximating the marginal P(d i ) via moment matching resulting in a Gaussian ˆP(d i ) with the same mean and variance as P(d i ). Then, we have ˆµ f di (d i ) = ˆP(d i ) µ di f (d i ). 15 / 17

16 Approximate Message Passing Since the messages 2 and 5 are approximate, iterate over all messages that are on the shortest path between any two approximate marginals ˆP(d i ) until the convergence of marginals. 16 / 17

17 References Kschischang, F. R., Frey, B. J., & Loeliger, H. A. (2001). Factor graphs and the sum-product algorithm. IEEE Transactions on information theory, 47(2), Herbrich, R., Minka, T., & Graepel, T. (2007). TrueSkill TM : a Bayesian skill rating system. In Advances in neural information processing systems (pp ). 17 / 17