Online Forest Density Estimation

Transcription

1 Online Forest Density Estimation Frédéric Koriche CRIL - CNRS UMR 8188, Univ. Artois koriche@cril.fr UAI 16 1

2 Outline 1 Probabilistic Graphical Models 2 Online Density Estimation 3 Online Forest Density Estimation 4 Conclusions 2

3 X 8 X 9 X 3 X 12 X 11 X 7 X 2 X 10 X 6 X 1 X 4 X Graphical models encode high-dimensional distributions in a compact and intuitive way: Qualitative uncertainty (interdependencies) is captured by the structure Quantitative uncertainty (probabilities) is captured by the parameters 3

4 Graphical Models Bayesian Networks Sparse Bayesian Networks Bayesian Polytrees Bayesian Forests Markov Networks Sparse Markov Networks Bounded Tree-width MNs Markov Forests For an outcome space X R n, a class of graphical models is a pair M = G Θ, where G is space of n-dimensional graphs, and Θ is a space of d-dimensional vectors. G captures structural constraints (directed vs. undirected, sparse vs. dense, etc.) Θ captures parametric constraints (binomial, multinomial, Gaussian, etc.) 4

5 Outline 1 Probabilistic Graphical Models 2 Online Density Estimation 3 Online Forest Density Estimation 4 Conclusions 5

6 Online Density Estimation l ( M t,x t ) Environment M t = ( G t,θ t ) Learner Repeated game between the learner and its environment. During each trial t = 1,,T, the learner chooses (the structure and the parameters of) a model M t M ; the environment responds by an outcome x t X, and the learner incurs the log-loss l(m t,x t ) = lnp M t (x t ) 6

7 Online density estimation is particularly suited to: * Adaptive environments, where the target distribution can change over time; * Streaming applications, where all the data is not available in advance; * Large-scale datasets, by processing only one outcome at a time. In the literature of online density estimation (universal coding): uni-dimensional models (binomial, multinomial, exponential families) have been extensively studied Xie and Barron (2000); Takimoto and Warmuth (2000); Kotłowski and Grünwald (2011),... much less is known, however, about multi-dimensional models, especially graphical models, where both the structure and the parameters are updated at each iteration! 7

8 The performance of an online learning algorithm A is measured according to two metrics: Minimax Regret Defined by the maximum, over every sequence of outcomes x 1:T = (x 1,,x T ), of the cumulative relative loss between A and the best model in M : [ ] T R(A,T ) = max l(m t,x t T ) min l(m,x t ) x 1:T X T t=1 M M t=1 Per-round complexity Given by the amount of computational resources spent by A at each trial t, for choosing a model M t in M, and evaluating its log-loss l ( M t,x t ) = lnp M t (x t ). 8

9 Outline 1 Probabilistic Graphical Models 2 Online Density Estimation 3 Online Forest Density Estimation 4 Conclusions 9

10 Markov Forests X 3 X X 9 X 8 X 7 X 2 X 6 X 15 X 5 X X 3 X 13 X 10 X 4 X 12 X 11 X 14 For a set of n random variables defined over the discrete domain {0,1,,m 1}, the class of (m-ary n-dimensional) Markov Forests is given by the product F m,n = F n Θ m,n, where F n is the space of all acyclic graphs of order n; Θ m,n is the space of all parameter vectors mapping a probability table θ i [0,1] m to each candidate node i, and a probability table θ ij [0,1] m m to each candidate edge (i,j). 10

11 Markov Forests Two key properties For the class of Markov forests, The probability distribution associated with a Markov forest M = (f,θ) can be factorized into a closed-form: ( ) fij n θij (x i,x j ) P M (x) = θ i (x i ) θ i (x i )θ j (x j ) i=1 (i,j) ( n 2 ) So, the log-loss extended to convf n Θ m,n is an affine function of the structure: l(p,θ,x) = ψ(x) + p,φ(x),where ψ(x) = ( ) 1 θi ln θ i [n] i (x i ) and φ (x i )θ j (x j ) ij(x i,x j ) = ln θ ij (x i,x j ) The space F n of forest structures is a matroid; minimizing a linear function over F n can be done in quadratic time using the matroid greedy algorithm. 11

12 The Algorithm set θ 1 = U m,n set p 1 = 0 for each trial t = 1,,T play M t = (f t,θ t ), where f t = SWAP 1(p t ) receive x t set θ i t+1 (u) = t u + 1/2 for all n nodes t + m/2 set θ ij t+1 (u,v) = t uv + 1/2 t + m 2 /2 [ ] ( draw r t in 0, 1 n 2 ) β uniformly at random t Initialization Parameter Update for all ( n) (Jeffreys rule) 2 possible edges set f t+ 1 2 = argmin f F n f,r t + t s=1 φ s (x s ) set p t+ 1 2 = α t p t +(1 α t )f t+ 2 1 ( ) set p t+1 = SWAP k p t+ 1 2 Structure Update (Mixture of perturbed leaders) SWAP k uses random base exchanges to derive a convex mixture with at most k components. 12

13 Analysis Based on the closed-form expression of Markov forests, parameter updates and structure updates can be analyzed in an independent way: ( ) ( ) R M 1:T,x 1:T = R p 1:T,x 1:T + R (θ ) 1:T,x 1:T where ( ) R p 1:T,x 1:T T = l(p t,θ t,x t ) l(p,θ t,x t ) t=1 ( ) R θ 1:T,x 1:T T = l(p,θ t,x t ) l(p,θ,x t ) t=1 (Structural Regret) (Parametric Regret) 13

14 Analysis Parametric Regret Decomposable into local regrets, which can be bounded using universal coding techniques: ( ) R θ 1:T,x 1:T = n ln θ i i=1 θ i 1:T + ln (i,j) F + (i,j) F (x i 1:T ) (x i 1:T ) θ ij (x1:t ) ij θ ij 1:T (x ij 1:T ) ln θ1:t i θ i (x 1:T i (x 1:T i ) ) θ 1:T j θ j (x j 1:T ) (x j 1:T ) Univariate estimators Bivariate estimators Bivariate compensation Using symmetric Dirichlet mixtures for the parametric estimators, T θ 1:T (x 1:T ) = P λ (x t )p µ (λ)dλ = Γ(mµ) mv=1 Γ(t v + µ) t=1 Γ(µ) m Γ(t + mµ) the parametric regret for µ = 1 2 (Jeffreys mixture) is in O(lnT ). The per-round time complexity for parameter updates is in O(m 2 n 2 ). 14

15 Analysis Structural Regret Based on the telescopic decomposition (and using l t = φ t (x t )), ( ) R p 1:T,x 1:T T = p t,l t p t+ 1 2,l t 0 t=1 T + p t+ 1 2,l t f t+ 1 2,l t Convex mixture t=1 T + f t+ 1 2,l t p,l t Follow the Perturbed Leader t=1 (Kalai and Vempala, 2005) the structural regret is in O( T lnt ). The per-round time complexity for structure updates is in in O(n 2 logn + kn 2 ). 15

16 Preliminary Experiments Average log-loss CL CLT OFDE F OFDE T Average log-loss CL CLT OFDE F OFDE T KDD Cup 2000: Iterations MSNBC: Iterations The OFDE algorithm (with F for forests, and T for trees, k = lnn) was compared to batch algorithms (Chow-Liu (1968) for trees, and Chow-Liu with Thresholding (2011) for forests), which had the benefit of hindsight for the train set. The average log-loss was measured on the test set at the end of each iteration. 16

17 Outline 1 Probabilistic Graphical Models 2 Online Density Estimation 3 Online Forest Density Estimation 4 Conclusions 17

18 Conclusions Online density estimation has very attractive properties Designed for adversarial environments Naturally suited to streaming applications Can be applied to large-scale applications with massive amounts of data Online graphical density estimation is challenging We are faced with a tradeoff between minimax optimality and computational complexity: Minimax optimality often requires super-exponential time. Online approximation algorithms (Kakade et al., 2009) look promising for handling more expressive graphical models (ex: polytrees, bounded treewidth networks). 18

19 Thank You! (and thanks to the anonymous reviewers for helping to improve this paper!) 19

20 References C. K. Chow and C. N. Liu. Approximating discrete probability distributions with dependence trees. IEEE Transactions on Information Theory, 14(3): , S. Kakade, A. Kalai, and K. Ligett. Playing games with approximation algorithms. SIAM J. Comput., 39(3): , A. Kalai and S. Vempala. Efficient algorithms for online decision problems. Journal of Computer and System Sciences, 71(3): , W. Kotłowski and P. Grünwald. Maximum likelihood vs. sequential normalized maximum likelihood in on-line density estimation. In Proc. of COLT, pages , E. Takimoto and M. Warmuth. The last-step minimax algorithm. In Proc. of ALT, pages , V. Tan, A. Anandkumar, and A. Willsky. Learning high-dimensional Markov forest distributions: Analysis of error rates. Journal of Machine Learning Research, 12: , Q. Xie and A. Barron. Asymptotic minimax regret for data compression, gambling, and prediction. IEEE Transactions on Information Theory, 46(2): ,