Efficient Information Planning in Graphical Models

Transcription

1 Efficient Information Planning in Graphical Models computational complexity considerations John Fisher & Giorgos Papachristoudis, MIT VITALITE Annual Review 2013 September 9, 2013 J. Fisher (VITALITE Annual Review 2013) 9 Sep 13 1 / 14

2 Information Fusion Distributed Information Fusion J. Fisher (VITALITE Annual Review 2013) 9 Sep 13 2 / 14

3 Established Results Key Ideas 1 A broad class of information measures - f -divergences are fundamentally linked to bounds on risk. Bartlett et al. [2003], Nguyen et al. [2009] f -divergence φ-risk bound on excess risk 2 as applied to information measures is a key enabler. Krause and Guestrin [2005], Williams et al. [2007], Papachristoudis and Fisher III [2012] off-line and on-line performance bounds guarantees on tractable planning methods incorporations of inhomogenous resource constraints 3 Submodular properties are intimately related to the structure of graphical models. Williams et al. [2007] local properties (and computations) yield global properties J. Fisher (VITALITE Annual Review 2013) 9 Sep 13 3 / 14

4 Key Ideas Key Ideas Information planning posed as combinatorial selection problem over sequential consideration of groups of measurements 1 bounds apply to all sequences (visit paths) 2 information rewards vary across walks 3 evaluation of multiple walks leads to increased information rewards with diminishing probability 4 evaluation of multiple walks leads to tighter upper bound also with reduced probability J. Fisher (VITALITE Annual Review 2013) 9 Sep 13 4 / 14

5 Some Context Distributed Sensing Key Ideas z 1 z 2 xn xk x0 z Ns Computational Hurdles Evaluating information measures for complex sensors induces a computational bottleneck. Evaluating information measures for simple sensors and complex graphs (or even simple graphs) induces a computational bottleneck. Due to the branching structure (i.e.,, dependence on prior sensor actions), optimal plans are intractable due to exponential complexity. J. Fisher (VITALITE Annual Review 2013) 9 Sep 13 5 / 14

6 Inference versus Information VoI Inf Gain x z 1... z k... z N Bayesian Inference p(x z 1,,z k ) = p(x) p(z 1 x) p(z 1 ) Information Gain p(z 2 x) p(z 2 z 1 ) p(z k x) p(z k z 1,,z k 1 ) complementary information {}}{ I (x;z 1,,z k ) = I (x;z 1 )+I (x;z 2 ) I (z 1 ;z 2 ) + + I (x;z k ) I (z k ;z 1,,z k 1 ) }{{} common information J. Fisher (VITALITE Annual Review 2013) 9 Sep 13 6 / 14

7 VoI Submodularity Given a set V, a real-valued function f on 2 V is submodular if f (A) + f (B) f (A B) + f (A B) A,B V. Define the set increment function as ρ S (j) f (S j) f (S). Equivalently, a real-valued function is submodular if ρ A (j) ρ B (j) A B V and j / B that is, the incremental value of j is greater relative to A than to any B which contains A. Submodularity captures the notion of diminishing returns J. Fisher (VITALITE Annual Review 2013) 9 Sep 13 7 / 14

8 Submodularity VoI Monotonicity: A real-valued f is monotone if Greedy Selection Batch setting g j = argmax ρ G j 1(u) u V\G j 1 f (A) f (B) ; A B, or ρ S (j) 0 ; j V,S V Sequential setting g j = argmax ρ G j 1(u) u V wj \G j 1 The batch setting chooses from among all measurements conditioned on previous selections. The sequential setting is restricted to only those available at the current node in the visit walk. J. Fisher (VITALITE Annual Review 2013) 9 Sep 13 8 / 14

9 Preliminaries VoI Notation X = {X 1,,X n } denotes n latent inference variables. Z = {Z 1,,Z n } denotes n measurement vectors. Each Z t is comprised of N t measurements corresponding to variable X t. V t = {1,,N t } indicate measurement indices, i.e., observation sets. Z i Z j X : Measurements are independent given X. Reward function: f : 2 V R : a set function that captures the value of sensing actions. Cost function: c : 2 V R + : a nonnegative set function that quantifies the cost of a subset, and where costs are assumed to be additive over the elements of the subset.. c(s) = c j j S J. Fisher (VITALITE Annual Review 2013) 9 Sep 13 9 / 14

10 Sequential Setting VoI Goal: Choose k 1 from V 1,, k n from V n : n O = argmax f (S) where S = S t and S i S j = /0, i j. S 1 k 1, S n k n t=1 Z 1 1 Z2 1 Z2 2 Z N2 2 Z 1 T Z 2 T Z 1 1 Z2 1 Z2 2 Z N2 2 Z 1 T Z 2 T Z 2 1 Z N1 1 X1 X3 X2 X4 XT Z NT T Z 2 1 Z N1 1 X1 X3 X2 X4 XT Z NT T Z3 1 Z3 2 Z N3 3 Z 1 4 Z 2 4 Z N4 4 Z3 1 Z3 2 Z N3 3 Z 1 4 Z 2 4 Z N4 4 N t measurements for each hidden variable X t. Visit walk: define the M-length visit walk as the order {w 1,,w M } in which we visit observation sets V t during a selection process. J. Fisher (VITALITE Annual Review 2013) 9 Sep / 14

11 Sequential Setting VoI Goal: Choose k 1 from V 1,, k n from V n : n O = argmax f (S) where S = S t and S i S j = /0, i j. S 1 k 1, S n k n t=1 Z2 1 Z2 2 Z N2 2 Z 1 T t =1 t =2 t = T Z 1 1 Z 2 T Z 2 1 X1 X2 XT Z NT T Z N1 1 X3 X4 Z3 1 Z3 2 Z N3 3 Z 1 4 Z 2 4 Z N4 4 h 1 h 1 h 1 h 1 h 1 h 2 h 2 h 3 h 1 h 1 h 4 h 4 h 4 h 4 h 4 wj wj+1wj+2 wm Analysis specialized to Markov Chains and LQG models Extends to trees and polytrees J. Fisher (VITALITE Annual Review 2013) 9 Sep / 14

12 Gaussian Markov Chains VoI We consider Gaussian Markov chains for convenience in derivations. The underlying dynamical system is: X k = A k 1 X k 1 + V k 1 Y k = C k X k + W k, where X 0 N ( x 0, Σ 0 ),V k 1 N (0,Q k 1 ),W k N (0,R k ). (A Markov chain is shown in the upper right figure.) Results can be generalized to trees and polytrees. J. Fisher (VITALITE Annual Review 2013) 9 Sep / 14

13 VoI Sparsity Usually, measurements are obtained from a small subset of the underlying process. A hidden variable depends only on a restricted set of hidden variables of the previous time point. t = k t = k +1 J. Fisher (VITALITE Annual Review 2013) 9 Sep / 14

14 Emprical Results VoI 14.8 IG as a function of complexity IG random walks segm walks (len: 2) segm walks (len: 3) segm walks (len: 4) segm walks (len: 5) forward walk worst complexity walk maximum IG walk number of messages J. Fisher (VITALITE Annual Review 2013) 9 Sep / 14

15 References References I P. L. Bartlett, M. I. Jordan, and J. D. Mcauliffe. Convexity, classification, and risk bounds. Journal of the American Statistical Association, A. Krause and C. Guestrin. Near-optimal nonmyopic value of information in graphical models. In Uncertainty in Artificial Intelligence, July X. Nguyen, M. J. Wainwright, and M. I. Jordan. On surrogate loss functions and f-divergences. Annals of Statistics, G. Papachristoudis and J. W. Fisher III. Theoretical guarantees on penalized information gathering. In Proc. IEEE Workshop on Statistical Signal Processing, August URL publications/papers/papachristoudis12sspworkshop.pdf. **. J. L. Williams, J. W. Fisher III, and A. S. Willsky. Performance guarantees for information theoretic active inference. In M. Meila and X. Shen, editors, Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, pages , March URL publications/papers/wilfis07aistats.pdf. **Outgrowth of Supervised Student Research J. Fisher (VITALITE Annual Review 2013) 9 Sep / 14