Subproblem-Tree Calibration: A Unified Approach to Max-Product Message Passing - Presentation
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The document presents a unified approach to max-product message passing through the subproblem-tree calibration (STC) algorithm, which aims to improve MAP inference for Markov Random Fields (MRFs). By employing a flexible dual-decomposition strategy, the algorithm can achieve dual-optimality for various configurations of subproblem trees, subsuming existing algorithms like MPLP, MSD, and TRW-S. Experimental results demonstrate the effectiveness of STC across multiple benchmark tasks, highlighting its adaptability in message passing method design.
Subproblem-Tree Calibration: A Unified Approach to Max-Product Message Passing - Presentation
- 1. Outline Introduction Background STC Framework Experimental Results Conclusion Subproblem-Tree Calibration: A Unified Approach to Max-Product Message Passing Varad Meru, Prolok Sundaresan Department of Computer Science, Donald Bren School of Information and Computer Science, UC Irvine December 10th, 2014 Citation: Wang, Huayan, and Koller Daphne. ”Subproblem-tree calibration: A unified approach to max-product message passing.” In Proceedings of the 30th International Conference on Machine Learning (ICML-13), pp. 190-198. 2013.
- 2. Outline Introduction Background STC Framework Experimental Results Conclusion Outline Introduction Background MAP: Maximum a posteriori estimation. LP relaxation, and dual decomposition Bethe cluster graphs STC Framework Subproblem multi-graph and subproblem trees Max-consistency and dual-optimal on trees The STC algorithm Fixed-point characterization Choosing allocation weights General primal solutions Experimental Results Conclusion
- 3. Outline Introduction Background STC Framework Experimental Results Conclusion Introduction I MAP-MRF : Finding the most probable assignments for MRFs (MPE) NP-Hard Large family of methods based on solving a dual problem of an LP relaxation. Recent Advances. Convergent version of these algorithms can be interpreted as block coordinate descent (BCD) in the dual. Variants operate on small blocks - Max-product linear programming algorithm (MPLP), max-sum diffusion (MSD) and Tree-weighted max-product message passing (TRW-S). Given block of dual-variables: enforce some consistency constraint over the block. Observation Difficulties in generalizing these methods arise due to strong consistency constraint - which are sufficient but not necessary.
- 4. Outline Introduction Background STC Framework Experimental Results Conclusion Introduction II Aim Dual-optimality can be established on a much broader choices of the dual objective. Deriving a “unified” message passing algorithms in an arbitrary dual-decomposition. Properties of the Resulted Algorithm (subproblem-tree calibration, or STC) Message passing on graph-object (subproblem multi-graph, or SMG) Subsumes MPLP, MSD, and TRW-S Achieves dual-optimality on blocks with flexible choices.
- 5. Outline Introduction Background STC Framework Experimental Results Conclusion MAP Inference MAP Inference problem over X and graph strcuture G = {V, E} can be formulated as maximize X Θ(X) Where Θ(X) = α∈A θα(Xα); A is the set of MRF cliques. xi ∈ V al(Xi) and x = x1:N
- 6. Outline Introduction Background STC Framework Experimental Results Conclusion LP relaxation, Dual decomposition I Large family of MAP inference methods based on solving Linear Programming (LP) relaxation maximize µ∈M Θ · µ Where µ = {µi(xi), µij(xi, xj)|∀i, xi, (i, j), (xi, xj)}; Θ is all MRF parameters {θi, θij} concatenated in same ordering as µ A decomposition of Θ(X) into subproblems c ∈ C, parameterized by {Θc} ∀x, c∈C Θc (x|c) = Θ(x) Where x|c denotes restricting the joint assignment to the scope of subproblem c.
- 7. Outline Introduction Background STC Framework Experimental Results Conclusion LP relaxation, Dual decomposition II Enforcing constraint by expressing reparameterization in terms of messages Θc = Θc + c :Xc∩X c=∅ δc →c(Xc ∩ X c) where the messages satisfy δc →c = −δc→c Each subproblem has its own copy of variables Xc
- 8. Outline Introduction Background STC Framework Experimental Results Conclusion Bethe cluster (region) Graph I Bipartite structure: one layer of “factor” nodes and one layer of small (usually unary) nodes. Restricted Design due to historical concern of satisfying the ’running intersection property’. D( δf→i ) = i max Xi θi (Xi ) + f max Xf Θf (Xf ) where the messages are only defined between the two layers (Bipartite structure). The dual (mentioned earlier) becomes more restricted due to the requirement of satisfying the running intersection property.
- 9. Outline Introduction Background STC Framework Experimental Results Conclusion Bethe cluster (region) Graph II (a) Markov Random Field (b) Cluster Graph (not Bethe Cluster)
- 10. Outline Introduction Background STC Framework Experimental Results Conclusion Bethe cluster (region) Graph III (c) Bethe Cluster Graph Figure 1: Cluster and Bethe Graph
- 11. Outline Introduction Background STC Framework Experimental Results Conclusion SMG and subproblem-tree I Subproblem Multi-Graph/Tree Given C, the subproblem multi-graph (SMG) G = (V, E) has one node for each c ∈ C and one edge between c and c for each tuple (c,c ,ϕ), where ϕ ∈ V ∪ E is shared by c and c . A subproblem multi-graph (SMG) is a tree T ⊂ G If we include all unary subproblems into the decomposition, we would get a SMG similar to Fig: (c) but with extra edges among the non-unary subproblems. So a tree in the Bethe cluster graph (which we call a Bethe tree) is also a subproblem tree by definition.
- 12. Outline Introduction Background STC Framework Experimental Results Conclusion SMG and subproblem-tree II
- 13. Outline Introduction Background STC Framework Experimental Results Conclusion SMG and subproblem-tree III For each SMG edge (c, c , ϕ) ∈ E, we have messages δc →c = −δc→c . Therefore the block (of dual variables) associated with subproblem tree T is given by: BT = {δc →c(Xϕ) : (c, c , ϕ) ∈ T }. (1)
- 14. Outline Introduction Background STC Framework Experimental Results Conclusion Max-consistency and dual-optimal trees I Given a block BT associated with some subproblem tree T , we want to achieve dual-optimal w.r.t. that block Dual-optimal on T The subproblem potentials Θc are dual-optimal on T if we can not further decrease the dual objective by changing messages in BT . Message passing algorithm achieves dual-optimality by enforcing some Consistency Constraint. We first identify constraint that is equivalent to dual-optimal on T . Assignments agree on T Assignments to all subproblems {xc}c∈T agree on T , denoted as xc ∼ T , if for ∀(c, c , ϕ) ∈ T , we have xc ϕ = xc ϕ .
- 15. Outline Introduction Background STC Framework Experimental Results Conclusion Max-consistency and dual-optimal trees II Weak max-consistency on T {Θc }c∈T satisfies weak max-consistency if c∈T max Xc Θc (Xc ) = max {Xc}∼T c∈T Θc (Xc ) Maximizing each subproblem independently gets to the same optimal value as maximizing them while requiring the assignments to agree on the tree. Let Mc ϕ be the (log)-max-marginal of c on ϕ, then Mc ϕ(xϕ) = max Xc|ϕ=xϕ Θc (Xc ) if ϕ = (i, j) ∈ E, Xc|ϕ = xϕ means Xc i = xi and Xc j = xj
- 16. Outline Introduction Background STC Framework Experimental Results Conclusion Max-consistency and dual-optimal trees III Strong max-consistency on T {Θc }c∈T satisfies strong max-consistency if Mc ϕ = Mc ϕ ∀(c, c , ϕ) ∈ T The relations among these consistency constraints are: Proposition 1. For any Bethe tree T , MPLP max-consistency =⇒ Weak max-consistency For any subproblem tree T (including Bethe trees), Strong max-consistency =⇒ Weak max-consistency. Weak max-consistency ⇐⇒ Dual-optimal on T .
- 17. Outline Introduction Background STC Framework Experimental Results Conclusion Subproblem tree calibration algorithm I Algorithm calibrates a subproblem-tree by an upstream pass and a downstream pass Both update subproblem potentials “in place” without storing any message. (a) MRF (b) SMG (c) Spanning Tree of SMG Figure 2: Flow of the Algorithm: Start with (a) to generate (b) and randomly selected (c) and ”Calibrate”
- 18. Outline Introduction Background STC Framework Experimental Results Conclusion Subproblem tree calibration algorithm II Algorithm - 1. Given MRF (left figure) 2. Split into subproblems (dual decomposition) 3. Build a multi-graph with a node for each subproblem (middle figure) 4. Repeat a. Randomly choose a subproblem-tree (right figure) b. “Calibrate” the tree by max-product / min-sum message passing Properties 1 Each tree calibration is a block coordinate descent step for the dual problem. 2 The “block” corresponds to all edges in the subproblem-tree. 3 Subsumes MPLP, TRW-S, and max-sum diffusion as special cases. 4 Handles larger and more flexible “blocks” than these methods.
- 19. Outline Introduction Background STC Framework Experimental Results Conclusion Subproblem tree calibration algorithm III
- 20. Outline Introduction Background STC Framework Experimental Results Conclusion Choosing allocation weights After STC, for each subproblem c max Xc Θc (Xc ) = ac · max {X¯c}∼T ¯c∈T Θ¯c (X¯c ) The downstream pass allocate “energy” to all subproblems according to their allocation weights. ”Energy” = negative lograrithm of the probabilities. Helps in the case of very small values to avoid numerical underflow as well as making the computations easier to handle - moving from max-product to max-summations.
- 21. Outline Introduction Background STC Framework Experimental Results Conclusion General Primal solution Given subproblem potentials, solutions to the original MAP inference problem can be constructed in different ways Visit the variables (in the original MRF) in some ordering, for example, X, X, . . . XN . And for Xi we choose the assignment: xi = arg max c:i∈scope(c) max XcXi Θc (Xc |Xj = xj, ∀j < i) Visiting each Xi, we choose its assignment to maximize the sum of all max-marginals from all subproblems covering Xi. Fix Xi = xi in all subproblems.
- 22. Outline Introduction Background STC Framework Experimental Results Conclusion Experimental MAP inference tasks I 1 The protein design benchmark 20 largest problems from that dataset Number of Variables - 101 to 180 Number of Edges - 1973 to 3005 Variable Cardinality - 154 2 Synthetic 20-by-20 grid Potentials from N(0, 1) Variable Cardinality - 100 3 ”Object detection” task from PIC-2011 37 problem instances Number of Variables - 60 / problem instance Number of Edges - 1770 / problem instance Variable Cardinality - 11 - 21
- 23. Outline Introduction Background STC Framework Experimental Results Conclusion Experimental MAP inference tasks II We observe that different methods tend to “converge” to different dual objectives, Even though the dual objectives in each plot should have exactly the same optimal value.
- 24. Outline Introduction Background STC Framework Experimental Results Conclusion Experimental MAP inference tasks III
- 25. Outline Introduction Background STC Framework Experimental Results Conclusion Conclusion Two dimensions of flexibility in designing a message passing algorithm for MAP inference: Choosing blocks to update Choosing a dual state on a plateau in each BCD step. STC algorithm can be applied with extreme flexibility in these choices. Finding Principled and adaptive strategies in making these choices will help design much more powerful message passing algorithms.
- 26. Outline Introduction Background STC Framework Experimental Results Conclusion Thank You Questions?