1. Learning the Structure of Related Tasks Presented by Lihan He Machine Learning Reading Group Duke University 02/03/2006 A. Niculescu-Mizil, R. Caruana

2. Outline  Introduction  Learning single Bayes networks from data  Learning from related tasks  Experimental results  Conclusions

3. Introduction Graphical model: Node represents random variables; edge represents dependency. Undirected graphical model: Markov network Directed graphical model: Bayesian network x1x1 x2x2 x3x3 x4x4 Causal relationships between nodes; Directed acyclic graph (DAG) : No directed cycles allowed; B={ G,θ }

4. Introduction Goal: simultaneously learn Bayes Net structures for multiple tasks. Different tasks are related; Structures might be similar, but not identical. Example: gene expression data. 1) Learning one single structure from data. 2) Generalizing to multiple task learning by setting joint prior of structures.

5. Single Bayesian network learning from data Bayes Network B={ G, θ }, including a set of n random variables X ={ X 1, X 2,…, X n } Joint probability P ( X) can be factorized by Given dataset D={x 1, x 2, …, x m }, where x i = (x 1,x 2,…,x n ), we can learn structure G and parameter θ from the dataset D.

6. Single Bayesian network learning from data Model selection : find the highest P(G|D) for all possible G Searching for all possible G is impossible:  n=4, there are 543 possible DAGs  n=10, there are O(10 18 ) possible DAGs Question: How to search the best structure in the huge amount of possible DAGs?

7. Algorithm: 1) Randomly generate an initial DAG, evaluate its score; 2) Evaluate the scores of all the neighbors of current DAG; 3) while {some neighbors have higher scores than current DAG} move to the neighbor that has the highest score Evaluate the scores of all the neighbors of the new DAG; end 4) Repeat (1) - (3) a number of times starting from different DAG every time. Single Bayesian network learning from data

8. Neighbors of a structure G: the set of all the DAGs that can be obtained by adding, removing or reversing an edge in G Single Bayesian network learning from data  Must satisfy acyclic constraint x1x1 x2x2 x3x3 x4x4 x1x1 x2x2 x3x3 x4x4 x1x1 x2x2 x3x3 x4x4 x1x1 x2x2 x3x3 x4x4 x1x1 x2x2 x3x3 x4x4

9. Given iid dataset D 1, D 2, …, D k, Simultaneously learn the structure B 1 ={G 1, θ 1 },B 2 ={G 2, θ 2 },…,B k ={G k, θ k } Structures (G 1,G 2,…,G k ) – similar, but not identical Learning from related task

10. One more assumption: the parameters of different networks are independent: Not true, but make structure learning more efficient. Since we focus on structure learning, not parameter learning, this is acceptable.

11. Learning from related task Prior:  If structures are not related: G 1,…,G k are independent a priori Structures are learned independently for each task.  If structures are identical, Learning the same structure: Learning the single structure under the restriction that TSK is always the parent of all the other nodes. Common structure: remove node TSK and all the edges connected to it.

12. Learning from related task Prior:  Between independent and identical: Penalize each edge ( X i, X j ) that is different in two DAGs δ=0: independent δ=1: identical 0<δ<1 For the k task prior

13. Learning from related task Model selection : find the highest P(G 1,…,G k |D 1,…D k )  Same idea as single task structure learning.  Question: what is a neighbor of (G 1,…,G k ) ? Def 1: Size of neighbors: O( n 2k ) Def 2: Def1 + one more constraint: All the changes of edges happen between the same two nodes for all DAGs in ( G 1,…, G k ) Size of neighbors: O( n 2 3 k )

14. Learning from related task Acceleration : At each iteration, algorithm must find best score from a set of neighbors Not necessary search all the elements in The first i tasks are specified and the rest k-i tasks are not specified. where is the upper bound of the neighbor subset

15. Results  Original network, delete edges with probability P del, create 5 tasks.  1000 data points.  10 trials  Compute KL-divergence and editing distance between learned structure and true structure. KL-divergenceEditing distance

16. Learning from related task