1. HC-Search: Learning Heuristics and Cost Functions for Structured Prediction
Janardhan Rao (Jana) Doppa, Alan Fern, and Prasad Tadepalli
Structured Prediction
HC-Search Framework
Heuristic Function Learning:
Conduct searches on training examples using the true loss function as a heuristic (generally is a good way to produce high-quality outputs)
Key idea: Learn a heuristic function to imitate the observed search behavior with loss function
Illustration: Ranking examples for greedy search
Cost Function Learning:
Key Idea: Learn to rank the outputs generated by the learned heuristic function 𝑯 as per their losses
Given: a set of structured input-output pairs of the form 𝑥, 𝑦
Handwriting recognition
𝑥= 𝑦= s t r u c t u r e d
Image labeling
𝑥 = 𝑦=
Learn: a function 𝐹 :𝑋 →𝑌 to make predictions on new inputs
Evaluation: against a loss function 𝐿 𝑥,𝑦, 𝐹 𝑥 ∈ 𝑅 +
Hamming loss, F1 score, B3 score …
Key Elements:
Search space over structured outputs
Cost function 𝑪 to score the potential outputs
Time-bounded search procedure to find low cost outputs
Heuristic function 𝑯 to make the search efficient
Search-based Inference:
Run a time-bounded search procedure guided by the heuristic function 𝑯 toward low cost parts of the space
Return least cost output uncovered by the search procedure
Illustration:
Properties
Anytime predictions
Minimal restrictions on the complexity of heuristic function and cost function
Rank Learner
Heuristic function 𝑯
Key Challenge: “Argmin” inference
Experimental Results
Existing Approaches: CRFs, Structured SVM …
Learn parameters of linear models
𝜙 𝑥,𝑦 is n-dim feature vector over input-output pairs
w is n-dim parameter vector
F(x) = 𝒂𝒓𝒈 𝐦𝒊𝒏 𝒚∈𝒀 𝒘⋅𝝓(𝒙,𝒚)
“Argmin” inference:
Find the min. scoring output in an exponentially large set of possible outputs
Computationally hard (NP-hard) for all but simplest dependency structure of features 𝜙(𝑥,𝑦)
Approximate inference algorithms (e.g., Loopy BP) are generally used in practice
Violates exact inference assumptions of most learning algorithms, and behavior not well understood
Our Approach:
Specify a time-bounded search architecture for “Argmin” inference
Learn to do as well as we can within that architecture
Assumption: we can learn to do well within the selected architecture
Greedy search in Limited Discrepancy Search (LDS) space [Doppa et al., 2012]
HC-Search outperforms state-of-the-art algorithms including C-Search (our prior approach that learns a single function C to also serve as heuristic) [Doppa et al., 2012]
Loss Decomposition Results:
Selection loss 𝝐 𝑪|𝑯 contributes more to the overall loss for both C-Search and HC-Search
Improvement of HC-Search over C-Search is due to the improvement in the selection loss
Clearly shows the advantage of separating the roles of heuristic and cost function
Take Home Message: viewing structured prediction as a search problem is useful
When applying to new problems: design a “high quality” search space (e.g., LDS space), pick a suitable search procedure, and use HC-Search “loss decomposition” to train and debug
|Y| is generally exponentially large
HC-Search: Learning
Loss Decomposition:
𝝐 : Overall expected loss
𝝐 𝑯 : Expected loss due to not generating optimal output
𝝐 𝑪|𝑯 : Expected loss due to not picking the best generated output
Key idea: Greedy stage-wise minimization guided by the loss decomposition
Step 1: 𝐻 = arg 𝑚𝑖𝑛 𝐻∈𝑯 𝜖 𝐻 (heuristic training)
Step 2: 𝐶 = arg 𝑚𝑖𝑛 𝐶∈𝑪 𝜖 𝐶| 𝐻 (cost function training)