Online Prediction at the Limit of Zero Temperature
We design an online algorithm to classify the vertices of a graph. Underpinning the algorithm is the probability distribution of an Ising model isomorphic to the graph. Each classification is based on predicting the label with maximum marginal probability in the limit of zero-temperature with respect to the labels and vertices seen so far. Computing these classifications is unfortunately based on a #P -complete problem. This motivates us to develop an algorithm for which we give a sequential guarantee in the online mistake bound framework. Our algorithm is optimal when the graph is a tree matching the prior results in [1]. For a general graph, the algorithm exploits the additional connectivity over a tree to provide a per-cluster bound. The algorithm is efficient, as the cumulative time to sequentially predict all of the vertices of the graph is quadratic in the size of the graph.
Summary: -------- The paper studies prediction of labels on a graph, where the nodes are data points, the edges indicate similarity in label, and getting different labels on the sides of an edge is penalized. The authors analyze the problem in an online setting, and provide a general mistake bound (Thm. 4) that holds for any algorithm that has three properties: monotonicity (observing a label can only result in more predictions of that label), permutation-invariance (order of observations does not matter), and a "Markov" property (predictions on a node are influenced only by its immediate connections). Next, the authors provide two prediction strategies that satisfy these properties (Thm. 9), one of which (0-Ising) is computationally inefficient while the other (longest-path) has polynomial running time in the size of the graph. Finally, the authors show a per-cluster mistake bound for these two strategies (Thm. 10) that, combined with the general mistake bound of Thm. 4, results in a concrete bound for these two algorithms. The new bound retrieves the previous optimal bound on trees, while it still gives meaningful (in cases improved) bounds when the graph is not a tree.Quality and Clarity: -------------------- Generally, the paper is well-written: the setting, definitions, and logical steps are laid out and explained clearly, which makes the paper easy to read even for those who are not directly working in this area. There are some problems to fix, which I have mentioned at the end.The theory seems to work, although I have not verified all the proofs in detail.Originality and Significance: ----------------------------- The contribution that the paper lays out seems interesting and I think it will be useful for the community. However, I am not working in this area directly, so I leave a deeper comparison with the related work to other reviewers that are more familiar with the previous work in this particular area.Other issues: -------------- It is not specified what is the format of the max-flow returned in step 2 of Figure 1, which propagates to the definition of graph $$I$$ in step 3. In particular, in Figure 2(c), has 1->3->2 been in the max flow? If no, then why has 11->10->12 been there? If yes, then why are the edges 1-3 and 3-2 still in I? - On page 3, paragraph 2, line 7, please fix the definition of the set of edges of the quotient graph. - Perhaps the first sentence of Sec. 2 needs to be moved to the first page to become the first sentence of the last paragraph on that page. Then I think the phrase "this graph" on the third-to-last line on page 1 will have a close-by reference. - perhaps the references [3] and [4] on line 4 of the first paragraph are swapped.