We present a polynomial time algorithm for online maximization of @math -submodular maximization. For online (nonmonotone) @math -submodular maximization, our algorithm achieves a tight approximate factor in an approximate regret. For online monotone @math -submodular maximization, our approximate-regret matches to the best-known approximation ratio, which is tight asymptotically as @math tends to infinity. Our approach is based on the Blackwell approachability theorem and online linear optimization.
Online learning of discrete structure is called . Efficient online algorithms were developed for various discrete structures, such as shortest paths and matroid basis @cite_8 @cite_13 . Most of these studies focused on optimizing reward loss functions (under a constraint), whereas our paper studies functions (without constraint).
@cite_13 @cite_8
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We consider an online prediction problem of combinatorial concepts where each combinatorial concept is represented as a vertex of a polyhedron described by a submodular function (base polyhedron). In general, there are exponentially many vertices in the base polyhedron. We propose polynomial time algorithms with regret bounds. In particular, for cardinality-based submodular functions, we give O(n2)-time algorithms. Kernels are typically applied to linear algorithms whose weight vector is a linear combination of the feature vectors of the examples. On-line versions of these algorithms are sometimes called "additive updates" because they add a multiple of the last feature vector to the current weight vector.In this paper we have found a way to use special convolution kernels to efficiently implement "multiplicative" updates. The kernels are defined by a directed graph. Each edge contributes an input. The inputs along a path form a product feature and all such products build the feature vector associated with the inputs.We also have a set of probabilities on the edges so that the outflow from each vertex is one. We then discuss multiplicative updates on these graphs where the prediction is essentially a kernel computation and the update contributes a factor to each edge. After adding the factors to the edges, the total outflow out of each vertex is not one any more. However some clever algorithms re-normalize the weights on the paths so that the total outflow out of each vertex is one again. Finally, we show that if the digraph is built from a regular expressions, then this can be used for speeding up the kernel and re-normalization computations.We reformulate a large number of multiplicative update algorithms using path kernels and characterize the applicability of our method. The examples include efficient algorithms for learning disjunctions and a recent algorithm that predicts as well as the best pruning of a series parallel digraphs.