Online Dynamic Programming

We propose a general method for combinatorial online learning problems whose offline optimization problem can be solved efficiently via a dynamic programming algorithm defined by an arbitrary min-sum recurrence. Examples include online learning of Binary Search Trees, Matrix-Chain Multiplications, $k$-sets, Knapsacks, Rod Cuttings, and Weighted Interval Schedulings. For each of these problems we use the underlying graph of subproblems (called a multi-DAG) for defining a representation of the solutions of the dynamic programming problem by encoding them as a generalized version of paths (called multipaths). These multipaths encode each solution as a series of successive decisions or components over which the loss is linear. We then show that the dynamic programming algorithm for each problem leads to online algorithms for learning multipaths in the underlying multi-DAG. The algorithms maintain a distribution over the multipaths in a concise form as their hypothesis. More specifically we generalize the existing Expanded Hedge and Component Hedge algorithms for the online shortest path problem to learning multipaths. Additionally, we introduce a new and faster prediction technique for Component Hedge which in our case directly samples from a distribution over multipaths, bypassing the need to decompose the distribution over multipaths into a mixture with small support.