Abstract. We study the problem of online path learning with non-additive gains, which is a central problem appearing in several applications, including ensemble structured prediction. We present new online algorithms for path learning with non-additive count-based gains for the three settings of full information, semi-bandit and full bandit with very favorable regret guarantees. A key component of our algorithms is the definition and computation of an intermediate context-dependent automaton that enables us to use existing algorithms designed for additive gains. We further apply our methods to the important application of ensemble structured prediction. Finally, beyond count-based gains, we give an efficient implementation of the EXP3 algorithm for the full bandit setting with an arbitrary (non-additive) gain.
Abstract. The standard techniques for online learning of combinatorial objects perform multiplicative updates followed by projections into the convex hull of all the objects. However, this methodology can be expensive if the convex hull contains many facets. For example, the convex hull of n-symbol Huffman trees is known to have exponentially many facets. We get around this difficulty by exploiting extended formulations, which encode the polytope of combinatorial objects in a higher dimensional “extended” space with only polynomially many facets. We develop a general framework for converting extended formulations into efficient online algorithms with good relative loss bounds. We present applications of our framework to online learning of Huffman trees and permutations. The regret bounds of the resulting algorithms are within a factor of O(√log(n)) of the state-of-the-art specialized algorithms for permutations, and depending on the loss regimes, improve on or match the state-of-the-art for Huffman trees. Our method is general and can be applied to other combinatorial objects.
Abstract. We consider the problem of repeatedly solving a variant of the same dynamic programming problem in successive trials. An instance of the type of problems we consider is to find a good binary search tree in a changing environment. At the beginning of each trial, the learner probabilistically chooses a tree with the n keys at the internal nodes and the n + 1 gaps between keys at the leaves. The learner is then told the frequencies of the keys and gaps and is charged by the average search cost for the chosen tree. The problem is online because the frequencies can change between trials. The goal is to develop algorithms with the property that their total average search cost (loss) in all trials is close to the total loss of the best tree chosen in hindsight for all trials. The challenge, of course, is that the algorithm has to deal with exponential number of trees. We develop a general methodology for tackling such problems for a wide class of dynamic programming algorithms. Our framework allows us to extend online learning algorithms like Hedge and Component Hedge to a significantly wider class of combinatorial objects than was possible before.
Abstract. Deep Neural Networks (DNN) have demonstrated superior ability to extract high level embedding vectors from low level features. Despite the success, the serving time is still the bottleneck due to expensive run-time computation of multiple layers of dense matrices. GPGPU, FPGA, or ASIC-based serving systems require additional hardware that are not in the mainstream design of most commercial applications. In contrast, tree or forest-based models are widely adopted because of low serving cost, but heavily depend on carefully engineered features. This work proposes a Deep Embedding Forest model that benefits from the best of both worlds. The model consists of a number of embedding layers and a forest/tree layer. The former maps high dimensional (hundreds of thousands to millions) and heterogeneous low-level features to the lower dimensional (thousands) vectors, and the latter ensures fast serving. Built on top of a representative DNN model called Deep Crossing, and two forest/tree-based models including XGBoost and LightGBM, a two-step Deep Embedding Forest algorithm is demonstrated to achieve on-par or slightly better performance as compared with the DNN counterpart, with only a fraction of serving time on conventional hardware. After comparing with a joint optimization algorithm called partial fuzzification, also proposed in this paper, it is concluded that the two-step Deep Embedding Forest has achieved near optimal performance. Experiments based on large scale data sets (up to 1 billion samples) from a major sponsored search engine proves the efficacy of the proposed model.