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Register a new userSample Complexity of Tree Search Configuration: Cutting Planes and Beyond
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Added by
Ellen Vitercik
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Cutting-plane methods have enabled remarkable successes in integer programming over the last few decades. State-of-the-art solvers integrate a myriad of cutting-plane techniques to speed up the underlying tree-search algorithm used to find optimal solutions. In this paper we prove the first guarantees for learning high-performing cut-selection policies tailored to the instance distribution at hand using samples. We first bound the sample complexity of learning cutting planes from the canonical family of Chvatal-Gomory cuts. Our bounds handle any number of waves of any number of cuts and are fine tuned to the magnitudes of the constraint coefficients. Next, we prove sample complexity bounds for more sophisticated cut selection policies that use a combination of scoring rules to choose from a family of cuts. Finally, beyond the realm of cutting planes for integer programming, we develop a general abstraction of tree search that captures key components such as node selection and variable selection. For this abstraction, we bound the sample complexity of learning a good policy for building the search tree.
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Monte Carlo tree search (MCTS) is extremely popular in computer Go which determines each action by enormous simulations in a broad and deep search tree. However, human experts select most actions by pattern analysis and careful evaluation rather than brute search of millions of future nteractions. In this paper, we propose a computer Go system that follows experts way of thinking and playing. Our system consists of two parts. The first part is a novel deep alternative neural network (DANN) used to generate candidates of next move. Compared with existing deep convolutional neural network (DCNN), DANN inserts recurrent layer after each convolutional layer and stacks them in an alternative manner. We show such setting can preserve more contexts of local features and its evolutions which are beneficial for move prediction. The second part is a long-term evaluation (LTE) module used to provide a reliable evaluation of candidates rather than a single probability from move predictor. This is consistent with human experts nature of playing since they can foresee tens of steps to give an accurate estimation of candidates. In our system, for each candidate, LTE calculates a cumulative reward after several future interactions when local variations are settled. Combining criteria from the two parts, our system determines the optimal choice of next move. For more comprehensive experiments, we introduce a new professional Go dataset (PGD), consisting of 253233 professional records. Experiments on GoGoD and PGD datasets show the DANN can substantially improve performance of move prediction over pure DCNN. When combining LTE, our system outperforms most relevant approaches and open engines based on MCTS.
Monte Carlo tree search (MCTS) has achieved state-of-the-art results in many domains such as Go and Atari games when combining with deep neural networks (DNNs). When more simulations are executed, MCTS can achieve higher performance but also requires enormous amounts of CPU and GPU resources. However, not all states require a long searching time to identify the best action that the agent can find. For example, in 19x19 Go and NoGo, we found that for more than half of the states, the best action predicted by DNN remains unchanged even after searching 2 minutes. This implies that a significant amount of resources can be saved if we are able to stop the searching earlier when we are confident with the current searching result. In this paper, we propose to achieve this goal by predicting the uncertainty of the current searching status and use the result to decide whether we should stop searching. With our algorithm, called Dynamic Simulation MCTS (DS-MCTS), we can speed up a NoGo agent trained by AlphaZero 2.5 times faster while maintaining a similar winning rate. Also, under the same average simulation count, our method can achieve a 61% winning rate against the original program.
Learning-based methods are growing prominence for planning purposes. However, there are very few approaches for learning-assisted constrained path-planning on graphs, while there are multiple downstream practical applications. This is the case for constrained path-planning for Autonomous Unmanned Ground Vehicles (AUGV), typically deployed in disaster relief or search and rescue applications. In off-road environments, the AUGV must dynamically optimize a source-destination path under various operational constraints, out of which several are difficult to predict in advance and need to be addressed on-line. We propose a hybrid solving planner that combines machine learning models and an optimal solver. More specifically, a graph convolutional network (GCN) is used to assist a branch and bound (B&B) algorithm in handling the constraints. We conduct experiments on realistic scenarios and show that GCN support enables substantial speedup and smoother scaling to harder problems.
This paper investigates the impact of query topology on the difficulty of answering conjunctive queries in the presence of OWL 2 QL ontologies. Our first contribution is to clarify the worst-case size of positive existential (PE), non-recursive Datalog (NDL), and first-order (FO) rewritings for various classes of tree-like conjunctive queries, ranging from linear queries to bounded treewidth queries. Perhaps our most surprising result is a superpolynomial lower bound on the size of PE-rewritings that holds already for linear queries and ontologies of depth 2. More positively, we show that polynomial-size NDL-rewritings always exist for tree-shaped queries with a bounded number of leaves (and arbitrary ontologies), and for bounded treewidth queries paired with bounded depth ontologies. For FO-rewritings, we equate the existence of polysize rewritings with well-known problems in Boolean circuit complexity. As our second contribution, we analyze the computational complexity of query answering and establish tractability results (either NL- or LOGCFL-completeness) for a range of query-ontology pairs. Combining our new results with those from the literature yields a complete picture of the succinctness and complexity landscapes for the considered classes of queries and ontologies.
The random k-SAT model is the most important and well-studied distribution over k-SAT instances. It is closely connected to statistical physics; it is used as a testbench for satisfiability algorithms, and average-case hardness over this distribution has also been linked to hardness of approximation via Feiges hypothesis. We prove that any Cutting Planes refutation for random k-SAT requires exponential size, for k that is logarithmic in the number of variables, in the (interesting) regime where the number of clauses guarantees that the formula is unsatisfiable with high probability.
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