We study the problem of diverse feature selection in linear regression: selecting a small subset of diverse features that can predict a given objective. Diversity is useful for several reasons such as interpretability, robustness to noise, etc. We propose several spectral regularizers that capture a notion of diversity of features and show that these are all submodular set functions. These regularizers, when added to the objective function for linear regression, result in approximately submodular functions, which can then be maximized approximately by efficient greedy and local search algorithms, with provable guarantees. We compare our algorithms to traditional greedy and \ell_1 -regularization schemes and show that we obtain a more diverse set of features that result in the regression problem being stable under perturbations.

State-of-the-art SAT solvers are nowadays able to handle huge real-world instances. The key to this success is the so-called Conflict-Driven Clause-Learning (CDCL) scheme, which encompasses a number of techniques that exploit the conflicts that are encountered during the search for a solution. In this article we extend these techniques to Integer Linear Programming (ILP), where variables may take general integer values instead of purely binary ones, constraints are more expressive than just propositional clauses, and there may be an objective function to optimise. We explain how these methods can be implemented efficiently, and discuss possible improvements. Our work is backed with a basic implementation that shows that, even in this far less mature stage, our techniques are already a useful complement to the state of the art in ILP solving.

The Gradient Descent-Ascent (GDA) algorithm, designed to solve minimax optimization problems, takes the descent and ascent steps either simultaneously (Sim-GDA) or alternately (Alt-GDA). While Alt-GDA is commonly observed to converge faster, the performance gap between the two is not yet well understood theoretically, especially in terms of global convergence rates. To address this theory-practice gap, we present fine-grained convergence analyses of both algorithms for strongly-convex-strongly-concave and Lipschitz-gradient objectives. Our new iteration complexity upper bound of Alt-GDA is strictly smaller than the lower bound of Sim-GDA; i.e., Alt-GDA is provably faster. Moreover, we propose Alternating-Extrapolation GDA (Alex-GDA), a general algorithmic framework that subsumes Sim-GDA and Alt-GDA, for which the main idea is to alternately take gradients from extrapolations of the iterates. We show that Alex-GDA satisfies a smaller iteration complexity bound, identical to that of the Extra-gradient method, while requiring less gradient computations. We also prove that Alex-GDA enjoys linear convergence for bilinear problems, for which both Sim-GDA and Alt-GDA fail to converge at all.

In this paper, we propose a Minimax Trust Region (MINIMAX-TR) algorithm and a Minimax Trust Region Algorithm with Contractions and Expansions(MINIMAX-TRACE) algorithm for solving nonconvex-strongly concave minimax problems. Both algorithms can find an $(\epsilon, \sqrt{\epsilon})$-second order stationary point(SSP) within $\mathcal{O}(\epsilon^{-1.5})$ iterations, which matches the best well known iteration complexity.

This paper introduces a reinforcement learning approach to optimize the Stochastic Vehicle Routing Problem with Time Windows (SVRP), focusing on reducing travel costs in goods delivery. We develop a novel SVRP formulation that accounts for uncertain travel costs and demands, alongside specific customer time windows. An attention-based neural network trained through reinforcement learning is employed to minimize routing costs. Our approach addresses a gap in SVRP research, which traditionally relies on heuristic methods, by leveraging machine learning. The model outperforms the Ant-Colony Optimization algorithm, achieving a 1.73% reduction in travel costs. It uniquely integrates external information, demonstrating robustness in diverse environments, making it a valuable benchmark for future SVRP studies and industry application.

The application of contemporary artificial intelligence techniques to address geometric problems and automated deductive proof has always been a grand challenge to the interdiscipline field of mathematics and artificial Intelligence. This is the fourth article in a series of our works, in our previous work, we established of a geometric formalized system known as FormalGeo. Moreover we annotated approximately 7000 geometric problems, forming the FormalGeo7k dataset. Despite the FGPS (Formal Geometry Problem Solver) can achieve interpretable algebraic equation solving and human-like deductive reasoning, it often experiences timeouts due to the complexity of the search strategy. In this paper, we introduced FGeo-TP (Theorem Predictor), which utilizes the language model to predict theorem sequences for solving geometry problems. We compared the effectiveness of various Transformer architectures, such as BART or T5, in theorem prediction, implementing pruning in the search process of FGPS, thereby improving its performance in solving geometry problems. Our results demonstrate a significant increase in the problem-solving rate of the language model-enhanced FGeo-TP on the FormalGeo7k dataset, rising from 39.7% to 80.86%. Furthermore, FGeo-TP exhibits notable reductions in solving time and search steps across problems of varying difficulty levels.

Feature selection is a crucial step in data mining to enhance model performance by reducing data dimensionality. However, the increasing dimensionality of collected data exacerbates the challenge known as the "curse of dimensionality", where computation grows exponentially with the number of dimensions. To tackle this issue, evolutionary computational (EC) approaches have gained popularity due to their simplicity and applicability. Unfortunately, the diverse designs of EC methods result in varying abilities to handle different data, often underutilizing and not sharing information effectively. In this paper, we propose a novel approach called PSO-based Multi-task Evolutionary Learning (MEL) that leverages multi-task learning to address these challenges. By incorporating information sharing between different feature selection tasks, MEL achieves enhanced learning ability and efficiency. We evaluate the effectiveness of MEL through extensive experiments on 22 high-dimensional datasets. Comparing against 24 EC approaches, our method exhibits strong competitiveness. Additionally, we have open-sourced our code on GitHub at https://github.com/wangxb96/MEL.

We study the Stochastic Shortest Path (SSP) problem with a linear mixture transition kernel, where an agent repeatedly interacts with a stochastic environment and seeks to reach certain goal state while minimizing the cumulative cost. Existing works often assume a strictly positive lower bound of the cost function or an upper bound of the expected length for the optimal policy. In this paper, we propose a new algorithm to eliminate these restrictive assumptions. Our algorithm is based on extended value iteration with a fine-grained variance-aware confidence set, where the variance is estimated recursively from high-order moments. Our algorithm achieves an $\tilde{\mathcal O}(dB_*\sqrt{K})$ regret bound, where $d$ is the dimension of the feature mapping in the linear transition kernel, $B_*$ is the upper bound of the total cumulative cost for the optimal policy, and $K$ is the number of episodes. Our regret upper bound matches the $\Omega(dB_*\sqrt{K})$ lower bound of linear mixture SSPs in Min et al. (2022), which suggests that our algorithm is nearly minimax optimal.

Monte-Carlo tree search (MCTS) is an effective anytime algorithm with a vast amount of applications. It strategically allocates computational resources to focus on promising segments of the search tree, making it a very attractive search algorithm in large search spaces. However, it often expends its limited resources on reevaluating previously explored regions when they remain the most promising path. Our proposed methodology, denoted as AmEx-MCTS, solves this problem by introducing a novel MCTS formulation. Central to AmEx-MCTS is the decoupling of value updates, visit count updates, and the selected path during the tree search, thereby enabling the exclusion of already explored subtrees or leaves. This segregation preserves the utility of visit counts for both exploration-exploitation balancing and quality metrics within MCTS. The resultant augmentation facilitates in a considerably broader search using identical computational resources, preserving the essential characteristics of MCTS. The expanded coverage not only yields more precise estimations but also proves instrumental in larger and more complex problems. Our empirical evaluation demonstrates the superior performance of AmEx-MCTS, surpassing classical MCTS and related approaches by a substantial margin.

Bayesian optimization is a methodology to optimize black-box functions. Traditionally, it focuses on the setting where you can arbitrarily query the search space. However, many real-life problems do not offer this flexibility; in particular, the search space of the next query may depend on previous ones. Example challenges arise in the physical sciences in the form of local movement constraints, required monotonicity in certain variables, and transitions influencing the accuracy of measurements. Altogether, such transition constraints necessitate a form of planning. This work extends Bayesian optimization via the framework of Markov Decision Processes, iteratively solving a tractable linearization of our objective using reinforcement learning to obtain a policy that plans ahead over long horizons. The resulting policy is potentially history-dependent and non-Markovian. We showcase applications in chemical reactor optimization, informative path planning, machine calibration, and other synthetic examples.