The fundamental goal assignment problem for a multi-robot application aims to assign a unique goal to each robot while ensuring collision-free paths, minimizing the total movement cost. A plausible algorithmic solution to this NP-hard problem involves an iterative process that integrates a task planner to compute the goal assignment while ignoring the collision possibilities among the robots and a multi-agent path-finding algorithm to find the collision-free trajectories for a given assignment. This procedure involves a method for computing the next best assignment given the current best assignment. A naive way of computing the next best assignment, as done in the state-of-the-art solutions, becomes a roadblock to achieving scalability in solving the overall problem. To obviate this bottleneck, we propose an efficient conflict-guided method to compute the next best assignment. Additionally, we introduce two more optimizations to the algorithm -- first for avoiding the unconstrained path computations between robot-goal pairs wherever possible, and the second to prevent duplicate constrained path computations for multiple robot-goal pairs. We extensively evaluate our algorithm for up to a hundred robots on several benchmark workspaces. The results demonstrate that the proposed algorithm achieves nearly an order of magnitude speedup over the state-of-the-art algorithm, showcasing its efficacy in real-world scenarios.

The standard approach to Computational Learning Theory is usually formulated within the so-called frequentist approach to Statistics. Within this paradigm one is interested in constructing an estimator, based on a flnite sample, which possesses a small loss (generalization error). While many algorithms have been constructed and analyzed within this context, it is not clear how these approaches relate to standard optimality criteria within the frequentist framework. Two classic optimality criteria within the latter approach are the minimax and admissibility criteria, which charac- terize optimality of estimators in a rigorous and precise fashion [9]. Except in some special cases [12], it is not known whether any of the approaches used within the Learning community lead to optimality in either of the above senses of the word. On the other hand, it is known that under certain regularity conditions, Bayesian estimators lead to either minimax or admissible estimators, and thus to well-deflned optimality in the classical (frequentist) sense. In fact, it can be shown that Bayes estimators are essentially the only estimators which can achieve optimality in the above senses [9]. This optimality feature provides strong motivation for the study of Bayesian approaches in a frequentist setting.

Heuristics are crucial in SAT solvers, while no heuristic rules are suitable for all problem instances. Therefore, it typically requires to refine specific solvers for specific problem instances. In this context, we present AutoSAT, a novel framework for automatically optimizing heuristics in SAT solvers. AutoSAT is based on Large Large Models (LLMs) which is able to autonomously generate code, conduct evaluation, then utilize the feedback to further optimize heuristics, thereby reducing human intervention and enhancing solver capabilities. AutoSAT operates on a plug-and-play basis, eliminating the need for extensive preliminary setup and model training, and fosters a Chain of Thought collaborative process with fault-tolerance, ensuring robust heuristic optimization. Extensive experiments on a Conflict-Driven Clause Learning (CDCL) solver demonstrates the overall superior performance of AutoSAT, especially in solving some specific SAT problem instances.

Neural networks offer good approximation to many tasks but consistently fail to reach perfect generalization, even when theoretical work shows that such perfect solutions can be expressed by certain architectures. Using the task of formal language learning, we focus on one simple formal language and show that the theoretically correct solution is in fact not an optimum of commonly used objectives -- even with regularization techniques that according to common wisdom should lead to simple weights and good generalization (L1, L2) or other meta-heuristics (early-stopping, dropout). However, replacing standard targets with the Minimum Description Length objective (MDL) results in the correct solution being an optimum.

We consider the problem of Neyman-Pearson classification which models unbalanced classification settings where error w.r.t. a distribution $\mu_1$ is to be minimized subject to low error w.r.t. a different distribution $\mu_0$. Given a fixed VC class $\mathcal{H}$ of classifiers to be minimized over, we provide a full characterization of possible distribution-free rates, i.e., minimax rates over the space of all pairs $(\mu_0, \mu_1)$. The rates involve a dichotomy between hard and easy classes $\mathcal{H}$ as characterized by a simple geometric condition, a three-points-separation condition, loosely related to VC dimension.

Due to statistical lower bounds on the learnability of many function classes under privacy constraints, there has been recent interest in leveraging public data to improve the performance of private learning algorithms. In this model, algorithms must always guarantee differential privacy with respect to the private samples while also ensuring learning guarantees when the private data distribution is sufficiently close to that of the public data. Previous work has demonstrated that when sufficient public, unlabelled data is available, private learning can be made statistically tractable, but the resulting algorithms have all been computationally inefficient. In this work, we present the first computationally efficient, algorithms to provably leverage public data to learn privately whenever a function class is learnable non-privately, where our notion of computational efficiency is with respect to the number of calls to an optimization oracle for the function class. In addition to this general result, we provide specialized algorithms with improved sample complexities in the special cases when the function class is convex or when the task is binary classification.

Consider a multi-class labelling problem, where the labels can take values in $[k]$, and a predictor predicts a distribution over the labels. In this work, we study the following foundational question: Are there notions of multi-class calibration that give strong guarantees of meaningful predictions and can be achieved in time and sample complexities polynomial in $k$? Prior notions of calibration exhibit a tradeoff between computational efficiency and expressivity: they either suffer from having sample complexity exponential in $k$, or needing to solve computationally intractable problems, or give rather weak guarantees. Our main contribution is a notion of calibration that achieves all these desiderata: we formulate a robust notion of projected smooth calibration for multi-class predictions, and give new recalibration algorithms for efficiently calibrating predictors under this definition with complexity polynomial in $k$. Projected smooth calibration gives strong guarantees for all downstream decision makers who want to use the predictor for binary classification problems of the form: does the label belong to a subset $T \subseteq [k]$: e.g. is this an image of an animal? It ensures that the probabilities predicted by summing the probabilities assigned to labels in $T$ are close to some perfectly calibrated binary predictor for that task. We also show that natural strengthenings of our definition are computationally hard to achieve: they run into information theoretic barriers or computational intractability. Underlying both our upper and lower bounds is a tight connection that we prove between multi-class calibration and the well-studied problem of agnostic learning in the (standard) binary prediction setting.

Expert demonstrations have proven an easy way to indirectly specify complex tasks. Recent algorithms even support extracting unambiguous formal specifications, e.g. deterministic finite automata (DFA), from demonstrations. Unfortunately, these techniques are generally not sample efficient. In this work, we introduce $L^*LM$, an algorithm for learning DFAs from both demonstrations and natural language. Due to the expressivity of natural language, we observe a significant improvement in the data efficiency of learning DFAs from expert demonstrations. Technically, $L^*LM$ leverages large language models to answer membership queries about the underlying task. This is then combined with recent techniques for transforming learning from demonstrations into a sequence of labeled example learning problems. In our experiments, we observe the two modalities complement each other, yielding a powerful few-shot learner.

In this article, we show that the completion problem, i.e. the decision problem whether a partial structure can be completed to a full structure, is NP-complete for many combinatorial structures. While the gadgets for most reductions in literature are found by hand, we present an algorithm to construct gadgets in a fully automated way. Using our framework which is based on SAT, we present the first thorough study of the completion problem on sign mappings with forbidden substructures by classifying thousands of structures for which the completion problem is NP-complete. Our list in particular includes interior triple systems, which were introduced by Knuth towards an axiomatization of planar point configurations. Last but not least, we give an infinite family of structures generalizing interior triple system to higher dimensions for which the completion problem is NP-complete.

To enhance the efficiency and practicality of federated bandit learning, recent advances have introduced incentives to motivate communication among clients, where a client participates only when the incentive offered by the server outweighs its participation cost. However, existing incentive mechanisms naively assume the clients are truthful: they all report their true cost and thus the higher cost one participating client claims, the more the server has to pay. Therefore, such mechanisms are vulnerable to strategic clients aiming to optimize their own utility by misreporting. To address this issue, we propose an incentive compatible (i.e., truthful) communication protocol, named Truth-FedBan, where the incentive for each participant is independent of its self-reported cost, and reporting the true cost is the only way to achieve the best utility. More importantly, Truth-FedBan still guarantees the sub-linear regret and communication cost without any overheads. In other words, the core conceptual contribution of this paper is, for the first time, demonstrating the possibility of simultaneously achieving incentive compatibility and nearly optimal regret in federated bandit learning. Extensive numerical studies further validate the effectiveness of our proposed solution.