Recent large language models (LLMs) have witnessed significant advancement in various tasks, including mathematical reasoning and theorem proving. As these two tasks require strict and formal multi-step inference, they are appealing domains for exploring the reasoning ability of LLMs but still face important challenges. Previous studies such as Chain-of-Thought (CoT) have revealed the effectiveness of intermediate steps guidance. However, such step-wise annotation requires heavy labor, leading to insufficient training steps for current benchmarks. To fill this gap, this work introduces MUSTARD, a data generation framework that masters uniform synthesis of theorem and proof data of high quality and diversity. MUSTARD synthesizes data in three stages: (1) It samples a few mathematical concept seeds as the problem category. (2) Then, it prompts a generative language model with the sampled concepts to obtain both the problems and their step-wise formal solutions. (3) Lastly, the framework utilizes a proof assistant (e.g., Lean Prover) to filter the valid proofs. With the proposed MUSTARD, we present a theorem-and-proof benchmark MUSTARDSAUCE with 5,866 valid data points. Each data point contains an informal statement, an informal proof, and a translated formal proof that passes the prover validation. We perform extensive analysis and demonstrate that MUSTARD generates validated high-quality step-by-step data. We further apply the MUSTARDSAUCE for fine-tuning smaller language models. The fine-tuned Llama 2-7B achieves a 15.41% average relative performance gain in automated theorem proving, and 8.18% in math word problems. Codes and data are available at https://github.com/Eleanor-H/MUSTARD.
We consider the multi-agent spatial navigation problem of computing the socially optimal order of play, i.e., the sequence in which the agents commit to their decisions, and its associated equilibrium in an N-player Stackelberg trajectory game. We model this problem as a mixed-integer optimization problem over the space of all possible Stackelberg games associated with the order of play's permutations. To solve the problem, we introduce Branch and Play (B&P), an efficient and exact algorithm that provably converges to a socially optimal order of play and its Stackelberg equilibrium. As a subroutine for B&P, we employ and extend sequential trajectory planning, i.e., a popular multi-agent control approach, to scalably compute valid local Stackelberg equilibria for any given order of play. We demonstrate the practical utility of B&P to coordinate air traffic control, swarm formation, and delivery vehicle fleets. We find that B&P consistently outperforms various baselines, and computes the socially optimal equilibrium.
Since the advent of Large Language Models a few years ago, they have often been considered the de facto solution for many AI problems. However, in addition to the many deficiencies of LLMs that prevent them from broad industry adoption, such as reliability, cost, and speed, there is a whole class of common real world problems that Large Language Models perform poorly on, namely, constraint satisfaction and optimization problems. These problems are ubiquitous and current solutions are highly specialized and expensive to implement. At Elemental Cognition, we developed our EC AI platform which takes a neuro-symbolic approach to solving constraint satisfaction and optimization problems. The platform employs, at its core, a precise and high performance logical reasoning engine, and leverages LLMs for knowledge acquisition and user interaction. This platform supports developers in specifying application logic in natural and concise language while generating application user interfaces to interact with users effectively. We evaluated LLMs against systems built on the EC AI platform in three domains and found the EC AI systems to significantly outperform LLMs on constructing valid and optimal solutions, on validating proposed solutions, and on repairing invalid solutions.
This paper investigated the potential of a multivariate Transformer model to forecast the temporal trajectory of the Fraction of Absorbed Photosynthetically Active Radiation (FAPAR) for short (1 month) and long horizon (more than 1 month) periods at the regional level in Europe and North Africa. The input data covers the period from 2002 to 2022 and includes remote sensing and weather data for modelling FAPAR predictions. The model was evaluated using a leave one year out cross-validation and compared with the climatological benchmark. Results show that the transformer model outperforms the benchmark model for one month forecasting horizon, after which the climatological benchmark is better. The RMSE values of the transformer model ranged from 0.02 to 0.04 FAPAR units for the first 2 months of predictions. Overall, the tested Transformer model is a valid method for FAPAR forecasting, especially when combined with weather data and used for short-term predictions.
Predictive models make mistakes. Hence, there is a need to quantify the uncertainty associated with their predictions. Conformal inference has emerged as a powerful tool to create statistically valid prediction regions around point predictions, but its naive application to regression problems yields non-adaptive regions. New conformal scores, often relying upon quantile regressors or conditional density estimators, aim to address this limitation. Although they are useful for creating prediction bands, these scores are detached from the original goal of quantifying the uncertainty around an arbitrary predictive model. This paper presents a new, model-agnostic family of methods to calibrate prediction intervals for regression problems with local coverage guarantees. Our approach is based on pursuing the coarsest partition of the feature space that approximates conditional coverage. We create this partition by training regression trees and Random Forests on conformity scores. Our proposal is versatile, as it applies to various conformity scores and prediction settings and demonstrates superior scalability and performance compared to established baselines in simulated and real-world datasets. We provide a Python package clover that implements our methods using the standard scikit-learn interface.
In machine learning, model calibration and predictive inference are essential for producing reliable predictions and quantifying uncertainty to support decision-making. Recognizing the complementary roles of point and interval predictions, we introduce Self-Calibrating Conformal Prediction, a method that combines Venn-Abers calibration and conformal prediction to deliver calibrated point predictions alongside prediction intervals with finite-sample validity conditional on these predictions. To achieve this, we extend the original Venn-Abers procedure from binary classification to regression. Our theoretical framework supports analyzing conformal prediction methods that involve calibrating model predictions and subsequently constructing conditionally valid prediction intervals on the same data, where the conditioning set or conformity scores may depend on the calibrated predictions. Real-data experiments show that our method improves interval efficiency through model calibration and offers a practical alternative to feature-conditional validity.
Scientific modeling applications often require estimating a distribution of parameters consistent with a dataset of observations - an inference task also known as source distribution estimation. This problem can be ill-posed, however, since many different source distributions might produce the same distribution of data-consistent simulations. To make a principled choice among many equally valid sources, we propose an approach which targets the maximum entropy distribution, i.e., prioritizes retaining as much uncertainty as possible. Our method is purely sample-based - leveraging the Sliced-Wasserstein distance to measure the discrepancy between the dataset and simulations - and thus suitable for simulators with intractable likelihoods. We benchmark our method on several tasks, and show that it can recover source distributions with substantially higher entropy than recent source estimation methods, without sacrificing the fidelity of the simulations. Finally, to demonstrate the utility of our approach, we infer source distributions for parameters of the Hodgkin-Huxley model from experimental datasets with thousands of single-neuron measurements. In summary, we propose a principled method for inferring source distributions of scientific simulator parameters while retaining as much uncertainty as possible.
Due to recent booming of UAVs technologies, these are being used in many fields involving complex tasks. Some of them involve a high risk to the vehicle driver, such as fire monitoring and rescue tasks, which make UAVs excellent for avoiding human risks. Mission Planning for UAVs is the process of planning the locations and actions (loading/dropping a load, taking videos/pictures, acquiring information) for the vehicles, typically over a time period. These vehicles are controlled from Ground Control Stations (GCSs) where human operators use rudimentary systems. This paper presents a new Multi-Objective Genetic Algorithm for solving complex Mission Planning Problems (MPP) involving a team of UAVs and a set of GCSs. A hybrid fitness function has been designed using a Constraint Satisfaction Problem (CSP) to check if solutions are valid and Pareto-based measures to look for optimal solutions. The algorithm has been tested on several datasets optimizing different variables of the mission, such as the makespan, the fuel consumption, distance, etc. Experimental results show that the new algorithm is able to obtain good solutions, however as the problem becomes more complex, the optimal solutions also become harder to find.
This paper proposes a distributed controller synthesis framework for safe navigation of multi-agent systems. We leverage control barrier functions to formulate collision avoidance with obstacles and teammates as constraints on the control input for a state-dependent network optimization problem that encodes team formation and the navigation task. Our algorithmic solution is valid for general nonlinear control dynamics and optimization problems. The resulting controller is distributed, satisfies the safety constraints at all times, and is asymptotically optimal. We illustrate its performance in a team of differential-drive robots in a variety of complex environments, both in simulation and in hardware.
A dataset is confounded if it is most easily solved via a spurious correlation, which fails to generalize to new data. In this work, we show that, in a continual learning setting where confounders may vary in time across tasks, the challenge of mitigating the effect of confounders far exceeds the standard forgetting problem normally considered. In particular, we provide a formal description of such continual confounders and identify that, in general, spurious correlations are easily ignored when training for all tasks jointly, but it is harder to avoid confounding when they are considered sequentially. These descriptions serve as a basis for constructing a novel CLEVR-based continually confounded dataset, which we term the ConCon dataset. Our evaluations demonstrate that standard continual learning methods fail to ignore the dataset's confounders. Overall, our work highlights the challenges of confounding factors, particularly in continual learning settings, and demonstrates the need for developing continual learning methods to robustly tackle these.
Large language models (LLMs) exhibit advanced reasoning skills, enabling robots to comprehend natural language instructions and strategically plan high-level actions through proper grounding. However, LLM hallucination may result in robots confidently executing plans that are misaligned with user goals or, in extreme cases, unsafe. Additionally, inherent ambiguity in natural language instructions can induce task uncertainty, particularly in situations where multiple valid options exist. To address this issue, LLMs must identify such uncertainty and proactively seek clarification. This paper explores the concept of introspective planning as a systematic method for guiding LLMs in forming uncertainty--aware plans for robotic task execution without the need for fine-tuning. We investigate uncertainty quantification in task-level robot planning and demonstrate that introspection significantly improves both success rates and safety compared to state-of-the-art LLM-based planning approaches. Furthermore, we assess the effectiveness of introspective planning in conjunction with conformal prediction, revealing that this combination yields tighter confidence bounds, thereby maintaining statistical success guarantees with fewer superfluous user clarification queries. Code is available at https://github.com/kevinliang888/IntroPlan.
An open scientific challenge is how to classify events with reliable measures of uncertainty, when we have a mechanistic model of the data-generating process but the distribution over both labels and latent nuisance parameters is different between train and target data. We refer to this type of distributional shift as generalized label shift (GLS). Direct classification using observed data $\mathbf{X}$ as covariates leads to biased predictions and invalid uncertainty estimates of labels $Y$. We overcome these biases by proposing a new method for robust uncertainty quantification that casts classification as a hypothesis testing problem under nuisance parameters. The key idea is to estimate the classifier's receiver operating characteristic (ROC) across the entire nuisance parameter space, which allows us to devise cutoffs that are invariant under GLS. Our method effectively endows a pre-trained classifier with domain adaptation capabilities and returns valid prediction sets while maintaining high power. We demonstrate its performance on two challenging scientific problems in biology and astroparticle physics with data from realistic mechanistic models.
Estimating robot pose from RGB images is a crucial problem in computer vision and robotics. While previous methods have achieved promising performance, most of them presume full knowledge of robot internal states, e.g. ground-truth robot joint angles. However, this assumption is not always valid in practical situations. In real-world applications such as multi-robot collaboration or human-robot interaction, the robot joint states might not be shared or could be unreliable. On the other hand, existing approaches that estimate robot pose without joint state priors suffer from heavy computation burdens and thus cannot support real-time applications. This work introduces an efficient framework for real-time robot pose estimation from RGB images without requiring known robot states. Our method estimates camera-to-robot rotation, robot state parameters, keypoint locations, and root depth, employing a neural network module for each task to facilitate learning and sim-to-real transfer. Notably, it achieves inference in a single feed-forward pass without iterative optimization. Our approach offers a 12-time speed increase with state-of-the-art accuracy, enabling real-time holistic robot pose estimation for the first time. Code and models are available at https://github.com/Oliverbansk/Holistic-Robot-Pose-Estimation.
Large Language Models (LLMs) play a pivotal role in generating vast arrays of narratives, facilitating a systematic exploration of their effectiveness for communicating life events in narrative form. In this study, we employ a zero-shot structured narrative prompt to generate 24,000 narratives using OpenAI's GPT-4. From this dataset, we manually classify 2,880 narratives and evaluate their validity in conveying birth, death, hiring, and firing events. Remarkably, 87.43% of the narratives sufficiently convey the intention of the structured prompt. To automate the identification of valid and invalid narratives, we train and validate nine Machine Learning models on the classified datasets. Leveraging these models, we extend our analysis to predict the classifications of the remaining 21,120 narratives. All the ML models excelled at classifying valid narratives as valid, but experienced challenges at simultaneously classifying invalid narratives as invalid. Our findings not only advance the study of LLM capabilities, limitations, and validity but also offer practical insights for narrative generation and natural language processing applications.
Action-constrained reinforcement learning (ACRL) is a popular approach for solving safety-critical and resource-allocation related decision making problems. A major challenge in ACRL is to ensure agent taking a valid action satisfying constraints in each RL step. Commonly used approach of using a projection layer on top of the policy network requires solving an optimization program which can result in longer training time, slow convergence, and zero gradient problem. To address this, first we use a normalizing flow model to learn an invertible, differentiable mapping between the feasible action space and the support of a simple distribution on a latent variable, such as Gaussian. Second, learning the flow model requires sampling from the feasible action space, which is also challenging. We develop multiple methods, based on Hamiltonian Monte-Carlo and probabilistic sentential decision diagrams for such action sampling for convex and non-convex constraints. Third, we integrate the learned normalizing flow with the DDPG algorithm. By design, a well-trained normalizing flow will transform policy output into a valid action without requiring an optimization solver. Empirically, our approach results in significantly fewer constraint violations (upto an order-of-magnitude for several instances) and is multiple times faster on a variety of continuous control tasks.
Graph generation is a fundamental problem in various domains, including chemistry and social networks. Recent work has shown that molecular graph generation using recurrent neural networks (RNNs) is advantageous compared to traditional generative approaches which require converting continuous latent representations into graphs. One issue which arises when treating graph generation as sequential generation is the arbitrary order of the sequence which results from a particular choice of graph flattening method. In this work we propose using RNNs, taking into account the non-sequential nature of graphs by adding an Orderless Regularization (OLR) term that encourages the hidden state of the recurrent model to be invariant to different valid orderings present under the training distribution. We demonstrate that sequential graph generation models benefit from our proposed regularization scheme, especially when data is scarce. Our findings contribute to the growing body of research on graph generation and provide a valuable tool for various applications requiring the synthesis of realistic and diverse graph structures.
Autonomous robots are often employed for data collection due to their efficiency and low labour costs. A key task in robotic data acquisition is planning paths through an initially unknown environment to collect observations given platform-specific resource constraints, such as limited battery life. Adaptive online path planning in 3D environments is challenging due to the large set of valid actions and the presence of unknown occlusions. To address these issues, we propose a novel deep reinforcement learning approach for adaptively replanning robot paths to map targets of interest in unknown 3D environments. A key aspect of our approach is a dynamically constructed graph that restricts planning actions local to the robot, allowing us to react to newly discovered static obstacles and targets of interest. For replanning, we propose a new reward function that balances between exploring the unknown environment and exploiting online-discovered targets of interest. Our experiments show that our method enables more efficient target discovery compared to state-of-the-art learning and non-learning baselines. We also showcase our approach for orchard monitoring using an unmanned aerial vehicle in a photorealistic simulator. We open-source our code and model at: https://github.com/dmar-bonn/ipp-rl-3d.
This study investigates leveraging stochastic gradient descent (SGD) to learn operators between general Hilbert spaces. We propose weak and strong regularity conditions for the target operator to depict its intrinsic structure and complexity. Under these conditions, we establish upper bounds for convergence rates of the SGD algorithm and conduct a minimax lower bound analysis, further illustrating that our convergence analysis and regularity conditions quantitatively characterize the tractability of solving operator learning problems using the SGD algorithm. It is crucial to highlight that our convergence analysis is still valid for nonlinear operator learning. We show that the SGD estimator will converge to the best linear approximation of the nonlinear target operator. Moreover, applying our analysis to operator learning problems based on vector-valued and real-valued reproducing kernel Hilbert spaces yields new convergence results, thereby refining the conclusions of existing literature.
Human de-occlusion, which aims to infer the appearance of invisible human parts from an occluded image, has great value in many human-related tasks, such as person re-id, and intention inference. To address this task, this paper proposes a dynamic mask-aware transformer (DMAT), which dynamically augments information from human regions and weakens that from occlusion. First, to enhance token representation, we design an expanded convolution head with enlarged kernels, which captures more local valid context and mitigates the influence of surrounding occlusion. To concentrate on the visible human parts, we propose a novel dynamic multi-head human-mask guided attention mechanism through integrating multiple masks, which can prevent the de-occluded regions from assimilating to the background. Besides, a region upsampling strategy is utilized to alleviate the impact of occlusion on interpolated images. During model learning, an amodal loss is developed to further emphasize the recovery effect of human regions, which also refines the model's convergence. Extensive experiments on the AHP dataset demonstrate its superior performance compared to recent state-of-the-art methods.
This paper introduces an algorithm for approximating the invariant set of closed-loop controlled dynamical systems identified using ReLU neural networks or piecewise affine PWA functions, particularly addressing the challenge of providing safety guarantees for ReLU networks commonly used in safety-critical applications. The invariant set of PWA dynamical system is estimated using ReLU networks or its equivalent PWA function. This method entails formulating the barrier function as a PWA function and converting the search process into a linear optimization problem using vertices. We incorporate a domain refinement strategy to increase flexibility in case the optimization does not find a valid barrier function. Moreover, the objective of optimization is to maximize the invariant set based on the current partition. Our experimental results demonstrate the effectiveness and efficiency of our approach, demonstrating its potential for ensuring the safety of PWA dynamical systems.