A classifier is considered interpretable if each of its decisions has an explanation which is small enough to be easily understood by a human user. A DNF formula can be seen as a binary classifier $κ$ over boolean domains. The size of an explanation of a positive decision taken by a DNF $κ$ is bounded by the size of the terms in $κ$, since we can explain a positive decision by giving a term of $κ$ that evaluates to true. Since both positive and negative decisions must be explained, we consider that interpretable DNFs are those $κ$ for which both $κ$ and $\overlineκ$ can be expressed as DNFs composed of terms of bounded size. In this paper, we study the family of $k$-DNFs whose complements can also be expressed as $k$-DNFs. We compare two such families, namely depth-$k$ decision trees and nested $k$-DNFs, a novel family of models. Experiments indicate that nested $k$-DNFs are an interesting alternative to decision trees in terms of interpretability and accuracy.

Large language models (LLMs) have exhibited extraordinary performance in a variety of tasks while it remains challenging for them to solve complex multi-step tasks as agents. In practice, agents sensitive to the outcome of certain key steps which makes them likely to fail the task because of a subtle mistake in the planning trajectory. Recent approaches resort to calibrating the reasoning process through reinforcement learning. They reward or penalize every reasoning step with process supervision, as known as Process Reward Models (PRMs). However, PRMs are difficult and costly to scale up with a large number of next action candidates since they require extensive computations to acquire the training data through the per-step trajectory exploration. To mitigate this issue, we focus on the relative reward trend across successive reasoning steps and propose maintaining an increasing reward in the collected trajectories for process supervision, which we term Reward Rising Optimization (RRO). Specifically, we incrementally augment the process supervision until identifying a step exhibiting positive reward differentials, i.e. rising rewards, relative to its preceding iteration. This method dynamically expands the search space for the next action candidates, efficiently capturing high-quality data. We provide mathematical groundings and empirical results on the WebShop and InterCode-SQL benchmarks, showing that our proposed RRO achieves superior performance while requiring much less exploration cost.

Electronic Health Records (EHR) offer rich real-world data for personalized medicine, providing insights into disease progression, treatment responses, and patient outcomes. However, their sparsity, heterogeneity, and high dimensionality make them difficult to model, while the lack of standardized ground truth further complicates predictive modeling. To address these challenges, we propose SCORE, a semi-supervised representation learning framework that captures multi-domain disease profiles through patient embeddings. SCORE employs a Poisson-Adapted Latent factor Mixture (PALM) Model with pre-trained code embeddings to characterize codified features and extract meaningful patient phenotypes and embeddings. To handle the computational challenges of large-scale data, it introduces a hybrid Expectation-Maximization (EM) and Gaussian Variational Approximation (GVA) algorithm, leveraging limited labeled data to refine estimates on a vast pool of unlabeled samples. We theoretically establish the convergence of this hybrid approach, quantify GVA errors, and derive SCORE's error rate under diverging embedding dimensions. Our analysis shows that incorporating unlabeled data enhances accuracy and reduces sensitivity to label scarcity. Extensive simulations confirm SCORE's superior finite-sample performance over existing methods. Finally, we apply SCORE to predict disability status for patients with multiple sclerosis (MS) using partially labeled EHR data, demonstrating that it produces more informative and predictive patient embeddings for multiple MS-related conditions compared to existing approaches.

Knowledge representation and reasoning in neural networks have been a long-standing endeavor which has attracted much attention recently. The principled integration of reasoning and learning in neural networks is a main objective of the area of neurosymbolic Artificial Intelligence (AI). In this chapter, a simple energy-based neurosymbolic AI system is described that can represent and reason formally about any propositional logic formula. This creates a powerful combination of learning from data and knowledge and logical reasoning. We start by positioning neurosymbolic AI in the context of the current AI landscape that is unsurprisingly dominated by Large Language Models (LLMs). We identify important challenges of data efficiency, fairness and safety of LLMs that might be addressed by neurosymbolic reasoning systems with formal reasoning capabilities. We then discuss the representation of logic by the specific energy-based system, including illustrative examples and empirical evaluation of the correspondence between logical reasoning and energy minimization using Restricted Boltzmann Machines (RBM). Learning from data and knowledge is also evaluated empirically and compared with a symbolic, neural and a neurosymbolic system. Results reported in this chapter in an accessible way are expected to reignite the research on the use of neural networks as massively-parallel models for logical reasoning and promote the principled integration of reasoning and learning in deep networks. We conclude the chapter with a discussion of the importance of positioning neurosymbolic AI within a broader framework of formal reasoning and accountability in AI, discussing the challenges for neurosynbolic AI to tackle the various known problems of reliability of deep learning.

This paper presents a new approach for jointly calibrating magnetometers and inertial measurement units, focusing on improving calibration accuracy and computational efficiency. The proposed method formulates the calibration problem as a maximum a posteriori estimation problem, treating both the calibration parameters and orientation trajectory of the sensors as unknowns. This formulation enables efficient optimization with closed-form derivatives. The method is compared against two state-of-the-art approaches in terms of computational complexity and estimation accuracy. Simulation results demonstrate that the proposed method achieves lower root mean square error in calibration parameters while maintaining competitive computational efficiency. Further validation through real-world experiments confirms the practical benefits of our approach: it effectively reduces position drift in a magnetic field-aided inertial navigation system by more than a factor of two on most datasets. Moreover, the proposed method calibrated 30 magnetometers in less than 2 minutes. The contributions include a new calibration method, an analysis of existing methods, and a comprehensive empirical evaluation. Datasets and algorithms are made publicly available to promote reproducible research.

Reinforcement learning (RL) holds significant promise for training LLM agents to handle complex, goal-oriented tasks that require multi-step interactions with external environments. However, a critical challenge when applying RL to these agentic tasks arises from delayed rewards: feedback signals are typically available only after the entire task is completed. This makes it non-trivial to assign delayed rewards to earlier actions, providing insufficient guidance regarding environmental constraints and hindering agent training. In this work, we draw on the insight that the ultimate completion of a task emerges from the cumulative progress an agent makes across individual steps. We propose Stepwise Progress Attribution (SPA), a general reward redistribution framework that decomposes the final reward into stepwise contributions, each reflecting its incremental progress toward overall task completion. To achieve this, we train a progress estimator that accumulates stepwise contributions over a trajectory to match the task completion. During policy optimization, we combine the estimated per-step contribution with a grounding signal for actions executed in the environment as the fine-grained, intermediate reward for effective agent training. Extensive experiments on common agent benchmarks (including Webshop, ALFWorld, and VirtualHome) demonstrate that SPA consistently outperforms the state-of-the-art method in both success rate (+2.5\% on average) and grounding accuracy (+1.9\% on average). Further analyses demonstrate that our method remarkably provides more effective intermediate rewards for RL training. Our code is available at https://github.com/WangHanLinHenry/SPA-RL-Agent.

We present a theoretical analysis of the Elo rating system, a popular method for ranking skills of players in an online setting. In particular, we study Elo under the Bradley-Terry-Luce model and, using techniques from Markov chain theory, show that Elo learns the model parameters at a rate competitive with the state-of-the-art. We apply our results to the problem of efficient tournament design and discuss a connection with the fastest-mixing Markov chain problem.

We consider 1-dimensional location estimation, where we estimate a parameter \lambda from n samples \lambda \eta_i, with each \eta_i drawn i.i.d. For fixed f the maximum-likelihood estimate (MLE) is well-known to be optimal in the limit as n \to \infty: it is asymptotically normal with variance matching the Cramer-Rao lower bound of \frac{1}{n\mathcal{I}}, where \mathcal{I} is the Fisher information of f . However, this bound does not hold for finite n, or when f varies with n . We show for arbitrary f and n that one can recover a similar theory based on the Fisher information of a smoothed version of f, where the smoothing radius decays with n .

Much of Bayesian inference centers around the design of estimators for inverse problems which are optimal assuming the data comes from a known prior. But what do these optimality guarantees mean if the prior is unknown? In recent years, algorithm unrolling has emerged as deep learning's answer to this age-old question: design a neural network whose layers can in principle simulate iterations of inference algorithms and train on data generated by the unknown prior. Despite its empirical success, however, it has remained unclear whether this method can provably recover the performance of its optimal, prior-aware counterparts.In this work, we prove the first rigorous learning guarantees for neural networks based on unrolling approximate message passing (AMP). For compressed sensing, we prove that when trained on data drawn from a product prior, the layers of the network approximately converge to the same denoisers used in Bayes AMP. We also provide extensive numerical experiments for compressed sensing and rank-one matrix estimation demonstrating the advantages of our unrolled architecture --- in addition to being able to obliviously adapt to general priors, it exhibits improvements over Bayes AMP in more general settings of low dimensions, non-Gaussian designs, and non-product priors.

We propose a new framework for formulating optimal transport distances between Markov chains. Previously known formulations studied couplings between the entire joint distribution induced by the chains, and derived solutions via a reduction to dynamic programming (DP) in an appropriately defined Markov decision process. This formulation has, however, not led to particularly efficient algorithms so far, since computing the associated DP operators requires fully solving a static optimal transport problem, and these operators need to be applied numerous times during the overall optimization process. In this work, we develop an alternative perspective by considering couplings between a flattened'' version of the joint distributions that we call discounted occupancy couplings, and show that calculating optimal transport distances in the full space of joint distributions can be equivalently formulated as solving a linear program (LP) in this reduced space. This LP formulation formulation allows us to port several algorithmic ideas from other areas of optimal transport theory.