While adversarial robustness and generalization have individually received substantial attention in the recent literature on quantum machine learning, their interplay is much less explored. In this chapter, we address this interplay for variational quantum models, which were recently proposed as function approximators in supervised learning. We discuss recent results quantifying both robustness and generalization via Lipschitz bounds, which explicitly depend on model parameters. Thus, they give rise to a regularization-based training approach for robust and generalizable quantum models, highlighting the importance of trainable data encoding strategies. The practical implications of the theoretical results are demonstrated with an application to time series analysis.

Mathematical reasoning has long represented one of the most fundamental and challenging frontiers in artificial intelligence research. In recent years, large language models (LLMs) have achieved significant advances in this area. This survey examines the development of mathematical reasoning abilities in LLMs through two high-level cognitive phases: comprehension, where models gain mathematical understanding via diverse pretraining strategies, and answer generation, which has progressed from direct prediction to step-by-step Chain-of-Thought (CoT) reasoning. We review methods for enhancing mathematical reasoning, ranging from training-free prompting to fine-tuning approaches such as supervised fine-tuning and reinforcement learning, and discuss recent work on extended CoT and "test-time scaling". Despite notable progress, fundamental challenges remain in terms of capacity, efficiency, and generalization. To address these issues, we highlight promising research directions, including advanced pretraining and knowledge augmentation techniques, formal reasoning frameworks, and meta-generalization through principled learning paradigms. This survey tries to provide some insights for researchers interested in enhancing reasoning capabilities of LLMs and for those seeking to apply these techniques to other domains.

Leveraging more test-time computation has proven to be an effective way to boost the reasoning capabilities of large language models (LLMs). Among various methods, the verify-and-improve paradigm stands out for enabling dynamic solution exploration and feedback incorporation. However, existing approaches often suffer from restricted feedback spaces and lack of coordinated training of different parties, leading to suboptimal performance. To address this, we model this multi-turn refinement process as a Markov Decision Process and introduce DPSDP (Direct Policy Search by Dynamic Programming), a reinforcement learning algorithm that trains an actor-critic LLM system to iteratively refine answers via direct preference learning on self-generated data. Theoretically, DPSDP can match the performance of any policy within the training distribution. Empirically, we instantiate DPSDP with various base models and show improvements on both in- and out-of-distribution benchmarks. For example, on benchmark MATH 500, majority voting over five refinement steps increases first-turn accuracy from 58.2% to 63.2% with Ministral-based models. An ablation study further confirms the benefits of multi-agent collaboration and out-of-distribution generalization.

We present LeanTutor, a Large Language Model (LLM)-based tutoring system for math proofs. LeanTutor interacts with the student in natural language, formally verifies student-written math proofs in Lean, generates correct next steps, and provides the appropriate instructional guidance. LeanTutor is composed of three modules: (i) an autoformalizer/proof-checker, (ii) a next-step generator, and (iii) a natural language feedback generator. The first module faithfully autoformalizes student proofs into Lean and verifies proof accuracy via successful code compilation. If the proof has an error, the incorrect step is identified. The next-step generator module outputs a valid next Lean tactic for incorrect proofs via LLM-based candidate generation and proof search. The feedback generator module leverages Lean data to produce a pedagogically-motivated natural language hint for the student user. To evaluate our system, we introduce PeanoBench, a human-written dataset derived from the Natural Numbers Game, consisting of 371 Peano Arithmetic proofs, where each natural language proof step is paired with the corresponding logically equivalent tactic in Lean. The Autoformalizer correctly formalizes 57% of tactics in correct proofs and accurately identifies the incorrect step in 30% of incorrect proofs. In generating natural language hints for erroneous proofs, LeanTutor outperforms a simple baseline on accuracy and relevance metrics.

While existing benchmarks probe the reasoning abilities of large language models (LLMs) across diverse domains, they predominantly assess passive reasoning, providing models with all the information needed to reach a solution. By contrast, active reasoning-where an LLM must interact with external systems to acquire missing evidence or data-has received little systematic attention. To address this shortfall, we present AR-Bench, a novel benchmark designed explicitly to evaluate an LLM's active reasoning skills. AR-Bench comprises three task families-detective cases, situation puzzles, and guessing numbers-that together simulate real-world, agentic scenarios and measure performance across commonsense, logical, and symbolic reasoning challenges. Empirical evaluation on AR-Bench demonstrates that contemporary LLMs exhibit pronounced difficulties with active reasoning: they frequently fail to acquire or leverage the information needed to solve tasks. This gap highlights a stark divergence between their passive and active reasoning abilities. Moreover, ablation studies indicate that even advanced strategies, such as tree-based searching or post-training approaches, yield only modest gains and fall short of the levels required for real-world deployment. Collectively, these findings highlight the critical need to advance methodology for active reasoning, e.g., incorporating interactive learning, real-time feedback loops, and environment-aware objectives for training. The benchmark is publicly available at: https://github.com/tmlr-group/AR-Bench.

Large Language Models (LLMs) have shown impressive performance in mathematical reasoning tasks when guided by Chain-of-Thought (CoT) prompting. However, they tend to produce highly confident yet incorrect outputs, which poses significant risks in domains like education, where users may lack the expertise to assess reasoning steps. To address this, we propose a structured framework that models stepwise confidence as a temporal signal and evaluates it using Signal Temporal Logic (STL). In particular, we define formal STL-based constraints to capture desirable temporal properties and compute robustness scores that serve as structured, interpretable confidence estimates. Our approach also introduces a set of uncertainty reshaping strategies to enforce smoothness, monotonicity, and causal consistency across the reasoning trajectory. Experiments show that our approach consistently improves calibration metrics and provides more reliable uncertainty estimates than conventional confidence aggregation and post-hoc calibration.

Large language models(LLMs) like Gemini are becoming common tools for supporting student writing. But most of their feedback is based only on the final essay missing important context about how that text was written. In this paper, we explore whether using writing process data, collected through keystroke logging and periodic snapshots, can help LLMs give feedback that better reflects how learners think and revise while writing. We built a digital writing tool that captures both what students type and how their essays evolve over time. Twenty students used this tool to write timed essays, which were then evaluated in two ways: (i) LLM generated feedback using both the final essay and the full writing trace, and (ii) After the task, students completed surveys about how useful and relatable they found the feedback. Early results show that learners preferred the process-aware LLM feedback, finding it more in tune with their own thinking. We also found that certain types of edits, like adding new content or reorganizing paragraphs, aligned closely with higher scores in areas like coherence and elaboration. Our findings suggest that making LLMs more aware of the writing process can lead to feedback that feels more meaningful, personal, and supportive.

This monograph explores the design and analysis of correlated noise mechanisms for differential privacy (DP), focusing on their application to private training of AI and machine learning models via the core primitive of estimation of weighted prefix sums. While typical DP mechanisms inject independent noise into each step of a stochastic gradient (SGD) learning algorithm in order to protect the privacy of the training data, a growing body of recent research demonstrates that introducing (anti-)correlations in the noise can significantly improve privacy-utility trade-offs by carefully canceling out some of the noise added on earlier steps in subsequent steps. Such correlated noise mechanisms, known variously as matrix mechanisms, factorization mechanisms, and DP-Follow-the-Regularized-Leader (DP-FTRL) when applied to learning algorithms, have also been influential in practice, with industrial deployment at a global scale.

--In recent years, with rising concerns for data privacy, Federated Learning has gained prominence, as it enables collaborative training without the aggregation of raw data from participating clients. However, much of the current focus has been on parametric gradient-based models, while nonparametric counterparts such as decision tree are relatively understudied. Existing methods for adapting decision trees to Federated Learning generally combine a greedy tree-building algorithm with differential privacy to produce a global model for all clients. These methods are limited to classification trees and categorical data due to the constraints of differential privacy. In this paper, we explore an alternative approach that utilizes Genetic Algorithm to facilitate the construction of personalized decision trees and accommodate categorical and numerical data, thus allowing for both classification and regression trees. Comprehensive experiments demonstrate that our method surpasses decision trees trained solely on local data and a benchmark algorithm. With rapid advancement of AI and machine learning, there are many concerns about data usage and privacy. Lawmakers worldwide have attempted to create incentives for companies to focus more on privacy in their model development, with key examples including the General Data Protection Regulations implemented by the European Union and the California Consumer Privacy Act.Federated Learning (FL) was introduced by Google as an approach for mobile devices to collaboratively solve a machine learning problem without sharing user's local data [14], [17]. In the FL framework, multiple clients contribute to solve a machine learning problem while maintaining their data locally. A global server helps aggregate information that clients deem fit to share, such as model weights, and construct an improved model. The two main scenarios of data distribution in FL are horizontal and vertical. In the former, clients have the same features but different set of samples while in the latter, clients have different features but the same set of samples. Currently, the main focus of the FL research community is on parametric, gradient-based models, yet there is an expanding body of literature that explores the use of decision tree models [25], [26] [7], [19].

Original Manuscript Submitted: 05/05/2025; Final Draft Received: mm/dd/yyyy Federated learning is a machine learning paradigm that leverages edge computing on client devices to optimize models while maintaining user privacy by ensuring that local data remains on the device. However, since all data is collected by clients, federated learning is susceptible to latent noise in local datasets. Factors such as limited measurement capabilities or human errors may introduce inaccuracies in client data. T o address this challenge, we propose the use of a stochastic neural network as the local model within the federated learning framework. Stochastic neural networks not only facilitate the estimation of the true underlying states of the data but also enable the quantification of latent noise. We refer to our federated learning approach, which incorporates stochastic neural networks as local models, as Federated stochastic neural networks. We will present numerical experiments demonstrating the performance and effectiveness of our method, particularly in handling non-independent and identically distributed data. KEY WORDS: Machine Learning, Federated Learning, Neural Network 1. INTRODUCTION The fundamental principles of federated learning can be traced back to earlier advancements in distributed computing and privacy-preserving machine learning techniques. Before federated learning was introduced in McMahan et al. (2016), distributed machine learning primarily focused on executing training processes in parallel across multiple nodes within a data center. Notable frameworks, such as MapReduce (Dean and Ghemawat (2004)) and AllReduce, were designed to aggregate data from different computational units, perform global aggregation using predefined operators, and subsequently redistribute the outcomes to all participating units.