In this book, written in Portuguese, we discuss what ill-posed problems are and how the regularization method is used to solve them. In the form of questions and answers, we reflect on the origins and future of regularization, relating the similarities and differences of its meaning in different areas, including inverse problems, statistics, machine learning, and deep learning.

Parallel Tempering (PT) is a classical MCMC algorithm designed for leveraging parallel computation to sample efficiently from high-dimensional, multimodal or otherwise complex distributions via annealing. One limitation of the standard formulation of PT is the growth of computational resources required to generate high-quality samples, as measured by effective sample size or round trip rate, for increasingly challenging distributions. To address this issue, we propose the framework: Generalised Parallel Tempering (GePT) which allows for the incorporation of recent advances in modern generative modelling, such as normalising flows and diffusion models, within Parallel Tempering, while maintaining the same theoretical guarantees as MCMC-based methods. For instance, we show that this allows us to utilise diffusion models in a parallelised manner, bypassing the usual computational cost of a large number of steps to generate quality samples. Further, we empirically demonstrate that GePT can improve sample quality and reduce the growth of computational resources required to handle complex distributions over the classical algorithm. Sampling from a complex probability distribution π over a state-space X, whose density π(x) is only known up to a normalising constant, is a fundamental task in modern statistical inference.

Class imbalance remains a major challenge in machine learning, especially in multi-class problems with long-tailed distributions. Existing methods, such as data resampling, cost-sensitive techniques, and logistic loss modifications, though popular and often effective, lack solid theoretical foundations. As an example, we demonstrate that cost-sensitive methods are not Bayes consistent. This paper introduces a novel theoretical framework for analyzing generalization in imbalanced classification. We propose a new class-imbalanced margin loss function for both binary and multi-class settings, prove its strong $H$-consistency, and derive corresponding learning guarantees based on empirical loss and a new notion of class-sensitive Rademacher complexity. Leveraging these theoretical results, we devise novel and general learning algorithms, IMMAX (Imbalanced Margin Maximization), which incorporate confidence margins and are applicable to various hypothesis sets. While our focus is theoretical, we also present extensive empirical results demonstrating the effectiveness of our algorithms compared to existing baselines.

In this paper, we consider combinatorial reinforcement learning with preference feedback, where a learning agent sequentially offers an action--an assortment of multiple items to--a user, whose preference feedback follows a multinomial logistic (MNL) model. This framework allows us to model real-world scenarios, particularly those involving long-term user engagement, such as in recommender systems and online advertising. However, this framework faces two main challenges: (1) the unknown value of each item, unlike traditional MNL bandits that only address single-step preference feedback, and (2) the difficulty of ensuring optimism while maintaining tractable assortment selection in the combinatorial action space with unknown values. In this paper, we assume a contextual MNL preference model, where the mean utilities are linear, and the value of each item is approximated by a general function. We propose an algorithm, MNL-VQL, that addresses these challenges, making it both computationally and statistically efficient. As a special case, for linear MDPs (with the MNL preference feedback), we establish the first regret lower bound in this framework and show that MNL-VQL achieves nearly minimax-optimal regret. To the best of our knowledge, this is the first work to provide statistical guarantees in combinatorial RL with preference feedback.

Temporal predictive models have the potential to improve decisions in health care, public services, and other domains, yet they often fail to effectively support decision-makers. Prior literature shows that many misalignments between model behavior and decision-makers' expectations stem from issues of model specification, namely how, when, and for whom predictions are made. However, model specifications for predictive tasks are highly technical and difficult for non-data-scientist stakeholders to interpret and critique. To address this challenge we developed Tempo, an interactive system that helps data scientists and domain experts collaboratively iterate on model specifications. Using Tempo's simple yet precise temporal query language, data scientists can quickly prototype specifications with greater transparency about pre-processing choices. Moreover, domain experts can assess performance within data subgroups to validate that models behave as expected. Through three case studies, we demonstrate how Tempo helps multidisciplinary teams quickly prune infeasible specifications and identify more promising directions to explore.

The rapid expansion of the Internet of Things (IoT) ecosystem has transformed various sectors but has also introduced significant cybersecurity challenges. Traditional centralized security methods often struggle to balance privacy preservation and real-time threat detection in IoT networks. To address these issues, this study proposes a Federated Learning-Driven Cybersecurity Framework designed specifically for IoT environments. The framework enables decentralized data processing by training models locally on edge devices, ensuring data privacy. Secure aggregation of these locally trained models is achieved using homomorphic encryption, allowing collaborative learning without exposing sensitive information. The proposed framework utilizes recurrent neural networks (RNNs) for anomaly detection, optimized for resource-constrained IoT networks. Experimental results demonstrate that the system effectively detects complex cyber threats, including distributed denial-of-service (DDoS) attacks, with over 98% accuracy. Additionally, it improves energy efficiency by reducing resource consumption by 20% compared to centralized approaches. This research addresses critical gaps in IoT cybersecurity by integrating federated learning with advanced threat detection techniques. The framework offers a scalable and privacy-preserving solution adaptable to various IoT applications. Future work will explore the integration of blockchain for transparent model aggregation and quantum-resistant cryptographic methods to further enhance security in evolving technological landscapes.

Modern machine learning pipelines are increasingly combining and mixing data from diverse and disparate sources, e.g., pre-training large language models. Yet, finding the optimal data mixture is a challenging and open problem. We formalize this data mixing problem as a bi-level objective: the best mixture is the one that would lead to the best model for a downstream objective. Unfortunately, this objective is generally intractable. In this paper, we make the observation that the bi-level data mixing objective becomes convex as our model class becomes larger. We develop and study a gradient-based approach for optimizing this convex objective, which we call MixMin, and test it on language modeling and chemistry tasks. MixMin was the only method that uniformly improved the data mixture in all our experiments. With MixMin, we improved the data mixture using less than 0.2% additional compute for a pythia-410M model trained on 8.2B tokens, resulting between 1-5% relative improvement to negative log likelihood on PIQA, ARC Easy, SciQ, and OpenWebMath. Crucially, we found that MixMin mixtures for smaller models improved training of larger models, suggesting that MixMin mixtures may be scale-invariant. When mixing bioassay data to train an XGBoost model, we saw improvements to average precision scores of 0.03-0.15.

Approximating a probability distribution using a set of particles is a fundamental problem in machine learning and statistics, with applications including clustering and quantization. Formally, we seek a finite weighted mixture of Dirac measures that best approximates the target distribution. While much existing work relies on the Wasserstein distance to quantify approximation errors, maximum mean discrepancy (MMD) has received comparatively less attention, especially when allowing for variable particle weights. We study the quantization problem from the perspective of minimizing MMD via gradient flow in the Wasserstein-Fisher-Rao (WFR) geometry. This gradient flow yields an ODE system from which we further derive a fixed-point algorithm called mean shift interacting particles (MSIP). We show that MSIP extends the (non-interacting) mean shift algorithm, widely used for identifying modes in kernel density estimates. Moreover, we show that MSIP can be interpreted as preconditioned gradient descent, and that it acts as a relaxation of Lloyd's algorithm for clustering. Our numerical experiments demonstrate that MSIP and the WFR ODEs outperform other algorithms for quantization of multi-modal and high-dimensional targets.

Diffusion models generate samples by estimating the score function of the target distribution at various noise levels. The model is trained using samples drawn from the target distribution, progressively adding noise. In this work, we establish the first (nearly) dimension-free sample complexity bounds for learning these score functions, achieving a double exponential improvement in dimension over prior results. A key aspect of our analysis is the use of a single function approximator to jointly estimate scores across noise levels, a critical feature of diffusion models in practice which enables generalization across timesteps. Our analysis introduces a novel martingale-based error decomposition and sharp variance bounds, enabling efficient learning from dependent data generated by Markov processes, which may be of independent interest. Building on these insights, we propose Bootstrapped Score Matching (BSM), a variance reduction technique that utilizes previously learned scores to improve accuracy at higher noise levels. These results provide crucial insights into the efficiency and effectiveness of diffusion models for generative modeling.

The learning coefficient plays a crucial role in analyzing the performance of information criteria, such as the Widely Applicable Information Criterion (WAIC) and the Widely Applicable Bayesian Information Criterion (WBIC), which Sumio Watanabe developed to assess model generalization ability. In regular statistical models, the learning coefficient is given by d/2, where d is the dimension of the parameter space. More generally, it is defined as the absolute value of the pole order of a zeta function derived from the Kullback-Leibler divergence and the prior distribution. However, except for specific cases such as reduced-rank regression, the learning coefficient cannot be derived in a closed form. Watanabe proposed a numerical method to estimate the learning coefficient, which Imai further refined to enhance its convergence properties. These methods utilize the asymptotic behavior of WBIC and have been shown to be statistically consistent as the sample size grows. In this paper, we propose a novel numerical estimation method that fundamentally differs from previous approaches and leverages a new quantity, "Empirical Loss," which was introduced by Watanabe. Through numerical experiments, we demonstrate that our proposed method exhibits both lower bias and lower variance compared to those of Watanabe and Imai. Additionally, we provide a theoretical analysis that elucidates why our method outperforms existing techniques and present empirical evidence that supports our findings.