This paper studies the safe reinforcement learning problem formulated as an episodic finite-horizon tabular constrained Markov decision process with an unknown transition kernel and stochastic reward and cost functions. We propose a model-based algorithm based on novel cost and reward function estimators that provide tighter cost pessimism and reward optimism. While guaranteeing no constraint violation in every episode, our algorithm achieves a regret upper bound of $\widetilde{\mathcal{O}}((\bar C - \bar C_b)^{-1}H^{2.5} S\sqrt{AK})$ where $\bar C$ is the cost budget for an episode, $\bar C_b$ is the expected cost under a safe baseline policy over an episode, $H$ is the horizon, and $S$, $A$ and $K$ are the number of states, actions, and episodes, respectively. This improves upon the best-known regret upper bound, and when $\bar C- \bar C_b=\Omega(H)$, it nearly matches the regret lower bound of $\Omega(H^{1.5}\sqrt{SAK})$. We deduce our cost and reward function estimators via a Bellman-type law of total variance to obtain tight bounds on the expected sum of the variances of value function estimates. This leads to a tighter dependence on the horizon in the function estimators. We also present numerical results to demonstrate the computational effectiveness of our proposed framework.

In this study, we propose a robust set of features derived from a thorough research of contemporary practices in voice pathology detection. The feature set is based on the combination of acoustic handcrafted features. Additionally, we introduce pitch difference as a novel feature. We combine this feature set, containing data from the publicly available Saarbr\"ucken Voice Database (SVD), with preprocessing using the K-Means Synthetic Minority Over-Sampling Technique algorithm to address class imbalance. Moreover, we applied multiple ML models as binary classifiers. We utilized support vector machine, k-nearest neighbors, naive Bayes, decision tree, random forest and AdaBoost classifiers. To determine the best classification approach, we performed grid search on feasible hyperparameters of respective classifiers and subsections of features. Our approach has achieved the state-of-the-art performance, measured by unweighted average recall in voice pathology detection on SVD database. We intentionally omit accuracy as it is highly biased metric in case of unbalanced data compared to aforementioned metrics. The results are further enhanced by eliminating the potential overestimation of the results with repeated stratified cross-validation. This advancement demonstrates significant potential for the clinical deployment of ML methods, offering a valuable tool for an objective examination of voice pathologies. To support our claims, we provide a publicly available GitHub repository with DOI 10.5281/zenodo.13771573. Finally, we provide REFORMS checklist.

When a robot encounters a novel object, how should it respond$\unicode{x2014}$what data should it collect$\unicode{x2014}$so that it can find the object in the future? In this work, we present a method for learning image features of an unknown number of novel objects. To do this, we use active coverage with respect to latent uncertainties of the novel descriptions. We apply ergodic stability and PAC-Bayes theory to extend statistical guarantees for VAEs to embodied agents. We demonstrate the method in hardware with a robotic arm; the pipeline is also implemented in a simulated environment. Algorithms and simulation are available open source, see http://sites.google.com/u.northwestern.edu/embodied-learning-hardware .

Bayesian optimisation for real-world problems is often performed interactively with human experts, and integrating their domain knowledge is key to accelerate the optimisation process. We consider a setup where experts provide advice on the next query point through binary accept/reject recommendations (labels). Experts' labels are often costly, requiring efficient use of their efforts, and can at the same time be unreliable, requiring careful adjustment of the degree to which any expert is trusted. We introduce the first principled approach that provides two key guarantees. (1) Handover guarantee: similar to a no-regret property, we establish a sublinear bound on the cumulative number of experts' binary labels. Initially, multiple labels per query are needed, but the number of expert labels required asymptotically converges to zero, saving both expert effort and computation time. (2) No-harm guarantee with data-driven trust level adjustment: our adaptive trust level ensures that the convergence rate will not be worse than the one without using advice, even if the advice from experts is adversarial. Unlike existing methods that employ a user-defined function that hand-tunes the trust level adjustment, our approach enables data-driven adjustments. Real-world applications empirically demonstrate that our method not only outperforms existing baselines, but also maintains robustness despite varying labelling accuracy, in tasks of battery design with human experts.

Reliable estimation of predictive uncertainty is crucial for machine learning applications, particularly in high-stakes scenarios where hedging against risks is essential. Despite its significance, a consensus on the correct measurement of predictive uncertainty remains elusive. In this work, we return to first principles to develop a fundamental framework of information-theoretic predictive uncertainty measures. Our proposed framework categorizes predictive uncertainty measures according to two factors: (I) The predicting model (II) The approximation of the true predictive distribution. Examining all possible combinations of these two factors, we derive a set of predictive uncertainty measures that includes both known and newly introduced ones. We empirically evaluate these measures in typical uncertainty estimation settings, such as misclassification detection, selective prediction, and out-of-distribution detection. The results show that no single measure is universal, but the effectiveness depends on the specific setting. Thus, our work provides clarity about the suitability of predictive uncertainty measures by clarifying their implicit assumptions and relationships.

Nonlinear time series models with exogenous regressors are essential in econometrics, queuing theory, and machine learning, though their statistical analysis remains incomplete. Key results, such as the law of large numbers and the functional central limit theorem, are known for weakly dependent variables. We demonstrate the transfer of mixing properties from the exogenous regressor to the response via coupling arguments. Additionally, we study Markov chains in random environments with drift and minorization conditions, even under non-stationary environments with favorable mixing properties, and apply this framework to single-server queuing models.

We study the mixing time of two popular discrete time Markov chains in continuous space, the unadjusted Langevin algorithm and the proximal sampler, which are discretizations of the Langevin dynamics. We extend mixing time analyses for these Markov chains to hold in $\Phi$-divergence. We show that any $\Phi$-divergence arising from a twice-differentiable strictly convex function $\Phi$ converges to $0$ exponentially fast along these Markov chains, under the assumption that their stationary distributions satisfies the corresponding $\Phi$-Sobolev inequality. Our rates of convergence are tight and include as special cases popular mixing time regimes, namely the mixing in chi-squared divergence under a Poincar\'e inequality, and the mixing in relative entropy under a log-Sobolev inequality. Our results follow by bounding the contraction coefficients arising in the appropriate strong data processing inequalities.

The aim of the notes is to demonstrate the potential for ideas in machine learning to impact on the fields of inverse problems and data assimilation. The perspective is one that is primarily aimed at researchers from inverse problems and/or data assimilation who wish to see a mathematical presentation of machine learning as it pertains to their fields. As a by-product of the presentation we present a succinct mathematical treatment of various topics in machine learning. The material on machine learning, along with some other related topics, is summarized in Part III, Appendix. Part I of the notes is concerned with inverse problems, employing material from Part III; Part II of the notes is concerned with data assimilation, employing material from Parts I and III.

The logit transform is arguably the most widely-employed link function beyond linear settings. This transformation routinely appears in regression models for binary data and provides, either explicitly or implicitly, a core building-block within state-of-the-art methodologies for both classification and regression. Its widespread use, combined with the lack of analytical solutions for the optimization of general losses involving the logit transform, still motivates active research in computational statistics. Among the directions explored, a central one has focused on the design of tangent lower bounds for logistic log-likelihoods that can be tractably optimized, while providing a tight approximation of these log-likelihoods. Although progress along these lines has led to the development of effective minorize-maximize (MM) algorithms for point estimation and coordinate ascent variational inference schemes for approximate Bayesian inference under several logit models, the overarching focus in the literature has been on tangent quadratic minorizers. In fact, it is still unclear whether tangent lower bounds sharper than quadratic ones can be derived without undermining the tractability of the resulting minorizer. This article addresses such a challenging question through the design and study of a novel piece-wise quadratic lower bound that uniformly improves any tangent quadratic minorizer, including the sharpest ones, while admitting a direct interpretation in terms of the classical generalized lasso problem. As illustrated in a ridge logistic regression, this unique connection facilitates more effective implementations than those provided by available piece-wise bounds, while improving the convergence speed of quadratic ones.

The vector quantization is a widely used method to map continuous representation to discrete space and has important application in tokenization for generative mode, bottlenecking information and many other tasks in machine learning. Vector Quantized Variational Autoencoder (VQ-VAE) is a type of variational autoencoder using discrete embedding as latent. We generalize the technique further, enriching the probabilistic framework with a Gaussian mixture as the underlying generative model. This framework leverages a codebook of latent means and adaptive variances to capture complex data distributions. This principled framework avoids various heuristics and strong assumptions that are needed with the VQ-VAE to address training instability and to improve codebook utilization. This approach integrates the benefits of both discrete and continuous representations within a variational Bayesian framework. Furthermore, by introducing the \textit{Aggregated Categorical Posterior Evidence Lower Bound} (ALBO), we offer a principled alternative optimization objective that aligns variational distributions with the generative model. Our experiments demonstrate that GM-VQ improves codebook utilization and reduces information loss without relying on handcrafted heuristics.