The notion of inference on function spaces is essential to the physical sciences and engineering, where the governing equations are frequently partial differential equations (PDEs) describing the evolution of functions in space and time. In particular, it is often desirable to infer the values of a function everywhere in a physical domain given a sparse number of observation points. There are numerous types of problems in which functional regression plays an important role, such as inverse problems, time series forecasting, data imputation/assimilation. Functional regression problems can be particularly challenging for real world datasets because the underlying stochastic process is often unknown. Much of the work on functional regression and inference has relied on Gaussian processes (GPs) (Rasmussen and Williams, 2006), a specific type of stochastic process in which any finite collection of points has a multivariate Gaussian distribution. Some of the earliest applications focused on analyzing geological data, such as the locations of valuable ore deposits, to identify where new deposits might be found (Chiles and Delfiner, 2012). GP regression (GPR) provides several advantages for functional inference including robustness and mathematical tractability for various problems. This has led to the use of GPR in an assortment of scientific and engineering fields, where precision and reliability in predictions and inferences can significantly impact outcomes (Deringer et al., 2021; Aigrain and Foreman-Mackey, 2023). Despite widespread adoption, the assumption of a GP prior for functional inference problems can be rather limiting, particularly in scenarios where the data exhibit heavy-tailed or multimodal distributions, e.g.

This work aims to capture the effect and benefits of depth for supervised learning via information-theoretic generalization bounds. We first derive two hierarchical bounds on the generalization error in terms of the Kullback-Leibler (KL) divergence or the 1-Wasserstein distance between the train and test distributions of the network internal representations. The KL divergence bound shrinks as the layer index increases, while the Wasserstein bound implies the existence of a layer that serves as a generalization funnel, which attains a minimal 1-Wasserstein distance. Analytic expressions for both bounds are derived under the setting of binary Gaussian classification with linear DNNs. To quantify the contraction of the relevant information measures when moving deeper into the network, we analyze the strong data processing inequality (SDPI) coefficient between consecutive layers of three regularized DNN models: Dropout, DropConnect, and Gaussian noise injection. This enables refining our generalization bounds to capture the contraction as a function of the network architecture parameters. Specializing our results to DNNs with a finite parameter space and the Gibbs algorithm reveals that deeper yet narrower network architectures generalize better in those examples, although how broadly this statement applies remains a question.

The Rational Speech Act (RSA) model provides a flexible framework to model pragmatic reasoning in computational terms. However, state-of-the-art RSA models are still fairly distant from modern machine learning techniques and present a number of limitations related to their interpretability and scalability. Here, we introduce a new RSA framework for metaphor understanding that addresses these limitations by providing an explicit formula - based on the mutually shared information between the speaker and the listener - for the estimation of the communicative goal and by learning the rationality parameter using gradient-based methods. The model was tested against 24 metaphors, not limited to the conventional $\textit{John-is-a-shark}$ type. Results suggest an overall strong positive correlation between the distributions generated by the model and the interpretations obtained from the human behavioral data, which increased when the intended meaning capitalized on properties that were inherent to the vehicle concept. Overall, findings suggest that metaphor processing is well captured by a typicality-based Bayesian model, even when more scalable and interpretable, opening up possible applications to other pragmatic phenomena and novel uses for increasing Large Language Models interpretability. Yet, results highlight that the more creative nuances of metaphorical meaning, not strictly encoded in the lexical concepts, are a challenging aspect for machines.

With the widespread adoption of Large Language Models (LLMs), the prevalence of iterative interactions among these models is anticipated to increase. Notably, recent advancements in multi-round self-improving methods allow LLMs to generate new examples for training subsequent models. At the same time, multi-agent LLM systems, involving automated interactions among agents, are also increasing in prominence. Thus, in both short and long terms, LLMs may actively engage in an evolutionary process. We draw parallels between the behavior of LLMs and the evolution of human culture, as the latter has been extensively studied by cognitive scientists for decades. Our approach involves leveraging Iterated Learning (IL), a Bayesian framework that elucidates how subtle biases are magnified during human cultural evolution, to explain some behaviors of LLMs. This paper outlines key characteristics of agents' behavior in the Bayesian-IL framework, including predictions that are supported by experimental verification with various LLMs. This theoretical framework could help to more effectively predict and guide the evolution of LLMs in desired directions.

Efficiently tackling multiple tasks within complex environment, such as those found in robot manipulation, remains an ongoing challenge in robotics and an opportunity for data-driven solutions, such as reinforcement learning (RL). Model-based RL, by building a dynamic model of the robot, enables data reuse and transfer learning between tasks with the same robot and similar environment. Furthermore, data gathering in robotics is expensive and we must rely on data efficient approaches such as model-based RL, where policy learning is mostly conducted on cheaper simulations based on the learned model. Therefore, the quality of the model is fundamental for the performance of the posterior tasks. In this work, we focus on improving the quality of the model and maintaining the data efficiency by performing active learning of the dynamic model during a preliminary exploration phase based on maximize information gathering. We employ Bayesian neural network models to represent, in a probabilistic way, both the belief and information encoded in the dynamic model during exploration. With our presented strategies we manage to actively estimate the novelty of each transition, using this as the exploration reward. In this work, we compare several Bayesian inference methods for neural networks, some of which have never been used in a robotics context, and evaluate them in a realistic robot manipulation setup. Our experiments show the advantages of our Bayesian model-based RL approach, with similar quality in the results than relevant alternatives with much lower requirements regarding robot execution steps. Unlike related previous studies that focused the validation solely on toy problems, our research takes a step towards more realistic setups, tackling robotic arm end-tasks.

Collaborative machine learning involves training models on data from multiple parties but must incentivize their participation. Existing data valuation methods fairly value and reward each party based on shared data or model parameters but neglect the privacy risks involved. To address this, we introduce differential privacy (DP) as an incentive. Each party can select its required DP guarantee and perturb its sufficient statistic (SS) accordingly. The mediator values the perturbed SS by the Bayesian surprise it elicits about the model parameters. As our valuation function enforces a privacy-valuation trade-off, parties are deterred from selecting excessive DP guarantees that reduce the utility of the grand coalition's model. Finally, the mediator rewards each party with different posterior samples of the model parameters. Such rewards still satisfy existing incentives like fairness but additionally preserve DP and a high similarity to the grand coalition's posterior. We empirically demonstrate the effectiveness and practicality of our approach on synthetic and real-world datasets.

We consider convex optimization with non-smooth objective function and log-concave sampling with non-smooth potential (negative log density). In particular, we study two specific settings where the convex objective/potential function is either semi-smooth or in composite form as the finite sum of semi-smooth components. To overcome the challenges caused by non-smoothness, our algorithms employ two powerful proximal frameworks in optimization and sampling: the proximal point framework for optimization and the alternating sampling framework (ASF) that uses Gibbs sampling on an augmented distribution. A key component of both optimization and sampling algorithms is the efficient implementation of the proximal map by the regularized cutting-plane method. We establish the iteration-complexity of the proximal map in both semi-smooth and composite settings. We further propose an adaptive proximal bundle method for non-smooth optimization. The proposed method is universal since it does not need any problem parameters as input. Additionally, we develop a proximal sampling oracle that resembles the proximal map in optimization and establish its complexity using a novel technique (a modified Gaussian integral). Finally, we combine this proximal sampling oracle and ASF to obtain a Markov chain Monte Carlo method with non-asymptotic complexity bounds for sampling in semi-smooth and composite settings.

The effectiveness of Voting Advice Applications (VAA) is often compromised by the length of their questionnaires. To address user fatigue and incomplete responses, some applications (such as the Swiss Smartvote) offer a condensed version of their questionnaire. However, these condensed versions can not ensure the accuracy of recommended parties or candidates, which we show to remain below 40%. To tackle these limitations, this work introduces an adaptive questionnaire approach that selects subsequent questions based on users' previous answers, aiming to enhance recommendation accuracy while reducing the number of questions posed to the voters. Our method uses an encoder and decoder module to predict missing values at any completion stage, leveraging a two-dimensional latent space reflective of political science's traditional methods for visualizing political orientations. Additionally, a selector module is proposed to determine the most informative subsequent question based on the voter's current position in the latent space and the remaining unanswered questions. We validated our approach using the Smartvote dataset from the Swiss Federal elections in 2019, testing various spatial models and selection methods to optimize the system's predictive accuracy. Our findings indicate that employing the IDEAL model both as encoder and decoder, combined with a PosteriorRMSE method for question selection, significantly improves the accuracy of recommendations, achieving 74% accuracy after asking the same number of questions as in the condensed version.

Zero-shot In-context learning is the phenomenon where models can perform the task simply given the instructions. However, pre-trained large language models are known to be poorly calibrated for this task. One of the most effective approaches to handling this bias is to adopt a contrastive decoding objective, which accounts for the prior probability of generating the next token by conditioning on some context. This work introduces an Anti-Language Model objective with a decay factor designed to address the weaknesses of In-context Machine Translation. We conduct our experiments across 3 model types and sizes, 3 language directions, and for both greedy decoding and beam search ($B=5$). The proposed method outperforms other state-of-art decoding objectives, with up to $20$ BLEU point improvement from the default objective observed in some settings.

Many statistical analyses, in both observational data and randomized control trials, ask: how does the outcome of interest vary with combinations of observable covariates? How do various drug combinations affect health outcomes, or how does technology adoption depend on incentives and demographics? Our goal is to partition this factorial space into ``pools'' of covariate combinations where the outcome differs across the pools (but not within a pool). Existing approaches (i) search for a single ``optimal'' partition under assumptions about the association between covariates or (ii) sample from the entire set of possible partitions. Both these approaches ignore the reality that, especially with correlation structure in covariates, many ways to partition the covariate space may be statistically indistinguishable, despite very different implications for policy or science. We develop an alternative perspective, called Rashomon Partition Sets (RPSs). Each item in the RPS partitions the space of covariates using a tree-like geometry. RPSs incorporate all partitions that have posterior values near the maximum a posteriori partition, even if they offer substantively different explanations, and do so using a prior that makes no assumptions about associations between covariates. This prior is the $\ell_0$ prior, which we show is minimax optimal. Given the RPS we calculate the posterior of any measurable function of the feature effects vector on outcomes, conditional on being in the RPS. We also characterize approximation error relative to the entire posterior and provide bounds on the size of the RPS. Simulations demonstrate this framework allows for robust conclusions relative to conventional regularization techniques. We apply our method to three empirical settings: price effects on charitable giving, chromosomal structure (telomere length), and the introduction of microfinance.