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We characterize a notion of confidence that arises in learning or updating beliefs: the amount of trust one has in incoming information and its impact on the belief state. This *learner's confidence* can be used alongside (and is easily mistaken for) probability or likelihood, but it is fundamentally a different concept---one that captures many familiar concepts in the literature, including learning rates and number of training epochs, Shafer's weight of evidence, and Kalman gain. We formally axiomatize what it means to learn with confidence, give two canonical ways of measuring confidence on a continuum, and prove that confidence can always be represented in this way. Under additional assumptions, we derive more compact representations of confidence-based learning in terms of vector fields and loss functions. These representations induce an extended language of compound "parallel" observations. We characterize *Bayesian* learning as the special case of an optimizing learner whose loss representation is a linear expectation.
Many researchers construct directed acyclic graph (DAG) models manually based on domain knowledge. Although numerous causal discovery algorithms were developed to automatically learn DAGs and other causal models from data, these remain challenging to use due to their tendency to produce results that contradict domain knowledge, among other issues. Here we propose a hybrid, iterative structure learning approach that combines domain knowledge with data-driven insights to assist researchers in constructing DAGs. Our method leverages conditional independence testing to iteratively identify variable pairs where an edge is either missing or superfluous. Based on this information, we can choose to add missing edges with appropriate orientation based on domain knowledge or remove unnecessary ones. We also give a method to rank these missing edges based on their impact on the overall model fit. In a simulation study, we find that this iterative approach to leverage domain knowledge already starts outperforming purely data-driven structure learning if the orientation of new edge is correctly determined in at least two out of three cases. We present a proof-of-concept implementation using a large language model as a domain expert and a graphical user interface designed to assist human experts with DAG construction.
We introduce a class of algorithms, termed proximal interacting particle Langevin algorithms (PIPLA), for inference and learning in latent variable models whose joint probability density is non-differentiable. Leveraging proximal Markov chain Monte Carlo techniques and interacting particle Langevin algorithms, we propose three algorithms tailored to the problem of estimating parameters in a non-differentiable statistical model. We prove non-asymptotic bounds for the parameter estimates produced by the different algorithms in the strongly log-concave setting and provide comprehensive numerical experiments on various models to demonstrate the effectiveness of the proposed methods. In particular, we demonstrate the utility of our family of algorithms for sparse Bayesian logistic regression, training of sparse Bayesian neural networks or neural networks with non-differentiable activation functions, image deblurring, and sparse matrix completion. Our theory and experiments together show that PIPLA family can be the de facto choice for parameter estimation problems in non-differentiable latent variable models.
Efficient global optimization and sampling are fundamental challenges, particularly in fields such as robotics and reinforcement learning, where gradients may be unavailable or unreliable. In this context, jointly optimizing multiple solutions is a promising approach to avoid local optima. While Stein Variational Gradient Descent (SVGD) provides a powerful framework for sampling diverse solutions, its reliance on first-order information limits its applicability to differentiable objectives. Existing gradient-free SVGD variants often suffer from slow convergence, and poor scalability. To improve gradient-free sampling and optimization, we propose Stein Variational CMA-ES, a novel gradient-free SVGD-like method that combines the efficiency of evolution strategies with SVGD-based repulsion forces. We perform an extensive empirical evaluation across several domains, which shows that the integration of the ES update in SVGD significantly improves the performance on multiple challenging benchmark problems. Our findings establish SV-CMA-ES as a scalable method for zero-order sampling and blackbox optimization, bridging the gap between SVGD and evolution strategies.
Multiple Instance Regression (MIR) and Learning from Label Proportions (LLP) are useful learning frameworks, where the training data is partitioned into disjoint sets or bags, and only an aggregate label, i.e., bag-label for each bag is available to the learner. In the case of MIR, the bag-label is the label of an undisclosed instance from the bag, while in LLP, the bag-label is the mean of the bag's labels. In this paper, we study for various loss functions in MIR and LLP, what is the optimal way to partition the dataset into bags such that the utility for downstream tasks like linear regression is maximized. We theoretically provide utility guarantees, and show that in each case, the optimal bagging strategy (approximately) reduces to finding an optimal clustering of the feature vectors and/or the labels with respect to natural objectives such as k-means. We also show that our bagging mechanisms can be made label-differentially private, incurring an additional utility error. We then generalize our results to the setting of Generalized Linear Models (GLMs). Finally, we experimentally validate our theoretical results.
Training machine learning models inherently involves a resource-intensive and noisy iterative learning procedure that allows epoch-wise monitoring of the model performance. However, the insights gained from the iterative learning procedure typically remain underutilized in multi-objective hyperparameter optimization scenarios. Despite the limited research in this area, existing methods commonly identify the trade-offs only at the end of model training, overlooking the fact that trade-offs can emerge at earlier epochs in cases such as overfitting. To bridge this gap, we propose an enhanced multi-objective hyperparameter optimization problem that treats the number of training epochs as a decision variable, rather than merely an auxiliary parameter, to account for trade-offs at an earlier training stage. To solve this problem and accommodate its iterative learning, we then present a trajectory-based multi-objective Bayesian optimization algorithm characterized by two features: 1) a novel acquisition function that captures the improvement along the predictive trajectory of model performances over epochs for any hyperparameter setting and 2) a multi-objective early stopping mechanism that determines when to terminate the training to maximize epoch efficiency. Experiments on synthetic simulations and hyperparameter tuning benchmarks demonstrate that our algorithm can effectively identify the desirable trade-offs while improving tuning efficiency.
There is an increasing interest in analyzing the behavior of machine learning systems against adversarial attacks. However, most of the research in adversarial machine learning has focused on studying weaknesses against evasion or poisoning attacks to predictive models in classical setups, with the susceptibility of Bayesian predictive models to attacks remaining underexplored. This paper introduces a general methodology for designing optimal evasion attacks against such models. We investigate two adversarial objectives: perturbing specific point predictions and altering the entire posterior predictive distribution. For both scenarios, we propose novel gradient-based attacks and study their implementation and properties in various computational setups.
Machine learning (ML) models are increasingly used as decision-support tools in high-risk domains. Evaluating the causal impact of deploying such models can be done with a randomized controlled trial (RCT) that randomizes users to ML vs. control groups and assesses the effect on relevant outcomes. However, ML models are inevitably updated over time, and we often lack evidence for the causal impact of these updates. While the causal effect could be repeatedly validated with ongoing RCTs, such experiments are expensive and time-consuming to run. In this work, we present an alternative solution: using only data from a prior RCT, we give conditions under which the causal impact of a new ML model can be precisely bounded or estimated, even if it was not included in the RCT. Our assumptions incorporate two realistic constraints: ML predictions are often deterministic, and their impacts depend on user trust in the model. Based on our analysis, we give recommendations for trial designs that maximize our ability to assess future versions of an ML model. Our hope is that our trial design recommendations will save practitioners time and resources while allowing for quicker deployments of updates to ML models.
Adversarially robust optimization (ARO) has emerged as the *de facto* standard for training models that hedge against adversarial attacks in the test stage. While these models are robust against adversarial attacks, they tend to suffer severely from overfitting. To address this issue, some successful methods replace the empirical distribution in the training stage with alternatives including *(i)* a worst-case distribution residing in an ambiguity set, resulting in a distributionally robust (DR) counterpart of ARO; *(ii)* a mixture of the empirical distribution with a distribution induced by an auxiliary (*e.g.*, synthetic, external, out-of-domain) dataset. Inspired by the former, we study the Wasserstein DR counterpart of ARO for logistic regression and show it admits a tractable convex optimization reformulation. Adopting the latter setting, we revise the DR approach by intersecting its ambiguity set with another ambiguity set built using the auxiliary dataset, which offers a significant improvement whenever the Wasserstein distance between the data generating and auxiliary distributions can be estimated. We study the underlying optimization problem, develop efficient solution algorithms, and demonstrate that the proposed method outperforms benchmark approaches on standard datasets.
In safety-critical applications, guaranteeing the satisfaction of constraints over continuous environments is crucial, e.g., an autonomous agent should never crash over obstacles or go off-road. Neural models struggle in the presence of these constraints, especially when they involve intricate algebraic relationships. To address this, we introduce a differentiable probabilistic layer that guarantees the satisfaction of non-convex algebraic constraints over continuous variables. This probabilistic algebraic layer (PAL) can be seamlessly plugged into any neural architecture and trained via maximum likelihood without requiring approximations. PAL defines a distribution over conjunctions and disjunctions of linear inequalities, parametrized by polynomials. This formulation enables efficient and exact renormalization via symbolic integration, which can be amortized across different data points and easily parallelized on a GPU. We showcase PAL and our integration scheme on a number of benchmarks for algebraic constraint integration and on real-world trajectory data.
Conformal prediction methods create prediction bands with distribution-free guarantees but do not explicitly capture epistemic uncertainty, which can lead to overconfident predictions in data-sparse regions. Although recent conformal scores have been developed to address this limitation, they are typically designed for specific tasks, such as regression or quantile regression. Moreover, they rely on particular modeling choices for epistemic uncertainty, restricting their applicability. We introduce EPICSCORE, a model-agnostic approach that enhances any conformal score by explicitly integrating epistemic uncertainty. Leveraging Bayesian techniques such as Gaussian Processes, Monte Carlo Dropout, or Bayesian Additive Regression Trees, EPICSCORE adaptively expands predictive intervals in regions with limited data while maintaining compact intervals where data is abundant. As with any conformal method, it preserves finite-sample marginal coverage. Additionally, it also achieves asymptotic conditional coverage. Experiments demonstrate its good performance compared to existing methods. Designed for compatibility with any Bayesian model, but equipped with distribution-free guarantees, EPICSCORE provides a general-purpose framework for uncertainty quantification in prediction problems.
Density regression models allow a comprehensive understanding of data by modeling the complete conditional probability distribution. While flexible estimation approaches such as normalizing flows (NFs) work particularly well in multiple dimensions, interpreting the input-output relationship of such models is often difficult, due to the black-box character of deep learning models. In contrast, existing statistical methods for multivariate outcomes such as multivariate conditional transformation models (MCTMs) are restricted in flexibility and are often not expressive enough to represent complex multivariate probability distributions. In this paper, we combine MCTMs with state-of-the-art and autoregressive NFs to leverage the transparency of MCTMs for modeling interpretable feature effects on the marginal distributions in the first step and the flexibility of neural-network-based NFs techniques to account for complex and non-linear relationships in the joint data distribution. We demonstrate our method's versatility in various numerical experiments and compare it with MCTMs and other NF models on both simulated and real-world data.
Maximal Ancestral Graphs (MAGs) provide an abstract representation of Directed Acyclic Graphs (DAGs) with latent (selection) variables. These graphical objects encode information about ancestral relations and d-separations of the DAGs they represent. This abstract representation has been used amongst others to prove the soundness and completeness of the FCI algorithm for causal discovery, and to derive a do-calculus for its output. One significant inherent limitation of MAGs is that they rule out the possibility of cyclic causal relationships. In this work, we address that limitation. We introduce and study a class of graphical objects that we coin "σ-Maximal Ancestral Graphs" ("σ-MAGs"). We show how these graphs provide an abstract representation of (possibly cyclic) Directed Graphs (DGs) with latent (selection) variables, analogously to how MAGs represent DAGs. We study the properties of these objects and provide a characterization of their Markov equivalence classes.
Formal explainability is an emerging field that aims to provide mathematically guaranteed explanations for the predictions made by machine learning models. Recent work in this area focuses on computing “probabilistic explanations” for the predictions made by classifiers based on specific data instances. The goal of this paper is to extend the concept of probabilistic explanations to the regression setting, treating the target regressor as a black box function. The class of probabilistic explanations consists of linear functions that meet a sparsity constraint, alongside a hyperplane constraint defined for the data instance being explained. While minimizing the precision error of such explanations is generally NPPP-hard, we demonstrate that it can be approximated by substituting the precision measure with a fidelity measure. Optimal explanations based on this fidelity objective can be effectively approached using Mixed Integer Programming (MIP). Moreover, we show that for certain distributions used to define the precision measure, explanations with approximation guarantees can be computed in polynomial time using a variant of Iterative Hard Thresholding (IHT). Experiments conducted on various datasets indicate that both the MIP and IHT approaches outperform the state-of-the-art LIME and MAPLE explainers.
Variable importance is one of the most widely used measures for interpreting machine learning with significant interest from both statistics and machine learning communities. Recently, increasing attention has been directed toward uncertainty quantification in these metrics. Current approaches largely rely on one-step procedures, which, while asymptotically efficient, can present higher sensitivity and instability in finite sample settings. To address these limitations, we propose a novel method by employing the targeted learning (TL) framework, designed to enhance robustness in inference for variable importance metrics. Our approach is particularly suited for conditional permutation variable importance. We show that it (i) retains the asymptotic efficiency of traditional methods, (ii) maintains comparable computational complexity, and (iii) delivers improved accuracy, especially in finite sample contexts. We further support these findings with numerical experiments that illustrate the practical advantages of our method and validate the theoretical results.
New proposals for causal discovery algorithms are typically evaluated using simulations and a few selected real data examples with known data generating mechanisms. However, there does not exist a general guideline for how such evaluation studies should be designed, and therefore, comparing results across different studies can be difficult. In this article, we propose to use negative controls as a common evaluation baseline by posing the question: Are we doing better than random guessing? For the task of graph skeleton estimation, we derive exact distributional results under random guessing for the expected behavior of a range of typical causal discovery evaluation metrics, including precision and recall. We show that these metrics can achieve very favorable values under random guessing in certain scenarios, and hence warn against using them without also reporting negative control results, i.e., performance under random guessing. We also propose an exact test of overall skeleton fit, and showcase its use on a real data application. Finally, we propose a general pipeline for using negative controls beyond the skeleton estimation task, and apply it both in a simulated example and a real data application.
Estimating causal effects from observational data is a key problem in causal inference, often addressed through covariate adjustment sets that enable unbiased estimation of interventional means. This paper tackles the challenge of finding optimal covariate adjustment sets under budget constraints, a practical concern in many applications. We present algorithms for enumerating valid and minimal adjustment sets up to a specified cost, ordered by their proximity to outcome variables, which coincides with estimator variance. Our approach builds on existing graphical criteria and extends them to accommodate budgetary considerations, providing a useful tool for addressing resource limitations.
The quality of probabilistic forecasts is crucial for decision-making under uncertainty. While proper scoring rules incentivize truthful reporting of precise forecasts, they fall short when forecasters face epistemic uncertainty about their beliefs, limiting their use in safety-critical domains where decision-makers~(DMs) prioritize proper uncertainty management. To address this, we propose a framework for scoring \emph{imprecise forecasts}---forecasts given as a set of beliefs. Despite existing impossibility results for deterministic scoring rules, we enable truthful elicitation by drawing connection to social choice theory and introducing a two-way communication framework where DMs first share their aggregation rules (e.g., averaging or min-max) used in downstream decisions for resolving forecast ambiguity. This, in turn, helps forecasters resolve indecision during elicitation. We further show that truthful elicitation of imprecise forecasts is achievable using proper scoring rules randomized over the aggregation procedure. Our approach allows DM to elicit and integrate the forecaster's epistemic uncertainty into their decision-making process, improving the credibility of downstream decisions.
Quantization [Alistarh et al., 2017] is an important (stochastic) compression technique that reduces the volume of transmitted bits during each communication round in distributed model training. Suresh et al. [2022] introduce correlated quantizers and show their advantages over independent counterparts by analyzing distributed SGD communication complexity. We analyze the fore- front distributed non-convex optimization algorithm MARINA [Gorbunov et al., 2022] utilizing the proposed correlated quantizers and show that it outperforms the original MARINA and distributed SGD of Suresh et al. [2022] with regard to the communication complexity. We significantly re- fine the original analysis of MARINA without any additional assumptions using the weighted Hessian variance [Tyurin et al., 2022], and then we expand the theoretical framework of MARINA to accommodate a substantially broader range of potentially correlated and biased compressors, thus dilating the applicability of the method beyond the conventional independent unbiased compressor setup. Extensive experimental results corroborate our theoretical findings.
We address the optimization problem of simultaneously minimizing multiple objective functionals over a family of probability distributions. This type of Multi-Objective Distributional Optimization commonly arises in machine learning and statistics, with applications in areas such as multiple target sampling, multi-task learning, and multi-objective generative modeling. To solve this problem, we propose an iterative particle-based algorithm, which we call Muliple Wasserstein Gradient Descent (MWGraD), which constructs a flow of intermediate empirical distributions, each being represented by a set of particles, which gradually minimize the multiple objective functionals simultaneously. Specifically, MWGraD consists of two key steps at each iteration. First, it estimates the Wasserstein gradient for each objective functional based on the current particles. Then, it aggregates these gradients into a single Wasserstein gradient using dynamically adjusted weights and updates the particles accordingly. In addition, we provide theoretical analysis and present experimental results on both synthetic and real-world datasets, demonstrating the effectiveness of MWGraD.