We derive novel concentration inequalities that bound the statistical error for a large class of stochastic optimization problems, focusing on the case of unbounded objective functions. Our derivations utilize the following tools: 1) A new form of McDiarmid's inequality that is based on sample dependent one component difference bounds and which leads to a novel uniform law of large numbers result for unbounded functions. 2) A Rademacher complexity bound for families of functions that satisfy an appropriate local Lipschitz property. As an application of these results, we derive statistical error bounds for denoising score matching (DSM), an application that inherently requires one to consider unbounded objective functions, even when the data distribution has bounded support. In addition, our results establish the benefit of sample reuse in algorithms that employ easily sampled auxiliary random variables in addition to the training data, e.g., as in DSM, which uses auxiliary Gaussian random variables.

Scientists often need to analyze the samples in a study that responded to treatment in order to refine their hypotheses and find potential causal drivers of response. Natural variation in outcomes makes teasing apart responders from non-responders a statistical inference problem. To handle latent responses, we introduce the causal two-groups (C2G) model, a causal extension of the classical two-groups model. The C2G model posits that treated samples may or may not experience an effect, according to some prior probability. We propose two empirical Bayes procedures for the causal two-groups model, one under semi-parametric conditions and another under fully nonparametric conditions. The semi-parametric model assumes additive treatment effects and is identifiable from observed data. The nonparametric model is unidentifiable, but we show it can still be used to test for response in each treated sample. We show empirically and theoretically that both methods for selecting responders control the false discovery rate at the target level with near-optimal power. We also propose two novel estimands of interest and provide a strategy for deriving estimand intervals in the unidentifiable nonparametric model. On a cancer immunotherapy dataset, the nonparametric C2G model recovers clinically-validated predictive biomarkers of both positive and negative outcomes. Code is available at https://github.com/tansey-lab/causal2groups.

Bayesian nonparametric methods are naturally suited to the problem of out-of-distribution (OOD) detection. However, these techniques have largely been eschewed in favor of simpler methods based on distances between pre-trained or learned embeddings of data points. Here we show a formal relationship between Bayesian nonparametric models and the relative Mahalanobis distance score (RMDS), a commonly used method for OOD detection. Building on this connection, we propose Bayesian nonparametric mixture models with hierarchical priors that generalize the RMDS. We evaluate these models on the OpenOOD detection benchmark and show that Bayesian nonparametric methods can improve upon existing OOD methods, especially in regimes where training classes differ in their covariance structure and where there are relatively few data points per class.

User-level differentially private stochastic convex optimization (DP-SCO) has garnered significant attention due to the paramount importance of safeguarding user privacy in modern large-scale machine learning applications. Current methods, such as those based on differentially private stochastic gradient descent (DP-SGD), often struggle with high noise accumulation and suboptimal utility due to the need to privatize every intermediate iterate. In this work, we introduce a novel linear-time algorithm that leverages robust statistics, specifically the median and trimmed mean, to overcome these challenges. Our approach uniquely bounds the sensitivity of all intermediate iterates of SGD with gradient estimation based on robust statistics, thereby significantly reducing the gradient estimation noise for privacy purposes and enhancing the privacy-utility trade-off. By sidestepping the repeated privatization required by previous methods, our algorithm not only achieves an improved theoretical privacy-utility trade-off but also maintains computational efficiency. We complement our algorithm with an information-theoretic lower bound, showing that our upper bound is optimal up to logarithmic factors and the dependence on $\epsilon$. This work sets the stage for more robust and efficient privacy-preserving techniques in machine learning, with implications for future research and application in the field.

Sparse variational Gaussian processes (GPs) construct tractable posterior approximations to GP models. At the core of these methods is the assumption that the true posterior distribution over training function values ${\bf f}$ and inducing variables ${\bf u}$ is approximated by a variational distribution that incorporates the conditional GP prior $p({\bf f} | {\bf u})$ in its factorization. While this assumption is considered as fundamental, we show that for model training we can relax it through the use of a more general variational distribution $q({\bf f} | {\bf u})$ that depends on $N$ extra parameters, where $N$ is the number of training examples. In GP regression, we can analytically optimize the evidence lower bound over the extra parameters and express a tractable collapsed bound that is tighter than the previous bound. The new bound is also amenable to stochastic optimization and its implementation requires minor modifications to existing sparse GP code. Further, we also describe extensions to non-Gaussian likelihoods. On several datasets we demonstrate that our method can reduce bias when learning the hyperpaparameters and can lead to better predictive performance.

The Weak-form Estimation of Non-linear Dynamics (WENDy) algorithm is extended to accommodate systems of ordinary differential equations that are nonlinear-in-parameters (NiP). The extension rests on derived analytic expressions for a likelihood function, its gradient and its Hessian matrix. WENDy makes use of these to approximate a maximum likelihood estimator based on optimization routines suited for non-convex optimization problems. The resulting parameter estimation algorithm has better accuracy, a substantially larger domain of convergence, and is often orders of magnitude faster than the conventional output error least squares method (based on forward solvers). The WENDy.jl algorithm is efficiently implemented in Julia. We demonstrate the algorithm's ability to accommodate the weak form optimization for both additive normal and multiplicative log-normal noise, and present results on a suite of benchmark systems of ordinary differential equations. In order to demonstrate the practical benefits of our approach, we present extensive comparisons between our method and output error methods in terms of accuracy, precision, bias, and coverage.

Transfer learning is a popular strategy to leverage external knowledge and improve statistical efficiency, particularly with a limited target sample. We propose a novel knowledge-guided Wasserstein Distributionally Robust Optimization (KG-WDRO) framework that adaptively incorporates multiple sources of external knowledge to overcome the conservativeness of vanilla WDRO, which often results in overly pessimistic shrinkage toward zero. Our method constructs smaller Wasserstein ambiguity sets by controlling the transportation along directions informed by the source knowledge. This strategy can alleviate perturbations on the predictive projection of the covariates and protect against information loss. Theoretically, we establish the equivalence between our WDRO formulation and the knowledge-guided shrinkage estimation based on collinear similarity, ensuring tractability and geometrizing the feasible set. This also reveals a novel and general interpretation for recent shrinkage-based transfer learning approaches from the perspective of distributional robustness. In addition, our framework can adjust for scaling differences in the regression models between the source and target and accommodates general types of regularization such as lasso and ridge. Extensive simulations demonstrate the superior performance and adaptivity of KG-WDRO in enhancing small-sample transfer learning.

Two important and closely related problems in statistical learning are the problems of estimating a partial correlation network and the inverse covariance matrix, also known as the precision matrix, from data. Partial correlation networks, which generalize the Gaussian graphical model, are used to model the relationships between variables while conditioning on all other variables, and are useful for inferring causal relationships between variables. Partial correlation networks are used in a plethora of applications, such as in the analysis of gene expression data, where the goal is to infer the regulatory relationships between genes (de la Fuente et al., 2004), and psychological data, where networks are used to model the relationships between psychological variables such as mood and attitude (Epskamp and Fried, 2018). The precision matrix, from which we can obtain the partial correlation network, is also of interest in its own right, as it also appears in linear discriminant analysis (Hastie et al., 2009) and in Markowitz portfolio selection (Markowitz, 1952). However, due to the high-dimensionality of the problem, estimating a precision or partial correlation matrix is often challenging as the number of parameters are on the order of the squared number of features. For this reason, classical methods such as using the inverse of the sample covariance matrix, are known to perform poorly whenever the number of observation is not extremely large. Additionally they produce estimates which are almost surely dense. This makes regularization crucial, since in many applications we typically only have a moderate number of observations, and in particular we are most often seeking a sparse estimate which gives rise to a more parsimonious and interpretable network model.

We propose a new method to extract discriminant and explainable features from a particular machine learning model, i.e., a combination of the scattering transform and the multiclass logistic regression. Although this model is well-known for its ability to learn various signal classes with high classification rate, it remains elusive to understand why it can generate such successful classification, mainly due to the nonlinearity of the scattering transform. In order to uncover the meaning of the scattering transform coefficients selected by the multiclass logistic regression (with the Lasso penalty), we adopt zeroth-order optimization algorithms to search an input pattern that maximizes the class probability of a class of interest given the learned model. In order to do so, it turns out that imposing sparsity and smoothness of input patterns is important. We demonstrate the effectiveness of our proposed method using a couple of synthetic time-series classification problems.

Variational data assimilation estimates the dynamical system states by minimizing a cost function that fits the numerical models with observational data. The widely used method, four-dimensional variational assimilation (4D-Var), has two primary challenges: (1) computationally demanding for complex nonlinear systems and (2) relying on state-observation mappings, which are often not perfectly known. Deep learning (DL) has been used as a more expressive class of efficient model approximators to address these challenges. However, integrating such models into 4D-Var remains challenging due to their inherent nonlinearities and the lack of theoretical guarantees for consistency in assimilation results. In this paper, we propose Tensor-Var to address these challenges using kernel Conditional Mean Embedding (CME). Tensor-Var improves optimization efficiency by characterizing system dynamics and state-observation mappings as linear operators, leading to a convex cost function in the feature space. Furthermore, our method provides a new perspective to incorporate CME into 4D-Var, offering theoretical guarantees of consistent assimilation results between the original and feature spaces. To improve scalability, we propose a method to learn deep features (DFs) using neural networks within the Tensor-Var framework. Experiments on chaotic systems and global weather prediction with real-time observations show that Tensor-Var outperforms conventional and DL hybrid 4D-Var baselines in accuracy while achieving efficiency comparable to the static 3D-Var method.