The process of learning new behaviors over time is a problem of great interest in both neuroscience and artificial intelligence. However, most standard analyses of animal training data either treat behavior as fixed or track only coarse performance statistics (e.g., accuracy, bias), providing limited insight into the evolution of the policies governing behavior. To overcome these limitations, we propose a dynamic psychophysical model that efficiently tracks trial-to-trial changes in behavior over the course of training. Our model consists of a dynamic logistic regression model, parametrized by a set of time-varying weights that express dependence on sensory stimuli as well as task-irrelevant covariates, such as stimulus, choice, and answer history.

We present the very first robust Bayesian Online Changepoint Detection algorithm through General Bayesian Inference (GBI) with $\beta$-divergences. The resulting inference procedure is doubly robust for both the predictive and the changepoint (CP) posterior, with linear time and constant space complexity. We provide a construction for exponential models and demonstrate it on the Bayesian Linear Regression model. In so doing, we make two additional contributions: Firstly, we make GBI scalable using Structural Variational approximations that are exact as $\beta \to 0$. Secondly, we give a principled way of choosing the divergence parameter $\beta$ by minimizing expected predictive loss on-line.

A machine learning model may exhibit discrimination when used to make decisions involving people. One potential cause for such outcomes is that the model uses a statistical proxy for a protected demographic attribute. In this paper we formulate a definition of proxy use for the setting of linear regression and present algorithms for detecting proxies. Our definition follows recent work on proxies in classification models, and characterizes a model's constituent behavior that: 1) correlates closely with a protected random variable, and 2) is causally influential in the overall behavior of the model. We show that proxies in linear regression models can be efficiently identified by solving a second-order cone program, and further extend this result to account for situations where the use of a certain input variable is justified as a ``business necessity''. Finally, we present empirical results on two law enforcement datasets that exhibit varying degrees of racial disparity in prediction outcomes, demonstrating that proxies shed useful light on the causes of discriminatory behavior in models.

Subsampling is a common and often effective method to deal with the computational challenges of large datasets. However, for most statistical models, there is no well-motivated approach for drawing a non-uniform subsample. We show that the concept of an asymptotically linear estimator and the associated influence function leads to asymptotically optimal sampling probabilities for a wide class of popular models. This is the only tight optimality result for subsampling we are aware of as other methods only provide probabilistic error bounds or optimal rates. Furthermore, for linear regression models, which have well-studied procedures for non-uniform subsampling, we empirically show our optimal influence function based method outperforms previous approaches even when using approximations to the optimal probabilities.

Bayesian optimization (BO) is a model-based approach for gradient-free black-box function optimization, such as hyperparameter optimization. Typically, BO relies on conventional Gaussian process (GP) regression, whose algorithmic complexity is cubic in the number of evaluations. As a result, GP-based BO cannot leverage large numbers of past function evaluations, for example, to warm-start related BO runs. We propose a multi-task adaptive Bayesian linear regression model for transfer learning in BO, whose complexity is linear in the function evaluations: one Bayesian linear regression model is associated to each black-box function optimization problem (or task), while transfer learning is achieved by coupling the models through a shared deep neural net. Experiments show that the neural net learns a representation suitable for warm-starting the black-box optimization problems and that BO runs can be accelerated when the target black-box function (e.g., validation loss) is learned together with other related signals (e.g., training loss). The proposed method was found to be at least one order of magnitude faster that methods recently published in the literature.

The lasso and elastic net linear regression models impose a double-exponential prior distribution on the model parameters to achieve regression shrinkage and variable selection, allowing the inference of robust models from large data sets. However, there has been limited success in deriving estimates for the full posterior distribution of regression coefficients in these models, due to a need to evaluate analytically intractable partition function integrals. Here, the Fourier transform is used to express these integrals as complex-valued oscillatory integrals over regression frequencies. This results in an analytic expansion and stationary phase approximation for the partition functions of the Bayesian lasso and elastic net, where the non-differentiability of the double-exponential prior has so far eluded such an approach. Use of this approximation leads to highly accurate numerical estimates for the expectation values and marginal posterior distributions of the regression coefficients, and allows for Bayesian inference of much higher dimensional models than previously possible.

Data normalisation, a common and often necessary preprocessing step in engineering and scientific applications, can severely distort the discovery of governing equations by magnitudebased sparse regression methods. This issue is particularly acute for the Sparse Identification of Nonlinear Dynamics (SINDy) framework, where the core assumption of sparsity is undermined by the interaction between data scaling and measurement noise. The resulting discovered models can be dense, uninterpretable, and physically incorrect. To address this critical vulnerability, we introduce the Sequential Thresholding of Coefficient of Variation (STCV), a novel, computationally efficient sparse regression algorithm that is inherently robust to data scaling. STCV replaces conventional magnitude-based thresholding with a dimensionless statistical metric, the Coefficient Presence (CP), which assesses the statistical validity and consistency of candidate terms in the model library. This shift from magnitude to statistical significance makes the discovery process invariant to arbitrary data scaling. Through comprehensive benchmarking on canonical dynamical systems and practical engineering problems, including a physical mass-spring-damper experiment, we demonstrate that STCV consistently and significantly outperforms standard Sequential Thresholding Least Squares (STLSQ) and Ensemble-SINDy (E-SINDy) on normalised, noisy datasets. The results show that STCV-based methods can successfully identify the correct, sparse physical laws even when other methods fail. By mitigating the distorting effects of normalisation, STCV makes sparse system identification a more reliable and automated tool for real-world applications, thereby enhancing model interpretability and trustworthiness.

In healthcare, predictive models increasingly inform patient-level decisions, yet little attention is paid to the variability in individual risk estimates and its impact on treatment decisions. For overparameterized models, now standard in machine learning, a substantial source of variability often goes undetected. Even when the data and model architecture are held fixed, randomness introduced by optimization and initialization can lead to materially different risk estimates for the same patient. This problem is largely obscured by standard evaluation practices, which rely on aggregate performance metrics (e.g., log-loss, accuracy) that are agnostic to individual-level stability. As a result, models with indistinguishable aggregate performance can nonetheless exhibit substantial procedural arbitrariness, which can undermine clinical trust. We propose an evaluation framework that quantifies individual-level prediction instability by using two complementary diagnostics: empirical prediction interval width (ePIW), which captures variability in continuous risk estimates, and empirical decision flip rate (eDFR), which measures instability in threshold-based clinical decisions. We apply these diagnostics to simulated data and GUSTO-I clinical dataset. Across observed settings, we find that for flexible machine-learning models, randomness arising solely from optimization and initialization can induce individual-level variability comparable to that produced by resampling the entire training dataset. Neural networks exhibit substantially greater instability in individual risk predictions compared to logistic regression models. Risk estimate instability near clinically relevant decision thresholds can alter treatment recommendations. These findings that stability diagnostics should be incorporated into routine model validation for assessing clinical reliability.

Data privacy is important in the AI era, and differential privacy (DP) is one of the golden solutions. However, DP is typically applicable only if data have a bounded underlying distribution. We address this limitation by leveraging second-moment information from a small amount of public data. We propose Public-moment-guided Truncation (PMT), which transforms private data using the public second-moment matrix and applies a principled truncation whose radius depends only on non-private quantities: data dimension and sample size. This transformation yields a well-conditioned second-moment matrix, enabling its inversion with a significantly strengthened ability to resist the DP noise. Furthermore, we demonstrate the applicability of PMT by using penalized and generalized linear regressions. Specifically, we design new loss functions and algorithms, ensuring that solutions in the transformed space can be mapped back to the original domain. We have established improvements in the models' DP estimation through theoretical error bounds, robustness guarantees, and convergence results, attributing the gains to the conditioning effect of PMT. Experiments on synthetic and real datasets confirm that PMT substantially improves the accuracy and stability of DP models.

Low-precision training is critical for optimizing the trade-off between model quality and training costs, necessitating the joint allocation of model size, dataset size, and numerical precision. While empirical scaling laws suggest that quantization impacts effective model and data capacities or acts as an additive error, the theoretical mechanisms governing these effects remain largely unexplored. In this work, we initiate a theoretical study of scaling laws for low-precision training within a high-dimensional sketched linear regression framework. By analyzing multiplicative (signal-dependent) and additive (signal-independent) quantization, we identify a critical dichotomy in their scaling behaviors. Our analysis reveals that while both schemes introduce an additive error and degrade the effective data size, they exhibit distinct effects on effective model size: multiplicative quantization maintains the full-precision model size, whereas additive quantization reduces the effective model size. Numerical experiments validate our theoretical findings. By rigorously characterizing the complex interplay among model scale, dataset size, and quantization error, our work provides a principled theoretical basis for optimizing training protocols under practical hardware constraints.