Classical asymptotic theory for statistical inference usually involves calibrating a statistic by fixing the dimension $d$ while letting the sample size $n$ increase to infinity. Recently, much effort has been dedicated towards understanding how these methods behave in high-dimensional settings, where $d$ and $n$ both increase to infinity together. This often leads to different inference procedures, depending on the assumptions about the dimensionality, leaving the practitioner in a bind: given a dataset with 100 samples in 20 dimensions, should they calibrate by assuming $n \gg d$, or $d/n \approx 0.2$? This paper considers the goal of dimension-agnostic inference; developing methods whose validity does not depend on any assumption on $d$ versus $n$. We introduce an approach that uses variational representations of existing test statistics along with sample splitting and self-normalization to produce a new test statistic with a Gaussian limiting distribution, regardless of how $d$ scales with $n$. The resulting statistic can be viewed as a careful modification of degenerate U-statistics, dropping diagonal blocks and retaining off-diagonal blocks. We exemplify our technique for some classical problems including one-sample mean and covariance testing, and show that our tests have minimax rate-optimal power against appropriate local alternatives. In most settings, our cross U-statistic matches the high-dimensional power of the corresponding (degenerate) U-statistic up to a $\sqrt{2}$ factor.
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Mathematical models are essential for understanding and making predictions about systems arising in nature and engineering. Yet, mathematical models are a simplification of true phenomena, thus making predictions subject to uncertainty. Hence, the ability to quantify uncertainties is essential to any modelling framework, enabling the user to assess the importance of certain parameters on quantities of interest and have control over the quality of the model output by providing a rigorous understanding of uncertainty. Peridynamic models are a particular class of mathematical models that have proven to be remarkably accurate and robust for a large class of material failure problems. However, the high computational expense of peridynamic models remains a major limitation, hindering outer-loop applications that require a large number of simulations, for example, uncertainty quantification. This contribution provides a framework to make such computations feasible. By employing a Multilevel Monte Carlo (MLMC) framework, where the majority of simulations are performed using a coarse mesh, and performing relatively few simulations using a fine mesh, a significant reduction in computational cost can be realised, and statistics of structural failure can be estimated. The results show a speed-up factor of 16x over a standard Monte Carlo estimator, enabling the forward propagation of uncertain parameters in a computationally expensive peridynamic model. Furthermore, the multilevel method provides an estimate of both the discretisation error and sampling error, thus improving the confidence in numerical predictions. The performance of the approach is demonstrated through an examination of the statistical size effect in quasi-brittle materials.
Statistical models are central to machine learning with broad applicability across a range of downstream tasks. The models are controlled by free parameters that are typically estimated from data by maximum-likelihood estimation or approximations thereof. However, when faced with real-world datasets many of the models run into a critical issue: they are formulated in terms of fully-observed data, whereas in practice the datasets are plagued with missing data. The theory of statistical model estimation from incomplete data is conceptually similar to the estimation of latent-variable models, where powerful tools such as variational inference (VI) exist. However, in contrast to standard latent-variable models, parameter estimation with incomplete data often requires estimating exponentially-many conditional distributions of the missing variables, hence making standard VI methods intractable. We address this gap by introducing variational Gibbs inference (VGI), a new general-purpose method to estimate the parameters of statistical models from incomplete data. We validate VGI on a set of synthetic and real-world estimation tasks, estimating important machine learning models such as VAEs and normalising flows from incomplete data. The proposed method, whilst general-purpose, achieves competitive or better performance than existing model-specific estimation methods.
Deep Neural Networks (DNN) are nowadays largely adopted in many application domains thanks to their human-like, or even superhuman, performance in specific tasks. However, due to unpredictable/unconsidered operating conditions, unexpected failures show up on field, making the performance of a DNN in operation very different from the one estimated prior to release. In the life cycle of DNN systems, the assessment of accuracy is typically addressed in two ways: offline, via sampling of operational inputs, or online, via pseudo-oracles. The former is considered more expensive due to the need for manual labeling of the sampled inputs. The latter is automatic but less accurate. We believe that emerging iterative industrial-strength life cycle models for Machine Learning systems, like MLOps, offer the possibility to leverage inputs observed in operation not only to provide faithful estimates of a DNN accuracy, but also to improve it through remodeling/retraining actions. We propose DAIC (DNN Assessment and Improvement Cycle), an approach which combines ''low-cost'' online pseudo-oracles and ''high-cost'' offline sampling techniques to estimate and improve the operational accuracy of a DNN in the iterations of its life cycle. Preliminary results show the benefits of combining the two approaches and integrating them in the DNN life cycle.
Clinical trials are an integral component of medical research. Trials require careful design to, for example, maintain the safety of participants, use resources efficiently and allow clinically meaningful conclusions to be drawn. Adaptive clinical trials (i.e. trials that can be altered based on evidence that has accrued) are often more efficient, informative and ethical than standard or non-adaptive trials because they require fewer participants, target more promising treatments, and can stop early with sufficient evidence of effectiveness or harm. The design of adaptive trials requires the pre-specification of adaptions that are permissible throughout the conduct of the trial. Proposed adaptive designs are then usually evaluated through simulation which provides indicative metrics of performance (e.g. statistical power and type-1 error) under different scenarios. Trial simulation requires assumptions about the data generating process to be specified but correctly specifying these in practice can be difficult, particularly for new and emerging diseases. To address this, we propose an approach to design adaptive clinical trials without needing to specify the complete data generating process. To facilitate this, we consider a general Bayesian framework where inference about the treatment effect on a time-to-event outcome can be performed via the partial likelihood. As a consequence, the proposed approach to evaluate trial designs is robust to the specific form of the baseline hazard function. The benefits of this approach are demonstrated through the redesign of a recent clinical trial to evaluate whether a third dose of a vaccine provides improved protection against gastroenteritis in Australian Indigenous infants.
We develop a toolbox for the error analysis of linear recurrences with constant or polynomial coefficients, based on generating series, Cauchy's method of majorants, and simple results from analytic combinatorics. We illustrate the power of the approach by several nontrivial application examples. Among these examples are a new worst-case analysis of an algorithm for computing Bernoulli numbers, and a new algorithm for evaluating differentially finite functions in interval arithmetic while avoiding interval blow-up.
We establish tightness of graph-based stochastic processes in the space $D[0+\epsilon,1-\epsilon]$ with $\epsilon >0$ that allows for discontinuities of the first kind. The graph-based stochastic processes are based on statistics constructed from similarity graphs. In this setting, the classic characterization of tightness is intractable, making it difficult to obtain convergence of the limiting distributions for graph-based stochastic processes. We take an alternative approach and study the behavior of the higher moments of the graph-based test statistics. We show that, under mild conditions of the graph, tightness of the stochastic process can be established by obtaining upper bounds on the graph-based statistics' higher moments. Explicit analytical expressions for these moments are provided. The results are applicable to generic graphs, including dense graphs where the number of edges can be of higher order than the number of observations.
We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. The core mathematical idea involves an efficient conversion of the strong form representation of a model to its weak form, and then solving a regression problem to perform parameter inference. The core statistical idea rests on the Errors-In-Variables framework, which necessitates the use of the iteratively reweighted least squares algorithm. Further improvements are obtained by using orthonormal test functions, created from a set of $C^{\infty}$ bump functions of varying support sizes. We demonstrate that WENDy is a highly robust and efficient method for parameter inference in differential equations. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy. For both higher dimensional systems and stiff systems, WENDy is typically both faster (often by orders of magnitude) and more accurate than forward solver-based approaches. We illustrate the method and its performance in some common population and neuroscience models, including logistic growth, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark model. Software and code for reproducing the examples is available at (//github.com/MathBioCU/WENDy).
Many problems arising in control require the determination of a mathematical model of the application. This has often to be performed starting from input-output data, leading to a task known as system identification in the engineering literature. One emerging topic in this field is estimation of networks consisting of several interconnected dynamic systems. We consider the linear setting assuming that system outputs are the result of many correlated inputs, hence making system identification severely ill-conditioned. This is a scenario often encountered when modeling complex cybernetics systems composed by many sub-units with feedback and algebraic loops. We develop a strategy cast in a Bayesian regularization framework where any impulse response is seen as realization of a zero-mean Gaussian process. Any covariance is defined by the so called stable spline kernel which includes information on smooth exponential decay. We design a novel Markov chain Monte Carlo scheme able to reconstruct the impulse responses posterior by efficiently dealing with collinearity. Our scheme relies on a variation of the Gibbs sampling technique: beyond considering blocks forming a partition of the parameter space, some other (overlapping) blocks are also updated on the basis of the level of collinearity of the system inputs. Theoretical properties of the algorithm are studied obtaining its convergence rate. Numerical experiments are included using systems containing hundreds of impulse responses and highly correlated inputs.
Multi-arm bandits are gaining popularity as they enable real-world sequential decision-making across application areas, including clinical trials, recommender systems, and online decision-making. Consequently, there is an increased desire to use the available adaptively collected datasets to distinguish whether one arm was more effective than the other, e.g., which product or treatment was more effective. Unfortunately, existing tools fail to provide valid inference when data is collected adaptively or require many untestable and technical assumptions, e.g., stationarity, iid rewards, bounded random variables, etc. Our paper introduces the design-based approach to inference for multi-arm bandits, where we condition the full set of potential outcomes and perform inference on the obtained sample. Our paper constructs valid confidence intervals for both the reward mean of any arm and the mean reward difference between any arms in an assumption-light manner, allowing the rewards to be arbitrarily distributed, non-iid, and from non-stationary distributions. In addition to confidence intervals, we also provide valid design-based confidence sequences, sequences of confidence intervals that have uniform type-1 error guarantees over time. Confidence sequences allow the agent to perform a hypothesis test as the data arrives sequentially and stop the experiment as soon as the agent is satisfied with the inference, e.g., the mean reward of an arm is statistically significantly higher than a desired threshold.
Since hardware resources are limited, the objective of training deep learning models is typically to maximize accuracy subject to the time and memory constraints of training and inference. We study the impact of model size in this setting, focusing on Transformer models for NLP tasks that are limited by compute: self-supervised pretraining and high-resource machine translation. We first show that even though smaller Transformer models execute faster per iteration, wider and deeper models converge in significantly fewer steps. Moreover, this acceleration in convergence typically outpaces the additional computational overhead of using larger models. Therefore, the most compute-efficient training strategy is to counterintuitively train extremely large models but stop after a small number of iterations. This leads to an apparent trade-off between the training efficiency of large Transformer models and the inference efficiency of small Transformer models. However, we show that large models are more robust to compression techniques such as quantization and pruning than small models. Consequently, one can get the best of both worlds: heavily compressed, large models achieve higher accuracy than lightly compressed, small models.
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