We present a novel and easy-to-use method for calibrating error-rate based confidence intervals to evidence-based support intervals. Support intervals are obtained from inverting Bayes factors based on a parameter estimate and its standard error. A $k$ support interval can be interpreted as "the observed data are at least $k$ times more likely under the included parameter values than under a specified alternative". Support intervals depend on the specification of prior distributions for the parameter under the alternative, and we present several types that allow different forms of external knowledge to be encoded. We also show how prior specification can to some extent be avoided by considering a class of prior distributions and then computing so-called minimum support intervals which, for a given class of priors, have a one-to-one mapping with confidence intervals. We also illustrate how the sample size of a future study can be determined based on the concept of support. Finally, we show how the bound for the type-I error rate of Bayes factors leads to a bound for the coverage of support intervals. An application to data from a clinical trial illustrates how support intervals can lead to inferences that are both intuitive and informative.
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Numerical predictions of quantities of interest measured within physical systems rely on the use of mathematical models that should be validated, or at best, not invalidated. Model validation usually involves the comparison of experimental data (outputs from the system of interest) and model predictions, both obtained at a specific validation scenario. The design of this validation experiment should be directly relevant to the objective of the model, that of predicting a quantity of interest at a prediction scenario. In this paper, we address two specific issues arising when designing validation experiments. The first issue consists in determining an appropriate validation scenario in cases where the prediction scenario cannot be carried out in a controlled environment. The second issue concerns the selection of observations when the quantity of interest cannot be readily observed. The proposed methodology involves the computation of influence matrices that characterize the response surface of given model functionals. Minimization of the distance between influence matrices allow one for selecting a validation experiment most representative of the prediction scenario. We illustrate our approach on two numerical examples. The first example considers the validation of a simple model based on an ordinary differential equation governing an object in free fall to put in evidence the importance of the choice of the validation experiment. The second numerical experiment focuses on the transport of a pollutant and demonstrates the impact that the choice of the quantity of interest has on the validation experiment to be performed.
Optimization algorithms such as projected Newton's method, FISTA, mirror descent, and its variants enjoy near-optimal regret bounds and convergence rates, but suffer from a computational bottleneck of computing ``projections'' in potentially each iteration (e.g., $O(T^{1/2})$ regret of online mirror descent). On the other hand, conditional gradient variants solve a linear optimization in each iteration, but result in suboptimal rates (e.g., $O(T^{3/4})$ regret of online Frank-Wolfe). Motivated by this trade-off in runtime v/s convergence rates, we consider iterative projections of close-by points over widely-prevalent submodular base polytopes $B(f)$. We first give necessary and sufficient conditions for when two close points project to the same face of a polytope, and then show that points far away from the polytope project onto its vertices with high probability. We next use this theory and develop a toolkit to speed up the computation of iterative projections over submodular polytopes using both discrete and continuous perspectives. We subsequently adapt the away-step Frank-Wolfe algorithm to use this information and enable early termination. For the special case of cardinality-based submodular polytopes, we improve the runtime of computing certain Bregman projections by a factor of $\Omega(n/\log(n))$. Our theoretical results show orders of magnitude reduction in runtime in preliminary computational experiments.
Making causal inferences from observational studies can be challenging when confounders are missing not at random. In such cases, identifying causal effects is often not guaranteed. Motivated by a real example, we consider a treatment-independent missingness assumption under which we establish the identification of causal effects when confounders are missing not at random. We propose a weighted estimating equation (WEE) approach for estimating model parameters and introduce three estimators for the average causal effect, based on regression, propensity score weighting, and doubly robust methods. We evaluate the performance of these estimators through simulations, and provide a real data analysis to illustrate our proposed method.
In recent years, implicit deep learning has emerged as a method to increase the effective depth of deep neural networks. While their training is memory-efficient, they are still significantly slower to train than their explicit counterparts. In Deep Equilibrium Models (DEQs), the training is performed as a bi-level problem, and its computational complexity is partially driven by the iterative inversion of a huge Jacobian matrix. In this paper, we propose a novel strategy to tackle this computational bottleneck from which many bi-level problems suffer. The main idea is to use the quasi-Newton matrices from the forward pass to efficiently approximate the inverse Jacobian matrix in the direction needed for the gradient computation. We provide a theorem that motivates using our method with the original forward algorithms. In addition, by modifying these forward algorithms, we further provide theoretical guarantees that our method asymptotically estimates the true implicit gradient. We empirically study this approach and the recent Jacobian-Free method in different settings, ranging from hyperparameter optimization to large Multiscale DEQs (MDEQs) applied to CIFAR and ImageNet. Both methods reduce significantly the computational cost of the backward pass. While SHINE has a clear advantage on hyperparameter optimization problems, both methods attain similar computational performances for larger scale problems such as MDEQs at the cost of a limited performance drop compared to the original models.
Machine learning (ML) models are costly to train as they can require a significant amount of data, computational resources and technical expertise. Thus, they constitute valuable intellectual property that needs protection from adversaries wanting to steal them. Ownership verification techniques allow the victims of model stealing attacks to demonstrate that a suspect model was in fact stolen from theirs. Although a number of ownership verification techniques based on watermarking or fingerprinting have been proposed, most of them fall short either in terms of security guarantees (well-equipped adversaries can evade verification) or computational cost. A fingerprinting technique introduced at ICLR '21, Dataset Inference (DI), has been shown to offer better robustness and efficiency than prior methods. The authors of DI provided a correctness proof for linear (suspect) models. However, in the same setting, we prove that DI suffers from high false positives (FPs) -- it can incorrectly identify an independent model trained with non-overlapping data from the same distribution as stolen. We further prove that DI also triggers FPs in realistic, non-linear suspect models. We then confirm empirically that DI leads to FPs, with high confidence. Second, we show that DI also suffers from false negatives (FNs) -- an adversary can fool DI by regularising a stolen model's decision boundaries using adversarial training, thereby leading to an FN. To this end, we demonstrate that DI fails to identify a model adversarially trained from a stolen dataset -- the setting where DI is the hardest to evade. Finally, we discuss the implications of our findings, the viability of fingerprinting-based ownership verification in general, and suggest directions for future work.
Although equivariant machine learning has proven effective at many tasks, success depends heavily on the assumption that the ground truth function is symmetric over the entire domain matching the symmetry in an equivariant neural network. A missing piece in the equivariant learning literature is the analysis of equivariant networks when symmetry exists only partially in the domain. In this work, we present a general theory for such a situation. We propose pointwise definitions of correct, incorrect, and extrinsic equivariance, which allow us to quantify continuously the degree of each type of equivariance a function displays. We then study the impact of various degrees of incorrect or extrinsic symmetry on model error. We prove error lower bounds for invariant or equivariant networks in classification or regression settings with partially incorrect symmetry. We also analyze the potentially harmful effects of extrinsic equivariance. Experiments validate these results in three different environments.
This work considers Gaussian process interpolation with a periodized version of the Mat{\'e}rn covariance function introduced by Stein (22, Section 6.7). Convergence rates are studied for the joint maximum likelihood estimation of the regularity and the amplitude parameters when the data is sampled according to the model. The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known. Finally, the case where the observed function is a fixed deterministic element of a Sobolev space of continuous functions is also considered, suggesting that bounding assumptions on some parameters can lead to different estimates.
The precision matrix that encodes conditional linear dependency relations among a set of variables forms an important object of interest in multivariate analysis. Sparse estimation procedures for precision matrices such as the graphical lasso (Glasso) gained popularity as they facilitate interpretability, thereby separating pairs of variables that are conditionally dependent from those that are independent (given all other variables). Glasso lacks, however, robustness to outliers. To overcome this problem, one typically applies a robust plug-in procedure where the Glasso is computed from a robust covariance estimate instead of the sample covariance, thereby providing protection against outliers. In this paper, we study such estimators theoretically, by deriving and comparing their influence function, sensitivity curves and asymptotic variances.
This PhD thesis contains several contributions to the field of statistical causal modeling. Statistical causal models are statistical models embedded with causal assumptions that allow for the inference and reasoning about the behavior of stochastic systems affected by external manipulation (interventions). This thesis contributes to the research areas concerning the estimation of causal effects, causal structure learning, and distributionally robust (out-of-distribution generalizing) prediction methods. We present novel and consistent linear and non-linear causal effects estimators in instrumental variable settings that employ data-dependent mean squared prediction error regularization. Our proposed estimators show, in certain settings, mean squared error improvements compared to both canonical and state-of-the-art estimators. We show that recent research on distributionally robust prediction methods has connections to well-studied estimators from econometrics. This connection leads us to prove that general K-class estimators possess distributional robustness properties. We, furthermore, propose a general framework for distributional robustness with respect to intervention-induced distributions. In this framework, we derive sufficient conditions for the identifiability of distributionally robust prediction methods and present impossibility results that show the necessity of several of these conditions. We present a new structure learning method applicable in additive noise models with directed trees as causal graphs. We prove consistency in a vanishing identifiability setup and provide a method for testing substructure hypotheses with asymptotic family-wise error control that remains valid post-selection. Finally, we present heuristic ideas for learning summary graphs of nonlinear time-series models.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
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