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State-of-the-art federated learning algorithms such as FedAvg require carefully tuned stepsizes to achieve their best performance. The improvements proposed by existing adaptive federated methods involve tuning of additional hyperparameters such as momentum parameters, and consider adaptivity only in the server aggregation round, but not locally. These methods can be inefficient in many practical scenarios because they require excessive tuning of hyperparameters and do not capture local geometric information. In this work, we extend the recently proposed stochastic Polyak stepsize (SPS) to the federated learning setting, and propose new locally adaptive and nearly parameter-free distributed SPS variants (FedSPS and FedDecSPS). We prove that FedSPS converges linearly in strongly convex and sublinearly in convex settings when the interpolation condition (overparametrization) is satisfied, and converges to a neighborhood of the solution in the general case. We extend our proposed method to a decreasing stepsize version FedDecSPS, that converges also when the interpolation condition does not hold. We validate our theoretical claims by performing illustrative convex experiments. Our proposed algorithms match the optimization performance of FedAvg with the best tuned hyperparameters in the i.i.d. case, and outperform FedAvg in the non-i.i.d. case.

Bayesian model comparison (BMC) offers a principled approach for assessing the relative merits of competing computational models and propagating uncertainty into model selection decisions. However, BMC is often intractable for the popular class of hierarchical models due to their high-dimensional nested parameter structure. To address this intractability, we propose a deep learning method for performing BMC on any set of hierarchical models which can be instantiated as probabilistic programs. Since our method enables amortized inference, it allows efficient re-estimation of posterior model probabilities and fast performance validation prior to any real-data application. In a series of extensive validation studies, we benchmark the performance of our method against the state-of-the-art bridge sampling method and demonstrate excellent amortized inference across all BMC settings. We then showcase our method by comparing four hierarchical evidence accumulation models that have previously been deemed intractable for BMC due to partly implicit likelihoods. In this application, we corroborate evidence for the recently proposed L\'evy flight model of decision-making and show how transfer learning can be leveraged to enhance training efficiency. We provide reproducible code for all analyses and an open-source implementation of our method.

Differential privacy guarantees allow the results of a statistical analysis involving sensitive data to be released without compromising the privacy of any individual taking part. Achieving such guarantees generally requires the injection of noise, either directly into parameter estimates or into the estimation process. Instead of artificially introducing perturbations, sampling from Bayesian posterior distributions has been shown to be a special case of the exponential mechanism, producing consistent, and efficient private estimates without altering the data generative process. The application of current approaches has, however, been limited by their strong bounding assumptions which do not hold for basic models, such as simple linear regressors. To ameliorate this, we propose $\beta$D-Bayes, a posterior sampling scheme from a generalised posterior targeting the minimisation of the $\beta$-divergence between the model and the data generating process. This provides private estimation that is generally applicable without requiring changes to the underlying model and consistently learns the data generating parameter. We show that $\beta$D-Bayes produces more precise inference estimation for the same privacy guarantees, and further facilitates differentially private estimation via posterior sampling for complex classifiers and continuous regression models such as neural networks for the first time.

Understanding the macroscopic characteristics of biological complexes demands precision and specificity in statistical ensemble modeling. One of the primary challenges in this domain lies in sampling from particular subsets of the state-space, driven either by existing structural knowledge or specific areas of interest within the state-space. We propose a method that enables sampling from distributions that rigorously adhere to arbitrary sets of geometric constraints in Euclidean spaces. This is achieved by integrating a constraint projection operator within the well-regarded architecture of Denoising Diffusion Probabilistic Models, a framework founded in generative modeling and probabilistic inference. The significance of this work becomes apparent, for instance, in the context of deep learning-based drug design, where it is imperative to maintain specific molecular profile interactions to realize the desired therapeutic outcomes and guarantee safety.

Model overconfidence and poor calibration are common in machine learning and difficult to account for when applying standard empirical risk minimization. In this work, we propose a novel method to alleviate these problems that we call odd-$k$-out learning (OKO), which minimizes the cross-entropy error for sets rather than for single examples. This naturally allows the model to capture correlations across data examples and achieves both better accuracy and calibration, especially in limited training data and class-imbalanced regimes. Perhaps surprisingly, OKO often yields better calibration even when training with hard labels and dropping any additional calibration parameter tuning, such as temperature scaling. We provide theoretical justification, establishing that OKO naturally yields better calibration, and provide extensive experimental analyses that corroborate our theoretical findings. We emphasize that OKO is a general framework that can be easily adapted to many settings and the trained model can be applied to single examples at inference time, without introducing significant run-time overhead or architecture changes.

In many applications in biology, engineering and economics, identifying similarities and differences between distributions of data from complex processes requires comparing finite categorical samples of discrete counts. Statistical divergences quantify the difference between two distributions. However, their estimation is very difficult and empirical methods often fail, especially when the samples are small. We develop a Bayesian estimator of the Kullback-Leibler divergence between two probability distributions that makes use of a mixture of Dirichlet priors on the distributions being compared. We study the properties of the estimator on two examples: probabilities drawn from Dirichlet distributions, and random strings of letters drawn from Markov chains. We extend the approach to the squared Hellinger divergence. Both estimators outperform other estimation techniques, with better results for data with a large number of categories and for higher values of divergences.

Multi-fidelity models are of great importance due to their capability of fusing information coming from different numerical simulations, surrogates, and sensors. We focus on the approximation of high-dimensional scalar functions with low intrinsic dimensionality. By introducing a low dimensional bias we can fight the curse of dimensionality affecting these quantities of interest, especially for many-query applications. We seek a gradient-based reduction of the parameter space through linear active subspaces or a nonlinear transformation of the input space. Then we build a low-fidelity response surface based on such reduction, thus enabling nonlinear autoregressive multi-fidelity Gaussian process regression without the need of running new simulations with simplified physical models. This has a great potential in the data scarcity regime affecting many engineering applications. In this work we present a new multi-fidelity approach that involves active subspaces and the nonlinear level-set learning method, starting from the preliminary analysis previously conducted in Romor et al. 2020. The proposed framework is tested on two high-dimensional benchmark functions, and on a more complex car aerodynamics problem. We show how a low intrinsic dimensionality bias can increase the accuracy of Gaussian process response surfaces.

Variable screening has been a useful research area that deals with ultrahigh-dimensional data. When there exist both marginally and jointly dependent predictors to the response, existing methods such as conditional screening or iterative screening often suffer from instability against the selection of the conditional set or the computational burden, respectively. In this article, we propose a new independence measure, named conditional martingale difference divergence (CMDH), that can be treated as either a conditional or a marginal independence measure. Under regularity conditions, we show that the sure screening property of CMDH holds for both marginally and jointly active variables. Based on this measure, we propose a kernel-based model-free variable screening method, which is efficient, flexible, and stable against high correlation among predictors and heterogeneity of the response. In addition, we provide a data-driven method to select the conditional set. In simulations and real data applications, we demonstrate the superior performance of the proposed method.

Statistical shape models (SSM) have been well-established as an excellent tool for identifying variations in the morphology of anatomy across the underlying population. Shape models use consistent shape representation across all the samples in a given cohort, which helps to compare shapes and identify the variations that can detect pathologies and help in formulating treatment plans. In medical imaging, computing these shape representations from CT/MRI scans requires time-intensive preprocessing operations, including but not limited to anatomy segmentation annotations, registration, and texture denoising. Deep learning models have demonstrated exceptional capabilities in learning shape representations directly from volumetric images, giving rise to highly effective and efficient Image-to-SSM. Nevertheless, these models are data-hungry and due to the limited availability of medical data, deep learning models tend to overfit. Offline data augmentation techniques, that use kernel density estimation based (KDE) methods for generating shape-augmented samples, have successfully aided Image-to-SSM networks in achieving comparable accuracy to traditional SSM methods. However, these augmentation methods focus on shape augmentation, whereas deep learning models exhibit texture bias results in sub-optimal models. This paper introduces a novel strategy for on-the-fly data augmentation for the Image-to-SSM framework by leveraging data-dependent noise generation or texture augmentation. The proposed framework is trained as an adversary to the Image-to-SSM network, augmenting diverse and challenging noisy samples. Our approach achieves improved accuracy by encouraging the model to focus on the underlying geometry rather than relying solely on pixel values.

Sufficient training data is normally required to train deeply learned models. However, the number of pedestrian images per ID in person re-identification (re-ID) datasets is usually limited, since manually annotations are required for multiple camera views. To produce more data for training deeply learned models, generative adversarial network (GAN) can be leveraged to generate samples for person re-ID. However, the samples generated by vanilla GAN usually do not have labels. So in this paper, we propose a virtual label called Multi-pseudo Regularized Label (MpRL) and assign it to the generated images. With MpRL, the generated samples will be used as supplementary of real training data to train a deep model in a semi-supervised learning fashion. Considering data bias between generated and real samples, MpRL utilizes different contributions from predefined training classes. The contribution-based virtual labels are automatically assigned to generated samples to reduce ambiguous prediction in training. Meanwhile, MpRL only relies on predefined training classes without using extra classes. Furthermore, to reduce over-fitting, a regularized manner is applied to MpRL to regularize the learning process. To verify the effectiveness of MpRL, two state-of-the-art convolutional neural networks (CNNs) are adopted in our experiments. Experiments demonstrate that by assigning MpRL to generated samples, we can further improve the person re-ID performance on three datasets i.e., Market-1501, DukeMTMCreID, and CUHK03. The proposed method obtains +6.29%, +6.30% and +5.58% improvements in rank-1 accuracy over a strong CNN baseline respectively, and outperforms the state-of-the- art methods.