We derive a family of loss functions to train models in the presence of sampling bias. Examples are when the prevalence of a pathology differs from its sampling rate in the training dataset, or when a machine learning practioner rebalances their training dataset. Sampling bias causes large discrepancies between model performance in the lab and in more realistic settings. It is omnipresent in medical imaging applications, yet is often overlooked at training time or addressed on an ad-hoc basis. Our approach is based on Bayesian risk minimization. For arbitrary likelihood models we derive the associated bias corrected loss for training, exhibiting a direct connection to information gain. The approach integrates seamlessly in the current paradigm of (deep) learning using stochastic backpropagation and naturally with Bayesian models. We illustrate the methodology on case studies of lung nodule malignancy grading.

Compartmental models are used for predicting the scale and duration of epidemics, estimating epidemiological parameters such as reproduction numbers, and guiding outbreak control measures [Brauer, 2008, O'Neill, 2010, Kucharski et al., 2020]. They are increasingly important because they allow joint modelling of disease dynamics and multimodal data, such as medical test results, cell phone and transport flow data [Rubrichi et al., 2018, Wu et al., 2020], census and demographic information [Prem et al., 2020]. However, statistical inference in stochastic variants of compartmental models is a major computational challenge [Bretó, 2018]. The likelihood function for model parameters is usually intractable because it involves summation over a prohibitively large number of configurations of latent variables representing counts of subpopulations in disease states which cannot be observed directly. This has lead to the recent development of sophisticated computational methods for approximate inference involving various forms of stochastic simulation [Funk and King, 2020].

We study preconditioned gradient-based optimization methods where the preconditioning matrix has block-diagonal form. Such a structural constraint comes with the advantage that the update computation is block-separable and can be parallelized across multiple independent tasks. Our main contribution is to demonstrate that the convergence of these methods can significantly be improved by a randomization technique which corresponds to repartitioning coordinates across tasks during the optimization procedure. We provide a theoretical analysis that accurately characterizes the expected convergence gains of repartitioning and validate our findings empirically on various traditional machine learning tasks. From an implementation perspective, block-separable models are well suited for parallelization and, when shared memory is available, randomization can be implemented on top of existing methods very efficiently to improve convergence.

Kernel functions are a powerful tool to enhance the $k$-means clustering algorithm via the kernel trick. It is known that the parameters of the chosen kernel function can have a dramatic impact on the result. In supervised settings, these can be tuned via cross-validation, but for clustering this is not straightforward and heuristics are usually employed. In this paper we study the impact of kernel parameters on kernel $k$-means. In particular, we derive a lower bound, tight up to constant factors, below which the parameter of the RBF kernel will render kernel $k$-means meaningless. We argue that grid search can be ineffective for hyperparameter search in this context and propose an alternative algorithm for this purpose. In addition, we offer an efficient implementation based on fast approximate exponentiation with provable quality guarantees. Our experimental results demonstrate the ability of our method to efficiently reveal a rich and useful set of hyperparameter values.

We consider machine learning, particularly regression, using locally-differentially private datasets. The Wasserstein distance is used to define an ambiguity set centered at the empirical distribution of the dataset corrupted by local differential privacy noise. The ambiguity set is shown to contain the probability distribution of unperturbed, clean data. The radius of the ambiguity set is a function of the privacy budget, spread of the data, and the size of the problem. Hence, machine learning with locally-differentially private datasets can be rewritten as a distributionally-robust optimization. For general distributions, the distributionally-robust optimization problem can relaxed as a regularized machine learning problem with the Lipschitz constant of the machine learning model as a regularizer. For linear and logistic regression, this regularizer is the dual norm of the model parameters. For Gaussian data, the distributionally-robust optimization problem can be solved exactly to find an optimal regularizer. This approach results in an entirely new regularizer for training linear regression models. Training with this novel regularizer can be posed as a semi-definite program. Finally, the performance of the proposed distributionally-robust machine learning training is demonstrated on practical datasets.

Continuous submodular functions are a category of generally non-convex/non-concave functions with a wide spectrum of applications. The celebrated property of this class of functions - continuous submodularity - enables both exact minimization and approximate maximization in poly. time. Continuous submodularity is obtained by generalizing the notion of submodularity from discrete domains to continuous domains. It intuitively captures a repulsive effect amongst different dimensions of the defined multivariate function. In this paper, we systematically study continuous submodularity and a class of non-convex optimization problems: continuous submodular function maximization. We start by a thorough characterization of the class of continuous submodular functions, and show that continuous submodularity is equivalent to a weak version of the diminishing returns (DR) property. Thus we also derive a subclass of continuous submodular functions, termed continuous DR-submodular functions, which enjoys the full DR property. Then we present operations that preserve continuous (DR-)submodularity, thus yielding general rules for composing new submodular functions. We establish intriguing properties for the problem of constrained DR-submodular maximization, such as the local-global relation. We identify several applications of continuous submodular optimization, ranging from influence maximization, MAP inference for DPPs to provable mean field inference. For these applications, continuous submodularity formalizes valuable domain knowledge relevant for optimizing this class of objectives. We present inapproximability results and provable algorithms for two problem settings: constrained monotone DR-submodular maximization and constrained non-monotone DR-submodular maximization. Finally, we extensively evaluate the effectiveness of the proposed algorithms.

In this paper, we introduce the Dissimilarity Mixture Autoencoder (DMAE), a novel neural network model that uses a dissimilarity function to generalize a family of density estimation and clustering methods. It is formulated in such a way that it internally estimates the parameters of a probability distribution through gradient-based optimization. Also, the proposed model can leverage from deep representation learning due to its straightforward incorporation into deep learning architectures, because, it consists of an encoder-decoder network that computes a probabilistic representation. Experimental evaluation was performed on image and text clustering benchmark datasets showing that the method is competitive in terms of unsupervised classification accuracy and normalized mutual information. The source code to replicate the experiments is publicly available at https://github.com/larajuse/DMAE

Sub-seasonal climate forecasting (SSF) focuses on predicting key climate variables such as temperature and precipitation in the 2-week to 2-month time scales. Skillful SSF would have immense societal value, in areas such as agricultural productivity, water resource management, transportation and aviation systems, and emergency planning for extreme weather events. However, SSF is considered more challenging than either weather prediction or even seasonal prediction. In this paper, we carefully study a variety of machine learning (ML) approaches for SSF over the US mainland. While atmosphere-land-ocean couplings and the limited amount of good quality data makes it hard to apply black-box ML naively, we show that with carefully constructed feature representations, even linear regression models, e.g., Lasso, can be made to perform well. Among a broad suite of 10 ML approaches considered, gradient boosting performs the best, and deep learning (DL) methods show some promise with careful architecture choices. Overall, suitable ML methods are able to outperform the climatological baseline, i.e., predictions based on the 30-year average at a given location and time. Further, based on studying feature importance, ocean (especially indices based on climatic oscillations such as El Nino) and land (soil moisture) covariates are found to be predictive, whereas atmospheric covariates are not considered helpful.

We are often interested in identifying the feasible subset of a decision space under multiple constraints to permit effective design exploration. If determining feasibility required computationally expensive simulations, the cost of exploration would be prohibitive. Bayesian search is data-efficient for such problems: starting from a small dataset, the central concept is to use Bayesian models of constraints with an acquisition function to locate promising solutions that may improve predictions of feasibility when the dataset is augmented. At the end of this sequential active learning approach with a limited number of expensive evaluations, the models can accurately predict the feasibility of any solution obviating the need for full simulations. In this paper, we propose a novel acquisition function that combines the probability that a solution lies at the boundary between feasible and infeasible spaces (representing exploitation) and the entropy in predictions (representing exploration). Experiments confirmed the efficacy of the proposed function.

In this paper, we consider non-convex optimization problems under \textit{unknown} yet safety-critical constraints. Such problems naturally arise in a variety of domains including robotics, manufacturing, and medical procedures, where it is infeasible to know or identify all the constraints. Therefore, the parameter space should be explored in a conservative way to ensure that none of the constraints are violated during the optimization process once we start from a safe initialization point. To this end, we develop an algorithm called Reliable Frank-Wolfe (Reliable-FW). Given a general non-convex function and an unknown polytope constraint, Reliable-FW simultaneously learns the landscape of the objective function and the boundary of the safety polytope. More precisely, by assuming that Reliable-FW has access to a (stochastic) gradient oracle of the objective function and a noisy feasibility oracle of the safety polytope, it finds an $\epsilon$-approximate first-order stationary point with the optimal ${\mathcal{O}}({1}/{\epsilon^2})$ gradient oracle complexity (resp. $\tilde{\mathcal{O}}({1}/{\epsilon^3})$ (also optimal) in the stochastic gradient setting), while ensuring the safety of all the iterates. Rather surprisingly, Reliable-FW only makes $\tilde{\mathcal{O}}(({d^2}/{\epsilon^2})\log 1/\delta)$ queries to the noisy feasibility oracle (resp. $\tilde{\mathcal{O}}(({d^2}/{\epsilon^4})\log 1/\delta)$ in the stochastic gradient setting) where $d$ is the dimension and $\delta$ is the reliability parameter, tightening the existing bounds even for safe minimization of convex functions. We further specialize our results to the case that the objective function is convex. A crucial component of our analysis is to introduce and apply a technique called geometric shrinkage in the context of safe optimization.