Gaussian processes remain popular as a flexible and expressive model class, but the computational cost of kernel hyperparameter optimization stands as a major limiting factor to their scaling and broader adoption. Recent work has made great strides combining stochastic estimation with iterative numerical techniques, essentially boiling down GP inference to the cost of (many) matrix-vector multiplies. Preconditioning -- a highly effective step for any iterative method involving matrix-vector multiplication -- can be used to accelerate convergence and thus reduce bias in hyperparameter optimization. Here, we prove that preconditioning has an additional benefit that has been previously unexplored. It not only reduces the bias of the $\log$-marginal likelihood estimator and its derivatives, but it also simultaneously can reduce variance at essentially negligible cost. We leverage this result to derive sample-efficient algorithms for GP hyperparameter optimization requiring as few as $\mathcal{O}(\log(\varepsilon^{-1}))$ instead of $\mathcal{O}(\varepsilon^{-2})$ samples to achieve error $\varepsilon$. Our theoretical results enable provably efficient and scalable optimization of kernel hyperparameters, which we validate empirically on a set of large-scale benchmark problems. There, variance reduction via preconditioning results in an order of magnitude speedup in hyperparameter optimization of exact GPs.
Large width limits have been a recent focus of deep learning research: modulo computational practicalities, do wider networks outperform narrower ones? Answering this question has been challenging, as conventional networks gain representational power with width, potentially masking any negative effects. Our analysis in this paper decouples capacity and width via the generalization of neural networks to Deep Gaussian Processes (Deep GP), a class of hierarchical models that subsume neural nets. In doing so, we aim to understand how width affects standard neural networks once they have sufficient capacity for a given modeling task. Our theoretical and empirical results on Deep GP suggest that large width is generally detrimental to hierarchical models. Surprisingly, we prove that even nonparametric Deep GP converge to Gaussian processes, effectively becoming shallower without any increase in representational power. The posterior, which corresponds to a mixture of data-adaptable basis functions, becomes less data-dependent with width. Our tail analysis demonstrates that width and depth have opposite effects: depth accentuates a model's non-Gaussianity, while width makes models increasingly Gaussian. We find there is a "sweet spot" that maximizes test set performance before the limiting GP behavior prevents adaptability, occurring at width = 1 or width = 2 for nonparametric Deep GP. These results make strong predictions about the same phenomenon in conventional neural networks: we show empirically that many neural network architectures need 10 - 500 hidden units for sufficient capacity - depending on the dataset - but further width degrades test performance.
Normalizing flows are invertible neural networks with tractable change-of-volume terms, which allows optimization of their parameters to be efficiently performed via maximum likelihood. However, data of interest is typically assumed to live in some (often unknown) low-dimensional manifold embedded in high-dimensional ambient space. The result is a modelling mismatch since -- by construction -- the invertibility requirement implies high-dimensional support of the learned distribution. Injective flows, mapping from low- to high-dimensional space, aim to fix this discrepancy by learning distributions on manifolds, but the resulting volume-change term becomes more challenging to evaluate. Current approaches either avoid computing this term entirely using various heuristics, or assume the manifold is known beforehand and therefore are not widely applicable. Instead, we propose two methods to tractably calculate the gradient of this term with respect to the parameters of the model, relying on careful use of automatic differentiation and techniques from numerical linear algebra. Both approaches perform end-to-end nonlinear manifold learning and density estimation for data projected onto this manifold. We study the trade-offs between our proposed methods, empirically verify that we outperform approaches ignoring the volume-change term by more accurately learning manifolds and the corresponding distributions on them, and show promising results on out-of-distribution detection.
Integrating methods for time-to-event prediction with diagnostic imaging modalities is of considerable interest, as accurate estimates of survival requires accounting for censoring of individuals within the observation period. New methods for time-to-event prediction have been developed by extending the cox-proportional hazards model with neural networks. In this paper, to explore the feasibility of these methods when applied to deep learning with echocardiography videos, we utilize the Stanford EchoNet-Dynamic dataset with over 10,000 echocardiograms, and generate simulated survival datasets based on the expert annotated ejection fraction readings. By training on just the simulated survival outcomes, we show that spatiotemporal convolutional neural networks yield accurate survival estimates.
Advances in computing power, deep learning architectures, and expert labelled datasets have spurred the development of medical imaging artificial intelligence systems that rival clinical experts in a variety of scenarios. The National Institutes of Health in 2018 identified key focus areas for the future of artificial intelligence in medical imaging, creating a foundational roadmap for research in image acquisition, algorithms, data standardization, and translatable clinical decision support systems. Among the key issues raised in the report: data availability, need for novel computing architectures and explainable AI algorithms, are still relevant despite the tremendous progress made over the past few years alone. Furthermore, translational goals of data sharing, validation of performance for regulatory approval, generalizability and mitigation of unintended bias must be accounted for early in the development process. In this perspective paper we explore challenges unique to high dimensional clinical imaging data, in addition to highlighting some of the technical and ethical considerations in developing high-dimensional, multi-modality, machine learning systems for clinical decision support.
Non-invasive and cost effective in nature, the echocardiogram allows for a comprehensive assessment of the cardiac musculature and valves. Despite progressive improvements over the decades, the rich temporally resolved data in echocardiography videos remain underutilized. Human reads of echocardiograms reduce the complex patterns of cardiac wall motion, to a small list of measurements of heart function. Furthermore, all modern echocardiography artificial intelligence (AI) systems are similarly limited by design - automating measurements of the same reductionist metrics rather than utilizing the wealth of data embedded within each echo study. This underutilization is most evident in situations where clinical decision making is guided by subjective assessments of disease acuity, and tools that predict disease onset within clinically actionable timeframes are unavailable. Predicting the likelihood of developing post-operative right ventricular failure (RV failure) in the setting of mechanical circulatory support is one such clinical example. To address this, we developed a novel video AI system trained to predict post-operative right ventricular failure (RV failure), using the full spatiotemporal density of information from pre-operative echocardiography scans. We achieve an AUC of 0.729, specificity of 52% at 80% sensitivity and 46% sensitivity at 80% specificity. Furthermore, we show that our ML system significantly outperforms a team of human experts tasked with predicting RV failure on independent clinical evaluation. Finally, the methods we describe are generalizable to any cardiac clinical decision support application where treatment or patient selection is guided by qualitative echocardiography assessments.
Scalable Gaussian Process methods are computationally attractive, yet introduce modeling biases that require rigorous study. This paper analyzes two common techniques: early truncated conjugate gradients (CG) and random Fourier features (RFF). We find that both methods introduce a systematic bias on the learned hyperparameters: CG tends to underfit while RFF tends to overfit. We address these issues using randomized truncation estimators that eliminate bias in exchange for increased variance. In the case of RFF, we show that the bias-to-variance conversion is indeed a trade-off: the additional variance proves detrimental to optimization. However, in the case of CG, our unbiased learning procedure meaningfully outperforms its biased counterpart with minimal additional computation.
Modern deep learning is primarily an experimental science, in which empirical advances occasionally come at the expense of probabilistic rigor. Here we focus on one such example; namely the use of the categorical cross-entropy loss to model data that is not strictly categorical, but rather takes values on the simplex. This practice is standard in neural network architectures with label smoothing and actor-mimic reinforcement learning, amongst others. Drawing on the recently discovered continuous-categorical distribution, we propose probabilistically-inspired alternatives to these models, providing an approach that is more principled and theoretically appealing. Through careful experimentation, including an ablation study, we identify the potential for outperformance in these models, thereby highlighting the importance of a proper probabilistic treatment, as well as illustrating some of the failure modes thereof.
Gaussian Processes (GPs) provide a powerful probabilistic framework for interpolation, forecasting, and smoothing, but have been hampered by computational scaling issues. Here we prove that for data sampled on one dimension (e.g., a time series sampled at arbitrarily-spaced intervals), approximate GP inference at any desired level of accuracy requires computational effort that scales linearly with the number of observations; this new theorem enables inference on much larger datasets than was previously feasible. To achieve this improved scaling we propose a new family of stationary covariance kernels: the Latent Exponentially Generated (LEG) family, which admits a convenient stable state-space representation that allows linear-time inference. We prove that any continuous integrable stationary kernel can be approximated arbitrarily well by some member of the LEG family. The proof draws connections to Spectral Mixture Kernels, providing new insight about the flexibility of this popular family of kernels. We propose parallelized algorithms for performing inference and learning in the LEG model, test the algorithm on real and synthetic data, and demonstrate scaling to datasets with billions of samples.
Simplex-valued data appear throughout statistics and machine learning, for example in the context of transfer learning and compression of deep networks. Existing models for this class of data rely on the Dirichlet distribution or other related loss functions; here we show these standard choices suffer systematically from a number of limitations, including bias and numerical issues that frustrate the use of flexible network models upstream of these distributions. We resolve these limitations by introducing a novel exponential family of distributions for modeling simplex-valued data - the continuous categorical, which arises as a nontrivial multivariate generalization of the recently discovered continuous Bernoulli. Unlike the Dirichlet and other typical choices, the continuous categorical results in a well-behaved probabilistic loss function that produces unbiased estimators, while preserving the mathematical simplicity of the Dirichlet. As well as exploring its theoretical properties, we introduce sampling methods for this distribution that are amenable to the reparameterization trick, and evaluate their performance. Lastly, we demonstrate that the continuous categorical outperforms standard choices empirically, across a simulation study, an applied example on multi-party elections, and a neural network compression task.