Score-based generative models learn a family of noise-conditional score functions corresponding to the data density perturbed with increasingly large amounts of noise. These perturbed data densities are tied together by the Fokker-Planck equation (FPE), a partial differential equation (PDE) governing the spatial-temporal evolution of a density undergoing a diffusion process. In this work, we derive a corresponding equation, called the score FPE that characterizes the noise-conditional scores of the perturbed data densities (i.e., their gradients). Surprisingly, despite impressive empirical performance, we observe that scores learned via denoising score matching (DSM) do not satisfy the underlying score FPE. We prove that satisfying the FPE is desirable as it improves the likelihood and the degree of conservativity. Hence, we propose to regularize the DSM objective to enforce satisfaction of the score FPE, and we show the effectiveness of this approach across various datasets.
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Motivated by applications in personalized medicine and individualized policy making, there is a growing interest in techniques for quantifying treatment effect heterogeneity in terms of the conditional average treatment effect (CATE). Some of the most prominent methods for CATE estimation developed in recent years are T-Learner, DR-Learner and R-Learner. The latter two were designed to improve on the former by being Neyman-orthogonal. However, the relations between them remain unclear, and likewise does the literature remain vague on whether these learners converge to a useful quantity or (functional) estimand when the underlying optimization procedure is restricted to a class of functions that does not include the CATE. In this article, we provide insight into these questions by discussing DR-learner and R-learner as special cases of a general class of Neyman-orthogonal learners for the CATE, for which we moreover derive oracle bounds. Our results shed light on how one may construct Neyman-orthogonal learners with desirable properties, on when DR-learner may be preferred over R-learner (and vice versa), and on novel learners that may sometimes be preferable to either of these. Theoretical findings are confirmed using results from simulation studies on synthetic data, as well as an application in critical care medicine.
Generalized linear mixed models are powerful tools for analyzing clustered data, where the unknown parameters are classically (and most commonly) estimated by the maximum likelihood and restricted maximum likelihood procedures. However, since the likelihood based procedures are known to be highly sensitive to outliers, M-estimators have become popular as a means to obtain robust estimates under possible data contamination. In this paper, we prove that, for sufficiently smooth general loss functions defining the M-estimators in generalized linear mixed models, the tail probability of the deviation between the estimated and the true regression coefficients have an exponential bound. This implies an exponential rate of consistency of these M-estimators under appropriate assumptions, generalizing the existing exponential consistency results from univariate to multivariate responses. We have illustrated this theoretical result further for the special examples of the maximum likelihood estimator and the robust minimum density power divergence estimator, a popular example of model-based M-estimators, in the settings of linear and logistic mixed models, comparing it with the empirical rate of convergence through simulation studies.
Diffusion models in the literature are optimized with various objectives that are special cases of a weighted loss, where the weighting function specifies the weight per noise level. Uniform weighting corresponds to maximizing the ELBO, a principled approximation of maximum likelihood. In current practice diffusion models are optimized with non-uniform weighting due to better results in terms of sample quality. In this work we expose a direct relationship between the weighted loss (with any weighting) and the ELBO objective. We show that the weighted loss can be written as a weighted integral of ELBOs, with one ELBO per noise level. If the weighting function is monotonic, then the weighted loss is a likelihood-based objective: it maximizes the ELBO under simple data augmentation, namely Gaussian noise perturbation. Our main contribution is a deeper theoretical understanding of the diffusion objective, but we also performed some experiments comparing monotonic with non-monotonic weightings, finding that monotonic weighting performs competitively with the best published results.
We study the computational complexity of multi-stage robust optimization problems. Such problems are formulated with alternating min/max quantifiers and therefore naturally fall into a higher stage of the polynomial hierarchy. Despite this, almost no hardness results with respect to the polynomial hierarchy are known. In this work, we examine the hardness of robust two-stage adjustable and robust recoverable optimization with budgeted uncertainty sets. Our main technical contribution is the introduction of a technique tailored to prove $\Sigma^p_3$-hardness of such problems. We highlight a difference between continuous and discrete budgeted uncertainty: In the discrete case, indeed a wide range of problems becomes complete for the third stage of the polynomial hierarchy; in particular, this applies to the TSP, independent set, and vertex cover problems. However, in the continuous case this does not happen and problems remain in the first stage of the hierarchy. Finally, if we allow the uncertainty to not only affect the objective, but also multiple constraints, then this distinction disappears and even in the continuous case we encounter hardness for the third stage of the hierarchy. This shows that even robust problems which are already NP-complete can still exhibit a significant computational difference between column-wise and row-wise uncertainty.
The enumeration of linear $\lambda$-terms has attracted quite some attention recently, partly due to their link to combinatorial maps. Zeilberger and Giorgetti (2015) gave a recursive bijection between planar linear normal $\lambda$-terms and planar maps, which, when restricted to 2-connected $\lambda$-terms (i.e., without closed sub-terms), leads to bridgeless planar maps. Inspired by this restriction, Zeilberger and Reed (2019) conjectured that 3-connected planar linear normal $\lambda$-terms have the same counting formula as bipartite planar maps. In this article, we settle this conjecture by giving a direct bijection between these two families. Furthermore, using a similar approach, we give a direct bijection between planar linear normal $\lambda$-terms and planar maps, whose restriction to 2-connected $\lambda$-terms leads to loopless planar maps. This bijection seems different from that of Zeilberger and Giorgetti, even after taking the map dual. We also explore enumerative consequences of our bijections.
As machine learning-enabled Text-to-Image (TTI) systems are becoming increasingly prevalent and seeing growing adoption as commercial services, characterizing the social biases they exhibit is a necessary first step to lowering their risk of discriminatory outcomes. This evaluation, however, is made more difficult by the synthetic nature of these systems' outputs; since artificial depictions of fictive humans have no inherent gender or ethnicity nor do they belong to socially-constructed groups, we need to look beyond common categorizations of diversity or representation. To address this need, we propose a new method for exploring and quantifying social biases in TTI systems by directly comparing collections of generated images designed to showcase a system's variation across social attributes -- gender and ethnicity -- and target attributes for bias evaluation -- professions and gender-coded adjectives. Our approach allows us to (i) identify specific bias trends through visualization tools, (ii) provide targeted scores to directly compare models in terms of diversity and representation, and (iii) jointly model interdependent social variables to support a multidimensional analysis. We use this approach to analyze over 96,000 images generated by 3 popular TTI systems (DALL-E 2, Stable Diffusion v 1.4 and v 2) and find that all three significantly over-represent the portion of their latent space associated with whiteness and masculinity across target attributes; among the systems studied, DALL-E 2 shows the least diversity, followed by Stable Diffusion v2 then v1.4.
In complex large-scale systems such as climate, important effects are caused by a combination of confounding processes that are not fully observable. The identification of sources from observations of system state is vital for attribution and prediction, which inform critical policy decisions. The difficulty of these types of inverse problems lies in the inability to isolate sources and the cost of simulating computational models. Surrogate models may enable the many-query algorithms required for source identification, but data challenges arise from high dimensionality of the state and source, limited ensembles of costly model simulations to train a surrogate model, and few and potentially noisy state observations for inversion due to measurement limitations. The influence of auxiliary processes adds an additional layer of uncertainty that further confounds source identification. We introduce a framework based on (1) calibrating deep neural network surrogates to the flow maps provided by an ensemble of simulations obtained by varying sources, and (2) using these surrogates in a Bayesian framework to identify sources from observations via optimization. Focusing on an atmospheric dispersion exemplar, we find that the expressive and computationally efficient nature of the deep neural network operator surrogates in appropriately reduced dimension allows for source identification with uncertainty quantification using limited data. Introducing a variable wind field as an auxiliary process, we find that a Bayesian approximation error approach is essential for reliable source inversion when uncertainty due to wind stresses the algorithm.
Diffusion models have shown incredible capabilities as generative models; indeed, they power the current state-of-the-art models on text-conditioned image generation such as Imagen and DALL-E 2. In this work we review, demystify, and unify the understanding of diffusion models across both variational and score-based perspectives. We first derive Variational Diffusion Models (VDM) as a special case of a Markovian Hierarchical Variational Autoencoder, where three key assumptions enable tractable computation and scalable optimization of the ELBO. We then prove that optimizing a VDM boils down to learning a neural network to predict one of three potential objectives: the original source input from any arbitrary noisification of it, the original source noise from any arbitrarily noisified input, or the score function of a noisified input at any arbitrary noise level. We then dive deeper into what it means to learn the score function, and connect the variational perspective of a diffusion model explicitly with the Score-based Generative Modeling perspective through Tweedie's Formula. Lastly, we cover how to learn a conditional distribution using diffusion models via guidance.
Designing and generating new data under targeted properties has been attracting various critical applications such as molecule design, image editing and speech synthesis. Traditional hand-crafted approaches heavily rely on expertise experience and intensive human efforts, yet still suffer from the insufficiency of scientific knowledge and low throughput to support effective and efficient data generation. Recently, the advancement of deep learning induces expressive methods that can learn the underlying representation and properties of data. Such capability provides new opportunities in figuring out the mutual relationship between the structural patterns and functional properties of the data and leveraging such relationship to generate structural data given the desired properties. This article provides a systematic review of this promising research area, commonly known as controllable deep data generation. Firstly, the potential challenges are raised and preliminaries are provided. Then the controllable deep data generation is formally defined, a taxonomy on various techniques is proposed and the evaluation metrics in this specific domain are summarized. After that, exciting applications of controllable deep data generation are introduced and existing works are experimentally analyzed and compared. Finally, the promising future directions of controllable deep data generation are highlighted and five potential challenges are identified.
With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.
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