NeurIPS 2024 Papers

Score matching estimators for point processes have gained widespread attention in recent years because they do not require the calculation of intensity integrals, thereby effectively addressing the computational challenges in maximum likelihood estimation (MLE). Some existing works have proposed score matching estimators for point processes. However, this work demonstrates that the incompleteness of the estimators proposed in those works renders them applicable only to specific problems, and they fail for more general point processes. To address this issue, this work introduces the weighted score matching estimator to point processes. Theoretically, we prove the consistency of the estimator we propose. Experimental results indicate that our estimator accurately estimates model parameters on synthetic data and yields results consistent with MLE on real data. In contrast, existing score matching estimators fail to perform effectively. Codes are publicly available at \url{https://github.com/KenCao2007/WSM_TPP}.

Recent methods are proposed to improve performance of domain adaptation by inferring domain index under an adversarial variational bayesian framework, where domain index is unavailable. However, existing methods typically assume that the global domain indices are sampled from a vanilla gaussian prior, overlooking the inherent structures among different domains. To address this challenge, we propose a Bayesian Domain Adaptation with Gaussian Mixture Domain-Indexing(GMDI) algorithm. GMDI employs a Gaussian Mixture Model for domain indices, with the number of component distributions in the "domain-themes'' space adaptively determined by a Chinese Restaurant Process. By dynamically adjusting the mixtures at the domain indices level, GMDI significantly improves domain adaptation performance. Our theoretical analysis demonstrates that GMDI achieves a more stringent evidence lower bound, closer to the log-likelihood. For classification, GMDI outperforms all approaches, and surpasses the state-of-the-art method, VDI, by up to 3.4%, reaching 99.3%. For regression, GMDI reduces MSE by up to 21% (from 3.160 to 2.493), achieving the lowest errors among all methods.

This work addresses the fundamental linear inverse problem in compressive sensing (CS) by introducing a new type of regularizing generative prior. Our proposed method utilizes ideas from classical dictionary-based CS and, in particular, sparse Bayesian learning (SBL), to integrate a strong regularization towards sparse solutions. At the same time, by leveraging the notion of conditional Gaussianity, it also incorporates the adaptability from generative models to training data. However, unlike most state-of-the-art generative models, it is able to learn from a few compressed and noisy data samples and requires no optimization algorithm for solving the inverse problem. Additionally, similar to Dirichlet prior networks, our model parameterizes a conjugate prior enabling its application for uncertainty quantification. We support our approach theoretically through the concept of variational inference and validate it empirically using different types of compressible signals.

Logical Credal Networks or LCNs were recently introduced as a powerful probabilistic logic framework for representing and reasoning with imprecise knowledge. Unlike many existing formalisms, LCNs have the ability to represent cycles and allow specifying marginal and conditional probability bounds on logic formulae which may be important in many realistic scenarios. Previous work on LCNs has focused exclusively on marginal inference, namely computing posterior lower and upper probability bounds on a query formula. In this paper, we explore abductive reasoning tasks such as solving MAP and Marginal MAP queries in LCNs given some evidence. We first formally define the MAP and Marginal MAP tasks for LCNs and subsequently show how to solve these tasks exactly using search-based approaches. We then propose several approximate schemes that allow us to scale MAP and Marginal MAP inference to larger problem instances. An extensive empirical evaluation demonstrates the effectiveness of our algorithms on both random LCN instances as well as LCNs derived from more realistic use-cases.

We propose a novel approach to sequential Bayesian inference based on variational Bayes (VB). The key insight is that, in the online setting, we do not need to add the KL term to regularize to the prior (which comes from the posterior at the previous timestep); instead we can optimize just the expected log-likelihood, performing a single step of natural gradient descent starting at the prior predictive. We prove this method recovers exact Bayesian inference if the model is conjugate. We also show how to compute an efficient deterministic approximation to the VB objective, as well as our simplified objective, when the variational distribution is Gaussian or a sub-family, including the case of a diagonal plus low-rank precision matrix. We show empirically that our method outperforms other online VB methods in the non-conjugate setting, such as online learning for neural networks, especially when controlling for computational costs.

Most applications of machine learning to classification assume a closed set of balanced classes. This is at odds with the real world, where class occurrence statistics often follow a long-tailed power-law distribution and it is unlikely that all classes are seen in a single sample. Nonparametric Bayesian models naturally capture this phenomenon, but have significant practical barriers to widespread adoption, namely implementation complexity and computational inefficiency. To address this, we present a method for extracting the inductive bias from a nonparametric Bayesian model and transferring it to an artificial neural network. By simulating data with a nonparametric Bayesian prior, we can metalearn a sequence model that performs inference over an unlimited set of classes. After training, this "neural circuit" has distilled the corresponding inductive bias and can successfully perform sequential inference over an open set of classes. Our experimental results show that the metalearned neural circuit achieves comparable or better performance than particle filter-based methods for inference in these models while being faster and simpler to use than methods that explicitly incorporate Bayesian nonparametric inference.

Spectral methods are widely used to estimate eigenvectors of a low-rank signal matrix subject to noise. These methods use the leading eigenspace of an observed matrix to estimate this low-rank signal. Typically, the entrywise estimation error of these methods depends on the coherence of the low-rank signal matrix with respect to the standard basis. In this work, we present a novel method for eigenvector estimation that avoids this dependence on coherence. Assuming a rank-one signal matrix, under mild technical conditions, the entrywise estimation error of our method provably has no dependence on the coherence under Gaussian noise (i.e., in the spiked Wigner model), and achieves the optimal estimation rate up to logarithmic factors. Simulations demonstrate that our method performs well under non-Gaussian noise and that an extension of our method to the case of a rank-$r$ signal matrix has little to no dependence on the coherence. In addition, we derive new metric entropy bounds for rank-$r$ singular subspaces under $\ell_{2,\infty}$ distance, which may be of independent interest. We use these new bounds to improve the best known lower bound for rank-$r$ eigenspace estimation under $\ell_{2,\infty}$ distance.

Scaling hyperparameter optimisation to very large datasets remains an open problem in the Gaussian process community. This paper focuses on iterative methods, which use linear system solvers, like conjugate gradients, alternating projections or stochastic gradient descent, to construct an estimate of the marginal likelihood gradient. We discuss three key improvements which are applicable across solvers: (i) a pathwise gradient estimator, which reduces the required number of solver iterations and amortises the computational cost of making predictions, (ii) warm starting linear system solvers with the solution from the previous step, which leads to faster solver convergence at the cost of negligible bias, (iii) early stopping linear system solvers after a limited computational budget, which synergises with warm starting, allowing solver progress to accumulate over multiple marginal likelihood steps. These techniques provide speed-ups of up to $72\times$ when solving to tolerance, and decrease the average residual norm by up to $7\times$ when stopping early.

To generate data from trained diffusion models, most inference algorithms, such as DDPM, DDIM, and other variants, rely on discretizing the reverse SDEs or their equivalent ODEs. In this paper, we view such approaches as decomposing the entire denoising diffusion process into several segments, each corresponding to a reverse transition kernel (RTK) sampling subproblem. Specifically, DDPM uses a Gaussian approximation for the RTK, resulting in low per-subproblem complexity but requiring a large number of segments (i.e., subproblems), which is conjectured to be inefficient. To address this, we develop a general RTK framework that enables a more balanced subproblem decomposition, resulting in $\tilde O(1)$ subproblems, each with strongly log-concave targets. We then propose leveraging two fast sampling algorithms, the Metropolis-Adjusted Langevin Algorithm (MALA) and Underdamped Langevin Dynamics (ULD), for solving these strongly log-concave subproblems. This gives rise to the RTK-MALA and RTK-ULD algorithms for diffusion inference. In theory, we further develop the convergence guarantees for RTK-MALA and RTK-ULD in total variation (TV) distance: RTK-ULD can achieve $\epsilon$ target error within $\tilde{\mathcal O}(d^{1/2}\epsilon^{-1})$ under mild conditions, and RTK-MALA enjoys a $\mathcal{O}(d^{2}\log(d/\epsilon))$ convergence rate under slightly stricter conditions. These theoretical results surpass the state-of-the-art convergence rates for diffusion inference and are well supported by numerical experiments.

Recent advancements in solving Bayesian inverse problems have spotlighted denoising diffusion models (DDMs) as effective priors. Although these have great potential, DDM priors yield complex posterior distributions that are challenging to sample from. Existing approaches to posterior sampling in this context address this problem either by retraining model-specific components, leading to stiff and cumbersome methods, or by introducing approximations with uncontrolled errors that affect the accuracy of the produced samples. We present an innovative framework, divide-and-conquer posterior sampling, which leverages the inherent structure of DDMs to construct a sequence of intermediate posteriors that guide the produced samples to the target posterior. Our method significantly reduces the approximation error associated with current techniques without the need for retraining. We demonstrate the versatility and effectiveness of our approach for a wide range of Bayesian inverse problems. The code is available at \url{https://github.com/Badr-MOUFAD/dcps}

Denoising Diffusion Probabilistic Models (DDPM) have recently gained significant attention. DDPMs compose a Markovian process that begins in the data domain and gradually adds noise until reaching pure white noise. DDPMs generate high-quality samples from complex data distributions by defining an inverse process and training a deep neural network to learn this mapping. However, these models are inefficient because they require many diffusion steps to produce aesthetically pleasing samples. Additionally, unlike generative adversarial networks (GANs), the latent space of diffusion models is less interpretable. In this work, we propose to generalize the denoising diffusion process into an Upsampling Diffusion Probabilistic Model (UDPM). In the forward process, we reduce the latent variable dimension through downsampling, followed by the traditional noise perturbation. As a result, the reverse process gradually denoises and upsamples the latent variable to produce a sample from the data distribution. We formalize the Markovian diffusion processes of UDPM and demonstrate its generation capabilities on the popular FFHQ, AFHQv2, and CIFAR10 datasets. UDPM generates images with as few as three network evaluations, whose overall computational cost is less than a single DDPM or EDM step while achieving an FID score of 6.86. This surpasses current state-of-the-art efficient diffusion models that use a single denoising step for sampling. Additionally, UDPM offers an interpretable and interpolable latent space, which gives it an advantage over traditional DDPMs. Our code is available online: \url{https://github.com/shadyabh/UDPM/}

In variational inference (VI), an approximation of the posterior distribution is selected from a family of distributions through numerical optimization. With the most common variational objective function, known as the evidence lower bound (ELBO), only convergence to a *local* optimum can be guaranteed. In this work, we instead establish the *global* convergence of a particular VI method. This VI method, which may be considered an instance of neural posterior estimation (NPE), minimizes an expectation of the inclusive (forward) KL divergence to fit a variational distribution that is parameterized by a neural network. Our convergence result relies on the neural tangent kernel (NTK) to characterize the gradient dynamics that arise from considering the variational objective in function space. In the asymptotic regime of a fixed, positive-definite neural tangent kernel, we establish conditions under which the variational objective admits a unique solution in a reproducing kernel Hilbert space (RKHS). Then, we show that the gradient descent dynamics in function space converge to this unique function. In ablation studies and practical problems, we demonstrate that our results explain the behavior of NPE in non-asymptotic finite-neuron settings, and show that NPE outperforms ELBO-based optimization, which often converges to shallow local optima.

We consider the design of practically-implementable schemes for the task of channel simulation. Existing methods do not scale with the number of simultaneous uses of the channel and are therefore unable to harness the amortization gains associated with simulating many uses of the channel at once. We show how techniques from the theory of error-correcting codes can be applied to achieve scalability and hence improved performance. As an exemplar, we focus on how polar codes can be used to efficiently simulate i.i.d. copies of a class of binary-output channels.

Contrastive learning methods, such as CLIP, leverage naturally paired data—for example, images and their corresponding text captions—to learn general representations that transfer efficiently to downstream tasks. While such approaches are generally applied to two modalities, domains such as robotics, healthcare, and video need to support many types of data at once. We show that the pairwise application of CLIP fails to capture joint information between modalities, thereby limiting the quality of the learned representations. To address this issue, we present Symile, a simple contrastive learning approach that captures higher-order information between any number of modalities. Symile provides a flexible, architecture-agnostic objective for learning modality-specific representations. To develop Symile's objective, we derive a lower bound on total correlation, and show that Symile representations for any set of modalities form a sufficient statistic for predicting the remaining modalities. Symile outperforms pairwise CLIP, even with modalities missing in the data, on cross-modal classification and retrieval across several experiments including on an original multilingual dataset of 33M image, text and audio samples and a clinical dataset of chest X-rays, electrocardiograms, and laboratory measurements. All datasets and code used in this work are publicly available at https://github.com/rajesh-lab/symile.

Pearson's $r$, the most widely-used correlation coefficient, is traditionally regarded as exclusively capturing linear dependence, leading to its discouragement in contexts involving nonlinear relationships. However, recent research challenges this notion, suggesting that Pearson's $r$ should not be ruled out a priori for measuring nonlinear monotone relationships. Pearson's $r$ is essentially a scaled covariance, rooted in the renowned Cauchy-Schwarz Inequality. Our findings reveal that different scaling bounds yield coefficients with different capture ranges, and interestingly, tighter bounds actually expand these ranges. We derive a tighter inequality than Cauchy-Schwarz Inequality, leverage it to refine Pearson's $r$, and propose a new correlation coefficient, i.e., rearrangement correlation. This coefficient is able to capture arbitrary monotone relationships, both linear and nonlinear ones. It reverts to Pearson's $r$ in linear scenarios. Simulation experiments and real-life investigations show that the rearrangement correlation is more accurate in measuring nonlinear monotone dependence than the three classical correlation coefficients, and other recently proposed dependence measures.

Laplace approximations are popular techniques for endowing deep networks with epistemic uncertainty estimates as they can be applied without altering the predictions of the trained network, and they scale to large models and datasets. While the choice of prior strongly affects the resulting posterior distribution, computational tractability and lack of interpretability of the weight space typically limit the Laplace approximation to isotropic Gaussian priors, which are known to cause pathological behavior as depth increases. As a remedy, we directly place a prior on function space. More precisely, since Lebesgue densities do not exist on infinite-dimensional function spaces, we recast training as finding the so-called weak mode of the posterior measure under a Gaussian process (GP) prior restricted to the space of functions representable by the neural network. Through the GP prior, one can express structured and interpretable inductive biases, such as regularity or periodicity, directly in function space, while still exploiting the implicit inductive biases that allow deep networks to generalize. After model linearization, the training objective induces a negative log-posterior density to which we apply a Laplace approximation, leveraging highly scalable methods from matrix-free linear algebra. Our method provides improved results where prior knowledge is abundant (as is the case in many scientific inference tasks). At the same time, it stays competitive for black-box supervised learning problems, where neural networks typically excel.

Overparameterized stochastic differential equation (SDE) models have achieved remarkable success in various complex environments, such as PDE-constrained optimization, stochastic control and reinforcement learning, financial engineering, and neural SDEs. These models often feature system evolution coefficients that are parameterized by a high-dimensional vector $\theta \in \mathbb{R}^n$, aiming to optimize expectations of the SDE, such as a value function, through stochastic gradient ascent. Consequently, designing efficient gradient estimators for which the computational complexity scales well with $n$ is of significant interest. This paper introduces a novel unbiased stochastic gradient estimator—the generator gradient estimator—for which the computation time remains stable in $n$. In addition to establishing the validity of our methodology for general SDEs with jumps, we also perform numerical experiments that test our estimator in linear-quadratic control problems parameterized by high-dimensional neural networks. The results show a significant improvement in efficiency compared to the widely used pathwise differentiation method: Our estimator achieves near-constant computation times, increasingly outperforms its counterpart as $n$ increases, and does so without compromising estimation variance. These empirical findings highlight the potential of our proposed methodology for optimizing SDEs in contemporary applications.

Conditional density estimation (CDE) goes beyond regression by modeling the full conditional distribution, providing a richer understanding of the data than just the conditional mean in regression. This makes CDE particularly useful in critical application domains. However, interpretable CDE methods are understudied. Current methods typically employ kernel-based approaches, using kernel functions directly for kernel density estimation or as basis functions in linear models. In contrast, despite their conceptual simplicity and visualization suitability, tree-based methods---which are arguably more comprehensible---have been largely overlooked for CDE tasks. Thus, we propose the Conditional Density Tree (CDTree), a fully non-parametric model consisting of a decision tree in which each leaf is formed by a histogram model. Specifically, we formalize the problem of learning a CDTree using the minimum description length (MDL) principle, which eliminates the need for tuning the hyperparameter for regularization. Next, we propose an iterative algorithm that, although greedily, searches the optimal histogram for every possible node split. Our experiments demonstrate that, in comparison to existing interpretable CDE methods, CDTrees are both more accurate (as measured by the log-loss) and more robust against irrelevant features. Further, our approach leads to smaller tree sizes than existing tree-based models, which benefits interpretability.