变分推断

An essential and challenging problem in causal inference is causal effect estimation from observational data. The problem becomes more difficult with the presence of unobserved confounding variables. The front-door adjustment is an approach for dealing with unobserved confounding variables. However, the restriction for the standard front-door adjustment is difficult to satisfy in practice. In this paper, we relax some of the restrictions by proposing the concept of conditional front-door (CFD) adjustment and develop the theorem that guarantees the causal effect identifiability of CFD adjustment. By leveraging the ability of deep generative models, we propose CFDiVAE to learn the representation of the CFD adjustment variable directly from data with the identifiable Variational AutoEncoder and formally prove the model identifiability. Extensive experiments on synthetic datasets validate the effectiveness of CFDiVAE and its superiority over existing methods. The experiments also show that the performance of CFDiVAE is less sensitive to the causal strength of unobserved confounding variables. We further apply CFDiVAE to a real-world dataset to demonstrate its potential application.

Integrals with discontinuous integrands are ubiquitous, arising from discrete structure in applications like topology optimization, graphics, and computational geometry. These integrals are often part of a forward model in an inverse problem where it is necessary to reason backwards about the parameters, ideally using gradient-based optimization. Monte Carlo methods are widely used to estimate the value of integrals, but this results in a non-differentiable approximation that is amenable to neither conventional automatic differentiation nor reparameterization-based gradient methods. This significantly disrupts efforts to integrate machine learning methods in areas that exhibit these discontinuities: physical simulation and robotics, design, graphics, and computational geometry. Although bespoke domain-specific techniques can handle special cases, a general methodology to wield automatic differentiation in these discrete contexts is wanting. We introduce a differentiable variant of the simple Monte Carlo estimator which samples line segments rather than points from the domain. We justify our estimator analytically as conditional Monte Carlo and demonstrate the diverse functionality of the method as applied to image stylization, topology optimization, and computational geometry.

We present a novel variational framework for performing inference in (neural) stochastic differential equations (SDEs) driven by Markov-approximate fractional Brownian motion (fBM). SDEs offer a versatile tool for modeling real-world continuous-time dynamic systems with inherent noise and randomness. Combining SDEs with the powerful inference capabilities of variational methods, enables the learning of representative distributions through stochastic gradient descent. However, conventional SDEs typically assume the underlying noise to follow a Brownian motion (BM), which hinders their ability to capture long-term dependencies. In contrast, fractional Brownian motion (fBM) extends BM to encompass non-Markovian dynamics, but existing methods for inferring fBM parameters are either computationally demanding or statistically inefficient. In this paper, building upon the Markov approximation of fBM, we derive the evidence lower bound essential for efficient variational inference of posterior path measures, drawing from the well-established field of stochastic analysis. Additionally, we provide a closed-form expression for optimal approximation coefficients and propose to use neural networks to learn the drift, diffusion and control terms within our variational posterior, leading to the variational training of neural-SDEs. In this framework, we also optimize the Hurst index, governing the nature of our fractional noise. Beyond validation on synthetic data, we contribute a novel architecture for variational latent video prediction,—an approach that, to the best of our knowledge, enables the first variational neural-SDE application to video perception.

Deep Ensembles (DEs) demonstrate improved accuracy, calibration and robustness to perturbations over single neural networks partly due to their functional diversity. Particle-based variational inference (ParVI) methods enhance diversity by formalizing a repulsion term based on a network similarity kernel. However, weight-space repulsion is inefficient due to over-parameterization, while direct function-space repulsion has been found to produce little improvement over DEs. To sidestep these difficulties, we propose First-order Repulsive Deep Ensemble (FoRDE), an ensemble learning method based on ParVI, which performs repulsion in the space of first-order input gradients. As input gradients uniquely characterize a function up to translation and are much smaller in dimension than the weights, this method guarantees that ensemble members are functionally different. Intuitively, diversifying the input gradients encourages each network to learn different features, which is expected to improve the robustness of an ensemble. Experiments on image classification datasets and transfer learning tasks show that FoRDE significantly outperforms the gold-standard DEs and other ensemble methods in accuracy and calibration under covariate shift due to input perturbations.

Normalising Flows are non-parametric statistical models known for their dual capabilities of density estimation and generation. They are distinguished by their inherently invertible architecture. However, the requirement of invertibility imposes constraints on their expressiveness, necessitating a large number of parameters and innovative architectural designs to achieve satisfactory outcomes. Whilst flow-based models predominantly rely on neural-network-based transformations for expressive designs, alternative transformation methods have received limited attention. In this work, we present Ferumal flow, a novel kernelised normalising flow paradigm that integrates kernels into the framework. Our results demonstrate that a kernelised flow can yield competitive or superior results compared to neural network-based flows whilst maintaining parameter efficiency. Kernelised flows excel especially in the low-data regime, enabling flexible non-parametric density estimation in applications with sparse data availability.

Learning disentangled representations for time series is a promising path to facilitate reliable generalization to in- and out-of distribution (OOD), offering benefits like feature derivation and improved interpretability and fairness, thereby enhancing downstream tasks. We focus on disentangled representation learning for home appliance electricity usage, enabling users to understand and optimize their consumption for a reduced carbon footprint. Our approach frames the problem as disentangling each attribute's role in total consumption. Unlike existing methods assuming attribute independence which leads to non-identiability, we acknowledge real-world time series attribute correlations, learned up to a smooth bijection using contrastive learning and a single autoencoder. To address this, we propose a Disentanglement under Independence-Of-Support via Contrastive Learning (DIOSC), facilitating representation generalization across diverse correlated scenarios. Our method utilizes innovative \textit{l}-variational inference layers with self-attention, effectively addressing temporal dependencies across bottom-up and top-down networks. We find that DIOSC can enhance the task of representation of time series electricity consumption. We introduce TDS (Time Disentangling Score) to gauge disentanglement quality. TDS reliably reflects disentanglement performance, making it a valuable metric for evaluating time series representations disentanglement. Code available at https://institut-polytechnique-de-paris.github.io/time-disentanglement-lib.

Distributed and federated learning algorithms and techniques associated primarily with minimization problems. However, with the increase of minimax optimization and variational inequality problems in machine learning, the necessity of designing efficient distributed/federated learning approaches for these problems is becoming more apparent. In this paper, we provide a unified convergence analysis of communication-efficient local training methods for distributed variational inequality problems (VIPs). Our approach is based on a general key assumption on the stochastic estimates that allows us to propose and analyze several novel local training algorithms under a single framework for solving a class of structured non-monotone VIPs. We present the first local gradient descent-accent algorithms with provable improved communication complexity for solving distributed variational inequalities on heterogeneous data. The general algorithmic framework recovers state-of-the-art algorithms and their sharp convergence guarantees when the setting is specialized to minimization or minimax optimization problems. Finally, we demonstrate the strong performance of the proposed algorithms compared to state-of-the-art methods when solving federated minimax optimization problems.

As is well known, both sampling from the posterior and computing the mean of the posterior in Gaussian process regression reduces to solving a large linear system of equations. We study the use of stochastic gradient descent for solving this linear system, and show that when done right---by which we mean using specific insights from the optimisation and kernel communities---stochastic gradient descent is highly effective. To that end, we introduce a particularly simple stochastic dual descent algorithm, explain its design in an intuitive manner and illustrate the design choices through a series of ablation studies. Further experiments demonstrate that our new method is highly competitive. In particular, our evaluations on the UCI regression tasks and on Bayesian optimisation set our approach apart from preconditioned conjugate gradients and variational Gaussian process approximations. Moreover, our method places Gaussian process regression on par with state-of-the-art graph neural networks for molecular binding affinity prediction.

We propose a new approach for propagating stable probability distributions through neural networks. Our method is based on local linearization, which we show to be an optimal approximation in terms of total variation distance for the ReLU non-linearity. This allows propagating Gaussian and Cauchy input uncertainties through neural networks to quantify their output uncertainties. To demonstrate the utility of propagating distributions, we apply the proposed method to predicting calibrated confidence intervals and selective prediction on out-of-distribution data. The results demonstrate a broad applicability of propagating distributions and show the advantages of our method over other approaches such as moment matching.

Inverse problems aim to determine parameters from observations, a crucial task in engineering and science. Lately, generative models, especially diffusion models, have gained popularity in this area for their ability to produce realistic solutions and their good mathematical properties. Despite their success, an important drawback of diffusion models is their sensitivity to the choice of variance schedule, which controls the dynamics of the diffusion process. Fine-tuning this schedule for specific applications is crucial but time-consuming and does not guarantee an optimal result. We propose a novel approach for learning the schedule as part of the training process. Our method supports probabilistic conditioning on data, provides high-quality solutions, and is flexible, proving able to adapt to different applications with minimum overhead. This approach is tested in two unrelated inverse problems: super-resolution microscopy and quantitative phase imaging, yielding comparable or superior results to previous methods and fine-tuned diffusion models. We conclude that fine-tuning the schedule by experimentation should be avoided because it can be learned during training in a stable way that yields better results. The code is available on https://github.com/casus/cvdm

Bayesian neural networks (BNNs) offer uncertainty quantification but come with the downside of substantially increased training and inference costs. Sparse BNNs have been investigated for efficient inference, typically by either slowly introducing sparsity throughout the training or by post-training compression of dense BNNs. The dilemma of how to cut down massive training costs remains, particularly given the requirement to learn about the uncertainty. To solve this challenge, we introduce Sparse Subspace Variational Inference (SSVI), the first fully sparse BNN framework that maintains a consistently sparse Bayesian model throughout the training and inference phases. Starting from a randomly initialized low-dimensional sparse subspace, our approach alternately optimizes the sparse subspace basis selection and its associated parameters. While basis selection is characterized as a non-differentiable problem, we approximate the optimal solution with a removal-and-addition strategy, guided by novel criteria based on weight distribution statistics. Our extensive experiments show that SSVI sets new benchmarks in crafting sparse BNNs, achieving, for instance, a 10-20× compression in model size with under 3\% performance drop, and up to 20× FLOPs reduction during training. Remarkably, SSVI also demonstrates enhanced robustness to hyperparameters, reducing the need for intricate tuning in VI and occasionally even surpassing VI-trained dense BNNs.

We introduce a deterministic variational formulation for training Bayesian last layer neural networks. This yields a sampling-free, single-pass model and loss that effectively improves uncertainty estimation. Our variational Bayesian last layer (VBLL) can be trained and evaluated with only quadratic complexity in last layer width, and is thus (nearly) computationally free to add to standard architectures. We experimentally investigate VBLLs, and show that they improve predictive accuracy, calibration, and out of distribution detection over baselines across both regression and classification. Finally, we investigate combining VBLL layers with variational Bayesian feature learning, yielding a lower variance collapsed variational inference method for Bayesian neural networks.

A newly-arising uncertainty estimation method named Evidential Deep Learning (EDL), which can obtain reliable predictive uncertainty in a single forward pass, has garnered increasing interest. Guided by the subjective logic theory, EDL obtains Dirichlet concentration parameters from deep neural networks, thus constructing a Dirichlet probability density function (PDF) to model the distribution of class probabilities. Despite its great success, we argue that EDL keeps nonessential settings in both stages of model construction and optimization. In this work, our analysis indicates that (1) in the construction of the Dirichlet PDF, a commonly ignored parameter termed prior weight governs the balance between leveraging the proportion of evidence and its magnitude in deriving predictive scores, and (2) in model optimization, a variance-minimized regularization term adopted by traditional EDL encourages the Dirichlet PDF to approach a Dirac delta function, potentially exacerbating overconfidence. Therefore, we propose the R-EDL (Relaxed-EDL) method by relaxing these nonessential settings. Specifically, R-EDL treats the prior weight as an adjustable hyper-parameter instead of a fixed scalar, and directly optimizes the expectation of the Dirichlet PDF provided to deprecate the variance-minimized regularization term. Extensive experiments and SOTA performances demonstrate the effectiveness of our method. Source codes are provided in Appendix E.

With the arrival of the Noisy Intermediate-Scale Quantum (NISQ) era, Variational Quantum Algorithms (VQAs) have emerged as popular approaches to obtain possible quantum advantage in the relatively near future. In particular, how to effectively incorporate the common symmetries in physical systems as hard constraints in VQAs remains a critical and open question. In this paper, we revisit the Hamming Weight (HW) preserving ansatz and establish the links from ansatz to various symmetries and constraints, which both enlarges the usage of HW preserving ansatz and provides a coherent solution for constrained VQAs. Meanwhile, we utilize the quantum optimal control theory and quantum overparameterization theory to analyze the capability and expressivity of HW preserving ansatz and verify these theoretical results on unitary approximation problem. We conduct detailed numerical experiments on two well-studied symmetry-preserving problems, namely ground state energy estimation and feature selection in machine learning. The superior performance demonstrates the efficiency and supremacy of the proposed HW preserving ansatz on constrained VQAs.

Estimating treatment effects has numerous real-world applications in various fields, such as epidemiology and political science. While much attention has been devoted to addressing the challenge using fully observational data, there has been comparatively limited exploration of this issue in cases when the treatment is not directly observed. In this paper, we tackle this problem by developing a general variational framework, which is flexible to integrate with advanced neural network-based approaches, to identify the average dose-response function (ADRF) with the continuously valued error-contaminated treatment. Our approach begins with the formulation of a probabilistic data generation model, treating the unobserved treatment as a latent variable. In this model, we leverage a learnable density estimation neural network to derive its prior distribution conditioned on covariates. This module also doubles as a generalized propensity score estimator, effectively mitigating selection bias arising from observed confounding variables. Subsequently, we calculate the posterior distribution of the treatment, taking into account the observed measurement and outcome. To mitigate the impact of treatment error, we introduce a re-parametrized treatment value, replacing the error-affected one, to make more accurate predictions regarding the outcome. To demonstrate the adaptability of our framework, we incorporate two state-of-the-art ADRF estimation methods and rigorously assess its efficacy through extensive simulations and experiments using semi-synthetic data.

The variational autoencoder (VAE) typically employs a standard normal prior as a regularizer for the probabilistic latent encoder. However, the Gaussian tail often decays too quickly to effectively accommodate the encoded points, failing to preserve crucial structures hidden in the data. In this paper, we explore the use of heavy-tailed models to combat over-regularization. Drawing upon insights from information geometry, we propose $t^3$VAE, a modified VAE framework that incorporates Student's t-distributions for the prior, encoder, and decoder. This results in a joint model distribution of a power form which we argue can better fit real-world datasets. We derive a new objective by reformulating the evidence lower bound as joint optimization of KL divergence between two statistical manifolds and replacing with $\gamma$-power divergence, a natural alternative for power families. $t^3$VAE demonstrates superior generation of low-density regions when trained on heavy-tailed synthetic data. Furthermore, we show that $t^3$VAE significantly outperforms other models on CelebA and imbalanced CIFAR-100 datasets.

Connecting optimal transport and variational inference, we present a principled and systematic framework for sampling and generative modelling centred around divergences on path space. Our work culminates in the development of the Controlled Monte Carlo Diffusion sampler (CMCD) for Bayesian computation, a score-based annealing technique that crucially adapts both forward and backward dynamics in a diffusion model. On the way, we clarify the relationship between the EM-algorithm and iterative proportional fitting (IPF) for Schroedinger bridges, deriving as well a regularised objective that bypasses the iterative bottleneck of standard IPF-updates. Finally, we show that CMCD has a strong foundation in the Jarzinsky and Crooks identities from statistical physics, and that it convincingly outperforms competing approaches across a wide array of experiments.

Deep Ensemble approach is a straightforward technique used to enhance the performance of deep neural networks by training them from different initial points, converging towards various local optima. However, a limitation of this methodology lies in its high computational overhead for inference, arising from the necessity to store numerous learned parameters and execute individual forward passes for each parameter during the inference stage.We propose a novel approach called Diffusion Bridge Network to address this challenge. Based on the theory of Schr\"odinger bridge, this method directly learns to simulate an Stochastic Differential Equation (SDE) that connects the output distribution of a single ensemble member to the output distribution of the ensembled model, allowing us to obtain ensemble prediction without having to invoke forward pass through all the ensemble models. By substituting the heavy ensembles with this lightweight neural network constructing DBN, we achieved inference with reduced computational cost while maintaining accuracy and uncertainty scores on benchmark datasets such as CIFAR-10, CIFAR-100, and TinyImageNet.

Maximizing the marginal log-likelihood is a crucial aspect of learning latent variable models, and variational inference (VI) stands as the commonly adopted method. However, VI can encounter challenges in achieving a high marginal log-likelihood when dealing with complicated posterior distributions. In response to this limitation, we introduce a novel variational importance sampling (VIS) approach that directly estimates and maximizes the marginal log-likelihood. VIS leverages the optimal proposal distribution, achieved by minimizing the forward $\chi^2$ divergence, to enhance marginal log-likelihood estimation. We apply VIS to various popular latent variable models, including mixture models, variational auto-encoders, and partially observable generalized linear models. Results demonstrate that our approach consistently outperforms state-of-the-art baselines, in terms of both log-likelihood and model parameter estimation. Code: \url{https://github.com/JerrySoybean/vis}.

As a sub-discipline of evolutionary and computational linguistics, emergent communication (EC) studies communication protocols, called emergent languages, arising in simulations where agents communicate. A key goal of EC is to give rise to languages that share statistical properties with natural languages. In this paper, we reinterpret Lewis's signaling game, a frequently used setting in EC, as beta-VAE and reformulate its objective function as ELBO. Consequently, we clarify the existence of prior distributions of emergent languages and show that the choice of the priors can influence their statistical properties. Specifically, we address the properties of word lengths and segmentation, known as Zipf's law of abbreviation (ZLA) and Harris's articulation scheme (HAS), respectively. It has been reported that the emergent languages do not follow them when using the conventional objective. We experimentally demonstrate that by selecting an appropriate prior distribution, more natural segments emerge, while suggesting that the conventional one prevents the languages from following ZLA and HAS.