Model Uncertainty in Evolutionary Optimization and Bayesian Optimization: A Comparative Analysis

Black-box optimization problems, which are common in many real-world applications, require optimization through input-output interactions without access to internal workings. This often leads to significant computational resources being consumed for simulations. Bayesian Optimization (BO) and Surrogate-Assisted Evolutionary Algorithm (SAEA) are two widely used gradient-free optimization techniques employed to address such challenges. Both approaches follow a similar iterative procedure that relies on surrogate models to guide the search process. This paper aims to elucidate the similarities and differences in the utilization of model uncertainty between these two methods, as well as the impact of model inaccuracies on algorithmic performance. A novel model-assisted strategy is introduced, which utilizes unevaluated solutions to generate offspring, leveraging the population-based search capabilities of evolutionary algorithm to enhance the effectiveness of model-assisted optimization. Experimental results demonstrate that the proposed approach outperforms mainstream Bayesian optimization algorithms in terms of accuracy and efficiency.

Enhancing SAEAs with Unevaluated Solutions: A Case Study of Relation Model for Expensive Optimization

Surrogate-assisted evolutionary algorithms (SAEAs) hold significant importance in resolving expensive optimization problems~(EOPs). Extensive efforts have been devoted to improving the efficacy of SAEAs through the development of proficient model-assisted selection methods. However, generating high-quality solutions is a prerequisite for selection. The fundamental paradigm of evaluating a limited number of solutions in each generation within SAEAs reduces the variance of adjacent populations, thus impacting the quality of offspring solutions. This is a frequently encountered issue, yet it has not gained widespread attention. This paper presents a framework using unevaluated solutions to enhance the efficiency of SAEAs. The surrogate model is employed to identify high-quality solutions for direct generation of new solutions without evaluation. To ensure dependable selection, we have introduced two tailored relation models for the selection of the optimal solution and the unevaluated population. A comprehensive experimental analysis is performed on two test suites, which showcases the superiority of the relation model over regression and classification models in the selection phase. Furthermore, the surrogate-selected unevaluated solutions with high potential have been shown to significantly enhance the efficiency of the algorithm.

SyMOT-Flow: Learning optimal transport flow for two arbitrary distributions with maximum mean discrepancy

Finding a transformation between two unknown probability distributions from samples is crucial for modeling complex data distributions and perform tasks such as density estimation, sample generation, and statistical inference. One powerful framework for such transformations is normalizing flow, which transforms an unknown distribution into a standard normal distribution using an invertible network. In this paper, we introduce a novel model called SyMOT-Flow that trains an invertible transformation by minimizing the symmetric maximum mean discrepancy between samples from two unknown distributions, and we incorporate an optimal transport cost as regularization to obtain a short-distance and interpretable transformation. The resulted transformation leads to more stable and accurate sample generation. We establish several theoretical results for the proposed model and demonstrate its effectiveness with low-dimensional illustrative examples as well as high-dimensional generative samples obtained through the forward and reverse flows.

NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems

Bayesian methods for solving inverse problems are a powerful alternative to classical methods since the Bayesian approach gives a probabilistic description of the problems and offers the ability to quantify the uncertainty in the solution. Meanwhile, solving inverse problems by data-driven techniques also proves to be successful, due to the increasing representation ability of data-based models. In this work, we try to incorporate the data-based models into a class of Langevin-based sampling algorithms in Bayesian inference. Loosely speaking, we introduce NF-ULA (Unadjusted Langevin algorithms by Normalizing Flows), which involves learning a normalizing flow as the prior. In particular, our algorithm only requires a pre-trained normalizing flow, which is independent of the considered inverse problem and the forward operator. We perform theoretical analysis by investigating the well-posedness of the Bayesian solution and the non-asymptotic convergence of the NF-ULA algorithm. The efficacy of the proposed NF-ULA algorithm is demonstrated in various imaging problems, including image deblurring, image inpainting, and limited-angle X-ray computed tomography (CT) reconstruction.

Dataset-free Deep learning Method for Low-Dose CT Image Reconstruction

Low-dose CT (LDCT) imaging attracted a considerable interest for the reduction of the object's exposure to X-ray radiation. In recent years, supervised deep learning has been extensively studied for LDCT image reconstruction, which trains a network over a dataset containing many pairs of normal-dose and low-dose images. However, the challenge on collecting many such pairs in the clinical setup limits the application of such supervised-learning-based methods for LDCT image reconstruction in practice. Aiming at addressing the challenges raised by the collection of training dataset, this paper proposed a unsupervised deep learning method for LDCT image reconstruction, which does not require any external training data. The proposed method is built on a re-parametrization technique for Bayesian inference via deep network with random weights, combined with additional total variational (TV) regularization. The experiments show that the proposed method noticeably outperforms existing dataset-free image reconstruction methods on the test data.

Graph Neural Network for Local Corruption Recovery

Graph neural networks (GNNs) have seen a surge of development for exploiting the relational information of input graphs. Nevertheless, messages propagating through a graph contain both interpretable patterns and small perturbations. Despite global noise could be distributed over the entire graph data, it is not uncommon that corruptions appear well-concealed and merely pollute local regions while still having a vital influence on the GNN learning and prediction performance. This work tackles the graph recovery problem from local poisons by a robustness representation learning. Our developed strategy identifies regional graph perturbations and formulates a robust hidden feature representation for GNNs. A mask function pinpointed the anomalies without prior knowledge, and an $\ell_{p,q}$ regularizer defends local poisonings through pursuing sparsity in the framelet domain while maintaining a conditional closeness between the observation and new representation. The proposed robust computational unit alleviates the inertial alternating direction method of multipliers to achieve an efficient solution. Extensive experiments show that our new model recovers graph representations from local pollution and achieves excellent performance.

Semi-Implicit Back Propagation

Neural network has attracted great attention for a long time and many researchers are devoted to improve the effectiveness of neural network training algorithms. Though stochastic gradient descent (SGD) and other explicit gradient-based methods are widely adopted, there are still many challenges such as gradient vanishing and small step sizes, which leads to slow convergence and instability of SGD algorithms. Motivated by error back propagation (BP) and proximal methods, we propose a semi-implicit back propagation method for neural network training. Similar to BP, the difference on the neurons are propagated in a backward fashion and the parameters are updated with proximal mapping. The implicit update for both hidden neurons and parameters allows to choose large step size in the training algorithm. Finally, we also show that any fixed point of convergent sequences produced by this algorithm is a stationary point of the objective loss function. The experiments on both MNIST and CIFAR-10 demonstrate that the proposed semi-implicit BP algorithm leads to better performance in terms of both loss decreasing and training/validation accuracy, compared to SGD and a similar algorithm ProxBP.

An Edge Driven Wavelet Frame Model for Image Restoration

Wavelet frame systems are known to be effective in capturing singularities from noisy and degraded images. In this paper, we introduce a new edge driven wavelet frame model for image restoration by approximating images as piecewise smooth functions. With an implicit representation of image singularities sets, the proposed model inflicts different strength of regularization on smooth and singular image regions and edges. The proposed edge driven model is robust to both image approximation and singularity estimation. The implicit formulation also enables an asymptotic analysis of the proposed models and a rigorous connection between the discrete model and a general continuous variational model. Finally, numerical results on image inpainting and deblurring show that the proposed model is compared favorably against several popular image restoration models.