In recent years, graph neural networks (GNNs) have become increasingly popular for solving NP-hard combinatorial optimization (CO) problems, such as maximum cut and maximum independent set. The core idea behind these methods is to represent a CO problem as a graph and then use GNNs to learn the node/graph embedding with combinatorial information. Although these methods have achieved promising results, given a specific CO problem, the design of GNN architectures still requires heavy manual work with domain knowledge. Existing automated GNNs are mostly focused on traditional graph learning problems, which is inapplicable to solving NP-hard CO problems. To this end, we present a new class of \textbf{AUTO}mated \textbf{G}NNs for solving \textbf{NP}-hard problems, namely \textbf{AutoGNP}. We represent CO problems by GNNs and focus on two specific problems, i.e., mixed integer linear programming and quadratic unconstrained binary optimization. The idea of AutoGNP is to use graph neural architecture search algorithms to automatically find the best GNNs for a given NP-hard combinatorial optimization problem. Compared with existing graph neural architecture search algorithms, AutoGNP utilizes two-hop operators in the architecture search space. Moreover, AutoGNP utilizes simulated annealing and a strict early stopping policy to avoid local optimal solutions. Empirical results on benchmark combinatorial problems demonstrate the superiority of our proposed model.

Quadratic programming (QP) is the most widely applied category of problems in nonlinear programming. Many applications require real-time/fast solutions, though not necessarily with high precision. Existing methods either involve matrix decomposition or use the preconditioned conjugate gradient method. For relatively large instances, these methods cannot achieve the real-time requirement unless there is an effective precondition. Recently, graph neural networks (GNNs) opened new possibilities for QP. Some promising empirical studies of applying GNNs for QP tasks show that GNNs can capture key characteristics of an optimization instance and provide adaptive guidance accordingly to crucial configurations during the solving process, or directly provide an approximate solution. Despite notable empirical observations, theoretical foundations are still lacking. In this work, we investigate the expressive or representative power of GNNs, a crucial aspect of neural network theory, specifically in the context of QP tasks, with both continuous and mixed-integer settings. We prove the existence of message-passing GNNs that can reliably represent key properties of quadratic programs, including feasibility, optimal objective value, and optimal solution. Our theory is validated by numerical results.

We present a fully probabilistic approach for solving binary optimization problems with black-box objective functions and with budget constraints. In the probabilistic approach, the optimization variable is viewed as a random variable and is associated with a parametric probability distribution. The original optimization problem is replaced with an optimization over the expected value of the original objective, which is then optimized over the probability distribution parameters. The resulting optimal parameter (optimal policy) is used to sample the binary space to produce estimates of the optimal solution(s) of the original binary optimization problem. The probability distribution is chosen from the family of Bernoulli models because the optimization variable is binary. The optimization constraints generally restrict the feasibility region. This can be achieved by modeling the random variable with a conditional distribution given satisfiability of the constraints. Thus, in this work we develop conditional Bernoulli distributions to model the random variable conditioned by the total number of nonzero entries, that is, the budget constraint. This approach (a) is generally applicable to binary optimization problems with nonstochastic black-box objective functions and budget constraints; (b) accounts for budget constraints by employing conditional probabilities that sample only the feasible region and thus considerably reduces the computational cost compared with employing soft constraints; and (c) does not employ soft constraints and thus does not require tuning of a regularization parameter, for example to promote sparsity, which is challenging in sensor placement optimization problems. The proposed approach is verified numerically by using an idealized bilinear binary optimization problem and is validated by using a sensor placement experiment in a parameter identification setup.

Chung's lemma is a classical tool for establishing asymptotic convergence rates of (stochastic) optimization methods under strong convexity-type assumptions and appropriate polynomial diminishing step sizes. In this work, we develop a generalized version of Chung's lemma, which provides a simple non-asymptotic convergence framework for a more general family of step size rules. We demonstrate broad applicability of the proposed generalized Chung's lemma by deriving tight non-asymptotic convergence rates for a large variety of stochastic methods. In particular, we obtain partially new non-asymptotic complexity results for stochastic optimization methods, such as stochastic gradient descent and random reshuffling, under a general $(\theta,\mu)$-Polyak-Lojasiewicz (PL) condition and for various step sizes strategies, including polynomial, constant, exponential, and cosine step sizes rules. Notably, as a by-product of our analysis, we observe that exponential step sizes can adapt to the objective function's geometry, achieving the optimal convergence rate without requiring exact knowledge of the underlying landscape. Our results demonstrate that the developed variant of Chung's lemma offers a versatile, systematic, and streamlined approach to establish non-asymptotic convergence rates under general step size rules.

In this study, we propose a machine learning method called Distributionally Robust Safe Sample Screening (DRSSS). DRSSS aims to identify unnecessary training samples, even when the distribution of the training samples changes in the future. To achieve this, we effectively combine the distributionally robust (DR) paradigm, which aims to enhance model robustness against variations in data distribution, with the safe sample screening (SSS), which identifies unnecessary training samples prior to model training. Since we need to consider an infinite number of scenarios regarding changes in the distribution, we applied SSS because it does not require model training after the change of the distribution. In this paper, we employed the covariate shift framework to represent the distribution of training samples and reformulated the DR covariate-shift problem as a weighted empirical risk minimization problem, where the weights are subject to uncertainty within a predetermined range. By extending the existing SSS technique to accommodate this weight uncertainty, the DRSSS method is capable of reliably identifying unnecessary samples under any future distribution within a specified range. We provide a theoretical guarantee for the DRSSS method and validate its performance through numerical experiments on both synthetic and real-world datasets.

Vehicular Metaverses are developed to enhance the modern automotive industry with an immersive and safe experience among connected vehicles and roadside infrastructures, e.g., RoadSide Units (RSUs). For seamless synchronization with virtual spaces, Vehicle Twins (VTs) are constructed as digital representations of physical entities. However, resource-intensive VTs updating and high mobility of vehicles require intensive computation, communication, and storage resources, especially for their migration among RSUs with limited coverages. To address these issues, we propose an attribute-aware auction-based mechanism to optimize resource allocation during VTs migration by considering both price and non-monetary attributes, e.g., location and reputation. In this mechanism, we propose a two-stage matching for vehicular users and Metaverse service providers in multi-attribute resource markets. First, the resource attributes matching algorithm obtains the resource attributes perfect matching, namely, buyers and sellers can participate in a double Dutch auction (DDA). Then, we train a DDA auctioneer using a generative pre-trained transformer (GPT)-based deep reinforcement learning (DRL) algorithm to adjust the auction clocks efficiently during the auction process. We compare the performance of social welfare and auction information exchange costs with state-of-the-art baselines under different settings. Simulation results show that our proposed GPT-based DRL auction schemes have better performance than others.

The field of image synthesis is currently flourishing due to the advancements in diffusion models. While diffusion models have been successful, their computational intensity has prompted the pursuit of more efficient alternatives. As a representative work, non-autoregressive Transformers (NATs) have been recognized for their rapid generation. However, a major drawback of these models is their inferior performance compared to diffusion models. In this paper, we aim to re-evaluate the full potential of NATs by revisiting the design of their training and inference strategies. Specifically, we identify the complexities in properly configuring these strategies and indicate the possible sub-optimality in existing heuristic-driven designs. Recognizing this, we propose to go beyond existing methods by directly solving the optimal strategies in an automatic framework. The resulting method, named AutoNAT, advances the performance boundaries of NATs notably, and is able to perform comparably with the latest diffusion models at a significantly reduced inference cost. The effectiveness of AutoNAT is validated on four benchmark datasets, i.e., ImageNet-256 & 512, MS-COCO, and CC3M. Our code is available at https://github.com/LeapLabTHU/ImprovedNAT.

The selection of the most appropriate algorithm to solve a given problem instance, known as algorithm selection, is driven by the potential to capitalize on the complementary performance of different algorithms across sets of problem instances. However, determining the optimal algorithm for an unseen problem instance has been shown to be a challenging task, which has garnered significant attention from researchers in recent years. In this survey, we conduct an overview of the key contributions to algorithm selection in the field of single-objective continuous black-box optimization. We present ongoing work in representation learning of meta-features for optimization problem instances, algorithm instances, and their interactions. We also study machine learning models for automated algorithm selection, configuration, and performance prediction. Through this analysis, we identify gaps in the state of the art, based on which we present ideas for further development of meta-feature representations.

Preference learning (PL) with large language models (LLMs) aims to align the LLMs' generations with human preferences. Previous work on reinforcement learning from human feedback (RLHF) has demonstrated promising results in in-distribution PL. However, due to the difficulty of obtaining human feedback, discretely training reward models for every encountered distribution is challenging. Thus, out-of-distribution (OOD) PL is practically useful for enhancing the generalization ability of LLMs with limited preference feedback. This work addresses OOD PL by optimizing a general reward model through a meta-learning approach. During meta-training, a bilevel optimization algorithm is utilized to learn a reward model capable of guiding policy learning to align with human preferences across various distributions. When encountering a test distribution, the meta-test procedure conducts regularized policy optimization using the learned reward model for PL. We theoretically demonstrate the convergence rate of the bilevel optimization algorithm under reasonable assumptions. Additionally, we conduct experiments on two text generation tasks across 20 held-out domains and outperform a variety of strong baselines across various evaluation metrics.

Low-rank adaptation (LoRA) is an efficient strategy for adapting latent diffusion models (LDMs) on a private dataset to generate specific images by minimizing the adaptation loss. However, the LoRA-adapted LDMs are vulnerable to membership inference (MI) attacks that can judge whether a particular data point belongs to the private dataset, thus leading to the privacy leakage. To defend against MI attacks, we first propose a straightforward solution: Membership-Privacy-preserving LoRA (MP-LoRA). MP-LoRA is formulated as a min-max optimization problem where a proxy attack model is trained by maximizing its MI gain while the LDM is adapted by minimizing the sum of the adaptation loss and the MI gain of the proxy attack model. However, we empirically find that MP-LoRA has the issue of unstable optimization, and theoretically analyze that the potential reason is the unconstrained local smoothness, which impedes the privacy-preserving adaptation. To mitigate this issue, we further propose a Stable Membership-Privacy-preserving LoRA (SMP-LoRA) that adapts the LDM by minimizing the ratio of the adaptation loss to the MI gain. Besides, we theoretically prove that the local smoothness of SMP-LoRA can be constrained by the gradient norm, leading to improved convergence. Our experimental results corroborate that SMP-LoRA can indeed defend against MI attacks and generate high-quality images. Our code is available at https://github.com/WilliamLUO0/StablePrivateLoRA.