Inexact Moreau Envelope Lagrangian Method for Non-Convex Constrained Optimization under Local Error Bound Conditions on Constraint Functions

In this paper, we study the inexact Moreau envelope Lagrangian (iMELa) method for solving smooth non-convex optimization problems over a simple polytope with additional convex inequality constraints. By incorporating a proximal term into the traditional Lagrangian function, the iMELa method approximately solves a convex optimization subproblem over the polyhedral set at each main iteration. Under the assumption of a local error bound condition for subsets of the feasible set defined by subsets of the constraints, we establish that the iMELa method can find an $\epsilon$-Karush-Kuhn-Tucker point with $\tilde O(\epsilon^{-2})$ gradient oracle complexity.

A stochastic smoothing framework for nonconvex-nonconcave min-sum-max problems with applications to Wasserstein distributionally robust optimization

Applications such as adversarially robust training and Wasserstein Distributionally Robust Optimization (WDRO) can be naturally formulated as min-sum-max optimization problems. While this formulation can be rewritten as an equivalent min-max problem, the summation of max terms introduces computational challenges, including increased complexity and memory demands, which must be addressed. These challenges are particularly evident in WDRO, where existing tractable algorithms often rely on restrictive assumptions on the objective function, limiting their applicability to state-of-the-art machine learning problems such as the training of deep neural networks. This study introduces a novel stochastic smoothing framework based on the \mbox{log-sum-exp} function, efficiently approximating the max operator in min-sum-max problems. By leveraging the Clarke regularity of the max operator, we develop an iterative smoothing algorithm that addresses these computational difficulties and guarantees almost surely convergence to a Clarke/directional stationary point. We further prove that the proposed algorithm finds an $\epsilon$-scaled Clarke stationary point of the original problem, with a worst-case iteration complexity of $\widetilde{O}(\epsilon^{-3})$. Our numerical experiments demonstrate that our approach outperforms or is competitive with state-of-the-art methods in solving the newsvendor problem, deep learning regression, and adversarially robust deep learning. The results highlight that our method yields more accurate and robust solutions in these challenging problem settings.

Universal Semantic Embeddings of Chemical Elements for Enhanced Materials Inference and Discovery

We present a framework for generating universal semantic embeddings of chemical elements to advance materials inference and discovery. This framework leverages ElementBERT, a domain-specific BERT-based natural language processing model trained on 1.29 million abstracts of alloy-related scientific papers, to capture latent knowledge and contextual relationships specific to alloys. These semantic embeddings serve as robust elemental descriptors, consistently outperforming traditional empirical descriptors with significant improvements across multiple downstream tasks. These include predicting mechanical and transformation properties, classifying phase structures, and optimizing materials properties via Bayesian optimization. Applications to titanium alloys, high-entropy alloys, and shape memory alloys demonstrate up to 23% gains in prediction accuracy. Our results show that ElementBERT surpasses general-purpose BERT variants by encoding specialized alloy knowledge. By bridging contextual insights from scientific literature with quantitative inference, our framework accelerates the discovery and optimization of advanced materials, with potential applications extending beyond alloys to other material classes.

A single-loop SPIDER-type stochastic subgradient method for expectation-constrained nonconvex nonsmooth optimization

Many real-world problems, such as those with fairness constraints, involve complex expectation constraints and large datasets, necessitating the design of efficient stochastic methods to solve them. Most existing research focuses on cases with no {constraint} or easy-to-project constraints or deterministic constraints. In this paper, we consider nonconvex nonsmooth stochastic optimization problems with expectation constraints, for which we build a novel exact penalty model. We first show the relationship between the penalty model and the original problem. Then on solving the penalty problem, we present a single-loop SPIDER-type stochastic subgradient method, which utilizes the subgradients of both the objective and constraint functions, as well as the constraint function value at each iteration. Under certain regularity conditions (weaker than Slater-type constraint qualification or strong feasibility assumed in existing works), we establish an iteration complexity result of $O(\epsilon^{-4})$ to reach a near-$\epsilon$ stationary point of the penalized problem in expectation, matching the lower bound for such tasks. Building on the exact penalization, an $(\epsilon,\epsilon)$-KKT point of the original problem is obtained. For a few scenarios, our complexity of either the {objective} sample subgradient or the constraint sample function values can be lower than the state-of-the-art results by a factor of $\epsilon^{-2}$. Moreover, on solving two fairness-constrained problems, our method is significantly (up to 466 times) faster than the state-of-the-art algorithms, including switching subgradient method and inexact proximal point methods.

PersonaMagic: Stage-Regulated High-Fidelity Face Customization with Tandem Equilibrium

Personalized image generation has made significant strides in adapting content to novel concepts. However, a persistent challenge remains: balancing the accurate reconstruction of unseen concepts with the need for editability according to the prompt, especially when dealing with the complex nuances of facial features. In this study, we delve into the temporal dynamics of the text-to-image conditioning process, emphasizing the crucial role of stage partitioning in introducing new concepts. We present PersonaMagic, a stage-regulated generative technique designed for high-fidelity face customization. Using a simple MLP network, our method learns a series of embeddings within a specific timestep interval to capture face concepts. Additionally, we develop a Tandem Equilibrium mechanism that adjusts self-attention responses in the text encoder, balancing text description and identity preservation, improving both areas. Extensive experiments confirm the superiority of PersonaMagic over state-of-the-art methods in both qualitative and quantitative evaluations. Moreover, its robustness and flexibility are validated in non-facial domains, and it can also serve as a valuable plug-in for enhancing the performance of pretrained personalization models.

Projected gradient methods for nonconvex and stochastic optimization: new complexities and auto-conditioned stepsizes

We present a novel class of projected gradient (PG) methods for minimizing a smooth but not necessarily convex function over a convex compact set. We first provide a novel analysis of the "vanilla" PG method, achieving the best-known iteration complexity for finding an approximate stationary point of the problem. We then develop an "auto-conditioned" projected gradient (AC-PG) variant that achieves the same iteration complexity without requiring the input of the Lipschitz constant of the gradient or any line search procedure. The key idea is to estimate the Lipschitz constant using first-order information gathered from the previous iterations, and to show that the error caused by underestimating the Lipschitz constant can be properly controlled. We then generalize the PG methods to the stochastic setting, by proposing a stochastic projected gradient (SPG) method and a variance-reduced stochastic gradient (VR-SPG) method, achieving new complexity bounds in different oracle settings. We also present auto-conditioned stepsize policies for both stochastic PG methods and establish comparable convergence guarantees.

Task-Oriented Diffusion Inversion for High-Fidelity Text-based Editing

Recent advancements in text-guided diffusion models have unlocked powerful image manipulation capabilities, yet balancing reconstruction fidelity and editability for real images remains a significant challenge. In this work, we introduce \textbf{T}ask-\textbf{O}riented \textbf{D}iffusion \textbf{I}nversion (\textbf{TODInv}), a novel framework that inverts and edits real images tailored to specific editing tasks by optimizing prompt embeddings within the extended \(\mathcal{P}^*\) space. By leveraging distinct embeddings across different U-Net layers and time steps, TODInv seamlessly integrates inversion and editing through reciprocal optimization, ensuring both high fidelity and precise editability. This hierarchical editing mechanism categorizes tasks into structure, appearance, and global edits, optimizing only those embeddings unaffected by the current editing task. Extensive experiments on benchmark dataset reveal TODInv's superior performance over existing methods, delivering both quantitative and qualitative enhancements while showcasing its versatility with few-step diffusion model.

Human Video Translation via Query Warping

In this paper, we present QueryWarp, a novel framework for temporally coherent human motion video translation. Existing diffusion-based video editing approaches that rely solely on key and value tokens to ensure temporal consistency, which scarifies the preservation of local and structural regions. In contrast, we aim to consider complementary query priors by constructing the temporal correlations among query tokens from different frames. Initially, we extract appearance flows from source poses to capture continuous human foreground motion. Subsequently, during the denoising process of the diffusion model, we employ appearance flows to warp the previous frame's query token, aligning it with the current frame's query. This query warping imposes explicit constraints on the outputs of self-attention layers, effectively guaranteeing temporally coherent translation. We perform experiments on various human motion video translation tasks, and the results demonstrate that our QueryWarp framework surpasses state-of-the-art methods both qualitatively and quantitatively.

DiffusionMat: Alpha Matting as Sequential Refinement Learning

In this paper, we introduce DiffusionMat, a novel image matting framework that employs a diffusion model for the transition from coarse to refined alpha mattes. Diverging from conventional methods that utilize trimaps merely as loose guidance for alpha matte prediction, our approach treats image matting as a sequential refinement learning process. This process begins with the addition of noise to trimaps and iteratively denoises them using a pre-trained diffusion model, which incrementally guides the prediction towards a clean alpha matte. The key innovation of our framework is a correction module that adjusts the output at each denoising step, ensuring that the final result is consistent with the input image's structures. We also introduce the Alpha Reliability Propagation, a novel technique designed to maximize the utility of available guidance by selectively enhancing the trimap regions with confident alpha information, thus simplifying the correction task. To train the correction module, we devise specialized loss functions that target the accuracy of the alpha matte's edges and the consistency of its opaque and transparent regions. We evaluate our model across several image matting benchmarks, and the results indicate that DiffusionMat consistently outperforms existing methods. Project page at~\url{https://cnnlstm.github.io/DiffusionMat

Damped Proximal Augmented Lagrangian Method for weakly-Convex Problems with Convex Constraints

We give a damped proximal augmented Lagrangian method (DPALM) for solving problems with a weakly-convex objective and convex linear/nonlinear constraints. Instead of taking a full stepsize, DPALM adopts a damped dual stepsize to ensure the boundedness of dual iterates. We show that DPALM can produce a (near) $\vareps$-KKT point within $O(\vareps^{-2})$ outer iterations if each DPALM subproblem is solved to a proper accuracy. In addition, we establish overall iteration complexity of DPALM when the objective is either a regularized smooth function or in a regularized compositional form. For the former case, DPALM achieves the complexity of $\widetilde{\mathcal{O}}\left(\varepsilon^{-2.5} \right)$ to produce an $\varepsilon$-KKT point by applying an accelerated proximal gradient (APG) method to each DPALM subproblem. For the latter case, the complexity of DPALM is $\widetilde{\mathcal{O}}\left(\varepsilon^{-3} \right)$ to produce a near $\varepsilon$-KKT point by using an APG to solve a Moreau-envelope smoothed version of each subproblem. Our outer iteration complexity and the overall complexity either generalize existing best ones from unconstrained or linear-constrained problems to convex-constrained ones, or improve over the best-known results on solving the same-structured problems. Furthermore, numerical experiments on linearly/quadratically constrained non-convex quadratic programs and linear-constrained robust nonlinear least squares are conducted to demonstrate the empirical efficiency of the proposed DPALM over several state-of-the art methods.