Nonconvex Optimization

We demonstrate that architectures which traditionally are considered to be ill-suited for a task can be trained using inductive biases from another architecture. We call a network untrainable when it overfits, underfits, or converges to poor results even when tuning their hyperparameters. For example, fully connected networks overfit on object recognition while deep convolutional networks without residual connections underfit. The traditional answer is to change the architecture to impose some inductive bias, although the nature of that bias is unknown. We introduce guidance, where a guide network steers a target network using a neural distance function. The target minimizes its task loss plus a layerwise representational similarity against the frozen guide. If the guide is trained, this transfers over the architectural prior and knowledge of the guide to the target. If the guide is untrained, this transfers over only part of the architectural prior of the guide. We show that guidance prevents FCN overfitting on ImageNet, narrows the vanilla RNN–Transformer gap, boosts plain CNNs toward ResNet accuracy, and aids Transformers on RNN-favored tasks. We further identify that guidance-driven initialization alone can mitigate FCN overfitting. Our method provides a mathematical tool to investigate priors and architectures, and in the long term, could automate architecture design.

Plug-and-play (PnP) methods with deep denoisers have shown impressive results in imaging problems. They typically require strong convexity or smoothness of the fidelity term and a (residual) non-expansive denoiser for convergence. These assumptions, however, are violated in Poisson inverse problems, and non-expansiveness can hinder denoising performance. To address these challenges, we propose a cocoercive conservative (CoCo) denoiser, which may be (residual) expansive, leading to improved denoising performance. By leveraging the generalized Helmholtz decomposition, we introduce a novel training strategy that combines Hamiltonian regularization to promote conservativeness and spectral regularization to ensure cocoerciveness. We prove that CoCo denoiser is a proximal operator of a weakly convex function, enabling a restoration model with an implicit weakly convex prior. The global convergence of PnP methods to a stationary point of this restoration model is established. Extensive experimental results demonstrate that our approach outperforms closely related methods in both visual quality and quantitative metrics.

Tools from control and dynamical systems have proven valuable for analyzing and developing optimization methods. In this paper, we establish rigorous theoretical foundations for using feedback linearization—a well-established nonlinear control technique—to solve constrained optimization problems. For equality-constrained optimization, we establish global convergence rates to first-order Karush-Kuhn-Tucker (KKT) points and uncover the close connection between the FL method and the Sequential Quadratic Programming (SQP) algorithm. Building on this relationship, we extend the FL approach to handle inequality-constrained problems. Furthermore, we introduce a momentum-accelerated feedback linearization algorithm and provide a rigorous convergence guarantee.

Adversarial attacks are widely used to evaluate model robustness, and in black-box scenarios, the transferability of these attacks becomes crucial. Existing generator-based attacks have excellent generalization and transferability due to their instance-agnostic nature. However, when training generators for multi-target tasks, the success rate of transfer attacks is relatively low due to the limitations of the model's capacity. To address these challenges, we propose a novel Dual-Flow framework for multi-target instance-agnostic adversarial attacks, utilizing Cascading Distribution Shift Training to develop an adversarial velocity function. Extensive experiments demonstrate that Dual-Flow significantly improves transferability over previous multi-target generative attacks. For example, it increases the success rate from Inception-v3 to ResNet-152 by 34.58%. Furthermore, our attack method shows substantially stronger robustness against defense mechanisms, such as adversarially trained models.

Evolutionary algorithms (EAs) have proven effective in exploring the vast solution spaces typical of graph-structured combinatorial problems. However, traditional encoding schemes, such as binary or numerical representations, often fail to straightforwardly capture the intricate structural properties of networks. Through employing the image-based encoding to preserve topological context, this study utilizes multimodal large language models (MLLMs) as evolutionary operators to facilitate structure-aware optimization over graph data. To address the visual clutter inherent in large-scale network visualizations, we leverage graph sparsification techniques to simplify structures while maintaining essential structural features. To further improve robustness and mitigate bias from different sparsification views, we propose a cooperative evolutionary optimization framework that facilitates cross-domain knowledge transfer and unifies multiple sparsified variants of diverse structures. Additionally, recognizing the sensitivity of MLLMs to network layout, we introduce an ensemble strategy that aggregates outputs from various layout configurations through consensus voting. Finally, experiments on real-world networks through various tasks demonstrate that our approach improves both the quality and reliability of solutions in MLLM-driven evolutionary optimization.

Low-Rank Adaptation (LoRA) lowers the computational and memory overhead of fine-tuning large models by updating a low-dimensional subspace of the pre-trained weight matrix. Albeit efficient, LoRA exhibits suboptimal convergence and noticeable performance degradation, due to inconsistent and imbalanced weight updates induced by its nonunique low-rank factorizations. To overcome these limitations, this article identifies the optimal low-rank factorization per step that minimizes an upper bound on the loss. The resultant refactored low-rank adaptation (RefLoRA) method promotes a flatter loss landscape, along with consistent and balanced weight updates, thus speeding up stable convergence. Extensive experiments evaluate RefLoRA on natural language understanding, and commonsense reasoning tasks with popular large language models including DeBERTaV3, LLaMA-7B, LLaMA2-7B and LLaMA3-8B. The numerical tests corroborate that RefLoRA converges faster, outperforms various benchmarks, and enjoys negligible computational overhead compared to state-of-the-art LoRA variants.

Physics sensing plays a central role in many scientific and engineering domains, which inherently involves two coupled tasks: reconstructing dense physical fields from sparse observations and optimizing scattered sensor placements to observe maximum information. While deep learning has made rapid advances in sparse-data reconstruction, existing methods generally omit optimization of sensor placements, leaving the mutual enhancement between reconstruction and placement on the shelf. To change this suboptimal practice, we propose PhySense, a synergistic two-stage framework that learns to jointly reconstruct physical fields and to optimize sensor placements, both aiming for accurate physics sensing. The first stage involves a flow-based generative model enhanced by cross-attention to adaptively fuse sparse observations. Leveraging the reconstruction feedback, the second stage performs sensor placement via projected gradient descent to satisfy spatial constraints. We further prove that the learning objectives of the two stages are consistent with classical variance-minimization principles, providing theoretical guarantees. Extensive experiments across three challenging benchmarks, especially a 3D geometry dataset, indicate PhySense achieves state-of-the-art physics sensing accuracy and discovers informative sensor placements previously unconsidered. Code is available at this repository: https://github.com/thuml/PhySense.

Recent advances in deep learning optimization have unveiled two intriguing phenomena under large learning rates: Edge of Stability (EoS) and Progressive Sharpening (PS), challenging classical Gradient Descent (GD) analyses. Current research approaches, using either generalist frameworks or minimalist examples, face significant limitations in explaining these phenomena. This paper advances the minimalist approach by introducing a two-layer network with a two-dimensional input, where one dimension is relevant to the response and the other is irrelevant. Through this model, we rigorously prove the existence of progressive sharpening and self-stabilization under large learning rates, and establish non-asymptotic analysis of the training dynamics and sharpness along the entire GD trajectory. Besides, we connect our minimalist example to existing works by reconciling the existence of a well-behaved "stable set" between minimalist and generalist analyses, and extending the analysis of Gradient Flow Solution sharpness to our two-dimensional input scenario. These findings provide new insights into the EoS phenomenon from both parameter and input data distribution perspectives, potentially informing more effective optimization strategies in deep learning practice.

Various forms of sparse attention have been explored to mitigate the quadratic computational and memory cost of the attention mechanism in transformers. We study sparse transformers not through a lens of efficiency but rather in terms of learnability and generalization. Empirically studying a range of attention mechanisms, we find that input-dependent sparse attention models appear to converge faster and generalize better than standard attention models, while input-agnostic sparse attention models show no such benefits -- a phenomenon that is robust across architectural and optimization hyperparameter choices. This can be interpreted as demonstrating that concentrating a model's "semantic focus" with respect to the tokens currently being considered (in the form of input-dependent sparse attention) accelerates learning. We develop a theoretical characterization of the conditions that explain this behavior. We establish a connection between the stability of the standard softmax and the loss function's Lipschitz properties, then show how sparsity affects the stability of the softmax and the subsequent convergence and generalization guarantees resulting from the attention mechanism. This allows us to theoretically establish that input-agnostic sparse attention does not provide any benefits. We also characterize conditions when semantic focus (input-dependent sparse attention) can provide improved guarantees, and we validate that these conditions are in fact met in our empirical evaluations.

Neural combinatorial optimization (NCO) has achieved remarkable performance, yet its learned model representations and decision rationale remain a black box. This impedes both academic research and practical deployment, since researchers and stakeholders require deeper insights into NCO models. In this paper, we take the first critical step towards interpreting NCO models by investigating their representations through various probing tasks. Moreover, we introduce a novel probing tool named Coefficient Significance Probing (CS-Probing) to enable deeper analysis of NCO representations by examining the coefficients and statistical significance during probing. Extensive experiments and analysis reveal that NCO models encode low-level information essential for solution construction, while capturing high-level knowledge to facilitate better decisions. Using CS-Probing, we find that prevalent NCO models impose varying inductive biases on their learned representations, uncover direct evidence related to model generalization, and identify key embedding dimensions associated with specific knowledge. These insights can be potentially translated into practice, for example, with minor code modifications, we improve the generalization of the analyzed model. Our work represents a first systematic attempt to interpret black-box NCO models, showcasing probing as a promising tool for analyzing their internal mechanisms and revealing insights for the NCO community. The source code is publicly available.

We introduce Parametric Density Path Optimization (PDPO), a novel method for computing action-minimizing paths between probability densities. The core idea is to represent the target probability path as the pushforward of a reference density through a parametric map, transforming the original infinite-dimensional optimization over densities to a finite-dimensional one over the parameters of the map. We derive a static formulation of the dynamic problem of action minimization and propose cubic spline interpolation of the path in parameter space to solve the static problem. Theoretically, we establish an error bound of the action under proper assumptions on the regularity of the parameter path. Empirically, we find that using 3–5 control points of the spline interpolation suffices to accurately resolve both multimodal and high-dimensional problems. We demonstrate that PDPO can flexibly accommodate a wide range of potential terms, including those modeling obstacles, mean-field interactions, stochastic control, and higher-order dynamics. Our method outperforms existing state-of-the-art approaches in benchmark tasks, demonstrating superior computational efficiency and solution quality.

We study Online Convex Optimization with adversarial constraints (COCO). At each round a learner selects an action from a convex decision set and then an adversary reveals a convex cost and a convex constraint function. The goal of the learner is to select a sequence of actions to minimize both regret and the cumulative constraint violation (CCV) over a horizon of length $T$. The best-known policy for this problem achieves $O(\sqrt{T})$ regret and $\tilde{O}(\sqrt{T})$ CCV. In this paper, we improve this by trading off regret to achieve substantially smaller CCV. This trade-off is especially important in safety-critical applications, where satisfying the safety constraints is non-negotiable. Specifically, for any bounded convex cost and constraint functions, we propose an online policy that achieves $\tilde{O}(\sqrt{dT}+ T^\beta)$ regret and $\tilde{O}(dT^{1-\beta})$ CCV, where $d$ is the dimension of the decision set and $\beta \in [0,1]$ is a tunable parameter. We begin with a special case, called the $\textsf{Constrained Expert}$ problem, where the decision set is a probability simplex and the cost and constraint functions are linear. Leveraging a new adaptive small-loss regret bound, we propose a computationally efficient policy for the $\textsf{Constrained Expert}$ problem, that attains $O(\sqrt{T\ln N}+T^{\beta})$ regret and $\tilde{O}(T^{1-\beta} \ln N)$ CCV for $N$ number of experts. The original problem is then reduced to the $\textsf{Constrained Expert}$ problem via a covering argument. Finally, with an additional $M$-smoothness assumption, we propose a computationally efficient first-order policy attaining $O(\sqrt{MT}+T^{\beta})$ regret and $\tilde{O}(MT^{1-\beta})$ CCV.

The substantial memory demands of pre-training and fine-tuning large language models (LLMs) require memory-efficient optimization algorithms. One promising approach is layer-wise optimization, which treats each transformer block as a single layer and optimizes it sequentially, while freezing the other layers to save optimizer states and activations. Although effective, these methods ignore the varying importance of the modules within each layer, leading to suboptimal performance. Moreover, layer-wise sampling provides only limited memory savings, as at least one full layer must remain active during optimization. To overcome these limitations, we propose **M**odule-wise **I**mportance **SA**mpling (**MISA**), a novel method that divides each layer into smaller modules and assigns importance scores to each module. MISA uses a weighted random sampling mechanism to activate modules, provably reducing gradient variance compared to layer-wise sampling. Additionally, we establish an $\mathcal{O}(1/\sqrt{K})$ convergence rate under non-convex and stochastic conditions, where $K$ is the total number of training steps, and provide a detailed memory analysis showcasing MISA's superiority over existing baseline methods. Experiments on diverse learning tasks validate the effectiveness of MISA.

While large language models (LLMs) show considerable promise across various fields, they have notable limitations in handling multi-document question answering (Multi-doc QA) tasks. The first challenge is long-range dependency modeling, where LLMs struggle to focus on key information in long texts, which weakens important semantic connections. Second, most LLMs suffer from the ''lost-in-the-middle'' issue, where they have difficulty processing information in the middle of long inputs. Current solutions either truncate global dependencies or demand costly finetuning, ultimately lacking a universal and simple solution for these challenges. To resolve these limitations, we propose Dual-Stage Adaptive Sharpening (DSAS) containing two modules. (i) The Contextual Gate Weighting (CGW) module alleviates ''lost-in-the-middle'' by assessing paragraph relevance through layer-wise attention tracking and position-aware weighting. (ii) The Reciprocal Attention Suppression (RAS) module enhances focus on critical paragraphs by suppressing information exchange between key and irrelevant texts, thus mitigating the limitations in long-range dependency modeling. Extensive experiments on four benchmarks demonstrate DSAS's efficacy across mainstream LLMs (Llama, Qwen, Mistral, and Deepseek), with an average F1-score improvement of 4.2% in Multi-doc QA tasks on Llama-3.1-8B-Instruct and Qwen2.5-14B-Instruct. Ablation studies confirm the essential contributions of both the CGW and RAS modules. In addition, detailed discussions in the Appendix further validate the robustness and scalability of DSAS.

Mathematical optimization is a fundamental tool for decision-making in a wide range of applications. However, in many real-world scenarios, the parameters of the optimization problem are not known a priori and must be predicted from contextual features. This gives rise to predict-then-optimize problems, where a machine learning model predicts problem parameters that are then used to make decisions via optimization. A growing body of work on decision-focused learning (DFL) addresses this setting by training models specifically to produce predictions that maximize downstream decision quality, rather than accuracy. While effective, DFL is computationally expensive, because it requires solving the optimization problem with the predicted parameters at each loss evaluation. In this work, we address this computational bottleneck for linear optimization problems, a common class of problems in both DFL literature and real-world applications. We propose a solver-free training method that exploits the geometric structure of linear optimization to enable efficient training with minimal degradation in solution quality. Our method is based on the insight that a solution is optimal if and only if it achieves an objective value that is at least as good as that of its adjacent vertices on the feasible polytope. Building on this, our method compares the estimated quality of the ground-truth optimal solution with that of its precomputed adjacent vertices, and uses this as loss function. Experiments demonstrate that our method significantly reduces computational cost while maintaining high decision quality.

We study nonlinearly preconditioned gradient methods for smooth nonconvex optimization problems, focusing on sigmoid preconditioners that inherently perform a form of gradient clipping akin to the widely used gradient clipping technique. Building upon this idea, we introduce a novel heavy ball-type algorithm and provide convergence guarantees under a generalized smoothness condition that is less restrictive than traditional Lipschitz smoothness, thus covering a broader class of functions. Additionally, we develop a stochastic variant of the base method and study its convergence properties under different noise assumptions. We compare the proposed algorithms with baseline methods on diverse tasks from machine learning including neural network training.

Forecasting multi-step user behavior trajectories requires reasoning over structured preferences across future actions, a challenge overlooked by traditional sequential recommendation. This problem is critical for applications such as personalized commerce and adaptive content delivery, where anticipating a user’s complete action sequence enhances both satisfaction and business outcomes. We identify an essential limitation of existing paradigms: their inability to capture global, listwise dependencies among sequence items. To address this, we formulate User Behavior Trajectory Prediction (UBTP) as a new task setting that explicitly models longterm user preferences. We introduce Listwise Preference Diffusion Optimization (LPDO), a diffusion-based training framework that directly optimizes structured preferences over entire item sequences. LPDO incorporates a Plackett–Luce supervision signal and derives a tight variational lower bound aligned with listwise ranking likelihoods, enabling coherent preference generation across denoising steps and overcoming the independent-token assumption of prior diffusion methods. To rigorously evaluate multi-step prediction quality, we propose the task-specific metric: Sequential Match (SeqMatch), which measures exact trajectory agreement, and adopt Perplexity (PPL), which assesses probabilistic fidelity. Extensive experiments on real-world user behavior benchmarks demonstrate that LPDO consistently outperforms state-of-the-art baselines, establishing a new benchmark for structured preference learning with diffusion models.

In this paper, we propose a novel tube-wise local tensor compressed sensing (CS) model, where sensing operators are independently applied to each tube of a third-order tensor. To recover the low-rank ground truth tensor, we minimize a non-convex objective via Burer–Monteiro factorization and solve it using gradient descent with spectral initialization. We prove that this approach achieves exact recovery with a linear convergence rate. Notably, our method attains provably lower sample complexity than existing TCS methods. Our proof leverages the leave-one-out technique to show that gradient descent generates iterates implicitly biased towards solutions with bounded incoherence, which ensures contraction of optimization error in consecutive iterates. Empirical results validate the effectiveness of GD in solving the proposed local TCS model.

Fairness and action smoothness are two crucial considerations in many online optimization problems, but they have yet to be addressed simultaneously. In this paper, we study a new and challenging setting of fairness-regularized smoothed online convex optimization with switching costs. First, to highlight the fundamental challenges introduced by the long-term fairness regularizer evaluated based on the entire sequence of actions, we prove that even without switching costs, no online algorithms can possibly achieve a sublinear regret or finite competitive ratio compared to the offline optimal algorithm as the problem episode length $T$ increases. Then, we propose **FairOBD** (Fairness-regularized Online Balanced Descent), which reconciles the tension between minimizing the hitting cost, switching cost, and fairness cost. Concretely, **FairOBD** decomposes the long-term fairness cost into a sequence of online costs by introducing an auxiliary variable and then leverages the auxiliary variable to regularize the online actions for fair outcomes. Based on a new approach to account for switching costs, we prove that **FairOBD** offers a worst-case asymptotic competitive ratio against a novel benchmark---the optimal offline algorithm with parameterized constraints---by considering $T\to\infty$. Finally, we run trace-driven experiments of dynamic computing resource provisioning for socially responsible AI inference to empirically evaluate **FairOBD**, showing that **FairOBD** can effectively reduce the total fairness-regularized cost and better promote fair outcomes compared to existing baseline solutions.

Open-weight large language model (LLM) zoos provide access to numerous high-quality models, but selecting the appropriate model for specific tasks remains challenging and requires technical expertise. Most users simply want factually correct, safe, and satisfying responses without concerning themselves with model technicalities, while inference service providers prioritize minimizing operating costs. These competing interests are typically mediated through service level agreements (SLAs) that guarantee minimum service quality. We introduce MESS+, a stochastic optimization algorithm for cost-optimal LLM request routing while providing rigorous SLA compliance guarantees. MESS+ learns request satisfaction probabilities of LLMs in real-time as users interact with the system, based on which model selection decisions are made by solving a per-request optimization problem. Our algorithm includes a novel combination of virtual queues and request satisfaction prediction, along with a theoretical analysis of cost optimality and constraint satisfaction. Across a wide range of state-of-the-art LLM benchmarks, MESS+ achieves an average of $2\times$ cost savings compared to existing LLM routing techniques.