Equilibria of realistic multiplayer games constitute a key solution concept both in practical applications, such as online advertising auctions and electricity markets, and in analytical frameworks used to study strategic voting in elections or assess policy impacts in integrated assessment models. However, efficiently computing these equilibria requires games to have a carefully designed structure and satisfy numerous restrictions; otherwise, the computational complexity becomes prohibitive. In particular, finding even approximate Nash equilibria in general normal-form games with three or more players is known to be PPAD-complete. Current state-of-the-art algorithms for computing Nash equilibria in multiplayer normal-form games either suffer from poor scalability due to their reliance on non-convex optimization solvers, or lack guarantees of convergence to a true equilibrium. In this paper, we propose a novel reformulation of the Nash equilibrium computation problem and develop a complete and sound spatial branch-and-bound algorithm based on this reformulation. We provide a qualitative analysis arguing why one should expect our approach to perform better than conventional formulation, and show the relationship between approximate solution to our reformulation and that of computing an approximate Nash equilibrium. Empirical evaluations demonstrate that our algorithm substantially outperforms existing complete methods.
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Given an undirected graph and a size parameter k, the Densest k-Subgraph (DkS) problem extracts the subgraph on k vertices with the largest number of induced edges. While DkS is NP--hard and difficult to approximate, penalty-based continuous relaxations of the problem have recently enjoyed practical success for real-world instances of DkS. In this work, we propose a scalable and exact continuous penalization approach for DkS using the error bound principle, which enables the design of suitable penalty functions. Notably, we develop new theoretical guarantees ensuring that both the global and local optima of the penalized problem match those of the original problem. The proposed penalized reformulation enables the use of first-order continuous optimization methods. In particular, we develop a non-convex proximal gradient algorithm, where the non-convex proximal operator can be computed in closed form, resulting in low per-iteration complexity. We also provide convergence analysis of the algorithm. Experiments on large-scale instances of the DkS problem and one of its variants, the Densest (k1, k2) Bipartite Subgraph (Dk1k2BS) problem, demonstrate that our method achieves a favorable balance between computation cost and solution quality.
Urban Network Security Games (UNSGs), which model the strategic allocation of limited security resources on city road networks, are critical for urban safety. However, finding a Nash Equilibrium (NE) in large-scale UNSGs is challenging due to their massive and combinatorial action spaces. One common approach to addressing these games is the Policy-Space Response Oracle (PSRO) framework, which requires computing best responses (BR) at each iteration. However, precisely computing exact BRs is impractical in large-scale games, and employing reinforcement learning to approximate BRs inevitably introduces errors that limit the overall effectiveness of the PSRO methods. Recent advancements in leveraging non-convex stochastic optimization to approximate an NE offer a promising alternative to the burdensome BR computation. However, utilizing existing stochastic optimization techniques with an unbiased loss function for UNSGs remains challenging because the action spaces are too vast to be effectively represented by neural networks. To address these issues, we introduce Tree-based Stochastic Optimization (TSO), a framework that bridges the gap between the stochastic optimization paradigm for NE-finding and the demands of UNSGs. Specifically, we employ the tree-based action representation that maps the whole action space onto a tree structure, addressing the challenge faced by neural networks in representing actions when the action space cannot be enumerated. We then incorporate this representation into the loss function and theoretically demonstrate its equivalence to the unbiased loss function. To further enhance the quality of the converged solution, we introduce a sample-and-prune mechanism that reduces the risk of being trapped in suboptimal local optima. Extensive experimental results indicate the superiority of TSO over other baseline algorithms in addressing the UNSGs.
Recent advances in 3D Gaussian Splatting (3DGS) have demonstrated great success in modeling reflective 3D objects and their interaction with the environment via deferred rendering (DR). However, existing methods often struggle with correctly reconstructing physical attributes such as albedo and reflectance, and therefore they do not support high-fidelity relighting. Observing that this limitation stems from the lack of shape and material information in RGB images, we present PhyGaP, a physically-grounded 3DGS method that leverages polarization cues to facilitate precise reflection decomposition and visually consistent relighting of reconstructed objects. Specifically, we design a polarimetric deferred rendering (PolarDR) process to model polarization by reflection, and a self-occlusion-aware environment map building technique (GridMap) to resolve indirect lighting of non-convex objects. We validate on multiple synthetic and real-world scenes, including those featuring only partial polarization cues, that PhyGaP not only excels in reconstructing the appearance and surface normal of reflective 3D objects ( 2 dB in PSNR and 45.7% in Cosine Distance better than existing RGB-based methods on average), but also achieves state-of-the-art inverse rendering and relighting capability.
Accurate camera viewpoint estimation under sparse-view conditions remains challenging, particularly in two-view scenarios. Recent approaches leverage diffusion models such as Zero123 to synthesize novel views conditioned on relative viewpoint, showing promising results when repurposed for viewpoint estimation via optimization with MSE loss. However, existing methods often suffer from non-convex loss landscape with numerous local minima, making them sensitive to initialization and reliant on naive multi-start strategies. We analyze these optimization challenges and visualize failure cases, showing that geometric ambiguities, such as symmetry and self-similarity, can mislead gradient-based updates toward incorrect viewpoints. To address these limitations, we propose a score-based method that reshapes the optimization landscape to guide updates toward the ground-truth viewpoint, followed by a refinement stage using a viewpoint-conditioned diffusion model. Experiments show that our method improves convergence, reduces reliance on brute-force sampling, and achieves competitive accuracy with higher sample-efficiency.
Federated learning (FL) has emerged as a widely adopted training paradigm for privacy-preserving machine learning. Despite the past success of SGD-based methods, they still suffer from severe data heterogeneity and the lack of adaptivity in practical applications. While several adaptive federated optimization methods (such as FedAdam) have been proposed and demonstrated to achieve faster convergence, they fail to show significant improvements in generalization performance under highly heterogeneous data distributions, and their optimization and generalization mechanisms remain insufficiently understood. To fill this gap, we introduce diffusion theory into the adaptive federated optimization framework and analyze the distinct effects of adaptive learning rate and global momentum from the perspectives of saddle-point escaping and flat-minima selection. Theoretical results show that although FedAdam outperforms FedAvg/FedAvgM in escaping saddle points, the latter escapes sharp minima more efficiently. The root cause lies in that adaptive learning rates, while enhancing saddle-point escape, weaken the preference for flat minima. Motivated by these insights, we propose FedAdamom, a new adaptive federated optimization algorithm that adapts the momentum hyperparameter rather than the learning rate. FedAdamom maintains strong saddle-point escaping capability while enhancing flat-minima selection. We further establish its convergence guarantees under non-convex objectives. Extensive experiments demonstrate that FedAdamom significantly outperforms existing adaptive federated optimization methods in terms of convergence speed, generalization performance, and preference for flat minima.
Dynamic multimodal fusion methods lack robust theoretical guidance for handling modal conflicts and inconsistent data quality. While recent theoretical work has associated modal weights with scalar quantities such as loss or confidence, this paradigm struggles to comprehensively address the risks caused by inconsistent predictive distributions across modalities. In this paper, we propose a **Co**nflict-driven **Ri**sk **M**inimization (**CoRiM**) dynamic fusion paradigm. Specifically, we redefine dynamic fusion as a principled, per-sample, direct risk minimization task. To this end, we first design a novel, differentiable Modality Conflict Risk (MCR) function, \mathcal R (w), which quantifies risk by directly modeling fused uncertainty and inter-modal consistency. Second, we identify that minimizing \mathcal R (w) is fundamentally a non-convex constrained optimization problem over the probabilistic simplex. To efficiently solve this specific challenge, we innovatively introduce the projection free Frank-Wolfe (FW) algorithm, as it is perfectly suited for optimization on the simplex.We prove that our designed \mathcal R (w) possesses L-smoothness, which provides theoretical guarantees for the convergence of the FW algorithm on our non-convex objective. Extensive experiments on multiple benchmark datasets demonstrate that CoRiM outperforms current state-of-the-art methods in high-conflict and noisy environments, validating the robustness of our method.
Muon has recently shown promising results in LLM training. In this work, we study how to further improve Muon. We argue that Muon’s orthogonalized update rule suppresses the emergence of heavy-tailed weight spectra and over-emphasizes the training along noise-dominated directions. Motivated by the Heavy-Tailed Self-Regularization (HT-SR) theory, we propose HTMuon. HTMuon preserves Muon’s ability to capture parameter interdependencies while producing heavier-tailed updates and inducing heavier-tailed weight spectra. Experiments on LLM pretraining and image classification show that HTMuon consistently improves performance over state-of-the-art baselines and can also serve as a plug-in on top of existing Muon variants. For example, on LLaMA pretraining on the C4 dataset, HTMuon reduces perplexity by up to 0.98 compared to Muon. We further theoretically show that HTMuon corresponds to steepest descent under the Schatten-q norm constraint and provide convergence analysis in smooth non-convex settings. The implementation of HTMuon is available at https://github.com/TDCSZ327/HTmuon.
Theory · Learning Theory
Normative theories allow one to elicit key parts of a ML algorithm from first principles, which is crucial at a time of championed scrutiny for ML work. Direct Preference Optimization (DPO) cleverly bypasses reward modeling by making an explicit link with a specific normative model of human choice. Our paper elevates this connection to the full generality of DPO's normative framework. Getting there requires reworking social choice theory's textbook path for a better RLHF/ML fit. It elevates the connection to a remarkably broad viewpoint on preference optimization, considering the current panorama of DPO follow-ups. It also unveils unexpected riches for ML, chief among which the support for *non-convex* losses, the fact that *any* compliant ML analytical choice can be embedded with *any* human choice model, and a normative framework's umbrella wide enough to safeguard DPO's *extensions* (margins, length correction, ...). A *toy* experiment ``far away'' from the DPO crowd is given.
Optimization · Non-Convex
The Densest $k$-Subgraph (D$k$S) is a fundamental combinatorial problem known for its theoretical hardness and breadth of applications. Recently, Lu et al. (AAAI 2025) introduced a penalty-based non-convex relaxation that achieves promising empirical performance; however, a rigorous theoretical understanding of its success remains unclear. In this work, we bridge this gap by providing a comprehensive theoretical analysis. We first establish the tightness of the relaxation, ensuring that the global maximum values of the original combinatorial problem and the relaxed problem coincide. Then we reveal the benign geometry of the optimization landscape by proving a strict dichotomy of stationary points: all integral stationary points are local maximizers, whereas all non-integral stationary points are strict saddles with explicit positive curvature. We propose a saddle-escaping Frank--Wolfe algorithm and prove that it achieves exact convergence to an integral local maximizer in a finite number of steps.
Optimization · Non-Convex
The training of neural networks often entails objective functions that are not globally $L$-smooth. For these functions, it is both theoretically and practically difficult to reply to the question: what is the largest possible step size that ensures the convergence of gradient descent (GD)? We address this longstanding open question in deep learning by providing a unifying definition of "large'' step sizes that requires only local Lipschitz (or even Hölder) continuity of the gradient. We design first-order adaptive methods that provably yield large step sizes and show that they operate at the edge of stability (EoS) right from the start of the training. In particular, the loss decreases nonmonotonically and the product between the step size and sharpness, i.e., the largest eigenvalue of the hessian, stays above the EoS threshold of 2 throughout training. Using our method, we are also able to minimize the sharpness all the way down to its global minimum. Contrary to expectation, we find that encountering globally-flat regions too early in the training may both slow down convergence and jeopardize the generalization ability of the network. Exploiting a self-stabilization argument, we allow GD to enter slightly sharper valleys and turn unsuccessful training runs into very successful ones.
Optimization · Non-Convex
In this paper, we study a class of non-convex optimization problems known as multi-affine quadratic equality constrained problems, which appear in various applications--from generating feasible force trajectories in robotic locomotion and manipulation to training neural networks. Although these problems are generally non-convex, they exhibit convexity or related properties when all variables except one are fixed. Under mild assumptions, we prove that the alternating direction method of multipliers (ADMM) converges when applied to this class of problems. Furthermore, when the "degree" of non-convexity in the constraints remains within certain bounds, we show that ADMM achieves a linear convergence rate. We validate our theoretical results through practical examples in robotic locomotion.
Optimization · Non-Convex
This paper addresses constrained smooth saddle-point problems in settings where projection onto the feasible sets is computationally expensive. We bridge the gap between projection-based and projection-free optimization by introducing a unified dual dynamic smoothing framework that enables the design of efficient single-loop algorithms. Within this framework, we establish convergence results for nonconvex-concave and nonconvex-strongly concave settings. Furthermore, we show that this framework is naturally applicable to convex-concave problems, providing a unified analysis across varying payoff structures. We propose and analyze three algorithmic variants based on the application of a linear minimization oracle over the minimization variable, the maximization variable, or both. Notably, our analysis yields anytime convergence guarantees without requiring a pre-specified iteration horizon. These results significantly narrow the performance gap between projection-free and projection-based methods for minimax optimization.
Optimization · Non-Convex
Sharpness-Aware Minimization (SAM) is widely used to seek flatter minima—often linked to better generalization. In its standard implementation, SAM updates the current iterate using the loss gradient evaluated at a point perturbed by distance $\rho$ along the normalized gradient direction. We show that, for some choices of $\rho$, SAM can stall at points where this shifted (perturbed-point) gradient vanishes despite a nonzero original gradient, and therefore, they are not stationary points of the original loss. We call these points hallucinated minimizers, prove their existence under simple nonconvex landscape conditions (e.g., the presence of a local minimizer and a local maximizer), and establish sufficient conditions for local convergence of the SAM iterates to them. We corroborate this failure mode in neural network training and observe that it aligns with SAM's performance degradation often seen at large $\rho$. Finally, as a practical safeguard, we find that a short initial SGD warm-start before enabling SAM mitigates this failure mode and reduces sensitivity to the choice of $\rho$.
Optimization · Non-Convex
The Gromov-Wasserstein (GW) distance provides a powerful framework for aligning structured data by comparing the intrinsic geometries of metric measure spaces, and has become a fundamental tool in machine learning. Most existing methods leverage entropy regularization to reduce the computational complexity to $\boldsymbol{\mathrm{O}}(n^3)$, where $n$ is the number of samples. However, this cubic time complexity remains a major bottleneck in large-scale applications, severely limiting the scalability. To address this challenge, we propose LoBCD-GW, an efficient GW optimization algorithm. Specifically, we reveal the data-dependent sparsity of large-magnitude updates to the coupling matrix and introduce a localized block coordinate selection strategy. This confines the optimization to a "selected set" of size $r$ (which is a parameter that depends on the given data set, and usually is much less than $n$), thereby reducing the complexity to $\boldsymbol{\mathrm{O}}(r^3)$. In addition, unlike prior acceleration methods often based on constraint relaxation, our method can guarantee the strict feasibility through a novel "marginal compensation mechanism" to synchronize local mass redistribution with global constraints. Finally, we conduct a set of experiments on various datasets, and the results demonstrate that our method achieves a $160\times$ speedup on large-scale graph alignment benchmarks, while maintaining state-of-the-art accuracy.
Optimization · Non-Convex
The Edge of Stability (EoS) is a phenomenon where the sharpness (largest eigenvalue) of the Hessian converges to $2/\eta$ during training with gradient descent (GD) with a step-size $\eta$. Despite violating classical smoothness assumptions, EoS has been widely observed in deep learning, but its theoretical foundations remain incomplete. We propose a framework for analyzing EoS of non-Euclidean GD using directional smoothness (Mishkin et al., 2024), which naturally extends to non-Euclidean norms. This approach allows us to characterize EoS beyond the standard Euclidean setting, encompassing methods such as $\ell_{\infty}$-descent, Block CD, Spectral GD, and Muon without momentum. We derive the appropriate measure of the generalized sharpness under an arbitrary norm. Our generalized sharpness measure includes previously studied vanilla GD and preconditioned GD as special cases. Through analytical results and experiments on neural networks, we show that non-Euclidean GD also exhibits progressive sharpening followed by oscillations around the threshold $2/\eta$. Practically, our framework provides a single, geometry-aware spectral measure that works across optimizers, bridging the gap between empirical observations and deep learning theory.
Optimization · Non-Convex
In this paper, we present the convergence analysis of the proximal Alternating Direction Method of Multipliers (ADMM) for problems with block anti-upper triangular constraints. While the linear constraints can be treated separately, most analyses of ADMM and its variants predominantly regard the linear constraints as one. Hence, it relies on assumptions related to the entire constraint matrix, such as the full column rank. However, some problems with block anti-upper triangular constraints that can be solved by ADMM do not satisfy these assumptions. To fill this gap, a new assumption is proposed and used to guarantee the global convergence of the proximal ADMM for nonconvex problems. In the strongly convex setting, we also prove the global convergence of the proximal ADMM and establish the linear convergence under four different scenarios. This work extends the theoretical understanding of the multi-block ADMM to more general cases with block anti-upper triangular constraints.
Optimization · Non-Convex
This paper studies second-order methods for nonconvex-strongly-convex bilevel optimization. We propose a novel fully second-order bilevel approximation method (FSBA) that achieves an iteration complexity of $\tilde{\mathcal{O}}(\epsilon^{-1.5})$ for finding the $(\epsilon, \mathcal{O}(\sqrt{\epsilon}))$ second-order stationary point of the hyper-objective function. Our results demonstrate that second-order methods can achieve an accelerated convergence rate than first-order methods in bilevel optimization. To address the heavy computational cost associated with the second-order oracle, we introduce a lazy variant of FSBA, called LFSBA, which reuses second-order information across several iterations. We prove that LFSBA exhibits better computational complexity than FSBA by a factor of $\sqrt{d}$, where $d$ is the dimension of the problem. We also apply a similar idea to nonconvex strongly-concave minimax optimization and propose the lazy minimax cubic-regularized Newton (LMCN) method with better computational complexity compared to existing second-order methods.
Social Aspects · Privacy
Differentially private (DP) stochastic optimization algorithms are widely used in privacy-preserving deep learning, where per-sample gradient clipping and noise injection protect sensitive information. However, these operations limit existing DP algorithms to converge within a constant-radius neighborhood of the first-order stationary point, leading to solution bias and the well-known privacy-utility trade-off. To enhance model utility, we propose a novel algorithmic framework called DP-C4, which is designed to be error-Consistently-vanishing, Coupledly-clipped, solution-Calibrated, and Convergence-guaranteed. Specifically, it incorporates a carefully designed coupled clipping scheme with shifted threshold strategy, ensuring that both clipping bias and noise variance asymptotically vanish, thereby eliminating the DP-induced solution bias. Moreover, we extend existing sensitivity analysis techniques and develop a tailored privacy budget allocation to guarantee the privacy of DP-C4. Compared with the well-recognized DP-SGD, our framework injects significantly less noise under the same privacy level. In addition, we prove that our framework converges to the optimum in strongly-convex case and to a diminishing neighborhood of the first-order stationary point in non-convex case. Experiments show that DP-C4 achieves superior privacy-utility trade-off over existing baselines across various tasks and datasets.
Optimization · Convex
Smoothness is crucial for attaining fast rates in first-order optimization. However, many optimization problems in modern machine learning involve non-smooth objectives. Recent studies relax the smoothness assumption by allowing the Lipschitz constant of the gradient to grow with respect to the gradient norm, which accommodates a broad range of objectives in practice. Despite this progress, existing generalizations of smoothness are restricted to Euclidean geometry with $\ell_2$-norm and only have theoretical guarantees for optimization in the Euclidean space. In this paper, we address this limitation by introducing a new $\ell*$-smoothness concept that measures the norm of Hessians in terms of a general norm and its dual, and establish convergence for mirror-descent-type algorithms, matching the rates under the classic smoothness. Notably, we propose a generalized self-bounding property that facilitates bounding the gradients via controlling suboptimality gaps, serving as a principal component for convergence analysis. Beyond deterministic optimization, we establish sharp convergence for stochastic mirror descent, matching state-of-the-art under classic smoothness. Our theory also extends to non-convex and composite optimization, which may shed light on practical usages of mirror descent, including pre-training and post-training of LLMs.