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Optimization · Neural Network Optimization

Anda Tang, Yiming Dong, Yutao Zeng, zhou Xun, Zhouchen Lin

The expanding computational costs and limited resources underscore the critical need for budgeted-iteration training, which aims to achieve optimal learning within predetermined iteration budgets. While learning rate schedules fundamentally govern the performance of different networks and tasks, particularly in budgeted-iteration scenarios, their design remains largely heuristic, lacking theoretical foundations. In addition, the optimal learning rate schedule requires extensive trial-and-error selection, making the training process inefficient. In this work, we propose the Unified Budget-Aware (UBA) schedule, a theoretically grounded learning rate schedule that consistently outperforms commonly-used schedules among diverse architectures and tasks under different constrained training budgets. First, we bridge the gap by constructing a novel training budget-aware optimization framework, which explicitly accounts for the robustness to landscape curvature variations. From this framework, we derive the UBA schedule, controlled by a single hyper-parameter $\varphi$ that provides a trade-off between flexibility and simplicity, eliminating the need for per-network numerical optimization. Moreover, we establish a theoretical connection between $\varphi$ and the condition number, adding interpretation and justification to our approach. Besides, we prove the convergence for different values of $\varphi$. We offer practical guidelines for $\varphi$ selection via theoretical analysis and empirical results. Extensive experimental results show that UBA $\textit{consistently surpasses}$ the commonly-used schedules across diverse vision and language tasks, spanning network architectures (e.g., ResNet, OLMo) and scales, under different training-iteration budgets.

Theory · Learning Theory

YuQing Xie, Tess Smidt

Equivariant neural networks have proven to be effective for tasks with known underlying symmetries. However, optimizing equivariant networks can be tricky and best training practices are less established than for standard networks. In particular, recent works have found small training benefits from relaxing equivariance constraints. This raises the question: do equivariance constraints introduce fundamental obstacles to optimization? Or do they simply require different hyperparameter tuning? In this work, we investigate this question through a theoretical analysis of the loss landscape geometry. We focus on networks built using permutation representations, which we can view as a subset of unconstrained MLPs. Importantly, we show that the parameter symmetries of the unconstrained model has nontrivial effects on the loss landscape of the equivariant subspace and under certain conditions can provably prevent learning of the global minima. Further, we empirically demonstrate in such cases, relaxing to an unconstrained MLP can sometimes solve the issue. Interestingly, the weights eventually found via relaxation corresponds to a different choice of group representation in the hidden layer. From this, we draw 3 key takeaways. (1) By viewing the unconstrained version of an architecture, we can uncover hidden parameter symmetries which were broken by choice of constraint enforcement (2) Hidden symmetries give important insights on loss landscapes and can induce critical points and even minima (3) Hidden symmetry induced minima can sometimes be escaped by constraint relaxation and we observe the network jumps to a different choice of constraint enforcement. Effective equivariance relaxation may require rethinking the fixed choice of group representation in the hidden layers.

Deep Learning · Graph Neural Networks

Wei Huang, Hanchen Wang, Dong Wen, SHAOZHEN MA, Wenjie Zhang, Xuemin Lin

Graph Edit Distance (GED) is a fundamental graph similarity metric widely used in various applications. However, computing GED is an NP-hard problem. Recent state-of-the-art hybrid GED solver has shown promising performance by formulating GED as a bipartite graph matching problem, then leveraging a generative diffusion model to predict node matching between two graphs, from which both the GED and its corresponding edit path can be extracted using a traditional algorithm. However, such methods typically rely heavily on ground-truth supervision, where the ground-truth node matchings are often costly to obtain in real-world scenarios. In this paper, we propose GEDRanker, a novel unsupervised GAN-based framework for GED computation. Specifically, GEDRanker consists of a matching-based GED solver and introduces an interpretable preference-aware discriminator. By leveraging preference signals over different node matchings derived from edit path lengths, the discriminator can guide the matching-based solver toward generating high-quality node matching without the need for ground-truth supervision. Extensive experiments on benchmark datasets demonstrate that our GEDRanker enables the matching-based GED solver to achieve near-optimal solution quality without any ground-truth supervision.

Optimization · Neural Network Optimization

Zhen Tian, Xin Zhao, Ji-Rong Wen

In this paper, we propose a novel optimizer state compression algorithm, namely \textbf{$\pi$-Quant}, which leverages the properties of irrational numbers (\eg $\pi$) for memory-efficient training. The core idea is based on our mathematical findings, which show that a pair of parameters can be represented by a single rotation angle using the complex rotation scheme. Building on this insight, we map the parameters into a complex space and perform quantization using the corresponding rotation angles. To efficiently integrate it into optimization process, we develop an efficient system of geometric equations that computes the precise rotation angles with linear complexity. We evaluate $\pi$-Quant on a wide range of tasks. Our experiments show that it can reduce the bit-width of parameters to 3.32-bit, achieving a 41.8\% decrease in GPU memory usage, all while maintaining full accuracy. \textcolor{blue}{We have submitted the code in supplementary materials}.

Deep Learning · Graph Neural Networks

Antonin Joly, Nicolas Keriven, Aline Roumy

Graph coarsening aims to diminish the size of a graph to lighten its memory footprint, and has numerous applications in graph signal processing and machine learning. It is usually defined using a reduction matrix and a lifting matrix, which, respectively, allows to project a graph signal from the original graph to the coarsened one and back. This results in a loss of information measured by the so-called Restricted Spectral Approximation (RSA). Most coarsening frameworks impose a fixed relationship between the reduction and lifting matrices, generally as pseudo-inverses of each other, and seek to define a coarsening that minimizes the RSA. In this paper, we remark that the roles of these two matrices are not entirely symmetric: indeed, putting constraints on the *lifting matrix alone* ensures the existence of important objects such as the coarsened graph's adjacency matrix or Laplacian. In light of this, in this paper, we introduce a more general notion of reduction matrix, that is *not* necessarily the pseudo-inverse of the lifting matrix. We establish a taxonomy of ``admissible'' families of reduction matrices, discuss the different properties that they must satisfy and whether they admit a closed-form description or not. We show that, for a *fixed* coarsening represented by a fixed lifting matrix, the RSA can be *further* reduced simply by modifying the reduction matrix. We explore different examples, including some based on a constrained optimization process of the RSA. Since this criterion has also been linked to the performance of Graph Neural Networks, we also illustrate the impact of this choices on different node classification tasks on coarsened graphs.

Computer Vision · Segmentation

Md Mostafijur Rahman, Radu Marculescu

U‑shaped networks output logits at multiple spatial scales, each capturing a different blend of coarse context and fine detail. Yet, training still treats these logits in isolation—either supervising only the final, highest‑resolution logits or applying deep supervision with identical loss weights at every scale—without exploring *mixed‑scale* combinations. Consequently, the decoder output misses the complementary cues that arise only when coarse and fine predictions are fused. To address this issue, we introduce LoMix ($\underline{Lo}$gits $\underline{Mix}$ing), a Neural Architecture Search (NAS)‑inspired, differentiable plug-and-play module that **generates** new mixed‑scale outputs and **learns** how exactly each of them should guide the training process. More precisely, LoMix mixes the multi-scale decoder logits with four lightweight fusion operators: addition, multiplication, concatenation, and attention-based weighted fusion, yielding a rich set of synthetic “mutant’’ maps. Every original or mutant map is given a softplus loss weight that is co‑optimized with network parameters, mimicking a one‑step architecture search that automatically discovers the most useful scales, mixtures, and operators. Plugging LoMix into recent U-shaped architectures (i.e., PVT‑V2‑B2 backbone with EMCAD decoder) on Synapse 8‑organ dataset improves DICE by +4.2\% over single‑output supervision, +2.2\% over deep supervision, and +1.5\% over equally weighted additive fusion, all with **zero** inference overhead. When training data are scarce (e.g., one or two labeled scans, 5\% of the trainset), the advantage grows to +9.23\%, underscoring LoMix’s data efficiency. Across four benchmarks and diverse U-shaped networks, LoMiX improves DICE by up to +13.5\% over single-output supervision, confirming that learnable weighted mixed‑scale fusion generalizes broadly while remaining data efficient, fully interpretable, and overhead-free at inference. Our implementation is available at https://github.com/SLDGroup/LoMix.

Theory · Reinforcement Learning and Planning

Jifeng Hu, Sili Huang, Zhejian Yang, Shengchao Hu, Li Shen, Hechang Chen, Lichao Sun, Yi Chang, Dacheng Tao

Conditional decision generation with diffusion models has shown powerful competitiveness in reinforcement learning (RL). Recent studies reveal the relation between energy-function-guidance diffusion models and constrained RL problems. The main challenge lies in estimating the intermediate energy, which is intractable due to the log-expectation formulation during the generation process. To address this issue, we propose the Analytic Energy-guided Policy Optimization (AEPO). Specifically, we first provide a theoretical analysis and the closed-form solution of the intermediate guidance when the diffusion model obeys the conditional Gaussian transformation. Then, we analyze the posterior Gaussian distribution in the log-expectation formulation and obtain the target estimation of the log-expectation under mild assumptions. Finally, we train an intermediate energy neural network to approach the target estimation of log-expectation formulation. We apply our method in 30+ offline RL tasks to demonstrate the effectiveness of our method. Extensive experiments illustrate that our method surpasses numerous representative baselines in D4RL offline reinforcement learning benchmarks.

Probabilistic Methods

Gerardo Duran-Martin, Leandro Sánchez-Betancourt, Alvaro Cartea, Kevin Murphy

We introduce scalable algorithms for online learning of neural network parameters and Bayesian sequential decision making. Unlike classical Bayesian neural networks, which induce predictive uncertainty through a posterior over model parameters, our methods adopt a predictive-first perspective based on martingale posteriors. In particular, we work directly with the one-step-ahead posterior predictive, which we parameterize with a neural network and update sequentially with incoming observations. This decouples Bayesian decision-making from parameter-space inference: we sample from the posterior predictive for decision making, and update the parameters of the posterior predictive via fast, frequentist Kalman-filter-like recursions. Our algorithms operate in a fully online, replay-free setting, providing principled uncertainty quantification without costly posterior sampling. Empirically, they achieve competitive performance–speed trade-offs in non-stationary contextual bandits and Bayesian optimization, offering 10–100 times faster inference than classical Thompson sampling while maintaining comparable or superior decision performance.

Optimization · Neural Network Optimization

Sen Bai, Chunqi Yang, Xin Bai, Xin Zhang, Zhengang Jiang

Binary (0-1) integer programming (BIP) is pivotal in scientific domains requiring discrete decision-making. As the advance of AI computing, recent works explore neural network-based solver for integer linear programming (ILP) problems. Yet, they lack scalability for tackling nonlinear challenges. To handle nonlinearities, state-of-the-art Branch-and-Cut solvers employ linear relaxations, leading to exponential growth in auxiliary variables and severe computation limitations. To overcome these limitations, we propose BIPNN, an unsupervised learning framework to solve BIP problems via hypergraph neural networks (HyperGNN). Specifically, (i) BIPNN reformulates BIPs-constrained, discrete, and nonlinear (sin, log, exp) optimization problems-into unconstrained, differentiable, and polynomial loss functions. The reformulation stems from the observation of a precise one-to-one mapping between polynomial BIP objectives and hypergraph structures, enabling the unsupervised training of HyperGNN to optimize BIP problems in an end-to-end manner. On this basis, (ii) we propose a GPU-accelerated and continuous-annealing-enhanced training pipeline for BIPNN. The pipeline enables BIPNN to optimize large-scale nonlinear terms in BIPs fully in parallel via straightforward gradient descent, thus significantly reducing the training cost while ensuring the generation of discrete, high-quality solutions. Extensive experiments on synthetic and real-world datasets highlight the superiority of our approach.

Deep Learning · Theory

Nicolas Zucchet, Francesco D'Angelo, Andrew Lampinen, Stephanie Chan

Emergence is a fascinating property of large language models and neural networks more broadly: as models scale and train for longer, they sometimes develop new abilities in sudden ways. Despite initial studies, we still lack a comprehensive understanding of how and when these abilities emerge. To address this gap, we study the emergence over training of sparse attention, a critical and frequently observed attention pattern in Transformers. By combining theoretical analysis of a toy model with empirical observations on small Transformers trained on a linear regression variant, we uncover the mechanics driving sparse attention emergence and reveal that emergence timing follows power laws based on task structure, architecture, and optimizer choice. We additionally find that repetition can greatly speed up emergence. Finally, we confirm these results on a well-studied in-context associative recall task. Our findings provide a simple, theoretically grounded framework for understanding how data distributions and model design influence the learning dynamics behind one form of emergence.

Deep Learning · Theory

Chuang Ma, Tomoyuki Obuchi, Toshiyuki Tanaka

A phenomenon known as ``Neural Collapse (NC)'' in deep classification tasks, in which the penultimate-layer features and the final classifiers exhibit an extremely simple geometric structure, has recently attracted considerable attention, with the expectation that it can deepen our understanding of how deep neural networks behave. The Unconstrained Feature Model (UFM) has been proposed to explain NC theoretically, and there emerges a growing body of work that extends NC to tasks other than classification and leverages it for practical applications. In this study, we investigate whether a similar phenomenon arises in deep Ordinal Regression (OR) tasks, via combining the cumulative link model for OR and UFM. We show that a phenomenon we call Ordinal Neural Collapse (ONC) indeed emerges and is characterized by the following three properties: (ONC1) all optimal features in the same class collapse to their within-class mean when regularization is applied; (ONC2) these class means align with the classifier, meaning that they collapse onto a one-dimensional subspace; (ONC3) the optimal latent variables (corresponding to logits or preactivations in classification tasks) are aligned according to the class order, and in particular, in the zero-regularization limit, a highly local and simple geometric relationship emerges between the latent variables and the threshold values. We prove these properties analytically within the UFM framework with fixed threshold values and corroborate them empirically across a variety of datasets. We also discuss how these insights can be leveraged in OR, highlighting the use of fixed thresholds.

Optimization · Neural Network Optimization

Shiva Sreeram, Alaa Maalouf, Pratyusha Sharma, Daniela Rus

Recently, Sharma et al. (2024) suggested a method called LAyer- SElective-Rank reduction (LASER) which demonstrated that pruning high‑order components of carefully chosen LLM’s weight matrices can boost downstream accuracy—without any gradient‑based fine‑tuning. Yet LASER’s exhaustive, per‑matrix search (each requiring full‑dataset forward passes) makes it impractical for rapid deployment. We demonstrate that this overhead can be removed and find that: (i) Only a small, carefully chosen subset of matrices needs to be inspected—eliminating the layer‑by‑layer sweep, (ii) The gradient of each matrix’s singular values pinpoints which matrices merit reduction, (iii) Increasing the factorization search space by allowing matrices rows to cluster around multiple subspaces and then decomposing each cluster separately further reduces overfitting on the original training data and further lifts accuracy by up to 24.6 percentage points, and finally, (iv) we discover that evaluating on just 100 samples rather than the full training data—both for computing the indicative gradients and for measuring the final accuracy—suffices to further reduce the search time; we explain that as adaptation to downstream tasks is dominated by prompting style, not dataset size. As a results, we show that combining these findings yields a fast and robust adaptation algorithm for downstream tasks. Overall, with a single gradient step on 100 examples and a quick scan of the top candidate layers and factorization techniques, we can adapt LLMs to new datasets—entirely without fine‑tuning.

Optimization · Everything Else

Sota Moriyama, Katsumi Inoue

The use of deep learning to solve fundamental AI problems such as Boolean Satisfiability (SAT) has been explored recently to develop robust and scalable reasoning systems. This work advances such neural-based reasoning approaches by developing a new Graph Neural Network (GNN) to differentiably solve (weighted) Maximum Satisfiability (MaxSAT). To this end, we propose SAT-based Graph Attention Networks (SGATs) as novel GNNs that are built on t-norm based attention and message passing mechanisms, and structurally designed to approximate greedy distributed local search. To demonstrate the effectiveness of our model, we develop a local search solver that uses SGATs to continuously solve any given MaxSAT problem. Experiments on (weighted) MaxSAT benchmark datasets demonstrate that SGATs significantly outperform existing neural-based architectures, and achieve state-of-the-art performance among continuous approaches, highlighting the strength of the proposed model.

Optimization

Gonzalo E. Constante, Hao Chen, Can Li

Deep learning models are increasingly deployed in safety-critical tasks where predictions must satisfy hard constraints, such as physical laws, fairness requirements, or safety limits. However, standard architectures lack built-in mechanisms to enforce such constraints, and existing approaches based on regularization or projection are often limited to simple constraints, computationally expensive, or lack feasibility guarantees. This paper proposes a model-agnostic framework for enforcing input-dependent linear equality and inequality constraints on neural network outputs. The architecture combines a task network trained for prediction accuracy with a safe network trained using decision rules from the stochastic and robust optimization literature to ensure feasibility across the entire input space. The final prediction is a convex combination of the two subnetworks, guaranteeing constraint satisfaction during both training and inference without iterative procedures or runtime optimization. We prove that the architecture is a universal approximator of constrained functions and derive computationally tractable formulations based on linear decision rules. Empirical results on benchmark regression tasks show that our method consistently satisfies constraints while maintaining competitive accuracy and low inference latency.

Optimization · Everything Else

Chengyu Lu, Zhenhua Li, Xi Lin, Ji Cheng, Qingfu Zhang

Multi-objective optimization problems (MOPs) are prevalent in numerous real-world applications. Recently, Pareto Set Learning (PSL) has emerged as a powerful paradigm for solving MOPs. PSL can produce a neural network for modeling the set of all Pareto optimal solutions. However, applying PSL to black-box objectives, particularly those exhibiting non-separability, high dimensionality, and/or other complex properties, remains very challenging. To address this issue, we propose leveraging evolution strategies (ESs), a class of specialized black-box optimization algorithms, within the PSL paradigm. Traditional ESs capture the complex dimensional dependencies less efficiently, which can significantly hinder their performance in PSL. To tackle this issue, we suggest encapsulating the dependencies within a neural network, which is then trained using a novel gradient estimation method. The proposed method, termed Neural-ES, is evaluated using a bespoke benchmark suite for black-box PSL. Experimental comparisons with other methods demonstrate the efficiency of Neural-ES, underscoring its ability to learn the Pareto sets of challenging black-box MOPs.

Social Aspects · Fairness, Equity, Justice and Safety

Mojtaba Kolahdouzi, Hatice Gunes, Ali Etemad

We study whether and how the choice of optimization algorithm can impact group fairness in deep neural networks. Through stochastic differential equation analysis of optimization dynamics in an analytically tractable setup, we demonstrate that the choice of optimization algorithm indeed influences fairness outcomes, particularly under severe imbalance. Furthermore, we show that when comparing two categories of optimizers, adaptive methods and stochastic methods, RMSProp (from the adaptive category) has a higher likelihood of converging to fairer minima than SGD (from the stochastic category). Building on this insight, we derive two new theoretical guarantees showing that, under appropriate conditions, RMSProp exhibits fairer parameter updates and improved fairness in a single optimization step compared to SGD. We then validate these findings through extensive experiments on three publicly available datasets, namely CelebA, FairFace, and MS-COCO, across different tasks as facial expression recognition, gender classification, and multi-label classification, using various backbones. Considering multiple fairness definitions including equalized odds, equal opportunity, and demographic parity, adaptive optimizers like RMSProp and Adam consistently outperform SGD in terms of group fairness, while maintaining comparable predictive accuracy. Our results highlight the role of adaptive updates as a crucial yet overlooked mechanism for promoting fair outcomes. We release the source code at: https://github.com/Mkolahdoozi/Some-Optimizers-Are-More-Equal.

Applications · Neuroscience, Cognitive Science

Qingqing Yang, Hsin-Hung Li

Working memory (WM) supports the temporary retention of task-relevant information. It is limited in capacity and inherently noisy. The ability to flexibly allocate WM resources is a hallmark of adaptive behavior. While it is well established that WM resources can be prioritized via selective attention, whether they can be allocated based on reward incentive alone remains under debate—raising open questions about whether humans can efficiently allocate WM resources based on utility. To address this, we conducted behavioral experiments using orientations as stimuli. Participants first learned stimulus–reward associations and then performed a delayed estimate WM task. We found that WM precision, indexed by the variability of memory reports, reflected both natural stimulus priors and utility-based allocation. The effects from reward and prior on memory variability both grew over time, indicating their effects in stabilizing memory representations. In contrast, memory bias was largely unaffected by time or reward. To interpret these findings, we extended efficient coding theory by incorporating time and reformulating the objective from minimizing estimation loss to maximizing expected utility. We showed that the behavioral results were consistent with an observer that efficiently allocates WM resources over time to maximize utility. Lastly, we trained recurrent neural networks (RNNs) to perform the same WM task under a 2×2 design: prior (uniform vs. natural) × reward policy (baseline vs. reward context). Human-like behaviors emerged in RNNs: memory was more stable (lower variability) for stimuli associated with higher probability or rewards, and these effects increased over time. Transfer learning showed that recurrent dynamics were crucial for adapting to different priors and reward policies. Together, these results provide converging behavioral and computational evidence that WM resource allocation is shaped by environmental statistics and rewards, offering insight into how intelligent systems can dynamically optimize memory for utility under resource constraints.

Optimization · Neural Network Optimization

Meyer Scetbon, Chao Ma, Wenbo Gong, Ted Meeds

Training large language models (LLMs) commonly relies on adaptive optimizers such as Adam (Kingma & Ba 2015), which accelerate convergence through moment estimates but incur substantial memory overhead. Recent stateless approaches such as SWAN (Ma et al., 2024) have shown that appropriate preprocessing of instantaneous gradient matrices can match the performance of adaptive methods without storing optimizer states. Building on this insight, we introduce \emph{gradient multi-normalization}, a principled framework for designing stateless optimizers that normalize gradients with respect to multiple norms simultaneously. Whereas standard first-order methods can be viewed as gradient normalization under a single norm (Bernstein & Newhouse, 2024), our formulation generalizes this perspective to a multi-norm setting. We derive an efficient alternating scheme that enforces these normalization constraints and show that our procedure can produce, up to an arbitrary precision, a fixed-point of the problem. This unifies and extends prior stateless optimizers, showing that SWAN arises as a specific instance with particular norm choices. Leveraging this principle, we develop SinkGD, a lightweight matrix optimizer that retains the memory footprint of SGD while substantially reducing computation relative to whitening-based methods. On the memory-efficient LLaMA training benchmark (Zhao et al., 2024), SinkGD achieves state-of-the-art performance, reaching the same evaluation perplexity as Adam using only 40\% of the training tokens.

Deep Learning · Generative Models and Autoencoders

Nina Vesseron, Louis Bethune, Marco Cuturi

The canonical approach in generative modeling is to split model fitting into two blocks: define first how to sample noise (e.g. Gaussian) and choose next what to do with it (e.g. using a single map or flows). We explore in this work an alternative route that ties sampling and mapping. We find inspiration in moment measures, a result that states that for any measure $\rho$, there exists a unique convex potential $u$ such that $\rho=\nabla u \sharp e^{-u}$. While this does seem to tie effectively sampling (from log-concave distribution $e^{-u}$) and action (pushing particles through $\nabla u$), we observe on simple examples (e.g., Gaussians or 1D distributions) that this choice is ill-suited for practical tasks. We study an alternative factorization, where $\rho$ is factorized as $\nabla w^* \sharp e^{-w}$, where $w^\ast$ is the convex conjugate of a convex potential $w$. We call this approach conjugate moment measures, and show far more intuitive results on these examples. Because $\nabla w^*$ is the Monge map between the log-concave distribution $e^{-w}$ and $\rho$, we rely on optimal transport solvers to propose an algorithm to recover $w$ from samples of $\rho$, and parameterize $w$ as an input-convex neural network. We also address the common sampling scenario in which the density of $\rho$ is known only up to a normalizing constant, and propose an algorithm to learn $w$ in this setting.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Marcus Lassila, Johan Oestman, Khac-Hoang Ngo, Alexandre Graell i Amat

We develop practical and theoretically grounded membership inference attacks (MIAs) against both independent and identically distributed (i.i.d.) data and graph-structured data. Building on the Bayesian decision-theoretic framework of Sabrayolles et al., we derive the Bayes-optimal membership inference rule for node-level MIAs against graph neural networks, addressing key open questions about optimal query strategies in the graph setting. We introduce BASE and G-BASE, tractable approximations of the Bayes-optimal membership inference. G-BASE achieves superior performance compared to previously proposed classifier-based node-level MIA attacks. BASE, which is also applicable to non-graph data, matches or exceeds the performance of prior state-of-the-art MIAs, such as LiRA and RMIA, at a significantly lower computational cost. Finally, we show that BASE and RMIA are equivalent under a specific hyperparameter setting, providing a principled, Bayes-optimal justification for the RMIA attack.